**6.2 Dependence of** *rngmin* **and** *memmax*

As we use reference values (i.e., *rngmin* and *memmax*) to evaluate the discount factors (i.e., *drng* and *dmem*), the ranking index, *Irank*, belongs to the second class of ranking indices according to the classification by [1]. In their axiomatic analysis, they have identified five indices of the second class, i.e., *J <sup>K</sup>*(), *K*(), *CHK*(), *W*() and *KPK*(), which all do not satisfy Axioms 5 and 6. Without listing counter-examples, *Irank* of the same class follows the same conclusion because they share a common feature of these ranking indices, i.e., use of reference values.

Why using reference values could violate Axioms 5 and 6? It is because the index values would depend on the information that is external to the fuzzy numbers themselves. For example, if a fuzzy number with a very small range is added to a set of fuzzy numbers for ranking (i.e., *FR*), this newly added fuzzy number will decrease the reference value, *rngmin*, and thus generally decrease the range discount values (*drng*) for the original set of fuzzy numbers. Then, all values of *Irank* would change because of adding a new fuzzy number to the set (i.e., *FR*).

While it seems undesirable by setting *rngmin* and *memmax* per individual sets of fuzzy numbers, can we simply set these two values as universal numbers that are applicable to all ranking problems (e.g., simply set *memmax* = 1)? Theoretically, it is a viable option. However, by doing so, we somehow lose our interpretation of "discount" factors that are relevant to a given set of fuzzy numbers that we want to rank in the problem. For example, it is not easy to interpret if the range of a fuzzy number, say 5, is large or small until we know a reference for comparison (e.g., if *rngmin* = 1, the range of 5 will quite large). In other words, the reference values, *rngmin* and *memmax*, provide a numerical context as relevant information for comparison.

To close this section, we want to make a note about the historical development of the Arrow's Impossibility Theorem, which proves that no voting method (or social welfare function) can satisfy a set of "reasonable" properties (or axioms) [41]. One famous "escaping route" is the information basis approach, which classifies the information content (or availability) for interpersonal comparisons with different axiomatic results [39, 40]. Back to our context, if representative x-value is taken as the only relevant information for FNR, the results by [28] are sufficient to design a ranking index that satisfies the six axioms in Section 2. However, if additional information is considered for FNR, the axiomatic properties cannot be guaranteed. To us, this tension seems fundamental.
