A new similarity function for generalized trapezoidal fuzzy numbers

Numerous authors have proposed functions to quantify the degree of similarity between two fuzzy numbers using various descriptive parameters, such as the geometric distance, the distance between the centers of gravity or the perimeter. However, these similarity functions have drawback for specific situations. We propose a new similarity measure for generalized trapezoidal fuzzy numbers aimed at overcoming such drawbacks. This new measure accounts for the distance between the centers of gravity and the geometric distance but also incorporates a new term based on the shared area between the fuzzy numbers. The proposed measure is compared against other measures in the literature

An improvised similarity measure for generalized fuzzy numbers

Similarity measure between two fuzzy sets is an important tool for comparing various characteristics of the fuzzy sets. It is a preferred approach as compared to distance methods as the defuzzification process in obtaining the distance between fuzzy sets will incur loss of information. Many similarity measures have been introduced but most of them are not capable to discriminate certain type of fuzzy numbers. In this paper, an improvised similarity measure for generalized fuzzy numbers that incorporate several essential features is proposed. The features under consideration are geometric mean averaging, Hausdorff distance, distance between elements, distance between center of gravity and the Jaccard index. The new similarity measure is validated using some benchmark sample sets. The proposed similarity measure is found to be consistent with other existing methods with an advantage of able to solve some discriminant problems that other methods cannot. Analysis of the advantages of the improvised similarity measure is presented and discussed. The proposed similarity measure can be incorporated in decision making procedure with fuzzy environment for ranking purposes

A Method to Construct Approximate Fuzzy Voronoi Diagram for Fuzzy Numbers of Dimension Two

In this paper, we propose an approximate "fuzzy Voronoi" diagram(FVD)for fuzzy numbers of dimension two (FNDT) by designing an extension ofcrisp Voronoi diagram for fuzzy numbers. The fuzzy Voronoi sites are defined asfuzzy numbers of dimension two. In this approach, the fuzzy numbers have a convexcontinuous differentiable shape. The proposed algorithm has two stages: in the firststage we use the Fortune’s algorithm in order to construct a "fuzzy Voronoi" diagramfor membership values of FNDTs that are equal to 1. In the second stage, we proposea new algorithm based on the Euclidean distance between two fuzzy numbers in orderto construct the approximate "fuzzy Voronoi" diagram for values of the membershipof FNDTs that are smaller than 1. The experimental results are presented for aparticular shape, the fuzzy ellipse numbers

Multiattribute Group Decision Making with Unknown Decision Expert Weights Information in the Framework of Interval Intuitionistic Trapezoidal Fuzzy Numbers

The aim of this paper is to investigate an approach to multiattribute group decision making with interval intuitionistic trapezoidal fuzzy numbers, in which the decision expert weights are unknown. First, we introduce a distance measure between two interval intuitionistic trapezoidal fuzzy matrixes, and based on the distance between individual matrix and extreme matrix, as well as the average matrix, we obtain the decision expert weights. Second, we utilize the interval intuitionistic trapezoidal fuzzy weighted geometric (IITFWG) operator and the interval intuitionistic trapezoidal fuzzy ordered weighted geometric (IITFOWG) operator to aggregate all individual interval intuitionistic trapezoidal fuzzy decision matrices into a collective interval intuitionistic trapezoidal fuzzy decision matrix and then derive the group overall evaluation values of the given alternatives. Finally, an illustrative example of emergency alternatives selection is given to demonstrate the effectiveness and superiority of the proposed method

Fuzzy logic controller for robot navigation in an unknown environment

This study aims to allow the robot to move safely without colliding with obstacles to reach a specified position in an unknown environment. To achieve the aim of the study, a fuzzy controller was proposed and employed in intelligent mobile robot navigation strategies within unknown environments. This fuzzy controller has four inputs (one target angle and three obstacle distance), two outputs (left and right speed) and 9 numbers of rules. A virtual mobile robot, E-puck robot in the Webots simulator was used to evaluate the performance of the proposed method. Few features such as time travelling, distance travelling of the output responses were analyzed. Comparisons are made between proposed fuzzy logic and Motlagh fuzzy controller. The simulation results were presented to verify the effectiveness of the proposed architectures in an unknown environment

Using the Fuzzy Grey Relational Analysis Method in Wastewater Treatment Process Selection

Due to the variety of treatment processes, the decision to choose the best treatment process is difficult. This paper describes a fuzzy grey relational analysis (GRA) method for selection of the optimal wastewater treatment process. The rating of all alternatives and the weight of each criterion is described by linguistic variables, which can be expressed in triangular fuzzy numbers. Then, a vertex method is used to calculate the distance between two triangular fuzzy numbers. According to the concept of the GRA, a fuzzy relative relational degree is defined to determine the ranking order of all alternatives by calculating the degree of fuzzy grey relational coefficient to both the fuzzy positive ideal solution (FPIS) and fuzzy negative ideal solution (FNIS) simultaneously. Furthermore, a case study is carried out and solved by both methods (i.e., GRA and fuzzy GRA) to show the feasibility and effectiveness of the proposed method. In the case study, five anaerobic wastewater treatment alternatives are evaluated and compared against technical, economic, environmental and administrative criteria and their sub-criteria. Finally, the related results of ranking alternatives from two methods are compared with each other's. By using both Fuzzy GRA and GRA, ABR process has been selected as the first priority and the best anaerobic process. The frequency count assessment of the Iran's industrial parks' WWTPs which have used this method and their performance, proved the priority of this method

Defuzzication through a Bi-Symmetrical Weighted Function

Abstract: In this endeavor, the researchers discuss the problem of defuzzification by minimizing the weighted distance between two fuzzy quantities. Also, this study obtains the nearest point with respect to a fuzzy number and shows that this point is unique relative to the weighted distance. By utilizing this point, a method is presented for effectively ranking various fuzzy numbers and their images to overcome the deficiencies of the previous techniques. Finally, several numerical examples following the procedure indicate the ranking results to be valid

Traffic Flow on Urban Networks with Fuzzy Information

Many methods of analysis of traffic on transport networks have been proposed which assume crisp travel time. Most of them are based on Wardrop's principle which says in particular “"The travel time on all routes actually used equal to and no greater than those which would be experienced by a single vehicle on any unused route.”" The principle represents a state of user equilibrium under the condition that drivers can get perfect traffic information. Though this principle reflects the definition of the crisp performance function used in traffic assignment, the drivers dynamically make decisions about route choice behaviour with their experience and given information. A method commonly used to represent such perception is stochastic traffic assignment. In the real world, however, the driver can only use fuzzy traffic information even if several types of information are available. The objective of the study is to formulate the fuzzy user equilibrium with fuzzy travel time and show the application of the techniques to an actual problem. First, a basic survey was carried out to ascertain the perception of drivers on an urban transportation network. The network includes the Hanshin Expressway and urban streets in the Osaka area. In the survey, the travel time for streets and expressways on the same O-D (Origin-Destination) are assumed to be Triangular Fuzzy Numbers (TFN). A TFN is simply defined as (Tl, To, Tr) which show the smallest value, centre of time values, and largest value respectively according to the perceived travel time T. Therefore, To is recognized to be an informed and physical travel time. Typical features of perception of travel time are summarized from the survey results. The membership function of the fuzzy number on travel time can be displayed once this database is constructed from the empirical survey. Second, the descriptive method of route choice behaviour is introduced to design the traffic assignment model. The crisp travel time for link a, Ta, is extended to fuzzy number TFa with the spread parameters described above. Two concepts of comparison among fuzzy travel times are introduced. They are the centre of gravity method and the generalized distance method based on compatibility. The former is the very simple concept that the centre of gravity point of a fuzzy number is adopted as a representative value of TFa. The latter method is based on the α-cut concept of fuzzy sets. The definition of generalized distance between fuzzy numbers is defined as the sum of successive intervals between numbers for each a value as it increases from zero to one. It is interesting that with TFNs, this can be carried out rapidly by adding the areas of the triangles in each case. Third, it is assumed that the state of user equilibrium is also generated even if fuzziness of travel time exists. Different results for user equilibrium are observed for conditions of fuzzy information when compared with those obtained under conditions of prefect information. In other words, the link performance function is extended in view of the concept of fuzzy numbers. In particular, the extension principle of fuzzy numbers allows that the comparison methods mentioned above are also valid when fuzzy travel time is applied on the route. Therefore, the fuzzified user equilibrium assignment model can be proposed with these concepts. In this section, the Fuzzified Frank-Wolfe(FFW) algorithm is introduced to obtain a fuzzy optimal solution as the solution of a conventional problem. In changing the algorithm, the modification is only to replace the crisp value of the travel time function t(x) with the representative values of the fuzzy travel time function tF(x). The fuzzified algorithm, therefore, is easily derived from a simple extension of the conventional algorithm. The results of a numerical example are presented to consider the stability of the algorithm. Different results of user equilibrium are observed when compared with those obtained under conditions of information. The relation between the width of the fuzzy travel time distribution and traffic flow is estimated on each link. It is observed that the user equilibrium flows shift according to the fuzzified link performance function. It is also mentioned that the idea can be applied to produce Fuzzy Incremental Assignment (FIA). Fourth, the application of the proposed method to a realistic problem is discussed. The information given to the drivers seems to change their perception of link travel time. In particular, this fact is usually observed when the change of the perceived width of fuzzy numbers according to the travel time information in TFN has an impact on the traffic flow on networks. Because the different definition of left and right spreads of travel time represents the change in human perception under different conditions of information, the impact can be evaluated as a change in traffic volume on the network. In conclusion, the relationship between information and traffic flow can be described by the proposed method. It becomes obvious that the traffic equilibrium flow changes according to traffic information which is available to the drivers. The results of traffic flow analysis under fuzzy information, therefore, become useful for the discussion of a future traffic information system

An integrated picture fuzzy ANP-TODIM multi-criteria decision-making approach for tourism attraction recommendation

In this paper, the picture fuzzy score and accuracy function are first defined. Then, a corresponding comparative method between two picture fuzzy numbers (PFNs) is developed. Next, a novel normalized picture fuzzy distance measure between two PFNs is disclosed, and part of the characteristics of the proposed distance measure are discussed. Afterwards, on the basis of the analytic network process (ANP) and an Acronym in Portuguese of Interactive and Multi-Criteria Decision-Making (TODIM) methods, an integrated ANP-TODIM approach is developed to resolve multi-criteria decision-making (MCDM) where the weights of the criteria are fully unknown. We use ANP approach to decide the weights of criteria on the basis of expert mean assessment method, and TODIM is utilized to obtain the ranking of alternatives. Finally, an illustrative example of an optimal tourism attraction recommendation is provided to testify applicability of the developed decision-making method and prove that its results are effective and reasonable. First published online 3 December 201

Fuzzy extension of the CODAS method for multi-criteria market segment evaluation

One of the important activities of a company that can increase its competitiveness is market segment evaluation and selection (mse/mss). We can usually consider mse/mss as a multi-criteria decision-making (mcdm) problem, and so we need to use an mcdm method to handle it. Uncertainty is one of the important factors that can affect the process of decision-making. Fuzzy mcdm approached have been designed to deal with the uncertainty of decision-making problems. In this study, a fuzzy extension of the codas (combinative distance-based assessment) method is proposed to solve multi-criteria group decision-making problems. We use linguistic variables and trapezoidal fuzzy numbers to extend the codas method. The proposed fuzzy codas method is applied to an example of market segment evaluation and selection problem under uncertainty. To validate the results, a comparison is performed between the fuzzy codas and two other mcdm methods (fuzzy edas and fuzzy topsis). A sensitivity analysis is also carried out to demonstrate the stability of the results of the fuzz codas. For this aim, ten sets of criteria weights are randomly generated and the example is solved using each set separately. The results of the comparison and the sensitivity analysis show that the proposed fuzzy codas method gives valid and stable results