In this paper, we first of all define the distance measure entitled generalized Hausdorff distance between two trapezoidal generalized fuzzy numbers (TGFNs) that has been introduced by Chen [10]. Then using a other distance and combining with generalized Hausdorff distance, we define the similarity measure. The basic properties of the above mentioned similarity measure are proved in detail. Finally we rank two generalized fuzzy numbers using distance measure and similarity measure between them.

Scott and Szewczyk in Technometrics, 2001, have introduced a similarity measure for two densities ¦1 and  ¦2 , bySIM(¦1 ¦2)=< ¦1 ¦2>/Ö<¦1 ¦2> <¦1 ¦2>Where<¦1 ¦2>ò+¥-¥¦1(c,q1) ¦2(c,q2)dc sim(¦1 ¦2) has some appropriate properties that can be suitable measures for the similarity of ¦1 and ¦2 . However, due to some restrictions on the value of parameters and the kind of densities, discrete or continuous, it cannot be used in general. The purpose of this article is to give some other measures, based on modified Scott's measure, and Kullback information, which may be better than sim (¦1 ¦2) in some cases. The properties of these new measures are studied and some examples are provided.

Sustainable tourism is one of today's rapidly growing economic areas. Promotional activities should be carried out to increase the income from sustainable tourism. To carry out these studies effectively, it is necessary to choose the best centre. Multi-Criteria Decision-Making (MCDM) approaches can be used to select the best tourism centre. In this paper, a new generalized similarity measure on intuitionistic trapezoidal fuzzy multi-numbers is developed for decision information. The desirable properties of this proposed measure are presented in detail. Further, we develop an approach to the MCDM problem based on the proposed measure. Finally, the effectiveness and applicability of our proposed MCDM model, as well as comparison analysis with other methods, are illustrated with a practical example.

generalized trapezoidal fuzzy numbers (GTFNs) have been widely applied in uncertain decision-making problems. The similarity between GTFNs plays an important part in solving such problems, while there are some limitations in existing similarity measure methods. Thus, based on the cosine similarity, a novel similarity measure of GTFNs is developed which is combined with the concepts of geometric distance, center of gravity, area and perimeter after analyzing the limitations of previous methods. Then comparative analysis is conducted with existing similarity measures, and the results show that the novel similarity measure has better distinguishability and lower invalidity. Furthermore, a general process, which combines the new similarity measure of GTFNs with compromise methods, is developed to deal with multi-attribute group decision making (MAGDM) problem. Finally, we combine fuzzy VIKOR with the general process as illustrated example, which proves the superiority of the developed similarity measure in solving MAGDM problem.

Document clustering has been widely used in information retrieval systems in order to improve the efficiency and also the effectiveness of ranked output systems using clustering hypothesis. Based on this hypothesis, documents relevant to a query tend to be highly similar in the context defined by the query. In this way, a pair of documents has an overall similarity (ignoring the query) and a specific similarity (similarity of a pair of documents given a query). A Query-Sensitive similarity measure (QSSM) is a mechanism to measure the similarity of two documents given a query. In this paper, in the first step, we identify the sources of information that may be used for this purpose. In the second step, we propose a QSSM based on these information sources. Finally, we propose a parametric QSSM that simultaneously makes use of the product and weighted sum to fuse the information from the identified sources. A genetic algorithm is used to learn the optimal values of parameters in this measure for a specific collection. The leave-one-out method is used to evaluate the proposed learning scheme. Our motivation for this is to see whether the learning scheme can perform significantly better than the measure proposed in the second step. Using several document collections, the performance of each measure is evaluated and the results are compared with other QSSMs proposed in the past research.

Recommender Systems (RS) provide personalized recommendation according to the user need by analyzing behavior of users and gathering their information. One of the algorithms used in recommender systems is user-based Collaborative Filtering (CF) method. The idea is that if users have similar preferences in the past, they will probably have similar preferences in the future. The important part of collaborative filtering algorithms is allocated to determine similarity between objects. Similarities between objects are classified to user-based similarity and item-based similarity. The most popular used similarity metrics in recommender systems are Pearson correlation coefficient, Spearman rank correlation, and Cosine similarity measure.Until now, little computation has been made for optimal similarity in collaborative filtering by researchers. For this reason, in this research, we propose an optimal similarity measure via a simple linear combination of values and ratio of ratings for user-based collaborative filtering by the use of Firefly algorithm; and we compare our experimental results with Pearson traditional similarity measure and optimal similarity measure based on genetic algorithm. Experimental results on real datasets show that proposed method not only improves recommendation accuracy significantly but also increases quality of prediction and recommendation performance.

The most important issue with trajectory analysis is calculating similarity between trajectories. In this paper a novel method for measuring similarity between trajectories based on the cost to match a set of trajectories segments was introduced. The similarity between two trajectories is defined as a minimum cost to match a trajectory to the other one. For this purpose, the segment based distance was introduced to as a cost of matching two trajectories segments. In addition, the dynamic programming technique is used to implement the time warp method. We performed some experiments to compare the proposed similarity measure with the similar approaches in the application of trajectory classification. The empirical quality of the proposed similarity measure was evaluated on 1-nearest neighbor (1-NN) classification task using 13 publicly available data sets. Compared to the other well-known similarity measures, the proposed method proved to be effective in the considered experiments based on the accuracy of classification.

Distance size, similarity size, and entropy size provide useful results for decision makers to make decisions about problems with uncertain data. The main focus of this research is to introduce a new measure for interval-valued intuitive fuzzy numbers. Many of the defined measures have shortcomings such as not being comprehensive, high volume of calculations and application in limited cases. Therefore, the main goal of this research is to introduce a measure of distance and similarity with a new and reduced approach for intuitive fuzzy numbers with a value interval. After presenting the structure and effective indicators in the proposed size, it can be seen that the amount of calculations is clearly reduced in the defined interval size. In addition, the proof that size properties hold for it is shown correctly. The presented size structure has the ability to be combined with the process related to multi-criteria decision making and medical diagnosis problems. For this purpose, while presenting hybrid algorithms, effective applications of it have been given by mentioning several prominent examples.