**6. Information content and axiomatic properties**

Since the pioneer work by Wang and Kerre [1], researchers have examined the axiomatic properties (i.e., the six axioms listed in Section 2) of fuzzy number ranking methods. The intent of this section is to discuss how the information content for ranking can influence the axiomatic properties in the context of our ranking approach. One key message is that the satisfaction of axioms depends on the selection of information that is deemed relevant to FNR. If more information is selected and considered for ranking, the ranking method is more likely to violate the axioms. This message is aligned with the topic of information basis in the analysis of the Arrow's Impossibility Theorem [39, 40].

#### **6.1 Analysis of Axiom 4**

Axiom 4 somewhat dictates the ranking of non-overlapping fuzzy numbers. If we only consider representative x-values for ranking (i.e., no range, membership ratio and discount factors), our ranking procedure will directly follow the results from Ban and Coroianu [28], and it will thus satisfy Axiom 4. However, if range and membership ratio are considered as relevant information for ranking, Axiom 4 can be violated, and the reason is given below.

Axiom 4 only focuses on the boundary values without considering any distributional information (e.g., range and membership). When multiple attributes are considered for ranking, Axiom 4 can be violated by strengthening the distributional aspect of the inferior fuzzy number (in view of Axiom 4). For example, we have *FO*2≽*FO*<sup>1</sup> in case o (see **Table 5**) according to Axiom 4, no matter how small of membership ratio of *FO*2. However, when we consider range and membership ratio, we obtain *FO*1≽*FO*<sup>2</sup> (see **Table 6**), which violates Axiom 4. Notably, the logic of such violation can be held whenever we deem membership ratio as relevant information for FNR, regardless of the details of the ranking procedures.

Notably, this discussion is not about rejecting Axiom 4. Instead, we want to explain one logical tension with Axiom 4. That is, Axiom 4 dictates some ranking of fuzzy numbers based on their boundary values only, and this opens a chance for the information of range and membership ratio to violate Axiom 4. Alternately, if we choose the index class by [28], representative x-values will be the only information considered for ranking, and the information of range (for example) will become irrelevant for FNR. In other words, if we consider that FNR should involve trade-off with multiple attributes in addition to representative x-value, Axiom 4 could be violated in some situations.
