fuzzy linear programming and applications abstract this paper presents a survey on methods for solving fuzzy linear programs first lp models with soft constraints are discussed then lp problems in which coefficients of constraints and or of the objective function may be fuzzy are outlined pivotal questions are the interpretation of the inequality relation in fuzzy constraints and the meaning of fuzzy objectives in addition to the commonly applied extended addition based on the min operator and used for the aggregation of the left hand sides of fuzzy constraints and fuzzy objectives a more flexible procedure based on yager s parametrized t norm t p is presented finally practical applications of fuzzy linear programs are listed

a review of some methods for ranking fuzzy subsets this paper deals with the problem of ranking n fuzzy subsets of the unit interval a number of methods suggested in the literature is reviewed and tested on a group of selected examples where the fuzzy sets can be nonnormal and or nonconvex the ranking is obtained from i the index of strict preference defined by watson ii three indexes proposed by yager iii the algorithm used by chang iv three versions of the a preference index suggested by adamo v the index defined by baas and kwakernaak vi three modified versions used by baldwin and guild vii the method proposed by kerre viii three forms of the index suggested by jain ix the four grades of dominance studied by dubois and prade in simple cases the results are good for all the methods with some exceptions in questionable cases where the decision must be probably modelled in accordance with the context in which it is imbedded the best indexes seem to be the dominances suggested by dubois and prade these indexes do not force any particular choice but clearly describe the situation hence allowing the decision maker himself to make his best choice

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

does increased regularity lower complexity intuitively the more regular a problem the easier it should be to solve examples drawn from ordinary and partial differential equations as well as from approximation support the intuition traub and woiniakowski conjectured that this is always the case in this paper we study linear problems we prove a weak form of the conjecture and show that this weak form cannot be strengthened to do this we consider what happens to the optimal error when regularity is increased if regularity is measured by a sobolev norm increasing the regularity improves the optimal error which allows us to establish the conjecture in the normed case on the other hand if regularity is measured by a sobolev seminorm it is no longer true that increasing the regularity improves the optimal error however a shifted version of this statement holds which enables us to establish the conjecture in the semi normed case 1 introduction we investigate the relation between regularity and complexity in this introduction we use words such as algorithm information cardinality and regularity without definition they are rigorously defined later based on a variety of examples traub and wozniakowski 6 conjectured that in general as the regularity of a class of problem elements increases the complexity decreases in this paper we consider linear problems we measure regularity by a sobolev norm or seminorm we prove a weak form of this conjecture and show that no stronger statement is possible to fix ideas we consider several examples example 1 1 consider the solution of the two point boundary value problem

on comparing interval numbers abstract this paper first presents a brief survey of the existing works on comparing and ranking any two interval numbers on the real line and then on the basis of this gives two approaches to compare any two interval numbers the first one describes a value judgement index along with a discussion on its strength and weakness over the other approaches this index has also been called an optimistic decision maker s preference index the other approach defines strict and fuzzy preference ordering between any two interval numbers from a pessimistic decision maker s point of view choice of different decision makers with different degrees of pessimism has also been considered

a multicriteria evaluation of methods for obtaining weights from ratio scale matrices abstract ratio scale matrices are commonly used as a vehicle to distinguish among competing alternatives or conflicting criteria the issue of extracting weights out of such matrices has attracted increasing attention in recent years and resulted in the development of several scaling methods this paper presents an attempt to evaluate some of the more known techniques in this area the evaluation procedure includes random generation of a large number of representative matrices which are used to evaluate the various scaling methods according to the specified criteria the performance of the different techniques is analyzed with respect to the proposed criteria and conclusions are drawn

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

a fuzzy extension of saaty s priority theory we present a fuzzy method for choosing among a number of alternatives under conflicting decision criteria a fuzzy version of saaty s pairwise comparison method 1980 extended by de graan 1980 and lootsma 1981 each ratio expressing the relative significance of a pair of factors is displayed in a matrix from which suitable weights can be extracted since these ratios are essentially fuzzy they express the opinion of a decision maker on the importance of a pair of factors we have adapted the above mentioned method in such a way that decision makers are asked to express their opinions in fuzzy numbers with triangular membership functions we apply the method at two distinct levels first to find fuzzy weights for the decision criteria and second to find fuzzy weights for the alternatives under each of the decision criteria by a suitable combination of these results we obtain fuzzy scores for the alternatives as well as their sensitivities using these fuzzy scores the decision makers should be able to make a choice for one of the alternatives

preference simulation and preference programming robustness issues in priority derivation abstract decision makers often resist having to make what appears to them as precise numerical judgements in fuzzy situations pairwise verbal comparisons used in the ahp are fuzzy in the sense that decision maker s need not relate verbal judgment to precise numbers because of the redundancy inherent in each set of judgments accurate priorities can be derived from such fuzzy verbal judgments another way of making fuzzy judgments is to express each judgment as a numerical interval this paper explores two new approaches for priority derivation when preferences are expressed as interval judgments one based on a simulation approach and the other based on mathematical programming the first approach assumes that the interval judgments are uniformly distributed and proceeds to derive the priority vectors and their underlying rank order by randomly sampling from these distribution this approach provides in addition to the priority vectors a measure of robustness given by the probability of rank reversal the second approach generates a region if one exists that encloses all priority vectors derived from inequalities representing the original interval judgments the two approaches are described and illustrated through a numerical example

approximate articulation of preference and priority derivation a comment abstract in a recent paper by a arbel 1989 the standard analytic hierarchy process ahp model is extended to the case where preference strength is expressed in terms of a range of scale values rather than a single value while this extension holds and is natural for the consistent case it is shown that the proposed solution for the more common inconsistent case is erroneous

uncertainty and rank order in the analytic hierarchy process abstract the analytic hierarchy process uses paired comparisons to derive a scale of relative importance for alternatives we investigate the effect of uncertainty in judgment on the stability of the rank order of alternatives the uncertainty experienced by decision makers in making comparisons is measured by associating with each judgment an interval of numerical values the approach leads to estimating the probability that an alternative or project exchanges rank with other projects these probabilities are then used to calculate the probability that project would change rank at all we combine the priority of importance of each project with the probability that it does not change rank to obtain the final ranking

testing against constant factor loading matrix with large panel high frequency data abstract in this paper we introduce a nonparametric test against the constancy of the factor loading matrix of a high dimensional continuous time factor model using high frequency data the central limit theorems on the test statistics with and without perturbation are established under the null hypothesis that the factor loading matrix is constant as time evolves the tests perform well in size and power interestingly the test statistic without perturbation converges at a rate that depends not only on the sample size but also on the dimension through the cross sectional dependence of the residual process which is a distinctive feature that contrasts with the low dimensional setting extensive numerical studies including monte carlo simulations and real data analysis validate the performance of our test

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

galerkin method and optimal error algorithms in hilbert spaces we consider operator equations of the formlu f wherel belongs to the class of linear bounded by a constantm and coercive with a constantm operators from a hilbert spacev onto its dualv andf belongs to a hilbert spacew v we study optimality of the galerkin methodp n lu n p n f whereu n v n v n is subspace ofv p n is the orthogonal projector ontov n andp n is dual top n we show that the galerkin method is quasi optimal independently of the choice of the subspacev n and the spacew ifm m in the casem m optimality of the method depends strongly on the choice ofv n andw therefore we define a new algorithm which is always optimal independently of the choice ofv n andw and relations betweenm andm

on fuzzy rank ordering in polyoptimization the application of polyoptimization in decision making gives the user a certain number of efficient points in general obtained by numerical methods for a final and reasonable solution for instance in projecting in chemical engineering this set must be reduced this can be done using additional information in the present paper we use statements on the pairwise comparison of importance between the several objective functions to facilitate reasonable decision making under consideration of given aspects such as distance measures regarding individual extrema at this we desire from the decision maker user only fuzzy information on the pairwise rank ordering between the objectives which often appears more reasonable than so called crisp quantitive statements especially for practical problems it is shown how to take advantage of zadeh s theory of fuzzy sets 1965 for our aims a numerical example is also investigated

interval estimation of priorities in the ahp this paper extends and modifies the analytic hierarchy process ahp and the synthetic hierarchy method shm of priority estimation to accommodate random data in the pairwise comparison matrices it employs a cauchy distribution to describe the pairwise comparison of alternatives in saaty matrices and shows how to modify these matrices in order to handle random data the use of random data yields saaty matrices that are not reciprocally symmetrical several variants of the ahp are then modified i to accommodate reciprocally asymmetric matrices and ii to allow each priority estimate to be expressed on an interval of possible values rather than as a single discrete point the merits of interval estimation are illustrated by an example

a fuzzy approach to deriving priorities from interval pairwise comparison judgements abstract in this paper we study the problem of priority elicitation in the analytic hierarchy process and propose a new approach to deriving crisp priorities from interval pairwise comparison judgements by introducing linear or non linear membership functions representing the decision maker s degree of satisfaction with various crisp priority vectors the interval judgements are transformed into fuzzy inequality constraints the interval prioritisation problem is then formulated as a fuzzy mathematical programming problem for obtaining an optimal crisp priority vector that maximises the overall degree of satisfaction the proposed approach yields linear or non linear mathematical programs capable of deriving priorities from consistent and inconsistent interval judgements the presence of a consistency index that measures the level of inconsistency of interval judgements is an attractive feature of our approach another feature which does not exist in the known prioritisation methods is the opportunity for additional prioritisation of the initial judgements numerical examples are given and comparisons with other interval prioritisation methods are carried out

a statistical approach to the analytic hierarchy process with interval judgements i distributions on feasible regions abstract this paper addresses the problem of extracting preferences for alternatives from interval judgement matrices in the analytic hierarchy process ahp the method of arbel for extracting such preferences which is based on the assumption that the interval judgements specify a feasible region in the weight space of the alternatives is critically appraised from a statistical perspective and some new ideas emanating from this approach are developed and discussed in particular it is proposed that a distribution for the weights on the feasible region which is both tractable and meaningful be adopted the mean of the distribution can then be used as an assessment of the overall ranking of the alternatives and quantities of interest such as probabilities and marginal distributions can immediately be quantified two specific distributions on the feasible region the uniform distribution and the distribution of random convex combinations with coefficients which are uniform spacings are examined in some detail and the ideas which emerge are illustrated by means of selected examples

a consistency test for rational weights in multi criterion decision analysis with fuzzy pairwise comparisons abstract we present some results and extensions of the use of fuzzy pairwise comparisons in multi criterion decision analysis we address the problem encountered in earlier articles by van laarhoven and pedrycz 1983 and boender et al 1989 in which normalized weights derived from estimates based on ratio scales can exhibit an irrational ordering we derive conditions for a rational ordering and illustrate that the irrational ordering is the result of relative preferences provided by the decision makers we provide a test for consistency that will insure a rational ordering of the normalized weights when using the methods of van laarhoven and pedrycz and boender et al for normalization

exact bayesian moment based inference for the distribution of the small time movements of an ito semimartingale abstract we modify the gallant and tauchen 1996 efficient method of moments emm method to perform exact bayesian inference where exact means no reliance on asymptotic approximations we use this modification to evaluate the empirical plausibility of recent predictions from high frequency financial theory regarding the small time movements of an ito semimartingale the theory indicates that the probability distribution of the small moves should be locally stable around the origin it makes no predictions regarding large rare jumps which get filtered out our exact bayesian procedure imposes support conditions on parameters as implied by this theory the empirical application uses s p index options extending over a wide range of moneyness including deep out of the money puts the evidence is consistent with a locally stable distribution valid over most of the support of the observed data while mildly failing in the extreme tails about which the theory makes no prediction we undertake diagnostic checks on all aspects of the procedure in particular we evaluate the distributional assumptions regarding a semi pivotal statistic and we test by monte carlo that the posterior distribution is properly centered with short credibility intervals taken together our results suggest a more important role than previously thought for pure jump like models with diminished if not absent diffusive component

optimal error properties of finite element methods for second order elliptic dirichlet problems we use the informational approach of traub and wozniakowski 9 to study the variational form of the second order elliptic dirichlet problem lu f on q c rn for f g hw q where r 1 a quasi uniform finite element method using n linear functionals fs f4 has h q norm error j n r 1 n we prove that it is asymptotically optimal among all methods using any information consisting of any n linear functionals an analogous result holds if l is of order 2m if f g hr q where r m then there is a finite element method whose ha q norm error is n 2m r a n for 0 n 2 we show that when n 1 there is no method using n function evaluations whose error is better than q n r thus for n 1 the finite element method with quadrature is asymptotically optimal among all methods using n evaluations of f