improving linguistic pairwise comparison consistency via linguistic discrete regions linguistic pairwise comparison matrices are widely used in decision making procedures however the matrices often give conflicting results when there are multiple criteria under consideration despite intensive research achieving consistency of such matrices remains a daunting task in this paper a novel approach based on linguistic discrete region is proposed to address the challenge unlike existing methods that require a single value for each comparison our approach allows a comparison to be expressed by a discrete region with multiple linguistic terms such front end gives users more freedom to express their opinions in the back end we propose an iterative searching algorithm that is able to achieve approximate optimal consistency for the comparison matrices with discrete region values the final results are single value matrices that not only guarantee approximate optimal consistency but comply with evaluators intentions a well as our approach does not modify any linguistic values like many existing methods we have conducted extensive evaluations and our empirical study confirms that the linguistic discrete region based approach significantly improves the consistency of linguistic pairwise comparison matrices

an exposition of the ahp in reply to the paper remarks on the analytic hierarchy process it is a fact that people make decisions and have been making decisions for a very long time contrary to what some of us who are interested in decision making may like to believe most people do not take seriously the existence of theories which purport to set their thinking and feeling right they claim to know their own value system and what they want they may wonder how anyone else can know well enough to tell them how best to organize their thinking in order to make better choices yet research has shown that complex decisions are beyond the capacity of the brain to synthesize intuitively and efficiently since decision making is a natural characteristic of people how do we describe what they do so that an ordinary mortal can understand what we are saying we do not wish to legislate the method with which people should make decisions but only to describe it even when it is prescribed by some method in the process we may learn things that can help people make better decisions how the analytic hierarchy process ahp forman et al harker 1986 harker and vargas 1987 saaty 1986 1988a b saaty and vargas 1987 xu 1988 golden et al 1989 saaty and alexander 1989 is a theory of measurement when applied in decision making it assists one to describe the general decision operation by decomposing a complex problem into a multi level hierarchic structure of objectives criteria subcriteria and alternatives the ahp provides a fundamental scale of relative magnitudes expressed in dominance units to represent judgments in the form of paired comparisons a ratio scale of relative magnitudes expressed in priority units is then derived from each set of comparisons an overall rcatio scale of priorities is then synthesized to obtain a ranking of the alternatives from its axioms to its procedures the ahp has turned out to be historically and theoretically a different and independent theory of decision making from utility theory much as a dialogue evolved in mathematics around the consistency of different geometries and around absolute and relative space and time in physics both to dispel absolute notions those who believe that only utility theory can tell us the absolute truth about man s decision making might take a close look at the ahp it has found varied and serious applications it also has a particular way of generating ratio scales and dealing with inconsistency in judgment that have contributed to its effectiveness in resource allocation and in the setting of priorities by a group of decision makers utility theory is a normative process the ahp as a descriptive theory encompasses procedures leading to outcomes as would be ranked by a normative theory but it must go beyond to deal with outcomes not accounted for by the demanding assumptions of a normative theory we must briefly describe the ahp to enable the reader to see that a practicable theory based on ratio scales need not dilute itself to satisfy expectations of people who derive their understanding from a theory based on interval scales this is particularly true if the rival theory in aspiring for generality also makes unrealistic assumptions for example about the transitivity and consistency of preferences and the difficult use of lotteries

an iterative approach for estimation of student performances based on linguistic evaluations in group decision analysis numerous approaches have been suggested in an attempt to solve the problem of aggregation of individual fuzzy opinions to form a group consensus as the basis of a group decision in this study an optimization model which reflects different points of view of many decision makers by weighting fuzzy opinions is proposed for the evaluation of student performances in student centered learning an iterative algorithm is provided for the solution of this model and the consequent theorem is proved experimental results show that the proposed iterative algorithm yields more efficient results than do the classical optimization methods moreover the wabl weighted averaging based on the levels method produces more accurate results than do the other frequently used defuzzification methods such as coa center of area and mom mean of maxima

how to make a decision the analytic hierarchy process abstract this paper serves as an introduction to the analytic hierarchy process a multicriteria decision making approach in which factors are arranged in a hierarchic structure the principles and the philosophy of the theory are summarized giving general background information of the type of measurement utilized its properties and applications

hesitant analytic hierarchy process abstract in traditional analytic hierarch process ahp decision makers dms are required to provide crisp judgments over paired comparisons of objectives to construct comparison matrices to enhance the modeling ability of traditional ahp we propose hesitant ahp h ahp that can consider the hesitancy experienced by the dms in decision h ahp is characterized by hesitant judgments where each hesitant judgment can be represented by several possible values different probability distributions can be used to further describe hesitant judgments according to the dms preferences based on a hesitant comparison matrix hcm that consists of hesitant judgments we define two indices to measure the consistency degree and the consensus degree of the hcm respectively from a stochastic point of view a new prioritization method is developed to derive priorities from hcms where the results are with probability interpretations we provide a step by step procedure for h ahp and demonstrate this new method with a real life decision making problem

the limited effectiveness of normal forms a critical review and extension of local bifurcation studies of the brusselator pde abstract the detection and unfolding of degenerate local bifuractions provides one of very few generally applicable analytical tools for studying complex dynamics in systems of arbitrarily high dimension using the brusselator partial differential equations pdes prigogine and lefever 1968 as motivation and main example we critically review this method we extend and correct previous calculations presenting explicit formulae from which normal forms accurate to third order may be computed and for the first time we carefully compare bifurcations and dynamics of these normal forms with those of the untransformed systems restricted to a center manifold and with galerkin and finite difference approximations of the original pde while judicious use of symbolic manipulations makes feasible such high order center manifold and normal form calculations we show that the conclusions drawn from them are of limited use in understanding spatio temporal complexity and chaos as guckenheimer 1981 argued the method permits proof of existence of quasi periodic motions and under mild genericity assumptions sil nikov chaos sub shifts of finite type but the parameter and phase space ranges in which these results may be applied are extremely small

influence of high order nonlinear fluctuations in the multivariate susceptible infectious recovered master equation we perform a high order analytical expansion of the epidemiological susceptible infectious recovered multivariate master equation and include terms up to and beyond single particle fluctuations it is shown that higher order approximations yield qualitatively different results than low order approximations which is incident to the influence of additional nonlinear fluctuations the fluctuations can be related to a meaningful physical parameter the basic reproductive number which is shown to dictate the rate of divergence in absolute terms from the ordinary differential equations more so than the total number of persons in the system in epidemiological terms the effect of single particle fluctuations ought to be taken into account as the reproductive number approaches unity

asymmetric hesitant fuzzy sigmoid preference relations in the analytic hierarchy process this study considers the practical phenomena in the process of preference elicitation and proposes an asymmetric sigmoid numerical scale asns based on a generalized sigmoid function it also offers proof of the scale s asymmetry variability consistency and diminishing utility properties further this study introduces the hesitant fuzzy preference format and defines the hesitant fuzzy continuous preference term based on this approach the asymmetric hesitant fuzzy sigmoid preference relation ahspr is developed and used in the analytic hierarchy process ahp the results show that the ahspr is a general and optimal preference relation additionally this study constructs a discrete fitting technology and an approximate translation method as the applied bases of the new scale and the preference relation following this a model framework of the ahspr in the ahp is provided finally this study re examines a well known numerical example in order to demonstrate the application and advantages of the proposed numerical scale the preference format and the modeling framework

a simulation approach for handling uncertainty in the analytic hierarchy process abstract uncertainty considerations are introduced into the analytic hierarchy process ahp the rank order of decision alternatives depends on two types of related uncertainties 1 uncertainty regarding the future characteristics of the decision making environment described by a set of scenarios and 2 uncertainty associated with the decision making judgment regarding each pairwise comparison a simulation approach for handling both types of related uncertainties in the ahp is described the example introduced by saaty and kearns 1985 is extended here to include uncertainty considerations

incorporating the uncertainty of decision judgements in the analytic hierarchy process abstract the uncertainty in the relative weights of a pairwise comparison matrix in the analytic hierarchy process ahp is caused by the uncertainty in our decision judgements and in many cases can not be avoided in this paper it is explicitly shown how such uncertainties can be incorporated within the framework of ahp and how the resulting uncertainties in the relative priorities of the decision alternatives can be computed the required algorithm and the computational procedures are also developed and illustrated with examples uncertainty is introduced as a fundamental concept independent of the concept of consistency with a view to extend the ahp as a decision analysis procedure

consequences of uncertainty in the analytic hierarchy process a simulation approach a methodology for the propagation of uncertainty in the analytic hierarchy process ahp is described the sole source of the uncertainty is assumed to lie in the elements of the preference matrices the uncertainty in decision alternative rankings and the probability of a rank reversal as functions of the number of alternatives and hierarchy depth are examined ranking uncertainty is found to decrease as the number of alternatives or the depth of the hierarchy is increased the observed behaviour of ranking uncertainty is partially explained using the central limit theorem

the analytic hierarchy process in an uncertain environment a simulation approach abstract traditionally decision makers were forced to converge ambiguous judgments to a single point estimate in order to describe a pairwise relationship between alternatives relative to some criterion for use in the analytic hierarchy process ahp since many circumstances exist which make such a convergence difficult confidence in the outcome of an ensuing ahp synthesis may be reduced likewise when a group of decision makers cannot arrive at a consensus regarding a judgment some members of the group may simply lose confidence in the overall synthesis if they lack faith in some of the judgments the ahp utilizes point estimates in order to derive the relative weights of criteria sub criteria and alternatives which govern a decision problem however when point estimates are difficult to determine distributions describing feasible judgments may be more appropriate using simulation we will demonstrate that levels of confidence can be developed expected weights can be calculated and expected ranks can be determined it will also be shown that the simulation approach is far more revealing than traditional sensitivity analysis

bifurcations of certain family of planar vector fields tangent to axes abstract the topological versality of the family v x x 1 x y y y 2 ax by cy 2 of vector fields on the plane is proven the asymptotic formulas of the bifurcational curves are given the proofs rely on an analysis of the limit cycles and the non abelian integrals along irrational curves

a unified approach to ahp with linking pins abstract recent work demonstrates the need for paired comparison estimates of criteria in the analytic hierarchy process ahp to follow prescribed rules and for the subsequent analysis to be contingent on these rules many different approaches are valid each reflecting a different rule for criteria comparison a generalized formula from which valid approaches may be generated is presented and the connection between these methods and saaty s supermatrix approach is explored

the theory of ratio scale estimation saaty s analytic hierarchy process the analytic hierarchy process developed by saaty saaty t l 1980 the analytic hierarchy process mcgraw hill new york has proven to be an extremely useful method for decision making and planning however some researchers in these areas have raised concerns over the theoretical basis underlying this process this paper addresses currently debated issues concerning the theoretical foundations of the analytic hierarchy process we also illustrate through proof and through examples the validity or fallaciousness of these criticisms

on the persistence of quasi periodic invariant tori for double hopf bifurcation of vector fields abstract we analyze the persistence of quasi periodic invariant 2 and 3 tori for the double hopf hopf hopf bifurcation by using the kam method we prove that in a sufficiently small neighborhood of the bifurcation point the full system has quasi periodic 2 tori for most of the parameter sets where its truncated normal form possesses 2 tori under appropriate conditions we obtain that the full system also has quasi periodic 3 tori for most parameters near the hopf bifurcation curve of its truncated normal form and along the direction of the bifurcation and these 3 tori bifurcate from invariant 2 tori we also give concrete formulas on the existence of quasi periodic invariant 2 and 3 tori which are based on coefficients of the truncated normal form

computing interval weights for incomplete pairwise comparison matrices of large dimension a weak consistency based approach multiple criteria decision making and evaluation problems dealing with a large number of objects are very demanding particularly when the use of pairwise comparison pc techniques is required a major drawback arises when it is not possible to obtain all the pcs due to time or cost limitations or to split the given problem into smaller subproblems in such cases two tools are needed to find acceptable weights of objects an efficient method for partially filling a pairwise comparison matrix pcm and a suitable method for deriving weights from this incomplete pcm this paper presents a novel interactive algorithm for large dimensional problems guided by two main ideas the sequential optimal choice of the pcs to be performed and the concept of weak consistency the proposed solution significantly reduces the number of needed pcs by adding information implied by the weak consistency after the input of each pc providing sets of feasible values for all missing pcs interval weights of objects are computed from the resulting incomplete weakly consistent pcm adapting the methodology for calculating fuzzy weights from fuzzy pcms the computed weight intervals thus cover all possible weakly consistent completions of the incomplete pcm the algorithm works both with saaty s pcms and fuzzy preference relations the performance of the algorithm is illustrated by a numerical example and a real life case study the performed simulation demonstrates that the proposed algorithm is capable of reducing the number of pcs required in pcms of dimension 15 and greater by more than 60 on average

remarks on the analytic hierarchy process the analytic hierarchy process ahp is flawed as a procedure for ranking alternatives in that the rankings produced by this procedure are arbitrary this paper provides a brief review of several areas of operational difficulty with the ahp and then focuses on the arbitrary rankings that occur when the principle of hierarchic composition is assumed this principle requires that the weights on the higher levels of a hierarchy can be determined independently of the weights on the lower levels virtually all of the published examples of the use of the ahp to evaluate alternatives relative to a set of criteria have assumed this principle the key to correcting this flaw is the synthesis of the ahp with the concepts of multiattribute utility theory

on some useful properties of the perron eigenvalue of a positive reciprocal matrix in the context of the analytic hierarchy process abstract a positive n x n matrix r rij is said to be reciprocal if its entries verify fji 1 rij 0 for all 1 2 decision alternatives on an arbitrary ratio scale saaty in journal of mathematical psychology 1977 proposed to use z ama x n n 1 0 a linear transform of the perron eigenvalue max of r as a measure of the cardinal consistency in an agent s responses and posed the problem of determining how it might vary as a function of the rij s he also suggested that an upper bound could be found for that consistency index when the entries of r are restricted to take their values in a bounded set both of these questions are answered here using classical results from linear algebra keywords analytic hierarchy process decision theory pairwise comparisons perron frobenius theo rem reciprocal matrices i introduction saaty 20 has introduced a procedure for pri oritising decision alternatives based on an n x n positive matrix r

an extended rule based inference for general decision making problems complex decision making problems are usually modeled in terms of a number of decisive variables that are related hierarchically pieces of evidence are aggregated in a bottom up way to determine the final decision goal for dealing with such problems ishizuka et al 5 6 developed a rule based inference procedure by using fuzzy sets to represent imprecise data and employed the dempster shafer theory to aggregate evidence for structural damage assessment this paper highlights the necessity of applying the rules with all possible consequent clauses in the procedure with the considerations of information quality and quantity when applying the procedure to more general decision making problems the use of such rules can provide high quality information which contains much less ignorance in the resultant information also it can afford more information due to the decrease in total ignorance a simple example with two cases is used to compare the results based on the two different presentations of rules in the inference procedure for illustrating the importance of considering the rules proposed in this paper

deterministic analysis of extrinsic and intrinsic noise in an epidemiological model we couple a stochastic collocation method with an analytical expansion of the canonical epidemiological master equation to analyze the effects of both extrinsic and intrinsic noise it is shown that depending on the distribution of the extrinsic noise the master equation yields quantitatively different results compared to using the expectation of the distribution for the stochastic parameter this difference is incident to the nonlinear terms in the master equation and we show that the deviation away from the expectation of the extrinsic noise scales nonlinearly with the variance of the distribution the method presented here converges linearly with respect to the number of particles in the system and exponentially with respect to the order of the polynomials used in the stochastic collocation calculation this makes the method presented here more accurate than standard monte carlo methods which suffer from slow nonmonotonic convergence in epidemiological terms the results show that extrinsic fluctuations should be taken into account since they effect the speed of disease breakouts and that the gamma distribution should be used to model the basic reproductive number

estimation of missing judgments in ahp pairwise matrices using a neural network based model selecting relevant features to make a decision and expressing the relationships between these features is not a simple task the decision maker must precisely define the alternatives and criteria which are more important for the decision making process the analytic hierarchy process ahp uses hierarchical structures to facilitate this process the comparison is realized using pairwise matrices which are filled in according to the decision maker judgments subsequently matrix consistency is tested and priorities are obtained by calculating the matrix principal eigenvector given an incomplete pairwise matrix two procedures must be performed first it must be completed with suitable values for the missing entries and second the matrix must be improved until a satisfactory level of consistency is reached several methods are used to fill in missing entries for incomplete pairwise matrices with correct comparison values additionally once pairwise matrices are complete and if comparison judgments between pairs are not consistent some methods must be used to improve the matrix consistency and therefore to obtain coherent results in this paper a model based on the multi layer perceptron mlp neural network is presented given an ahp pairwise matrix this model is capable of completing missing values and improving the matrix consistency at the same time

abelian integrals and bifurcation theory abstract conditions are given for uniqueness of limit cycles for autonomous equations in the plane the results are applicable to codimension two bifurcations near equilibrium points for vector fields

existence of at most 1 2 or 3 zeros of a melnikov function and limit cycles abstract we investigate the existence of at most one two or three limit cycles bifurcated from a periodic annulus of a hamiltonian system under a class of perturbations and obtain some sufficient conditions which ensure that the corresponding melnikov function has at most one two or three zeros in an open interval we also give applications to some systems which appear in codimension two bifurcations and to some lienard systems

simulation of fuzzy multiattribute models for grey relationships multiattribute decision making involves tradeoffs among alternative performances over multiple attributes the accuracy of performance measures are usually assumed to be accurate most multiattribute models also assume given values for the relative importance of weights for attributes however there is usually some uncertainty involved in both of these model inputs outranking multiattribute methods have always provided fuzzy input for performance scores many analysts have also recognized that weight estimates also involve some imprecision either through individual decision maker uncertainty or through aggregation of diverging group member preferences many fuzzy multiattribute models have been proposed but they have focused on identifying the expected value solution or extreme solutions this paper demonstrates how simulation can be used to reflect fuzzy inputs which allows more complete probabilistic interpretation of model results

a new fuzzy programming method to derive the priority vector from an interval reciprocal comparison matrix in this paper we discuss the interval reciprocal comparison matrices and suggest a new fuzzy programming method nfpm to derive the priority vector from an interval reciprocal comparison matrix based on mikhailov s membership function we first give a membership function which is used to measure the decision maker s satisfaction degree for the priority vector derived from each interval constraint then by using the membership function we propose a new fuzzy programming method to get the most satisfaction degree of each priority vector the nfpm has many prominent characteristics especially it is capable of deriving the priority vector from a consistent or inconsistent interval reciprocal comparison matrix moreover the nfpm can overcome some drawbacks of mikhailov s fuzzy programming method fpm for inconsistent interval reciprocal comparison matrices finally several examples are given to illustrate the applicability and advantage of the proposed method

diagnosis with dependent symptoms bayes theorem and the analytic hierarchy process judgments are needed in medical diagnosis to determine what tests to perform given certain symptoms for many diseases what information to gather on symptoms and what combination of symptoms lead to a given disease are not well known even when the number of symptoms is small the required number of experiments to generate adequate statistical data can be unmanageably large there is need in diagnosis for an integrative model that incorporates both statistical data and expert judgment when statistical data are present but no expert judgment is available one property of this model should be to reproduce results obtained through time honored procedures such as bayes theorem when expert judgment is also present it should be possible to combine judgment with statistical data to identify the disease that best describes the observed symptoms here we are interested in the analytic hierarchy process ahp framework that deals with dependence among the elements or clusters of a decision structure to combine statistical and judgmental information it is shown that the posterior probabilities derived from bayes theorem are part of this framework and hence that bayes theorem is a sufficient condition of a solution in the sense of the ahp an illustration is given as to how a purely judgment based model in the ahp can be used in medical diagnosis the application of the model to a case study demonstrates that both statistics and judgment can be combined to provide diagnostic support to medical practitioner colleagues with whom we have interacted in doing this work

applying fuzzy linguistic preference relations to the improvement of consistency of fuzzy ahp the lack of consistency in decision making can lead to inconsistent conclusions in fuzzy analytic hierarchy process fuzzy ahp method it is difficult to ensure a consistent pairwise comparison furthermore establishing a pairwise comparison matrix requires nx n 1 2 judgments for a level with n criteria alternatives the number of comparisons increases as the number of criteria increases therefore the decision makers judgments will most likely be inconsistent to alleviate inconsistencies this study applies fuzzy linguistic preference relations fuzzy linprera to construct a pairwise comparison matrix with additive reciprocal property and consistency in this study the fuzzy ahp method is reviewed and then the fuzzy linprera method is proposed finally the presented method is applied to the example addressed by kahraman et al c kahraman d ruan i dogan fuzzy group decision making for facility location selection information sciences 157 2003 135 153 this study reveals that the proposed method yields consistent decision rankings from only n 1 pairwise comparisons which is the same result as in kahraman et al research the presented fuzzy linguistic preference relations method is an easy and practical way to provide a mechanism for improving consistency in fuzzy ahp method

the wiener askey polynomial chaos for stochastic differential equations we present a new method for solving stochastic differential equations based on galerkin projections and extensions of wiener s polynomial chaos specifically we represent the stochastic processes with an optimum trial basis from the askey family of orthogonal polynomials that reduces the dimensionality of the system and leads to exponential convergence of the error several continuous and discrete processes are treated and numerical examples show substantial speed up compared to monte carlo simulations for low dimensional stochastic inputs

reconstruction methods for ahp pairwise matrices habitually decision makers are exposed to situations that require a lot of knowledge and expertise therefore they need tools to help them choose the best possible alternatives analytic hierarchical process ahp is one of those tools and it is widely used in many fields while the use of ahp is very simple there is a situation that becomes complex the consistency of the pairwise matrices in order to obtain the consistent pairwise matrix from the inconsistent one reconstruction methods can be used but they cannot guarantee getting the right matrix according to the judgments of the decision maker this situation does not allow proper evaluation of methods reliability i e it is not possible to obtain a reliable ranking of alternatives based on an inconsistent matrix in this work a new way to evaluate the reliability of matrix reconstruction methods is proposed this technique uses a novel measure for alternatives ranking comparison based on element positions and distances which is introduced in order to compare several matrix reconstruction methods finally in order to demonstrate the extensibility of this new reliability measure two reconstruction methods based on bio inspired models a genetic algorithm and the firefly algorithm are presented and evaluated by using the aforementioned reliability measure

solving chemical master equations by adaptive wavelet compression solving chemical master equations numerically on a large state space is known to be a difficult problem because the huge number of unknowns is far beyond the capacity of traditional methods we present an adaptive method which compresses the problem very efficiently by representing the solution in a sparse wavelet basis that is updated in each step the step size is chosen adaptively according to estimates of the temporal and spatial approximation errors numerical examples demonstrate the reliability of the error estimation and show that the method can solve large problems with bimodal solution profiles

an intelligent decomposition of pairwise comparison matrices for large scale decisions abstract a pairwise comparison matrix pcm has been used to compute for relative priorities of elements and are integral components in widely applied decision making tools the analytic hierarchy process ahp and its generalized form the analytic network process anp however pcms suffer from several issues limiting their applications to large scale decision problems these limitations can be attributed to the curse of dimensionality that is a large number of pairwise comparisons need to be elicited from a decision maker this issue results to inconsistent preferences due to the limited cognitive powers of decision makers to address these limitations this research proposes a pcm decomposition methodology that reduces the elicited pairwise comparisons a binary integer program is proposed to intelligently decompose a pcm into several smaller subsets using interdependence scores among elements since the subsets are disjoint the most independent pivot element is identified to connect all subsets to derive the global weights of the elements from the original pcm as a result the number of pairwise comparison is reduced and consistency is of the comparisons is improved the proposed decomposition methodology is applied to both ahp and anp to demonstrate its advantages

a codimension two bifurcation with a third order picard fuchs equation abstract using a third order picard fuchs equation we show that a certain two parameter family of planar vectorfields for parameter values in a certain cone has a unique limit cycle which is born from a hopf bifurcation and dies in a saddle connection this removes a superfluous hypothesis in theorem 3 2 chapter 13 of s n chow and j k hale methods of bifurcation theory springer verlag new york 1982

axiomatic foundation of the analytic hierarchy process this paper contains an axiomatic treatment of the analytic hierarchy process ahp the set of axioms corresponding to hierarchic structures are a special case of axioms for priority setting in systems with feedback which allow for a wide class of dependencies the axioms highlight 1 the reciprocal property that is basic in making paired comparisons 2 homogeneity that is characteristic of people s ability for making comparisons among things that are not too dissimilar with respect to a common property and hence the need for arranging them within an order preserving hierarchy 3 dependence of a lower level on the adjacent higher level 4 the idea that an outcome can only reflect expectations when the latter are well represented in the hierarchy the ahp neither assumes transitivity or the stronger condition of consistency nor does it include strong assumptions of the usual notions of rationality a number of facts are derived from these axioms providing an operational basis for the ahp

using analytic hierarchy process and fuzzy set theory to rate and rank the disability abstract any discrimination against qualified disabled people in all employment practices is violating the americans with disabilities act ada and it must be prohibited assigning qualified disabled people to a proper job or making reasonable accommodations for disabled employees is not only ethical but also necessary for reducing the social economic burden therefore in order to place qualified disabled people in appropriate job positions a multidimensional assessment method should be developed for measuring and interpreting the disabled people s residual capabilities with respect to their strengths weaknesses and their compatibility to job requirements and work environments thus the primary purpose of this study is to discuss how to structure a hierarchy related to the problem of deriving an overall disability index odi for measuring the individual s disability the analytic hierarchy process along with the entropy theory and fuzzy set theory is applied for eliciting the weights among the attributes and aggregating the multiple attributes into a single measurement we believe that the resultant odi is meaningful and useful for both the evaluator who supervises a rehabilitation program and the human resource director who engages in a hiring process

study of perturbed lotka volterra systems via abelian integrals abstract we consider the lotka volterra system as a hamiltonian one and study a special perturbation of it for this perturbed system we get results on the number of limit cycles the main tools used are abelian integrals and degenerate hopf bifurcation

new criteria for the monotonicity of the ratio of two abelian integrals abstract new criteria to determine the monotonicity of the ratio of two abelian integrals are given when two abelian integrals have the forms h f 1 x y d x and h f 2 x y d x or the forms h f 1 x y d x and h f 2 x y d x and h are ovals belonging to the level set x y h x y h where h x y has the form y 2 2 x or x y 2 2 x we give new criteria which are defined directly by the functions which appear in the above abelian integrals and prove that the monotonicity of the criteria implies the monotonicity of the ratios of the abelian integrals the new criteria are applicable in a large class of problems some of which simplify the existing proofs and some of which generalize known results

uniqueness of periodic orbits of some vector fields with codimension two singularities abstract we consider the question of uniqueness of limit cycles of some normal form equations of vector fields with singularities of codimension two by using suitable choices of scaling on the normal form equations we give simple proofs on uniqueness of periodic orbits

a monte carlo investigation of incomplete pairwise comparison matrices in ahp the analytic hierarchy process ahp is a decision analysis technique used to evaluate complex multiattribute alternatives among one or more decision makers it imposes a hierarchical structure on any complex multicriterion problem however a major drawback of the ahp is that a large number of pairwise comparisons is needed to calibrate the hierarchy when there are a few levels and sublevels the ahp can be applied in a straightforward manner to derive the weights relative preference for each alternative as the size of the hierarchy increases the number of pairwise comparisons increases rapidly it is well established in the marketing and consumer behavior literature that in a very long interview even under the best circumstances the respondent is likely to suffer from information overload recognition of this problem was the motivation which led to the investigation of a modification of ahp which required less data collection i e a reduction in the threat of information overload the first question to be answered is the effect on ahp weights due to different patterns of missing data likely to result from reallife data collection in this study a monte carlo simulation was conducted which uses the incomplete pairwise comparisons ipc algorithm 14 to investigate the effect of reduced sets of pairwise comparisons in the ahp data for the study were generated with known structure and comparisons made between complete and incomplete matrices the results of the simulation suggest that incomplete sets of pairwise comparison matrices can capture the attribute level weights without significant loss of accuracy and independent of decision model form and amount of error considered

an approach for solving nonlinear multi objective separable discrete optimization problem with one constraint we propose an exact solution approach for solving nonlinear multi objective optimization problems with separable discrete variables and a single constraint the approach converts the multi objective problem into a single objective problem by using surrogate multipliers from which we find all the solutions with objective values within a given range we call this the surrogate target problem which is solved by using an algorithm based on the modular approach computational experiments demonstrate the effectiveness of this approach in solving large scale problems a simple example is presented to illustrate an interactive decision making process

a thurstonian view of the analytic hierarchy process abstract the application of deterministic decision models in situations characterized by noise and uncertainty is likely to produce results of questionable value in this paper some very simple probabilistic models are developed and substituted for the deterministic scales used in the analytic hierarchy process ahp it is shown that the use of these probabilistic models can extend the domain of ahp to situations such as consensual or group decision making that possess significant amounts of uncertainty in addition explicit measures of the variation present in the evaluation of decision alternatives and attributes are obtained

an analysis of distributions of priority values from alternative comparison scales within ahp the analytic hierarchy process ahp introduced by t l saaty is a well known and popular method of multi criteria decision making central to this method are the pairwise comparisons between criteria and decision alternatives made using a 9 unit scale the appropriateness of saaty s original one to nine 1 9 scale has been the subject of much debate and cause for concern this paper contrasts the appropriateness of the 1 9 scale with other alternative 9 unit scales also used in ahp by looking at the probability distributions of the associated priority values for large problems estimated probability distributions are found for the priority values through using the method of parzen windows

a criterion for determining the monotonicity of the ratio of two abelian integrals abstract a new method to determine the monotonicity of the ratio of two abelian integrals is given we find two criterion functions defined directly by the functions which appear in the abelian integrals the monotonicities of the criterion functions imply the monotonicity of the ratio of the abelian integrals by using this criterion several known results can be deduced easier and some new results are obtained

asymmetric fuzzy preference relations based on the generalized sigmoid scale and their application in decision making involving risk appetites numerical preference relations constitute a useful decision making technique and can be classified into two types on the basis of the 0 1 0 9 and 1 9 scales however these numerical scales cannot fully describe some desired properties such as asymmetry consistency variability and diminishing utility in accordance with the preference relations to address this issue we first develop a generalized sigmoid function and define a continuous preference set on the basis of which we propose the generalized sigmoid scale then we introduce the optimal discrete fitting and risk preference selection approaches to solve the risk appetite parameters in the generalized sigmoid scale using these new methods we further propose the asymmetric fuzzy preference relation afpr examine its additive transitivity property and some weak transitivity properties and design an approximate consistency test the corresponding five step modeling process under an asymmetric fuzzy preference environment is constructed which can be applied in decision making involving different risk appetites finally two examples are used to demonstrate the properties and advantages of these new methods the first example is simple and shows the differences between the traditional fuzzy preference relations and the afprs while the second is a practical case and is used to illustrate the feasibility and reasonability of the afprs

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

inconsistency and non additive capacities the analytic hierarchy process in the framework of choquet integration we examine the ahp in the framework of choquet integration and we propose an extension of the standard ahp aggregation scheme on the basis of the shapley values associated with the criteria in our model a measure of dominance inconsistency between criteria is defined in terms of the totally inconsistent matrix associated with the main pairwise comparison matrix of the ahp the measure of dominance inconsistency is then used to construct a non additive capacity whose associated shapley values reduce to the standard ahp priority weights in the consistency case in the general inconsistency case however the extended aggregation scheme based on the shapley weighted mean tends to attenuate resp emphasize the priority weights of the criteria with higher resp lower average dominance inconsistency with respect to the other criteria

an adaptive wavelet method for the chemical master equation an adaptive wavelet method for the chemical master equation is constructed the method is based on the representation of the solution in a sparse haar wavelet basis the time integration by rothe s method and an iterative procedure which in each time step selects those degrees of freedom which are essential for propagating the solution the accuracy and efficiency of the approach is discussed and the performance of the adaptive wavelet method is demonstrated by numerical examples