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Numerous algorithms have been proposed to solve the shortest-pathproblem; many of them consider a single-mode network and crispcosts. Other attempts have addressed the problem of fuzzy costs ina single-mode network, the so-called fuzzy shortest-path problem(FSPP). The main contribution of the present work is to solve theoptimum path problem in a multimodal transportation network, inwhich the costs of the arcs are fuzzy values. Metropolitantransportation systems are multimodal in that they usually containmultiple modes, such as bus, metro, and monorail. The proposedalgorithm is based on the path algebra and dioid of $k$-shortestfuzzy paths. The approach considers the number of mode changes,the correct order of the modes used, and the modeling of two-waypaths. An advantage of the method is that there is no restrictionon the number and variety of the services to be considered. Totrack the algorithm step by step, it is applied to apseudo-multimodal network.

Due to the explosive growth of the world-wide web, automatictext summarization has become an essential tool for web users. In this paperwe present a novel approach for creating text summaries. Using fuzzy logicand word-net, our model extracts the most relevant sentences from an originaldocument. The approach utilizes fuzzy measures and inference on theextracted textual information from the document to find the most significantsentences. Experimental results reveal that the proposed approach extractsthe most relevant sentences when compared to other commercially availabletext summarizers. Text pre-processing based on word-net and fuzzy analysisis the main part of our work.

In this paper difference methods to solve "fuzzy partial differential equations" (FPDE) such as fuzzy hyperbolic and fuzzy parabolic equations are considered. The existence of the solution and stability of the method are examined in detail. Finally examples are presented to show that the Hausdorff distance between the exact solution and approximate solution tends to zero.

In this paper, we study the finitely many constraints of the fuzzyrelation inequality problem and optimize the linear objectivefunction on the region defined by the fuzzy max-product operator.Simplification operations have been given to accelerate theresolution of the problem by removing the components having noeffect on the solution process. Also, an algorithm and somenumerical and applied examples are presented to abbreviate andillustrate the steps of the problem resolution.

We prove a related fixed point theorem for n mappings which arenot necessarily continuous in n fuzzy metric spaces using an implicit relationone of them is a sequentially compact fuzzy metric space which generalizeresults of Aliouche, et al. [2], Rao et al. [14] and [15].

The main purpose of this paper is to consider the t-best simultaneousapproximation in fuzzy normed spaces. We develop the theory of t-bestsimultaneous approximation in quotient spaces. Then, we discuss the relationshipin t-proximinality and t-Chebyshevity of a given space and its quotientspace.

The aim of this paper is the study of fuzzy basis and dimensionof fuzzy hypervector spaces. In this regard, first the notions of fuzzy linearindependence and fuzzy basis are introduced and then some related results areobtained. In particular, it is shown that for a large class of fuzzy hypervectorspace the fuzzy basis exist. Finally, dimension of a fuzzy hypervector space isdefined and the basic properties of that are investigated.

In this paper, we introduce and study the prime, strongly prime,semiprime and irreducible fuzzy bi-ideals of a semigroup. We characterize thosesemigroups for which each fuzzy bi-ideal is semiprime. We also characterizethose semigroups for which each fuzzy bi-ideal is strongly prime.

In the present paper we define the notion of fuzzy inner productand study the properties of the corresponding fuzzy norm. In particular, it isshown that the Cauchy-Schwarz inequality holds. Moreover, it is proved thatevery such fuzzy inner product space can be imbedded in a complete one andthat every subspace of a fuzzy Hilbert space has a complementary subspace.Finally, the notions of fuzzy boundedness and operator norm are introducedand the relationship between continuity and boundedness are investigated. Itis shown also that the space of all fuzzy bounded operators is complete.

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