**5. Recommendations**

OFNs are a form of a record of imprecise information. The purpose of approximation of OFNs by TrOFNs was justified in the paper. Following that premise, six OFN approximation problems by the nearest TrOFN were proposed and discussed in the paper. It appeared that the two remaining approximation problems UC-approximation ad IC-approximation always deliver identical solutions. Additionally, the ICCI-approximation and ICCA-approximation problem always have identical solutions. Therefore, they should not be differentiated. Each of remaining four approximation methods has its advantages and disadvantages.

The main advantage of UC-approximation method is the fact that it is the only discussed method that delivers a solution of any OFN. IT is also the only one of the discussed approximation methods free of the occurrence of error propagation phenomenon of imprecision characteristics, that is, ambiguity index and entropy measure. Thanks to using the UC-approximation for processing complex sets of information, it is always safe. The drawbacks of UC-approximation method are the discrepancies between the characteristics of imprecision of approximated OFN and approximating TrOFN. This distorts the usefulness evaluation of processed information. From the point of view of algebra, UC-approximation is a linear operator of all OFN space.

All other approximation methods might lack the solution of incidental OFN. Such cases can be easily recognised utilising the fact that every TrOFN is characterised by a monotonic sequence of parameters. If, during implementation of a chosen approximation method of a given OFN we will obtain a non-monotonic sequence of parameters, then such an approximation problem has no solution. Attempts should be made to approximate a given OFN using another approximation method. There always exists at least one approximation method with a solution for a given OFN.

CA-approximation method approximates a given OFN by such TrOFN that its oriented energy index is equal to oriented energy index of OFN. This lets us monitor the usefulness of processed information more credibly. A more precise approximation method is ICCS-approximation, which approximates a given OFN by TrOFN of the identical support closure and equally oriented energy index. It allows one to monitor the usefulness of processed information more credibly. This means that CA-approximation should be used only in cases in which ICCA-approximation method does not deliver the solution. For both these approximation methods, there are no universal conditions restraining the set of approximated OFNs.

On the contrary, in case of CI-approximation method, the approximated OFNs must be characterised by a low level of entropy that satisfies condition (82). The CA-approximation method approximated a given OFN by TrOFN with an equally oriented energy index and entropy measure. It lets us monitor the usefulness of processed information more credibly than in case of the approximation obtained by the CA-approximation method.

A significant, common disadvantage of CA-, CI-, and ICCA-approximation methods is the occurrence of zero-error propagation phenomenon, described in Section 4. This usually leads to the conclusion that using such methods of approximation in cases of big, complex sets of information creates a threat for the credible monitoring of processed data usefulness.

In this case, the procedure of approximation method choice depends on individual preferences of the researcher who wants to approximate OFNs by TrOFNs. Nevertheless, it can be clarified by the following procedure:


The above-described procedure of the approximation method choice can be used separately for any number. It is acceptable to use different methods for different numbers in the same empirical implementation.

The results obtained in the paper let us, for instance, use the Oriented Behavioural Present Value [17,18] in portfolio models described in [19,21,22,24].
