notes on order preservation and consistency in ahp abstract the pairwise comparisons method is a convenient tool used when the relative order among different concepts alternatives needs to be determined one popular implementation of the method is based on solving an eigenvalue problem for the pairwise comparisons matrix in such cases the ranking result for the principal eigenvector of the pairwise comparisons matrix is adopted while the eigenvalue is used to determine the index of inconsistency a lot of research has been devoted to the critical analysis of the eigenvalue based approach one of them is the work of bana e costa and vansnick 2008 in their work the authors define the conditions of order preservation cop and show that even for sufficiently consistent pairwise comparisons matrices this condition cannot be met the presented work defines more precise criteria for determining when the cop is met to formulate the criteria an error factor is used describing how far the input to the ranking procedure is from the ranking result the relationship between the saaty consistency index and cop is also discussed

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

uncertainty index based interval assignment by interval ahp abstract in a multi attribute decision making problem indigenous values are assigned to attributes based on a decision maker s subjective judgments the given judgments are often uncertain because of the uncertainty of situations and intuitiveness of human judgments in order to reflect the uncertainty in the assigned values they are denoted as intervals whose widths represent the possibilities of attributes since it is difficult for a decision maker to assign values directly to attributes in case of more than two attributes he she gives a pairwise comparison matrix by comparing two attributes at one occasion the given matrix contains two kinds of uncertainty one is inconsistency among comparisons and the other is incompleteness of comparisons this paper proposes the models to obtain intervals of attributes from the given uncertain pairwise comparison matrix at first the uncertainty indexes of a set of intervals are defined from the viewpoints of entropy in probability sum or maximum of widths or ignorance then considering that too uncertain information is not useful the intervals of attributes are obtained by minimizing their uncertainty indexes

the mathematical equivalence of the spanning tree and row geometric mean preference vectors and its implications for preference analysis abstract pairwise comparison is a widely used approach to elicit comparative judgements from a decision maker dm and there are a number of methods that can be used to then subsequently derive a consistent preference vector from the dm s judgements while the most widely used method is the eigenvector method the row geometric mean approach has gained popularity due to its mathematical properties and its ease of implementation in this paper we discuss a spanning tree method and prove the mathematical equivalence of its preference vector to that of the row geometric mean approach this is an important finding due to the fact that it identifies an approach for generating a preference vector which has the mathematical properties of the row geometric mean preference vector and yet in its entirety the spanning tree method has more to offer than the row geometric mean method in that it is inherently applicable to incomplete sets of pairwise comparison judgements and also facilitates the use of statistical and visual techniques to gain insights into inconsistency in the dm s judgements

a clarification of remarks on the analytic hierarchy process the replies by saaty 1990 and by harker and vargas 1990 to my remarks on the analytic hierarchy process dyer 1990 are useful in focusing attention on areas of disagreement regarding the ahp and provide readers with sufficient information to form their own opinions the purpose of this response is not to extend the debate but to clarify the two methods proposed in my paper to resolve the problem of rank reversal in the ahp harker and vargas 1989 misinterpreted my first proposed method method 1 in their reply the key observation they missed is that both the weights on the criteria and the scores of the alternatives on the criteria must be normalized with respect to the same range of alternative values in method 1 the decision maker must specify the ranges over which he assumed the alternatives might vary on each criterion when he determined the weights an example may be useful in clarifying the key ideas suppose each alternative in the harker and vargas 1990 example represents the choice of a decision maker among sums of money deposited in two different bank accounts and each criterion is simply the amount of money deposited in a single bank account obviously the decision maker would use the sum of the accounts as the basis for ranking the alternatives and would select b from the three alternatives a b and c let us see how the appropriate use of method 1 would lead to this result without recourse to the supermatrix technique or to the concept of dependence of the criteria on the alternatives the decision maker would consider the deposits in each bank account to be equally important if he assumed that the ranges of the possible deposits were equal for example suppose the decision maker assumes that the values for each bank account criterion vary over the range 1 80 he responds that changes from 1 to 80 on each criterion are equally important so the normalized eigenvector for the ahp criterion weights is 1 2 1 2 using method 1 since the range of values on criterion c1 is 75 80 and the decision maker assumed a range of 1 80 in determining the equal criterion weights we must introduce a dummy alternative with the value of 1 on criterion c1 this dummy alternative does not need to be explicitly introduced with values on the other criteria but it could be and would not affect the results as long as its assumed values on the other criteria are in the range 1 80 similarly a dummy alternative with a value of 80 must be introduced for criterion c2 the normalized eigenvector for criterion c1 would become 80 235 79 235 75 235 1 235 for example where the last component 1 235 is the score for the dummy alternative we normalize each resulting eigenvector by subtracting its smallest component and then dividing by the largest component the appropriate scores for the three alternatives are

eigenvector ranking method as a measuring tool formulas for errors abstract the eigenvector method em is well known to derive information from pairwise comparison matrices in decision making processes however this method is logically incomplete since its actual numerical error is unknown and its reliability is doubted by such phenomena as right left asymmetry rank reversal and reversal of order of intensity of preference in this paper we associate em with some standard measuring procedure analyze this procedure from the viewpoint of measurement theory and find the actual em error we show that the above phenomena have the same cause and are eliminated when the em errors are taken into account the full decision support tool which has all components of a standard measuring tool is composed of pairwise comparisons as an initial measuring procedure em as a data processor and the obtained formulas for em errors as an error indicator we consider two versions of this tool based on the right and the left principal eigenvectors of a pairwise comparison matrix both versions are equally suitable to measure and rank any comparable elements with positive numerical values and have the same mean relative errors equal to the square root of the double saaty s consistency index using the mean relative error we find the simple upper estimate for maximum permissible errors not impeding the reliable ranking this estimate imposes tight restrictions on inconsistency of expert judgements in decision making processes with a large number of alternatives these restrictions are much stronger than previously thought

enhancing data consistency in decision matrix adapting hadamard model to mitigate judgment contradiction cardinal and ordinal inconsistencies are important and popular research topics in the study of decision making with pair wise comparison matrices pcms few of the currently employed tactics are capable of simultaneously dealing with both cardinal and ordinal inconsistency issues in one model and most are heavily dependent on the method chosen for weight priorities derivation or the obtained closest matrix by optimization method that may change many of the original values in this paper we propose a hadamard product induced bias matrix model which only requires the use of the data in the original matrix to identify and adjust the cardinally inconsistent element s in a pcm through graph theory and numerical examples we show that the adapted hadamard model is effective in identifying and eliminating the ordinal inconsistencies also for the most inconsistent element identified in the matrix we develop innovative methods to improve the consistency of a pcm the proposed model is only dependent on the original matrix is independent of the methods chosen to derive the priority vectors and preserves most of the original information in matrix a since only the most inconsistent element s need s to be modified our method is much easier to implement than any of the existing models and the values it recommends for replacement outperform those derived from the literature it significantly enhances matrix consistency and improves the reliability of pcm decision making

stable inversion of nonlinear non minimum phase systems output tracking control of non minimum phase systems is a highly challenging problem encountered in the control of flexible manipulators space structures and elsewhere classical inversion provides exact output tracking but leads to internal instability while recent nonlinear regulation provides stable asymptotic tracking but admits large transient errors as a first step to solve this problem this paper addresses the stable inversion of non minimum phase nonlinear systems using the notions of zero dynamics and stable unstable manifolds an invertibility condition is established for a class of systems a stable but non causal inverse is obtained offline that can be incorporated into a stabilizing controller for dead beat output tracking this inverse contrasts with the causal inverse proposed by hirschorn where unstable zero dynamics result in unbounded inverse solutions our results reduce to those of hirschorn for minimum phase systems however in a numerical example the stable inverse has achiev

some comments on saaty s ahp the purpose of this short paper is to help clarify some questions which have arisen with respect to the suitability or even the correctness of the way saaty s ahp method handles criteria weights sometimes causing the rank reversal phenomenon the position set forth in this paper is that this undesirable effect does not per se invalidate that method but it does make it necessary to identify the kind of situations in which the method is suitable

stochastic group preference modelling in the multiplicative ahp a model of group consensus this paper proposes a model for the representation of a consensus seeking group decision making process under the multiplicative variant of the analytic hierarchy process ahp in particular it proposes a group preference model which expresses the group s preference intensity judgements as random variables with associated probability distributions and the structure of the multiplicative ahp is exploited to derive interval judgements for the alternatives final impact scores as perceived by the group these interval judgements can be used to determine the probability that a rank reversal may occur amongst alternatives due to the group preference uncertainty i e to assess the stability of the final impact score vector which in turn offers information about the degree of consensus within the group about the decision

a semiring based study of judgment matrices properties and models in decision making and group decision making multiplicative reciprocal judgment matrices and additive reciprocal judgment matrices are used as two kinds of important preference information in this paper semirings are applied to the discussion of judgment matrix properties first two special semirings are constructed second the properties of the consistent judgment matrices are given as a set of equations all in the semiring sense which include idempotency equations and fixed point equations we find that there exists one and only one specially constrained fixed point as the priority vector of a consistent judgment matrix third optimization models for inconsistent judgment matrices are presented finally we offer some simple illustrative examples

a rank based approach to the sequential selection and assignment problem abstract in the classical sequential assignment problem machines are to be allocated sequentially to jobs so as to maximize the expected total return where the return from an allocation of job j to machine k is the product of the value xj of the job and the weight pk of the machine the set of m machines and their weights are given ahead of time but n jobs arrive in sequential order and their values are usually treated as independent identically distributed random variables from a known univariate probability distribution with known parameter values in the paper we consider a rank based version of the sequential selection and assignment problem that minimizes the sum of weighted ranks of jobs and machines the so called secretary problem is shown to be a special case of our sequential assignment problem i e m 1 due to its distribution free property our rank based assignment strategy can be successfully applied to various managerial decision problems such as machine scheduling job interview kidney allocations for transplant and emergency evacuation plan of patients in a mass casualty situation

aggregation of direct and indirect judgments in pairwise comparison matrices with a re examination of the criticisms by bana e costa and vansnick in a recent paper by bana e costa and vansnick c a bana e costa j c vansnick a critical analysis of the eigenvalue method used to derive priorities in ahp european journal of operational research 187 3 2008 1422 1428 analytic hierarchy process ahp particularly its eigenvector method em used for deriving priorities from pairwise comparison matrices was criticized for the violation of a so called condition of order preservation cop due to this violation the em was considered to have a serious fundamental weakness which makes the use of ahp as a decision support tool very problematic the consistency ratio cr index in the ahp was also criticized for its failure to act as an alert of this violation of cop in this paper we look into decision makers overall judgments which can be obtained through the aggregation of their direct and indirect judgments and then re examine bana e costa and vansnick s numerical examples with a detailed analysis to show the invalidity of their criticisms

avoiding rank reversal in ahp decision support models abstract extending the work of a previous paper which showed that rank reversal in ahp is caused by eigenvector normalization this paper shows why methods proposed for avoiding rank reversal work when they are unnecessary and when they are futile the paper also discusses the implicit often unwarranted assumption that the dm s preference structure can be represented by an additive value function and examines the ramifications when a nonadditive value function is required

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

understanding local ignorance and non specificity within the ds ahp method of multi criteria decision making abstract ds ahp is a nascent technique for multi criteria decision making inspired by the analytic hierarchy process and whose mathematical foundation is based on the dempster shafer theory of evidence dst one aspect a consequence of the utilisation of dst in ds ahp is the allowance of a decision maker to make preference judgements on groups of decision alternatives das over different criteria in this paper developments are undertaken on ds ahp with respect to the measurement of the level of local ignorance in the judgements made together with the elucidation of an associated measure of non specificity through the evaluation of the limits on these measures subsequent index values are constructed these index values are used to introduce the concept of a vagueness plot within ds ahp the vagueness plot is shown to aid in gauging the results of interactive changes in the preference judgements made the index values and the vagueness plot presented enable the notion of the anchoring and adjustment aspect to decision making in ds ahp to be investigated

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

multi attribute decision making a simulation comparison of select methods abstract several methods have been proposed for solving multi attribute decision making problems madm a major criticism of madm is that different techniques may yield different results when applied to the same problem the problem considered in this study consists of a decision matrix input of n criteria weights and ratings of l alternatives on each criterion the comparative performance of some methods has been investigated in a few mostly field studies in this simulation experiment we investigate the performance of eight methods electre topsis multiplicative exponential weighting mew simple additive weighting saw and four versions of ahp original vs geometric scale and right eigenvector vs mean transformation solution simulation parameters are the number of alternatives criteria and their distribution the solutions are analyzed using twelve measures of similarity of performance similarities and differences in the behavior of these methods are investigated dissimilarities in weights produced by these methods become stronger in problems with few alternatives however the corresponding final rankings of the alternatives vary across methods more in problems with many alternatives although less significant the distribution of criterion weights affects the methods differently in general all ahp versions behave similarly and closer to saw than the other methods electre is the least similar to saw except for closer matching the top ranked alternative followed by mew topsis behaves closer to ahp and differently from electre and mew except for problems with few criteria a similar rank reversal experiment produced the following performance order of methods saw and mew best followed by topsis ahps and electre it should be noted that the electre version used was adapted to the common madm problem and therefore it did not take advantage of the method s capabilities in handling problems with ordinal or imprecise information

implementation of the centroid method of solymosi and dombi multiple objective decision support either for selection among finite alternatives or for combining multiple functions in mathematical programming often involves estimating a set of weights of relative importance recently solymosi and dombi proposed a means of estimating such weights based upon ordinal preference information this method relies upon the centroid of bounded weights analytic hierarchy process uses ratio estimates of pairwise comparisons using only ordinal information the centroid of weights is developed as an alternative means of weighting hierarchical elements these weights could be used in selection among alternatives directly in concordance analysis or in multiple objective linear programming formulation for the centroid of feasible weight values is presented with a table of values for cases where the preference ranking includes no ties comparison of the technique on a student job selection decision indicates slight but insignificant decline in accuracy relative to the analytic hierarchy based method

computing random consistency indices and assessing priority vectors reliability abstract the paper deals with two crucial steps in multi criteria decision analysis that are consistency of the judgments and priority vectors for alternatives criteria from a side several consistency indices are proposed for measuring the consistency of a pairwise comparison matrix from another one conditions weaker than consistency such as transitivity and weak consistency are proposed for representing further levels of coherence of a decision maker when he she expresses his her preferences by means of reciprocal pairwise comparison matrix firstly in this paper a simulation is performed in order to establish a relation between random consistency index and coherence level then since weak consistency ensures reliability to priority vectors proposed in literature a second simulation is performed in order to measure in case of no weak consistency the reliability of these priority vectors

consistency analysis of triangular fuzzy reciprocal preference relations abstract in order to simulate the uncertainty associated with impression or vagueness a decision maker may give her his judgments by means of triangular fuzzy reciprocal preference relations in the process of decision making the study of their consistency becomes a very important aspect to avoid a misleading solution based on the reciprocity property this paper proposes a new definition of consistent triangular fuzzy reciprocal preference relations the new definition is different from that reduced by consistent fuzzy reciprocal preference relations proposed by buckley 1985 the properties of consistent triangular fuzzy reciprocal preference relations in the light of the new definition are studied in detail in addition the shortcomings of the proof procedure of the proposition given by wang and chen 2008 are pointed out and the proposition is reproved by using the new definition of consistent triangular fuzzy reciprocal preference relations finally using the n 1 restricted comparison ratios a method for obtaining consistent triangular fuzzy reciprocal preference relations is proposed and an algorithm is shown to make a consistent decision ranking numerical results are further calculated to illustrate the new definition and the obtained algorithm

decision making with the ahp why is the principal eigenvector necessary in this paper it is shown that the principal eigenvector is a necessary representation of the priorities derived from a positive reciprocal pairwise comparison judgment matrix a aij when a is a small perturbation of a consistent matrix when providing numerical judgments an individual attempts to estimate sequentially an underlying ratio scale and its equivalent consistent matrix of ratios near consistent matrices are essential because when dealing with intangibles human judgment is of necessity inconsistent and if with new information one is able to improve inconsistency to near consistency then that could improve the validity of the priorities of a decision in addition judgment is much more sensitive and responsive to large rather than to small perturbations and hence once near consistency is attained it becomes uncertain which coefficients should be perturbed by small amounts to transform a near consistent matrix to a consistent one if such perturbations were forced they could be arbitrary and thus distort the validity of the derived priority vector in representing the underlying decision

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

ranking fuzzy multicriteria alternatives with respect to a decision maker s fuzzy goal abstract a two stage method for ranking fuzzy multicriteria alternatives with respect to a decision maker s fuzzy goal is presented in the first stage the joint membership function of each fuzzy alternative and the membership function of the fuzzy goal are projected orthogonally from the multicriteria space to the one dimensional preference subspace of the decision maker in the second stage a ranking index is used to rank the fuzzy projections in the decision maker s preference subspace a numerical example demonstrates the method

comparison of the rembrandt system with analytic hierarchy process abstract this study compared the use of the rembrandt system to analytic hierarchy process ahp in a group multiple criteria selection problem the decision was to select an operations management text and the decision was the responsibility of a faculty group the primary purpose of this study is to apply rembrandt to the same problem that was supported by ahp there are no noticeable differences between the techniques from the perspective of the users as the same input is used important technical differences between ahp and rembrandt are demonstrated including 1 different ratio input scales 2 alternative calculation of impact scores and 3 a different aggregation procedure rembrandt was found to recommend the same decision as ahp when the geometric mean was used for aggregation but a different decision was given by conventional ahp using arithmetic mean aggregation

acceptability measurement and priority weight elicitation of triangular fuzzy multiplicative preference relations based on geometric consistency and uncertainty indices this paper investigates consistency and uncertainty measurements as well as acceptability checking for triangular fuzzy multiplicative preference relations tfmprs a geometric consistency index is proposed to measure the consistency degree of a tfmpr and thresholds of acceptable consistency are discussed for tfmprs uncertainty indices are introduced to measure indeterminacy degrees of triangular fuzzy judgments and of tfmprs and a notion of acceptable tfmprs are put forward by considering both acceptable consistency and acceptable uncertainty in tfmprs we also explore the elicitation of triangular fuzzy weights from acceptable tfmprs and the aggregation of local triangular fuzzy weights for solving hierarchical multi criteria decision making problems a geometric mean based equation and a logarithmic least square model are developed to elicit normalized acceptable triangular fuzzy multiplicative weights from an acceptable tfmpr an equation and a linear program are established to transform triangular fuzzy multiplicative weights into triangular fuzzy additive weights a geometric weighting based equation and a pair of linear programs are devised to generate global priority fuzzy weights from local triangular fuzzy multiplicative weights three numerical examples including a hierarchical multi criteria decision making problem are used to illustrate how to apply the developed framework and comparative studies are provided to show its availability and advantages

a finite energy property of stable inversion to nonminimum phase nonlinear systems stable inversion is a completely new approach to the output tracking control of nonminimum phase nonlinear systems it not only offers exact reproduction of a given reference output trajectory but also guarantees stability of all external and internal signals after a brief review of the stable inversion problem the paper establishes a finite energy property of stable inverses that is out of infinitely many possible inverse solutions the one provided by the stable inversion process is the only one that has finite energy or the spl lscr sub 2 spl infin spl infin norm the effectiveness of the stable inversion approach in output tracking as compared with the well known classic inversion and nonlinear regulation approaches is demonstrated by working out an example of a nonlinear nonminimum phase system

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

a comparative study of the numerical scales and the prioritization methods in ahp abstract in the analytic hierarchy process ahp a decision maker first gives linguistic pairwise comparisons then obtains numerical pairwise comparisons by selecting certain numerical scale to quantify them and finally derives a priority vector from the numerical pairwise comparisons in particular the validity of this decision making tool relies on the choice of numerical scale and the design of prioritization method by introducing a set of concepts regarding the linguistic variables and linguistic pairwise comparison matrices lpcms and by defining the deviation measures of lpcms we present two performance measure algorithms to evaluate the numerical scales and the prioritization methods using these performance measure algorithms we compare the most common numerical scales the saaty scale the geometrical scale the ma zheng scale and the salo hamalainen scale and the prioritization methods the eigenvalue method and the logarithmic least squares method in addition we also discuss the parameter of the geometrical scale develop a new prioritization method and construct an optimization model to select the appropriate numerical scales for the ahp decision makers the findings in this paper can help the ahp decision makers select suitable numerical scales and prioritization methods

on the efficacy of modeling multi attribute decision problems using ahp and sinarchy abstract ahp is a multi attribute decision making methodology widely used by both practitioners and researchers in the 1980s critics had raised questions regarding its proper use there were quite a few suggested modifications to overcome the supposed limitations of ahp these modifications are themselves limited as they typically impede the applicability of ahp in this paper we revisit some of the earlier criticisms we have two objectives 1 to articulate the proper use of ahp by highlighting the assumptions and implications underlying ahp and 2 to show that sinarchy can be used to address the earlier criticisms while maintaining the applicability of the ahp framework we identify that in ahp tradeoffs between criteria vary amongst individual alternatives and are dependent on the alternative s proportion of contribution towards each criterion for problems where tradeoffs between criteria are in terms of their relative measurements sinarchy should be used it is also shown that sinarchy can prevent rank reversal illustrative examples are included throughout

when is the condition of order preservation met abstract this article explores the relationship between inconsistency in the pairwise comparisons method and conditions of order preservation a pairwise comparisons matrix with elements from an alo group an abelian linearly ordered group is investigated this approach allows for a generalization of previous results sufficient conditions for order preservation based on the properties of elements of pairwise comparisons matrix are derived elucidating numerical examples are presented

post optimality analysis of priority vectors derived from interval comparison matrices by lexicographic goal programming abstract in a recent paper by wang y m wang on lexicographic goal programming method for generating weights from inconsistent interval pair wise comparison matrices applied mathematics and computation 173 2006 985 991 it is shown that lexicographic goal programming method results in different rankings for upper and lower triangular interval judgments arranged in a positive reciprocal pair wise comparison matrix this was unexpected from a prioritization point of view because upper and lower triangular entries in reciprocal interval pair wise comparison matrices essentially provide the same judgment information a research question was posed at the same article which ranking is true or more reliable motivated by this question in this paper we provide a post optimality analysis to detect the most reliable priority vector among m vectors derived by lexicographical goal programming for an inconsistent interval pair wise comparison matrix of order n the method is based on violations in the order of preferences among the resultant priorities and runs in o mn 2 time when there exists a tie between some vectors we also illustrate a tie breaking algorithm running in o mn 4 time based on violations in the order of the preference intensities among the resultant priorities we re visited the examples discussed in the afore mentioned article and validate the practical usability of this method

ds ahp method a mathematical analysis including an understanding of uncertainty abstract the analytic hierarchy process ahp was developed to aid decision makers to rank or sort information based on a number of criteria a recent advance is the ds ahp method which incorporates the dempster shafer theory of evidence with ahp this method allows judgements on groups of decision alternatives da to be made it also offers a measure of uncertainty in the final results in this paper a mathematical analysis of ds ahp is included constructing the functional form of the preference weightings given to groups of da these functions allow an understanding of the appropriateness of the rating scale values used in the ds ahp method through evaluating the range of uncertainty able to be expressed by the decision maker

willmore flow of planar networks abstract geometric gradient flows for elastic energies of willmore type play an important role in mathematics and in many applications the evolution of elastic curves has been studied in detail both for closed as well as for open curves although elastic flows for networks also have many interesting features they have not been studied so far from the point of view of mathematical analysis so far it was not even clear what are appropriate boundary conditions at junctions in this paper we give a well posedness result for willmore flow of networks in different geometric settings and hence lay a foundation for further mathematical analysis a main point in the proof is to check whether different proposed boundary conditions lead to a well posed problem in this context one has to check the lopatinskii shapiro condition in order to apply the solonnikov theory for linear parabolic systems in holder spaces which is needed in a fixed point argument we also show that the solution we get is unique in a purely geometric sense

modelling multicriteria value interactions with reasoning maps idiographic causal maps are extensively employed in operational research to support problem structuring and complex decision making processes they model means end or causal discourses as a network of concepts connected by links denoting influence thus enabling the representation of chains of arguments made by decision makers there have been proposals to employ such structures to support the structuring of multicriteria evaluation models within an additive value measurement framework however a drawback of this multi methodological modelling is the loss of richness of interactions along the means end chains when evaluating options this has led to the development of methods that make use of the structure of the map itself to evaluate options such as the reasoning maps method which employs ordinal scales and ordinal operators for such evaluation however despite their potential reasoning maps cannot model explicitly value interactions nor perform a quantitative ranking of options limiting their applicability and usefulness in this article we propose extending the reasoning maps approach through a multilinear evaluation model structure built with the macbeth multicriteria method the model explicitly captures the value interactions between concepts along the map and employs the macbeth protocol of questioning to assess the strength of influence for each means end link the feasibility of the proposed approach to evaluate options and to deal with multicriteria interactions is tested in a real world application to support the construction of a population health index

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

fuzzy cognitive network process comparisons with fuzzy analytic hierarchy process in new product development strategy fuzzy analytic hierarchy process f ahp has increasingly been applied in many areas however as the perception and cognition toward the semantic representation for the linguistic rating scale used by the fuzzy pairwise comparison in f ahp are still open to discuss f ahp very likely produces misapplications this research proposes the fuzzy cognitive network process f cnp as an ideal alternative to f ahp a new product development application using f ahp is revised using f cnp this study shows that the proposed f cnp yields better results due to the appropriate mathematical definition of the fuzzy paired interval scale for the human perception of paired difference

the analytic hierarchy process an exposition this exposition on the analytic hierarchy process ahp has the following objectives 1 to discuss why ahp is a general methodology for a wide variety of decision and other applications 2 to present brief descriptions of successful applications of the ahp and 3 to elaborate on academic discourses relevant to the efficacy and applicability of the ahp vis a vis competing methodologies we discuss the three primary functions of the ahp structuring complexity measurement on a ratio scale and synthesis as well as the principles and axioms underlying these functions two detailed applications are presented in a linked document athttp mdm gwu edu formangass pdf

a critical analysis of the eigenvalue method used to derive priorities in ahp abstract a lot of research has been devoted to the critical analysis of the analytic hierarchy process ahp from various perspectives however as far as we know no one has addressed a fundamental problem discussed in this paper concerning the meaning of the priority vector derived from the principal eigenvalue method used in ahp the role of ahp s consistency ratio is also analysed

acceptability analysis and priority weight elicitation for interval multiplicative comparison matrices existing research on acceptability of pairwise interval comparison matrices focuses on acceptable consistency by controlling their inconsistency levels to within a certain threshold however a perfectly consistent but highly indeterminate interval comparison matrix can be unacceptable as it contains little sometimes no useful decision information this paper first analyzes the current definition of acceptable consistency for interval multiplicative comparison matrices imcms and shows its technical deficiencies we then introduce a new notion of acceptable imcms considering both inconsistency and indeterminacy levels in imcms a geometric mean based index is proposed to measure the indeterminacy ratio of an imcm and useful properties are derived for consistent imcms and acceptable imcms an indeterminacy ratio and geometric mean based transformation equation is subsequently put forward to convert normalized acceptable interval multiplicative weights into an acceptable imcm with consistency by introducing an auxiliary constraint a logarithmic least square model is established to generate interval multiplicative weights from acceptable imcms a geometric mean based possibility degree formula is designed to compare and rank normalized interval multiplicative weights two numerical examples are presented to illustrate how to utilize the proposed framework

on the relativity of relative measures accommodating both rank preservation and rank reversals in the ahp abstract the analytic hierarchy process ahp has been criticized for allowing the introduction of new alternatives to cause rank reversals among existing alternatives while in many cases this is a perfectly valid phenomenon there are also many cases where rank should be preserved to address both contingencies the ahp now supports two types of synthesis procedures one that preserves rank and one that does not this paper proposes guidelines for the selection of one synthesis mode over the other it also proposes several modifications to software implementations of the ahp to better support both synthesis modes and to reduce user stress over the choice

multiple criteria robust interactive decision analysis mcrid for optimizing public policies abstract important public policy decisions are often very complex involving many factors large uncertainties and multiple conflicting objectives decision analysis which is a formal framework for dealing with such issues has been applied to a variety of public sector problems over the last two decades a limitation of using decision analysis is the often onerous assessment burden it places on the decision maker dm we have developed a novel interactive procedure and decision support system dss for performing decision analysis on the personal computer called robust interactive decision analysis rid which avoids the difficult problems of precisely assessing a dm s utility and state probability functions associated with traditional decision treee analysis the rid method permits a dm to voluntarily and interactively express strong viz sure binary preferences for actions partial decision functions and full decision functions and only imprecise probability and utility function assessments the inputs are used with various dominance operators via a mathematical programming framework to prune the state probability utility and decision space until there is convergence on a relatively small subset of efficient strategies or an optimal choice strategy conceptually the operation of the rid method can be regarded as a state pruning system and its computer implementation as a dss in this paper we extended the rid concept and dss to deal with multiple criteria decisions obviating the need to precisely and completely assess attribute weights tradeoffs consequences and or value utility functions the approach which we call multiple criteria robust interactive decision analysis mcrid is developed illustrated and shown to be effective in eliminating many inefficient alternatives however since the aggregate utility of each alternative is interval valued it then becomes difficult to intuitively choose an optimal alternative from among the remaining efficient ones since the intervals are not defined by any specific probability distributions we have tentatively shown that aggregate utility intervals ultimately converge toward a normal density function and can be so approximated with relatively few attributes hence stochastic dominance is employed to further filter out alternatives an interactive goal programming gp technique is then developed to assist the dm in choosing an optimal alternative

l 2 flow of elastic curves with clamped boundary conditions abstract in this paper we investigate the l 2 flow of elastic curves with clamped boundary conditions in n dimensional euclidean spaces the l 2 flow corresponds to a fourth order parabolic equation in the case of closed curves the long time existence of solutions of this evolution equation has been derived in the literature we extend this result to the case of open i e non closed curves with clamped boundary conditions

an application of incomplete pairwise comparison matrices for ranking top tennis players pairwise comparison is an important tool in multi attribute decision making pairwise comparison matrices pcm have been applied for ranking criteria and for scoring alternatives according to a given criterion our paper presents a special application of incomplete pcms ranking of professional tennis players based on their results against each other the selected 25 players have been on the top of the atp rankings for a shorter or longer period in the last 40 years some of them have never met on the court one of the aims of the paper is to provide ranking of the selected players however the analysis of incomplete pairwise comparison matrices is also in the focus the eigenvector method and the logarithmic least squares method were used to calculate weights from incomplete pcms in our results the top three players of four decades were nadal federer and sampras some questions have been raised on the properties of incomplete pcms and remains open for further investigation

a modified algorithm for invertibility in nonlinear systems the algorithm of hirschorn 1 for the construction of inverses of nonlinear systems is modified and new sufficient conditions for invertibility and inverse system for a larger class of systems are obtained for the systems which do not satisfy the invertibility conditions a characterization of input space on which the input output map is injective is derived

geometry of decision making and the vector space formulation of the analytic hierarchy process we extend the conventional analytic hierarchy process ahp to an euclidean vector space and develop formulations for aggregation of the alternative preferences with the criteria preferences relative priorities obtained from such a formulation are almost identical with the ones obtained using conventional ahp each decision is represented by a preference vector indicating the orientation of the decision maker s mind in the decision space spanned by the decision alternatives this adds a geometric meaning to the decision making processes we utilise the measure of similarity between any two decision makers and apply it for analysing decisions in a homogeneous group we propose an aggregation scheme for calculating the group preference from individual preferences using a simple vector addition procedure that satisfies pareto optimality condition the results agree very well with the ones of conventional ahp

the analytic hierarchy process with stochastic judgements the analytic hierarchy process ahp is a widely used method for multicriteria decision support based on the hierarchical decomposition of objectives evaluation of preferences through pairwise comparisons and a subsequent aggregation into global evaluations the current paper integrates the ahp with stochastic multicriteria acceptability analysis smaa an inverse preference method to allow the pairwise comparisons to be uncertain a simulation experiment is used to assess how the consistency of judgements and the ability of the smaa ahp model to discern the best alternative deteriorates as uncertainty increases across a range of simulated problems results indicate that according to conventional benchmarks judgements are likely to remain consistent unless uncertainty is severe but that the presence of uncertainty in almost any degree is sufficient to make the choice of best alternative unclear

the łojasiewicz simon gradient inequality for open elastic curves abstract in this paper we consider the elastic energy for open curves in euclidean space subject to clamped boundary conditions and obtain the ojasiewicz simon gradient inequality for this energy functional thanks to this inequality we can prove that a suitably reparametrized solution to the associated l 2 gradient flow converges for large time to an elastica that is to a critical point of the functional

curve shortening straightening flow for non closed planar curves with infinite length abstract we consider a motion of non closed planar curves with infinite length the motion is governed by a steepest descent flow for the geometric functional which consists of the sum of the length functional and the total squared curvature we call the flow shortening straightening flow in this paper first we prove a long time existence result for the shortening straightening flow for non closed planar curves with infinite length then we show that the solution converges to a stationary solution as time goes to infinity moreover we give a classification of the stationary solution