Ordered fuzzy numbers are defined by Kosiński. In this way, he was going to supplement a fuzzy number by orientation. A significant drawback of Kosiński's theory is that there exist such ordered fuzzy numbers which, in fact, are not fuzzy numbers. For this reason, a fully formalized correct definition of ordered fuzzy numbers is proposed here. Also, the arithmetic proposed by Kosiński has a significant disadvantage. The space of ordered fuzzy numbers is not closed under Kosiński's addition. On the other hand, many mathematical applications require the considered space be closed under used arithmetic operations. Therefore, the Kosinski's theory is modified in this way that the space of ordered fuzzy numbers is closed under revised arithmetic operations. In addition, it is shown that the multiple revised sum of finite sequence of ordered fuzzy numbers depends on its summands ordering.

The main purpose of this paper is to investigate the application potential of ordered fuzzy numbers (OFN) to support evaluation of negotiation offers. The Simple Additive Weighting (SAW) and the Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) methods are extended to the case when linguistic evaluations are represented by OFN. We study the applicability of OFN for linguistic evaluation negotiation options and also provide the theoretical foundations of SAW and TOPSIS for constructing a scoring function for negotiation offers. We show that the proposed framework allows us to represent the negotiation information in a more direct and adequate way, especially in ill-structured negotiation problems, allows for holistic evaluation of negotiation offers, and produces consistent rankings, even though new packages are added or removed. An example is presented in order to demonstrate the usefulness of presented fuzzy numerical approach in evaluation of negotiation offers.

No abstract

From the perspective of each evaluation criterion, any decision alternative is evaluated by means of trapezoidal ordered fuzzy numbers (TrOFN). This approach is justified in the way that some criteria are linguistically evaluated. In this paper, decision alternatives are evaluated using oriented fuzzy Simple Additive Weighting (OF-SAW) scoring function. The ranking of alternatives may be defined by means of a nonincreasing sequence of defuzzified values of a scoring function. Any defuzzification procedure distorts ordered fuzzy numbers in a way that information on imprecision and orientation is lost. This undermines the credibility of the determined alternatives' ranking. The main purpose of this paper is to avoid the defuzzification stage in the OF-SAW method. Thus, the OF-SAW method is equipped with fuzzy scoring order. This OF-SAW method is described as a negotiation scoring system. We study an empirical example of the OF-SAW application and rank some negotiation offers. Here, we focus on the effects of replacing the defuzzified scoring function by a fuzzy one. The obtained conclusions are generalized for the case of any decision alternatives.

Abstract. The return rate is considered here as a fuzzy probabilistic set. Then the expected return is obtained as a fuzzy subset in the real line. This result is a theoretical foundation for new investment strategies. All considered strategies result of comparison profit fuzzy index and limit value. In this way we obtain an imprecise investment recommendation. Financial equilibrium criteria are a special case of comparison of the profit index and the limit value. The following criteria are generalized here: the Sharpe's Ratio, the Jensen's Alpha and the Treynor's Ratio. Moreover, the safety-first criteria are generalized here for the fuzzy case. The Roy Criterion, the Kataoka Criterion and the Telser Criterion are also generalized. Obtained results show that proposed theory is useful for the investment applications.