Crisp Point Research Articles

Fuzzy analytic hierarchy process: A logarithmic fuzzy preference programming methodology

Fuzzy analytic hierarchy process (AHP) proves to be a very useful methodology for multiple criteria decision-making in fuzzy environments, which has found substantial applications in recent years. The vast majority of the applications use a crisp point estimate method such as the extent analysis or the fuzzy preference programming (FPP) based nonlinear method for fuzzy AHP priority derivation. The extent analysis has been revealed to be invalid and the weights derived by this method do not represent the relative importance of decision criteria or alternatives. The FPP-based nonlinear priority method also turns out to be subject to significant drawbacks, one of which is that it may produce multiple, even conflict priority vectors for a fuzzy pairwise comparison matrix, leading to entirely different conclusions. To address these drawbacks and provide a valid yet practical priority method for fuzzy AHP, this paper proposes a logarithmic fuzzy preference programming (LFPP) based methodology for fuzzy AHP priority derivation, which formulates the priorities of a fuzzy pairwise comparison matrix as a logarithmic nonlinear programming and derives crisp priorities from fuzzy pairwise comparison matrices. Numerical examples are tested to show the advantages of the proposed methodology and its potential applications in fuzzy AHP decision-making.

N-IDEALS OF SUBTRACTION ALGEBRAS

Using N-structures, the notion of an N-ideal in a subtrac- tion algebra is introduced. Characterizations of an N-ideal are discussed. Conditions for an N-structure to be an N-ideal are provided. The de- scription of a created N-ideal is established. A (crisp) set A in a universe X can be defined in the form of its characteris- tic function µA : X ! {0,1} yielding the value 1 for elements belonging to the set A and the value 0 for elements excluded from the set A. So far most of the generalization of the crisp set have been conducted on the unit interval (0,1) and they are consistent with the asymmetry observation. In other words, the generalization of the crisp set to fuzzy sets relied on spreading positive infor- mation that fit the crisp point {1} into the interval (0,1). Because no negative meaning of information is suggested, we now feel a need to deal with negative information. To do so, we also feel a need to supply mathematical tool. To attain such object, Jun et al. (4) introduced a new function which is called negative-valued function, and constructed N-structures. They discussed N- subalgebras and N-ideals in BCK/BCI-algebras. Schein (6) considered systems of the form (';-,\), where ' is a set of functions closed under the composition - of functions (and hence (';-) is a function semigroup) and the set theo- retic subtraction (and hence (';\) is a subtraction algebra in the sense of (1)). He proved that every subtraction semigroup is isomorphic to a dierence semigroup of invertible functions. Zelinka (7) discussed a problem proposed by Schein concerning the structure of multiplication in a subtraction semigroup. He solved the problem for subtraction algebras of a special type, called the atomic subtraction algebras. Jun et al. (2, 3) introduced the notion of ideals in subtraction algebras and discussed characterization of ideals. Jun et al. (5) provided conditions for an ideal to be irreducible. They introduced the notion of an order system in a subtraction algebra, and investigated related proper- ties. They provided relations between ideals and order systems, and dealt with

Genetic learning of fuzzy integrals accumulating human-reported environmental stress

In this paper, we develop models based on fuzzy integrals (both of the Choquet and Sugeno type) for accumulating annoyance by noise, odor or light caused by particular sources or activities. As underlying fuzzy measures, we have opted for k-maxitive measures (in particular 1-maxitive or 2-maxitive) as the best known crisp model points in this direction. The fuzzy measures are learnt from survey data and optimized using genetic algorithms. Attention is paid to several types of inconsistencies that typically arise in data sets collected through social surveys. Also, special care is taken to make sure that the Sugeno integral and the genetic algorithm that optimizes the associated fuzzy measure operates solely on the ordinal scale of linguistic labels.

Handling design perceptions: an axiomatic design perspective

This study develops a method for determining optimal design embodiments under the following assumptions: (1) the design approach involves the axiomatic design theory and (2) the design-relevant information refers to a designer’s intuition, expressible as f-granular information. The optimal design embodiments are then obtained by determining such optimal set of crisp points having definiteness positions close to each other. Here, a crisp point means a value of one of the parameters needed to specify the design. The definiteness position of a crisp point depends on two factors: (1) how much a designer knows about this point, and (2) how desirable it is to design the actual product. The usefulness of the proposed method is demonstrated by an example that involves both weight and dimension measures. As axiomatic-design-theoretic practices move further into the age of design automation, various machine intelligence techniques capable of computing qualitative information (f-granular information) rather than numbers (numerical or semi-numerical information) will be needed. This study is intended to further such investigations.

Analysis of the optimal configuration of energy transportation infrastructure in Asia with a linear programming energy system model : Fujii, Y. et al. International Journal of Global Energy Issues, 2002, 18, (1), 22–43

Fuzzy analytic hierarchy process (AHP) proves to be a very useful methodology for multiple criteria decision-making in fuzzy environments, which has found substantial applications in recent years. The vast majority of the applications use a crisp point estimate method such as the extent analysis or the fuzzy preference programming (FPP) based nonlinear method for fuzzy AHP priority derivation. The extent analysis has been revealed to be invalid and the weights derived by this method do not represent the relative importance of decision criteria or alternatives. The FPP-based nonlinear priority method also turns out to be subject to significant drawbacks, one of which is that it may produce multiple, even conflict priority vectors for a fuzzy pairwise comparison matrix, leading to entirely different conclusions. To address these drawbacks and provide a valid yet practical priority method for fuzzy AHP, this paper proposes a logarithmic fuzzy preference programming (LFPP) based methodology for fuzzy AHP priority derivation, which formulates the priorities of a fuzzy pairwise comparison matrix as a logarithmic nonlinear programming and derives crisp priorities from fuzzy pairwise comparison matrices. Numerical examples are tested to show the advantages of the proposed methodology and its potential applications in fuzzy AHP decision-making.

Nearness concepts in fuzzy neighbourhood spaces and in their fuzzy proximity spaces

The nearness concept giving rise to a fuzzy topological space is a fuzzy relation of closeness between fuzzy subsets and crisp points of the space. Lowen [8] introduced the category of fuzzy neighbourhood spaces, which we denote by FNS. His heuristic idea was that the fuzzy topologies in FNS should be characterized by means of fuzzy neighbourhoods of crisp points. It may be argued that the ‘fuzziness’ of some concepts related to FNS is rather restricted. But, besides the benefits gained from this restriction in that FNS is a category of ‘well behaved’ fuzzy topological spaces (cf. [1, 5, 8, 9, 14, 17, 18]), we argue in this paper that this restriction acts to reduce the above mentioned concept of nearness to the following one: The nearness concept giving rise to a fuzzy neighbourhood space is a fuzzy relation of closeness between the crisp subsets and the crisp points of the space. Such fuzzy relation may naturally arise when fuzziness is engrained in the definition of the points of the space; for instance in the fuzzy unit interval of Hutton [4]. Artico and Moresco [2] introduced a concept of fuzzy proximities. We show that this concept is particular to fuzzy neighbourhood spaces only. We then argue that the nearness concept giving rise to a fuzzy proximity space is a fuzzy relation of proximity defined on all pairs of crisp subsets of the space.

On fuzzy pseudo-normed vector spaces

We provide a method for introducing fuzzy pseudo-metric topologies on sets, and fuzzy pseudo-normed topologies on vector spaces over R or C which will be fuzzy linear topologies. We define fuzzy pseudo-metrics for pairs of crisp points, and fuzzy pseudo-norms for crisp points, as fuzzy real numbers ⩾ 0 (as defined by Hutton). We define the associated fuzzy open balls, with crisp points for their centres and fuzzy real numbers > 0 for their radii. These form a basis for the associated fuzzy pseudo-metric topology. The axioms governing fuzzy pseudo-metric and fuzzy pseudo-norm are straightforward extensions for the corresponding axioms in the crisp case. The formulation conforms with Zadeh's Extension Principle. We show that Katsaras' concept of fuzzy seminorm (Fuzzy Sets and Systems 12 (1984) 143–154) is equivalent to ours, in the sense that both concepts will result in same fuzzy linear topologies.

Dynamic topologies and their applications to crisp topologies, fuzzifications of crisp topologies, and fuzzy topologies on the crisp real line

We introduce and develop topologies whose members are dynamic sets à la Zeleny. This development spawns the notions of phase and co-phase spaces, fuzzy dual, dynamic dual, and co-fuzzy dual with the following results being obtained: Result A. The notion of fuzzy dual provides a unifying framework for the diverse fuzzification schemes of Hutton-Gantner-Steinlage-Warren, Klein, and Lowen. Result B. The dynamic duals of the fuzzy real lines are both homeostatic and coheterostatic. Result C. The fuzzy real lines, re-interpreted as fuzzy duals of R with the usual Euclidean topology T , induce on R , via the notion of co-fuzzy dual, a class of canonical, stratified fuzzy topologies {CO R L }. E.g., if L is a DeMorgan algebra and α ϵ L c -{1}, then the α-compactness of A ⊂ R is equivalent to the ordinary compactness of A w.r.t. T ; and if L is a Hutton algebra, ( R , L, CO R L ) is metrizable in the sense of Hutton and Erceg with Erceg metric d( L) such that d(L) (p r 1, p s 1)=¦r − s¦ (where p r is the crisp point or impulse with support r), i.e., d( L) extends the Euclidean metric on R .

Disciplined Party Voting: A Labor Innovation?

When the Labor Party entered Parliament in N.S.W. in 1891, it intended not to compete with the other two parties for ministerial positions but rather to act as a third party which would offer 'support in return for concessions' and use its numbers to maintain in office whichever party offered the more attractive programme.1 As Crisp points out, This meant that party discipline within the House had to be strict,2 and therefore the Labor Party devised rules and procedures designed to produce the necessary solidarity in divisions. The result, so historians have claimed, was to force the other parties to follow Labor's example of parliamentary organisation. J. D. B. Miller writes: