Ranking Fuzzy numbers(FNs) was a critical issue in Fuzzy computing field. Generally, triangular FNs, trapezoidal FNs, and even interval-valued FNs(IVFNs) were often expressed in ranking. However, ranking intuitionistic FNs(IFNs) were less mentioned due to the complicated components in membership functions. Herein, we will develop Fuzzy binary relation that is an extended Fuzzy preference relation(EFPR) to express the preference degree of two IFNs, and then the relation is improved to be a relative preference relation(RPR) used to rank a set of IFNs. Since EFPR on IFNs is a total ordering relation, RPR will be also a total ordering relation. Based on belonging and non-belonging components of membership functions in IFNs, using EFPR being also Fuzzy preference relation(FPR) is suitable to compare FNs on pairwise, but time complexity on Fuzzy operation of comparison computing is complicated. Hence, RPR is developed to avoid comparing on pairwise. Through yielding RPR values for a set of IFNs, IFNs are effectively and efficiently ranked to utilize related decision-making problems.

The purpose of this study has been concretized in terms of impact of mobile advertising on the purchase decision of the consumer and aimed to answer to research questions: which is the underlying structure of criteria in mobile advertising? And which criteria are important for consumers in evaluating mobile advertising? A sample of consumers in Taiwan was surveyed and using Fuzzy judgment to determine the vague perception of consumers. In addition, the proposed Fuzzy linguistic preference relation (FLPR) is used to express the subjective preferences of consumers with respect to the considered criteria of mobile advertising, and also use the criterion gauges to evaluate the implementation of mobile advertising strategies. Finally, an empirical study is illustrated to demonstrate that the proposed method is more suitable than the traditional method, especially when the consumer judgments are likely be inconsistent in pair-wise comparison. The presented Fuzzy linguistic preference relation method is an easy and practical way to provide a mechanism for improving consistency in perceptual measure of mobile advertising.

In this paper, we first use a Fuzzy preference relation with a membership function representing preference degree for comparing two interval-valued Fuzzy numbers and then utilize a relative preference relation improved from the Fuzzy preference relation to rank a set of interval-valued Fuzzy numbers. Since the Fuzzy preference relation is a total ordering relation that satisfies reciprocal and transitive laws on interval-valued Fuzzy numbers, the relative preference relation is also a total ordering relation. Practically, the Fuzzy preference relation is more reasonable on ranking interval-valued Fuzzy numbers than defuzzification because defuzzification does not present preference degree between Fuzzy numbers and loses messages. However, Fuzzy pair-wise comparison for the Fuzzy preference relation is more complex and difficult than defuzzification. To resolve Fuzzy pair-wise comparison tie, the relative preference relation takes the strengths of defuzzification and the Fuzzy preference relation into consideration. The relative preference relation expresses preference degrees of interval-valued Fuzzy numbers over average as the Fuzzy preference relation does, and ranks Fuzzy numbers by relative crisp values as defuzzification does. In fact, the application of relative preference relation was shown in traditional Fuzzy numbers, such as triangular and trapezoidal Fuzzy numbers, for previous approaches. In this paper, we extend and utilize the relative preference relation on interval-valued Fuzzy numbers, especially for triangular interval-valued Fuzzy numbers. Obviously, interval-valued Fuzzy numbers based on the relative preference relation are easily and quickly ranked, and able to reserve Fuzzy information.

The competition among shopping malls is getting increasingly more intense. In order to enhance corporate competitiveness and profitability, how to retain key loyal customers has become an important issue for the managers of shopping malls. Customer loyalty programs are a universal practice. This study proposes an analytical model for developing the strategic management of customer loyalty programs. Based on quality function deployment, our model constructs a Fuzzy relational matrix to integrate the customer benefits and the customer loyalty programs. The results contribute to a better understanding of the different importance of customer benefits with respect to “ Utilitarian benefits” , “ Hedonic benefits” and “ Symbolic benefits” . The study’ s findings provide prioritized customer loyalty programs, including “ Newest information transmissions, ” “ Communication service specialists, ” “ Exclusive free gift promotions, ” and “ VIP lounge service, ” which can be used to determine the directions for the improvement of shopping malls. The study has implications for the theory and practitioners. First, it allows the integration of customer benefits in customer loyalty programs and the consideration of VIP customers in the programs. Second, the requirement evaluation considering Fuzzy linguistic preferences allows for prioritizing the customer loyalty programs when dealing with uncertainty and vagueness. Finally, this approach helps to obtain more realistic results by integrating Fuzzy decision-making into QFD.

Dombi and Baczy nski presented a new approach to the problem of implication operation by introducing the preference implication, which has very advantageous properties. In this paper, it is presented how the preference implication is connected with soft inequalities and with sigmoid functions. Utilizing this connection the preference implication-based preference measure for two Fuzzy numbers is introduced and its key properties, including the reciprocity, are described. Then, the exact expression for computing the new preference measure for trapezoidal Fuzzy numbers is presented. Here, using the new preference measure, two crisp relations over trapezoidal Fuzzy numbers are introduced. It is shown that one of them is a strict (but not a total) order relation, and the other one is an equivalence relation. The strict order relation can be used to rank comparable Fuzzy numbers, while the equivalence relation, which we call the indifference relation, expresses that the order of some Fuzzy numbers is indifferent. These two crisp relations can be used to rank a collection of trapezoidal Fuzzy numbers. Lastly, the proposed ranking method is compared with some well-known existing Fuzzy number ranking methods.

It is well known fact that binary relations are generalized mathematical functions. Contrary to functions from domain to range, binary relations may assign to each element of domain two or more elements of range. Some basic operations on functions such as the inverse and composition are applicable to binary relations as well. Depending on the domain or range or both are Fuzzy value Fuzzy set, interval Fuzzy value Fuzzy set or Fuzzy number value Fuzzy set, define of the Fuzzy relation is different. Given a Fuzzy relation, its domain and range are Fuzzy number value Fuzzy sets. In this paper, initially we define Fuzzy number value Fuzzy sets and then propose Fuzzy number-valued Fuzzy relation (FN-VFR). We also introduce property of reflexive, symmetric, transitive and equivalence relation of FN-VFR. As follow, we prove some theorems for FN-VFR with property of reflexive, symmetric and transitive. Also, we show examples for FN-VFR.

If there is a Fuzzy relation $\widetilde{R}$ between two spaces $U$ and $V$, the Fuzzy approximations in both spaces based on $\widetilde{R}$ are widely studied, and they basically reflect only the influence from one space to another. In this paper, on each space of $U$ and $V$, two new Fuzzy relations are derived from $\widetilde{R}$, a positive low-value relation and a conservative high-value relation, to reflect the interaction and feedback between the two spaces. So, the Fuzzy approximations based on them can reflect the combination of the action and the reaction from one space to another. Therefore, two spaces $U$ and $V$ are closely accompanied, and $(U, V, \widetilde{R})$ is a whole, so it is called a Fuzzy accompanied approximation space (FAAS). In an FAAS, the properties of the Fuzzy approximation models on each space are studied, the relationships between Fuzzy approximation models of two spaces are researched, and examples to show how the approximation operator models in the FAAS to solve practical problems from multiple perspectives are also illustrated. More importantly, when the Fuzzy relation $\widetilde{R}$ is a binary relation $R$ or the two spaces are the same, the special cases of FAAS are investigated and some important new models and new results are obtained, which add new ideas and methods to the current research.

The purpose of this study is to prioritize leagility enablers in food and beverage industries in Qazvin Province with the hybrid method of FLin PreRa-FQFD. In doing so, it directly assessed the impact of leagility enablers on its attributes with fewer pairwise comparisons and yields consistent priority ranking which lead to improvement in decision-making. In this study, after reviewing literature and theoretical background of the leagility, its attributes and enablers are identified and a framework for prioritizing these indicators as well as major competitive advantage is designed. The study was based on 38 companies from the food and beverage industries, in Qazvin province. The findings indicate that competitive advantage "costs" is the most important competitive advantage in this industry. "Customer and market sensitiveness", is the most important attribute based on expert viewpoints. Also among the enablers, "rapid introduction of new products and production cycle time reduction" in achieved the highest weights and priorities.

This paper proposes a quadratic programming method (QPM) for ranking alternatives based on multiplicative preference relations (MPRs) and Fuzzy preference relations (FPRs). The proposed QPM can be used for deriving a ranking from either a MPR or a FPR, or a group of MPRs, or a group of FPRs, or their mixtures. The proposed approach is tested and examined with two numerical examples, and comparative analyses with the existing methods are provided to show the effectiveness and advantages of the QPM.