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A solid travelling salesman problem (STSP) is a travelling salesman problem (TSP) where the salesman visits all the cities only once in his tour using dierent conveyances to travel from one city to another. Costs and environmental eect factors for travelling between the cities using dierent conveyances are dierent. Goal of the problem is to nd a complete tour with minimum cost that damages the environment least. An ant colony optimization (ACO) algorithm is developed to solve the problem. Performance of the algorithm for the problem is compared with another soft computing algorithm, Genetic Algorithm(GA). Problems are solved with crisp as well as fuzzy costs. For fuzzy cost and environmental eect factors, cost function as well as environment constraints become fuzzy. As optimization of a fuzzy objective function is not well de ned, fuzzy possibility approach is used to get optimal decision. To test the eciency of the algorithm, the problem is solved considering only one conveyance facility ignoring the environmental eect constraint, i.e., a classical two dimensional TSP (taking standard data sets from TSPLIB for solving the problem). Dierent numerical examples are used for illustration.

Aggregation operators (AOs) have been studied by many schol- ars. As many AOs are proposed, there is still lacking approach to classify the categories of AO, and to select the appropriate AO within the AO candidates. In this research, each AO can be regarded as a cognitive style or individual dierence. A Cognitive Style and Aggregation Operator (CSAO) model is pro- posed to analyze the mapping relationship between the aggregation operators and the cognitive styles represented by the decision attitudes. Four algorithms are proposed for CSAO: CSAO-1, CSAO-2 and two selection strategies on the basis of CSAO-1 and CSAO-2. The numerical examples illustrate how the choice of the aggregation operators on the basis of the decision attitudes can be determined by the selection strategies of CSAO-1 and CSAO-2. The CSAO model can be applied to decision making systems with the selection problems of the appropriate aggregation operators with consideration of the cognitive styles of the decision makers.

The linear multiobjective transportation problem is a special type of vector minimum problem in which constraints are all equality type and the objectives are conicting in nature. This paper presents an application of fuzzy goal programming to the linear multiobjective transportation problem. In this paper, we use a special type of nonlinear (hyperbolic and exponential) membership functions to solve multiobjective transportation problem. It gives an optimal compromise solution. The obtained result has been compared with the solution obtained by using a linear membership function. To illustrate the methodology some numerical examples are presented.

In this paper, relations among the membership values of gener- alized fuzzy languages such as intuitionistic fuzzy language, interval-valued fuzzy language and vague language are studied. It will aid in studying the properties of one language when the properties of another are known. Further, existence of a minimized nite automaton with vague ( final) states for any vague regular language recognized by a nite automaton with vague ( final) states is shown in this paper. Finally, an ecient algorithm is given for minimizing the nite automaton with vague ( final) states. Similarly, it can be shown for intuitionistic fuzzy regular language. These may contribute to a better understanding of the role of nite automaton with vague ( final) states or the nite automaton with intuitionistic fuzzy ( final) states while studying lexical analysis, decision making etc.

Intuitionistic Fuzzy Modal Operator was defined by Atanassov in cite{at3}in 1999. In 2001, cite{at4}, he introduced the generalization of thesemodal operators. After this study, in 2004, Dencheva cite{dencheva} definedsecond extension of these operators. In 2006, the third extension of thesewas defined in cite{at6} by Atanassov. In 2007,cite{gc1}, the authorintroduced a new operator over Intuitionistic Fuzzy Sets which is ageneralization of Atanassov's and Dencheva's operators. At the same year,Atanassov defined an operator which is an extension of all the operatorsdefined until 2007. The diagram of One Type Modal Operators onIntuitionistic Fuzzy Sets was introduced first in 2007 by Atanassovcite{at10}. In 2008, Atanassov defined the most general operator and in2010 the author expanded the diagram of One Type Modal Operators onIntuitionistic Fuzzy Sets with the operator $Z_{alpha ,beta }^{omega }$.Some relationships among these operators were studied by several researchers%cite{at5}-cite{at8} cite{gc1}, cite{gc3}, cite{dencheva}- cite%{narayanan}.The aim of this paper is to expand the diagram of one type modal operatorsover intuitionistic fuzzy sets . For this purpose, we defined a new modaloparator $Z_{alpha ,beta }^{omega ,theta }$ over intuitionistic fuzzysets. It is shown that this oparator is the generalization of the operators$Z_{alpha ,beta }^{omega },E_{alpha ,beta },boxplus _{alpha ,beta},boxtimes _{alpha ,beta }.$

This paper investigates existence and uniqueness results for the first order fuzzy integro-differential equations. Then numerical results and error bound based on the left rectangular quadrature rule, trapezoidal rule and a hybrid of them are obtained. Finally an example is given to illustrate the performance of the methods.

In this paper we present some properties of set-norm exhaustive set multifunctions and also of atoms and pseudo-atoms of set multifunctions taking values in the family of non-empty subsets of a commutative semigroup with unity.

We de ne approximate xed point in fuzzy norm spaces and prove the existence theorems, we also consider approximate pair constructive map- ping and show its relation with approximate fuzzy xed point.

The objective of this paper is to deal with the fuzzy conic program- ming problems. The aim here is to derive weak and strong duality theorems for a general fuzzy conic programming. Toward this end, The convexity-like concept of fuzzy mappings is introduced and then a speci c ordering cone is established based on the parameterized representation of fuzzy numbers. Un- der this setting, duality theorems are extended from crisp conic optimization problems to fuzzy ones.

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