Oriented Fuzzy Numbers vs. Fuzzy Numbers

: A formal model of an imprecise number can be given as, inter alia, a fuzzy number or oriented fuzzy numbers. Are they formally equivalent models? Our main goal is to seek formal differences between fuzzy numbers and oriented fuzzy numbers. For this purpose, we examine algebraic structures composed of numerical spaces equipped with addition, dot multiplication, and subtraction determined in a usual way. We show that these structures are not isomorphic. It proves that oriented fuzzy numbers and fuzzy numbers are not equivalent models of an imprecise number. This is the ﬁrst original study of a problem of a dissimilarity between oriented fuzzy numbers and fuzzy numbers. Therefore, any theorems on fuzzy numbers cannot automatically be extended to the case of oriented fuzzy numbers. In the second part of the article, we study the purposefulness of a replacement of fuzzy numbers by oriented fuzzy numbers. We show that for a portfolio analysis, oriented fuzzy numbers are more useful than fuzzy numbers. Therefore, we conclude that oriented fuzzy numbers are an original and useful tool for real-world problems.


Introduction
An imprecise number is an approximation of a fixed value crisp number. A commonly accepted model of an imprecise number is a fuzzy number (FN) [1,2], determined as a fuzzy subset of the space of real numbers. The intuitive concept of ordered FN was introduced by Kosiński and his co-workers [3][4][5] as an extension of the FN concept. Ordered FNs' usefulness follows from the fact that it is interpreted as FN equipped with information about the position of the approximated number. Ordered FNs have already begun to find their use in modelling a real-world problems [6][7][8][9][10][11][12][13][14][15][16][17][18][19].
Unfortunately, the ordered FNs' theory has one significant drawback. Kosiński shows that there exist such ordered FNs that their sum cannot be represented by a membership function [20]. For this formal reason, the Kosiński' theory was revised in [21] in this way that a new definition of ordered FNs corresponds to the intuitive definition by Kosiński. An Ordered FN defined in the context of the revised theory is now called an oriented FN (OFN) [22]. OFNs are already used in decision analysis, economics, and finance [23][24][25][26][27][28][29].
In the field of fuzzy sets, many authors propose such new concepts and theories that are isomorphic with the existing elements of the general theory of fuzzy sets [30]. Klement and Mesiar show a lot of such examples in their paper [30]. For this reason, they recommend an in-depth confrontation of the newly proposed concepts with the existing historically shaped elements of the fuzzy set theory. Finally, they emphasize that any modification of an existing fuzzy concept may be accepted if it is more useful in certain applications. The authors of this article fully agree with the postulates expressed by Klement and Mesiar. The FN is a commonly accepted model of an imprecise number. In our work [21], we propose the concept of OFN as a new model of an imprecise number. Therefore, in line with Klement and Mesiar postulates [30], our main goal is to look for formal differences between the FN and OFN. To the best of our knowledge, it is the first study devoted to this

Fuzzy Sets-Basic Facts
The space of all declarative sentences is noted by the symbol P. Subjects of any cognitive-application activity are elements of a space X. The basic tool for classifying these elements is the concept of a set. For any predicate ϕ A : X → P , the set A ⊂ X can be determined in the following way The predicate ϕ A ∈ P X is called a predicate of the set A ⊂ X. Any set and its predicate are one-to-one link. For unique determination of a set form, it is necessary to determine the manner in which the relationship between the actual state of affairs and the information contained in the sentence about this state is given.
The starting point for a discussion on this topic is to reduce our considerations to the classical propositional calculus. The subject of the classical propositional calculus are only those declarative sentences that are true or false. Any sentence that meet this condition is called a logical sentence. The space of all logical sentences is noted by the symbol P 0 ⊂ P. For each true sentence p ∈ P 0 , we assign the truth value v(p) = 1 (2) For each false sentence p ∈ P 0 , we assign the truth value v(p) = 0.
In this way, we define the function of a logical evaluation v ∈ {0; 1} P 0 . Any set is a set (crisp set) described in the classical set theory. The family of all such sets is noted by the symbol B(X). For any set A ∈ B(X), we determine its characteristic function χ A ∈ {0, 1} X = 2 X given by the identity Mathematics 2021, 9,523 3 of 27 The characteristic function value χ A (x) is equal to the truth value of the sentence "x ∈ A". Two-valued logic has been criticized many times. Therefore, it was extended by Łukasiewicz [31,32] to multivalued logic. The subject of considerations in multivalued logic are those sentences for which the connected relation "no less true" is uniquely defined. The space of all sentences meeting this condition is noted by the symbol P 1 ⊂ P. We have P 0 ⊂ P 1 ⊂ P. Using multivalued logic, we assume that: • Any sentence is not less true than any false sentence, • Any true sentence is not less true than any sentence.
For each sequence p ∈ P 1 , we assign the truth value v(p) understood as a "degree in which the evaluated sentence is true". Because multivalued logic is an extension of two-valued logic, for any sentence p ∈ P 0 we have v(p) = v(p). (6) In this way, we define the function of logical evaluation v ∈ [0, 1] P 1 . Any set is a fuzzy set intuitively introduced by Zadeh [33], Menger [34,35], and Klaua [36,37] (see [38]). We denote the family of all fuzzy sets by the symbol F(X). For any fuzzy set A ∈ F(X), we determine its membership function µ A ∈ [0, 1] X given by the identity The membership function value µ A (x) is equal to the truth value of the sentence "x ∈ A".
We can define any fuzzy set in a more formal way. The spaces B(X) and 2 X are isomorphic. This isomorphism is determined by the increasing bijection Φ : 2 X → B(X) i.e., Then we have B(X) = Φ 2 X .
It means that the family B(X) of all crisp sets is determined by isomorphism Φ as the image of the family 2 X of all characteristic function on the space X. Because 2 X ⊂ [0, 1] X , we can extend the isomorphism Φ : 2 X → B(X) to an increasing injection Φ determined on [0, 1] X . For any membership function µ A ∈ [0, 1] X , the value is called a fuzzy set. Then, the space F(X) is determined in the following way Increasing bijection Φ used above is not uniquely defined. The identity (8) shows that the unique form of the isomorphism depends on the type of multivalued logic used. On the other hand, this multivalued logic determines the set operators in F(X). In our considerations, a set operators are determined in the following way Mathematics 2021, 9, 523 4 of 27 The development trends of fuzzy set theory are restricted by Zadeh's extension principle [39][40][41]. This principle can be formally described as follows.
Let the fixed notion Q be explicitly defined with the use of two-valued logic as some relationship between elements of space X. Then this notion is described by predicate ϕ Q ∈ P X 0 . Then Zadeh's extension principle states that any extension Q of the notion Q in the fuzzy case is described by the same predictor ϕ Q ∈ P X 1 evaluated with the use of multivalued logic.
Any fuzzy subset A ∈ F(X) may be characterized using the following crisp sets: • the α−cuts [A] α determined for each α ∈ [0, 1] as follows • the support closure [A] 0 + given in the following way • the core Core(A) distinguished with the use of the formula

Fuzzy Number-Basic Facts
An imprecise number may be considered as a family of real values belonging to it in a varying degree. For this reason, an imprecise number is usually represented by a FN defined as a fuzzy subset of the family R of all real numbers. The most general definition of FN was proposed by Dubois and Prade [1,2]. Definition 1. The fuzzy number (FN) is such a fuzzy subset L ∈ F(R) represented by its upper semi-continuous membership function µ L ∈ [0, 1] R satisfying the conditions: Core(L) = ∅, (19) ∀ (x,y,z)∈R 3 : x ≤ y ≤ z ⇒ µ L (y) ≥ min{µ L (x), µ L (z)}, (20) ∀ (a,d)∈R 2 : Sup(L ) = [a, d] The set of all FN we denote by the symbol F. Each FN L is interpreted as an imprecise number "about z" for any z ∈ Core(L). Understanding the phrase "about z" depends on the applied pragmatics of the natural language. Any real number a ∈ R is such FN Mathematics 2021, 9, x FOR PEER REVIEW ∈ ℙ 0 . Then Zadeh's extension principle states that any extension ̃ the fuzzy case is described by the same predictor ∈ ℙ 1 evaluated w tivalued logic.
Any fuzzy subset ∈ ℱ( ) may be characterized using the follow

Fuzzy Number-Basic Facts
An imprecise number may be considered as a family of real value a varying degree. For this reason, an imprecise number is usually re defined as a fuzzy subset of the family ℝ of all real numbers. The mos of FN was proposed by Dubois and Prade [1,2]. The set of all FN we denote by the symbol . Each FN ℒ is interp cise number "about " for any ∈ ℴ ℯ(ℒ). Understanding the phr pends on the applied pragmatics of the natural language. Any real num FN ⟦ ⟧ ∈ that It implies that ℝ ⊂ . On the other hand, any FN fulfilling (22) Moreover, we immediately obtain from conditions (22) that any num sented by its membership function ⟦ ⟧ ∈ [0,1] ℝ given by the identity Let symbol * denotes any arithmetic operation defined in ℝ. By note an extension of arithmetic operation * to . According to the Principle, for any pair ( , ℒ) ∈ 2 represented respectively by their tions , ∈

Fuzzy Number-Basic Facts
An imprecise number may be considered as a family of real values belonging to it in a varying degree. For this reason, an imprecise number is usually represented by a FN defined as a fuzzy subset of the family ℝ of all real numbers. The most general definition of FN was proposed by Dubois and Prade [1,2].
The set of all FN we denote by the symbol . Each FN ℒ is interpreted as an imprecise number "about " for any ∈ ℴ ℯ(ℒ). Understanding the phrase "about " depends on the applied pragmatics of the natural language. Any real number ∈ ℝ is such It implies that ℝ ⊂ . On the other hand, any FN fulfilling (22) is a real number. Moreover, we immediately obtain from conditions (22) that any number ⟦ ⟧ is represented by its membership function ⟦ ⟧ ∈ [0,1] ℝ given by the identity Let symbol * denotes any arithmetic operation defined in ℝ. By symbol ⊛ we denote an extension of arithmetic operation * to . According to the Zadeh's Extension Principle, for any pair ( , ℒ) ∈ 2 represented respectively by their membership functions , ∈ [0,1] ℝ , Dubois and Prade [2] define the FN by means of its membership function ∈ [0,1] ℝ given by the identity: In line with the above, we can extend basic arithmetic operators to a fuzzy case in a = Sup

Fuzzy Number-Basic Facts
An imprecise number may be considered as a family of real values belonging to it in a varying degree. For this reason, an imprecise number is usually represented by a FN defined as a fuzzy subset of the family ℝ of all real numbers. The most general definition of FN was proposed by Dubois and Prade [1,2].
The set of all FN we denote by the symbol . Each FN ℒ is interpreted as an imprecise number "about " for any ∈ ℴ ℯ(ℒ). Understanding the phrase "about " depends on the applied pragmatics of the natural language. Any real number ∈ ℝ is such It implies that ℝ ⊂ . On the other hand, any FN fulfilling (22) is a real number. Moreover, we immediately obtain from conditions (22) that any number ⟦ ⟧ is represented by its membership function ⟦ ⟧ ∈ [0,1] ℝ given by the identity Let symbol * denotes any arithmetic operation defined in ℝ. By symbol ⊛ we denote an extension of arithmetic operation * to . According to the Zadeh's Extension Principle, for any pair ( , ℒ) ∈ 2 represented respectively by their membership functions , ∈ [0,1] ℝ , Dubois and Prade [2] define the FN by means of its membership function ∈ [0,1] ℝ given by the identity: In line with the above, we can extend basic arithmetic operators to a fuzzy case in a = {a}.

Fuzzy Number-Basic Facts
An imprecise number may be considered as a family of real values belo a varying degree. For this reason, an imprecise number is usually represen defined as a fuzzy subset of the family ℝ of all real numbers. The most gene of FN was proposed by Dubois and Prade [1,2]. The set of all FN we denote by the symbol . Each FN ℒ is interpreted cise number "about " for any ∈ ℴ ℯ(ℒ). Understanding the phrase " pends on the applied pragmatics of the natural language. Any real number FN ⟦ ⟧ ∈ that ℴ ℯ(⟦ ⟧) = ̅̅̅̅̅̅ (⟦ ⟧ ) = { }.

Fuzzy Number-Basic Facts
An imprecise number may be considered as a family of real values belonging to it in a varying degree. For this reason, an imprecise number is usually represented by a FN defined as a fuzzy subset of the family ℝ of all real numbers. The most general definition of FN was proposed by Dubois and Prade [1,2].
The set of all FN we denote by the symbol . Each FN ℒ is interpreted as an imprecise number "about " for any ∈ ℴ ℯ(ℒ). Understanding the phrase "about " depends on the applied pragmatics of the natural language. Any real number ∈ ℝ is such FN ⟦ ⟧ ∈ that ℴ ℯ(⟦ ⟧) = ̅̅̅̅̅̅ (⟦ ⟧ ) = { }.
It implies that ℝ ⊂ . On the other hand, any FN fulfilling (22) is a real number. Moreover, we immediately obtain from conditions (22) that any number ⟦ ⟧ is represented by its membership function ⟦ ⟧ ∈ [0,1] ℝ given by the identity Let symbol * denotes any arithmetic operation defined in ℝ. By symbol ⊛ we denote an extension of arithmetic operation * to . According to the Zadeh's Extension Principle, for any pair ( , ℒ) ∈ 2 represented respectively by their membership functions , ∈ The set of all FN we denote by the symbol . Each FN ℒ is interpreted as an imprecise number "about " for any ∈ ℴ ℯ(ℒ). Understanding the phrase "about " depends on the applied pragmatics of the natural language. Any real number ∈ ℝ is such FN ⟦ ⟧ ∈ that ℴ ℯ(⟦ ⟧) = ̅̅̅̅̅̅ (⟦ ⟧ ) = { }.
It implies that ℝ ⊂ . On the other hand, any FN fulfilling (22) is a real number. Moreover, we immediately obtain from conditions (22) that any number ⟦ ⟧ is represented by its membership function ⟦ ⟧ ∈ [0,1] ℝ given by the identity Let symbol * denotes any arithmetic operation defined in ℝ. By symbol ⊛ we denote an extension of arithmetic operation * to . According to the Zadeh's Extension Principle, for any pair ( , ℒ) ∈ 2 represented respectively by their membership functions , ∈ [0,1] ℝ , Dubois and Prade [2] define the FN ℳ = ⊛ ℒ (24) Let symbol * denotes any arithmetic operation defined in R. By symbol we denote an extension of arithmetic operation * to F. According to the Zadeh's Extension Prin-ciple, for any pair (K, L) ∈ F 2 represented respectively by their membership functions µ K , µ L ∈ [0, 1] R , Dubois and Prade [2] define the FN by means of its membership function µ M ∈ [0, 1] R given by the identity: In line with the above, we can extend basic arithmetic operators to a fuzzy case in a following way:

•
For any pair (β, L ) ∈ R × F, the "dot multiplication" determined with the use of its membership function µ M ∈ [0, 1] R given by the identity • For any pair (K, L) ∈ F 2 , the "addition" determined with the use of its membership function µ M ∈ [0, 1] R given by the identity In this article, we will discuss an algebraic structure F, , ⊕ understood as the space F equipped with dot multiplication and addition ⊕. From the identity (29) we get that the addition ⊕ is associative and commutative. Moreover, we have  For any pair ( , ℒ) ∈ 2 , the "addition" determined with the use of its membership function ∈ [0,1] ℝ given by the identity ( ) = sup{min{ ( ), ( )} : = + , ( , ) ∈ ℝ 2 }.
In this article, we will discuss an algebraic structure 〈 ,⊙, ⨁〉 understood as the space equipped with dot multiplication ⊙ and addition ⨁. From the identity (29) we get that the addition ⨁ is associative and commutative. Moreover, we have Lemma 1. The number ⟦0⟧ is the additive identity (The additive identity is also sometimes called an additive neutral element) in the algebraic structure 〈 ,⊙, ⨁〉.
This together with the addition commutativity proves that


The unary minus operator "-" on ℝ extended to the minus operator ⊝ on by the identity


The subtraction "-" on ℝ extended to the subtraction ⊝ on by the identity The unary minus operator ⊖ and the subtraction ⊖ meet the condition (25) of Dubois and Prade definition.
Equations (25) and (29) are very difficult to apply. A great facilitation here is the fact that any FN can be equivalently defined as follows: In this article, we will discuss an algebraic structure 〈 ,⊙, ⨁〉 understood as the space equipped with dot multiplication ⊙ and addition ⨁. From the identity (29) we get that the addition ⨁ is associative and commutative. Moreover, we have

Lemma 1. The number ⟦0⟧
is the additive identity (The additive identity is also sometimes called an additive neutral element) in the algebraic structure 〈 ,⊙, ⨁〉.
This together with the addition commutativity proves that For the algebraic structure 〈 ,⊙, ⨁〉, we can determine following operators:


The unary minus operator "-" on ℝ extended to the minus operator ⊝ on by the identity


The subtraction "-" on ℝ extended to the subtraction ⊝ on by the identity The unary minus operator ⊖ and the subtraction ⊖ meet the condition (25) of Dubois and Prade definition.
Equations (25) and (29) are very difficult to apply. A great facilitation here is the fact that any FN can be equivalently defined as follows: is represented by its membership function µ M ∈ [0, 1] R given as follows determined with the use of its membership function given by the identity For any pair ( , ℒ) ∈ 2 , the "addition" determined with the use of its membership function given by the identity ( ) = sup{min{ ( ), ( )} : = + , ( , ) ∈ ℝ 2 }. (29) In this article, we will discuss an algebraic structure 〈 ,⊙, ⨁〉 understood as the space equipped with dot multiplication ⊙ and addition ⨁. From the identity (29) we get that the addition ⨁ is associative and commutative. Moreover, we have is the additive identity (The additive identity is also sometimes called an additive neutral element) in the algebraic structure 〈 ,⊙, ⨁〉.

Proof of Lemma 1. Let us take any FN ℒ ∈ represented by its membership function
The identities (23) and (29) imply that the sum is represented by its membership function given as follows This together with the addition commutativity proves that For the algebraic structure 〈 ,⊙, ⨁〉, we can determine following operators:


The unary minus operator "-" on ℝ extended to the minus operator ⊝ on by the identity


The subtraction "-" on ℝ extended to the subtraction ⊝ on by the identity The unary minus operator ⊖ and the subtraction ⊖ meet the condition (25) of Du- determined with the use of its membership function given by the identity For any pair ( , ℒ) ∈ 2 , the "addition" determined with the use of its membership function given by the identity ( ) = sup{min{ ( ), ( )} : = + , ( , ) ∈ ℝ 2 }. (29) In this article, we will discuss an algebraic structure 〈 ,⊙, ⨁〉 understood as the space equipped with dot multiplication ⊙ and addition ⨁. From the identity (29) we get that the addition ⨁ is associative and commutative. Moreover, we have is the additive identity (The additive identity is also sometimes called an additive neutral element) in the algebraic structure 〈 ,⊙, ⨁〉.

Proof of Lemma 1. Let us take any FN ℒ ∈ represented by its membership function
The identities (23) and (29) imply that the sum is represented by its membership function given as follows This together with the addition commutativity proves that For the algebraic structure 〈 ,⊙, ⨁〉, we can determine following operators:


The unary minus operator "-" on ℝ extended to the minus operator ⊝ on by the identity


The subtraction "-" on ℝ extended to the subtraction ⊝ on by the identity This together with the addition commutativity proves that determined with the use of its membership function given by the identity For any pair ( , ℒ) ∈ 2 , the "addition" determined with the use of its membership function given by the identity ( ) = sup{min{ ( ), ( )} : = + , ( , ) ∈ ℝ 2 }. (29) In this article, we will discuss an algebraic structure 〈 ,⊙, ⨁〉 understood as the space equipped with dot multiplication ⊙ and addition ⨁. From the identity (29) we get that the addition ⨁ is associative and commutative. Moreover, we have is the additive identity (The additive identity is also sometimes called an additive neutral element) in the algebraic structure 〈 ,⊙, ⨁〉.

Proof of Lemma 1. Let us take any FN ℒ ∈ represented by its membership function
The identities (23) and (29) imply that the sum is represented by its membership function given as follows This together with the addition commutativity proves that For the algebraic structure 〈 ,⊙, ⨁〉, we can determine following operators:


The unary minus operator "-" on ℝ extended to the minus operator ⊝ on by the identity determined with the use of its membership function given by the identity For any pair ( , ℒ) ∈ 2 , the "addition" determined with the use of its membership function given by the identity ( ) = sup{min{ ( ), ( )} : = + , ( , ) ∈ ℝ 2 }. (29) In this article, we will discuss an algebraic structure 〈 ,⊙, ⨁〉 understood as the space equipped with dot multiplication ⊙ and addition ⨁. From the identity (29) we get that the addition ⨁ is associative and commutative. Moreover, we have is the additive identity (The additive identity is also sometimes called an additive neutral element) in the algebraic structure 〈 ,⊙, ⨁〉.

Proof of Lemma 1. Let us take any FN ℒ ∈ represented by its membership function
The identities (23) and (29) imply that the sum is represented by its membership function given as follows This together with the addition commutativity proves that For the algebraic structure 〈 ,⊙, ⨁〉, we can determine following operators:


The unary minus operator "-" on ℝ extended to the minus operator ⊝ on by the identity .
For the algebraic structure F, , ⊕ , we can determine following operators: • The unary minus operator "-" on R extended to the minus operator on F by the identity Mathematics 2021, 9, 523 6 of 27

•
The subtraction "-" on R extended to the subtraction on F by the identity The unary minus operator and the subtraction meet the condition (25) of Dubois and Prade definition.
Equations (25) and (29) are very difficult to apply. A great facilitation here is the fact that any FN can be equivalently defined as follows: where the left reference function L L ∈ [0, 1] [a,b] and the right reference function R L ∈ [0, 1] [c,d] are upper semi-continuous monotonic ones meeting the condition (21).
In further considerations, we will use the following concepts.

Definition 5. For any bounded continuous and non-increasing function r
Using results obtained in [43], for any pair (L(a, b, c, d, L K , R K ), L(e, f , g, h, L M , R M )) ∈ F 2 we get the sum In an analogous way, for any pair (β, L(a, b, c, d, L K , R K )) ∈ R × F we get the dot product where Example 1. Let us calculate the dot product where the FN K = L(2, 4, 8, 10, L K , R k ) is characterized by its membership function It is very easy to check that reference functions L K and R K are strictly monotonic. Using dependences (44)-(48), we get where Example 2. Let us calculate the sum where FN K is characterized by its membership function (50) and FN M = L(5, 6, 11, 15, L M , R M ) is determined by membership function In addition, reference functions L M and R M are strictly monotonic. Using dependences (39)-(43), we get where Example 2 shows a very high level of formal complexity of any FNs addition. Therefore, in many applications researchers limit their considerations to the following kind of FNs. Definition 6. For any nondecreasing sequence (a, b, c, d) ⊂ R, the trapezoidal fuzzy number (TrFN) is the FN Tr(a, b, c, d) = T ∈ F determined explicitly by its membership functions The space of all TrFNs is denoted by the symbol F Tr . For any a ∈ R The TrFN we have ∈ ℙ 0 . Then Zadeh's extension principle states that any extension ̃ of the notion in the fuzzy case is described by the same predictor ∈ ℙ 1 evaluated with the use of multivalued logic.
Any fuzzy subset ∈ ℱ( ) may be characterized using the following crisp sets :  the support closure [ ] 0 + given in the following way  the core ℴ ℯ( ) distinguished with the use of the formula

Fuzzy Number-Basic Facts
An imprecise number may be considered as a family of real values belonging to it in a varying degree. For this reason, an imprecise number is usually represented by a FN defined as a fuzzy subset of the family ℝ of all real numbers. The most general definition of FN was proposed by Dubois and Prade [1,2].

Definition 1. The fuzzy number (FN) is such a fuzzy subset ℒ ∈ ℱ(ℝ) represented by its upper semicontinuous membership function
∈ [0,1] ℝ satisfying the conditions: The set of all FN we denote by the symbol . Each FN ℒ is interpreted as an imprecise number "about " for any ∈ ℴ ℯ(ℒ). Understanding the phrase "about " depends on the applied pragmatics of the natural language. Any real number ∈ ℝ is such It implies that ℝ ⊂ . On the other hand, any FN fulfilling (22) is a real number. Moreover, we immediately obtain from conditions (22) that any number ⟦ ⟧ is represented by its membership function ⟦ ⟧ ∈ [0,1] ℝ given by the identity Let symbol * denotes any arithmetic operation defined in ℝ. By symbol ⊛ we denote an extension of arithmetic operation * to . According to the Zadeh's Extension Principle, for any pair ( , ℒ) ∈ 2 represented respectively by their membership functions , ∈ [0,1] ℝ , Dubois and Prade [2] define the FN .
Therefore, we can write R ⊂ F Tr ⊂ F. Using identities (39)- (43), for any pair (Tr(a, b, c, d), Tr(e, f , g, h)) ∈ F 2 Tr we get the sum  (Tr(a, b, c, d), Tr(e, f , g, h)) ∈ F 2 Tr we get the subtraction The main disadvantage of FN arithmetic is described by lemma below.

Lemma 2.
The subtraction is not an inverse operator to addition ⊕.
Proof of Lemma 2. Let us take into account TrFN Tr(a, b, c, d) / ∈ R. In accordance with (23), we have a < d. Using (71), we get In this article, we will discuss an algebraic structure 〈 ,⊙, ⨁ space equipped with dot multiplication ⊙ and addition ⨁. From get that the addition ⨁ is associative and commutative. Moreover, Lemma 1. The number ⟦0⟧ is the additive identity (The additive identity i additive neutral element) in the algebraic structure 〈 ,⊙, ⨁〉.
Proof of Lemma 1. Let us take any FN ℒ ∈ represented by its mem [0,1] ℝ . The identities (23) and (29) imply that the sum is represented by its membership function ∈ [0,1] ℝ given as follows This together with the addition commutativity proves that

□.
For the algebraic structure 〈 ,⊙, ⨁〉, we can determine followin  The unary minus operator "-" on ℝ extended to the minus the identity The subtraction "-" on ℝ extended to the subtraction ⊝ on The unary minus operator ⊖ and the subtraction ⊖ meet the bois and Prade definition.
Equations (25) and (29) are very difficult to apply. A great facil that any FN can be equivalently defined as follows: This raises problems concerning the solution of fuzzy linear equations and with the interpretation of specific improper fuzzy arithmetic results.

Kosiński's Theory
The notion of ordered FN is intuitively introduced by Kosiński and his co-writers [3][4][5] as such a model of an imprecise number and its arithmetic that subtraction is the inverse operator to addition. Kosiński was going to define ordered FN as an extension of a concept of FN. The original definition of ordered FNs was formulated in the following way.
In the Kosiński's theory, definition (74) replaces definition (25) proposed by Dubois and Prade [2]. Kosiński has developed his theory without using the results obtained by Goetschel and Voxman [43]. Nevertheless, the Kosiński's definition of addition is coherent with identity (29) describing addition ⨁ of FNs.
In the Kosiński's theory, definition (74) replaces definition (25) proposed by Dubois and Prade [2]. Kosiński has developed his theory without using the results obtained by Goetschel and Voxman [43]. Nevertheless, the Kosiński's definition of addition is coherent with identity (29) describing addition ⊕ of FNs.
Any sequence (a, b, c, d) ⊂ R meets exactly one of the following conditions If the condition (75) Figure 1c. This graph has an arrow denoting the orientation, which provides additional information. A positively oriented ordered FN is interpreted as an imprecise number, which may increase. Some example of membership function of FN determined by a positively oriented ordered FN is presented in Figure 2a.  Figure 1c. This graph has an arrow denoting the orientation, which provides additional information. A positively oriented ordered FN is interpreted as an imprecise number, which may increase. Some example of membership function of FN determined by a positively oriented ordered FN is presented in Figure 2a. If the condition (76) is fulfilled then the ordered FN ̿ ( , , , , , ) is negatively oriented. A negatively oriented ordered FN is interpreted as such an imprecise number, which may decrease. Some example of a membership function of FN determined by a negatively oriented ordered FN is presented in the Figure 2b.
̿ ( )  a, b, c, d, f s , g s ) is not related to any FN. This disadvantage of original Kosiński's theory will be discussed below.
If the sequence (a, b, c, d) ⊂ R is not monotonic then using the identity (34) we get the membership relation µ L · a, b, c, d, f −1 , which is not a function. Therefore, this relation cannot be considered as a membership function of any fuzzy set. For this reason, for any non-monotonic sequence (a, b, c, d) ⊂ R the ordered FN = S(a, b, c, d, f s , g s ) is called an improper one [4]. Some example of a membership relation determined by a negatively oriented improper ordered FN is presented in Figure 3. The remaining ordered FNs are called proper ones. Some examples of proper ordered FNs are presented in Figure 2. If the condition (76) is fulfilled then the ordered FN ̿ ( , , , , , ) is negatively oriented. A negatively oriented ordered FN is interpreted as such an imprecise number, which may decrease. Some example of a membership function of FN determined by a negatively oriented ordered FN is presented in the Figure 2b.
If the condition (77) is fulfilled then the ordered FN ̿ ( , , , , , ) is not related to any FN. This disadvantage of original Kosiński's theory will be discussed below.
Therefore, this relation cannot be considered as a membership function of any fuzzy set. For this reason, for any non-monotonic sequence ( , , , ) ⊂ ℝ the ordered FN ̿ ( , , , , , ) is called an improper one [4]. Some example of a membership relation determined by a negatively oriented improper ordered FN is presented in Figure 3. The remaining ordered FNs are called proper ones. Some examples of proper ordered FNs are presented in Figure 2. We note above that for the case (77), the ordered FN ̿ ( , , , , , ) is undefined. We can remove the disadvantage by means of the generalization of Kosiński's theory to the case when an up-function and a down-function are monotonic continuous surjections. Thanks to the use of Goetschel-Voxman results [43], we can generalize the ordered FN theory in such a manner that fully corresponds to the intuitive Kosiński's approach to the notion of ordered FN. We agree with other scientists [13,15] that the ordered FN should We note above that for the case (77), the ordered FN = S(a, b, c, d, f s , g s ) is undefined. We can remove the disadvantage by means of the generalization of Kosiński's theory to the case when an up-function and a down-function are monotonic continuous surjections. Thanks to the use of Goetschel-Voxman results [43], we can generalize the ordered FN theory in such a manner that fully corresponds to the intuitive Kosiński's approach to the notion of ordered FN. We agree with other scientists [13,15] that the ordered FN should be called the Kosiński's number. For this reason, the generalized ordered FN will be called Kosiński's numbers. In [21], it is shown that we have: In Theorem 2, we use a modified notation of numerical intervals, which is explained In Theorem 2, we use a modified notation of numerical intervals, which is explained Appendix A. The space of all KN is denoted by the symbol ̌. Any KN  ̿ ( , , , , , ) fulfilling the condition (75) is positively oriented. Any KN ̿ ( , , , , , ) fulfilling the condition (76) is negatively oriented. Any KN  ̿ ( , , , , , ) is an unoriented FN ⟦ ⟧ ∈ fulfilling the condition (22). Then, we have entities ( ) = ( ) = . (79) Of course, then KN represents the crisp number ∈ ℝ. Let symbol * denote any arithmetic operation defined in ℝ. By * we denote an xtension of arithmetic operation * to ̌. In the Kosiński's theory, the arithmetic operaons * are defined by (74). In this way we obtain Kosiński's arithmetic.
Using (74), for any pair ( , ̿ ( , , , , , )) ∈ ℝ ×̌ we get dot product The condition (74) implies that the unary operator minus ⊟ and the subtraction may by expressed for ̌ in the same way as for . Kosiński [20] has shown that: The addition ⊞ is commutative and associative, The number ⟦0⟧ is additive identity in the algebraic structure 〈̌, ⊡ , ⊞ 〉, The subtraction ⊟ is inverse operator to addition ⊞ .
Therefore, we can say that KNs fulfil the postulates put formulated by Kosiński. It is portant advantages of Kosiński's theory. On the other hand, let us look at the following xample.
∈ F fulfilling the condition (22). Then, we have identities Of course, then KN represents the crisp number d ∈ R. Let symbol * denote any arithmetic operation defined in R. By * K we denote an extension of arithmetic operation * toǨ. In the Kosiński's theory, the arithmetic operations * K are defined by (74). In this way we obtain Kosiński's arithmetic.
Using (74) Comparing (44) with (80), we notice that the dot product and the dot product K are different arithmetic operations. It implies that the Dubois-Prade definition (25) and the Kosiński's definition (74) are not equivalent.

In an analogous way, for any pair
The condition (74) implies that the unary operator minus K and the subtraction K may by expressed forǨ in the same way as for F. Kosiński [20] has shown that:

•
The addition K is commutative and associative, • The number In this article, we will discuss an algebraic structure 〈 ,⊙, ⨁〉 understood as the space equipped with dot multiplication ⊙ and addition ⨁. From the identity (29) we get that the addition ⨁ is associative and commutative. Moreover, we have Lemma 1. The number ⟦0⟧ is the additive identity (The additive identity is also sometimes called an additive neutral element) in the algebraic structure 〈 ,⊙, ⨁〉. (23) and (29) imply that the sum


The unary minus operator "-" on ℝ extended to the minus operator ⊝ on by the identity


The subtraction "-" on ℝ extended to the subtraction ⊝ on by the identity The unary minus operator ⊖ and the subtraction ⊖ meet the condition (25) of Dubois and Prade definition.
Equations (25) and (29) are very difficult to apply. A great facilitation here is the fact that any FN can be equivalently defined as follows: is additive identity in the algebraic structure Ǩ , K , K , • The subtraction K is inverse operator to addition K .
Therefore, we can say that KNs fulfil the postulates put formulated by Kosiński. It is important advantages of Kosiński's theory. On the other hand, let us look at the following example.
The KN = M is an improper one because the sequence (17,15,14,15) is not monotonic. The above example shows that the sum of proper KNs may be an improper KN. This fact was already known to Kosiński [20].
On the other hand, this property of addition K results in the fact that the improper KNs cannot be omitted from the Kosiński theory. This is a major inconvenience because improper numbers cannot be interpreted in the context of the fuzzy set theory. Such an assessment of the Kosinski theory indicates the need of revision.

Oriented Fuzzy Numbers
Under the influence of the above argumentation, Kosinski's theory was revised in [21]. In the revised theory, KNs are replaced by a following kind of ordered FNs. is interpreted as an imprecise number "about or slightly below z". By symbol K − we denote the space of all negatively oriented OFNs. Understanding the phrases "about or slightly above z" and "about or slightly below z" depends on the used language pragmatics. If a = d, OFN ↔ L (a, a, a, a, S L , E L ) = ∈ ℙ 0 . Then Zadeh's extension principle states that any extension ̃ of the notion in the fuzzy case is described by the same predictor ∈ ℙ 1 evaluated with the use of multivalued logic.
Any fuzzy subset ∈ ℱ( ) may be characterized using the following crisp sets :

Fuzzy Number-Basic Facts
An imprecise number may be considered as a family of real values belonging to it in a varying degree. For this reason, an imprecise number is usually represented by a FN defined as a fuzzy subset of the family ℝ of all real numbers. The most general definition of FN was proposed by Dubois and Prade [1,2].

Definition 1. The fuzzy number (FN) is such a fuzzy subset ℒ ∈ ℱ(ℝ) represented by its upper semicontinuous membership function
∈ [0,1] ℝ satisfying the conditions: The set of all FN we denote by the symbol . Each FN ℒ is interpreted as an imprecise number "about " for any ∈ ℴ ℯ(ℒ). Understanding the phrase "about " depends on the applied pragmatics of the natural language. Any real number ∈ ℝ is such FN ⟦ ⟧ ∈ that ℴ ℯ(⟦ ⟧) = ̅̅̅̅̅̅ (⟦ ⟧ ) = { }.
It implies that ℝ ⊂ . On the other hand, any FN fulfilling (22) is a real number. Moreover, we immediately obtain from conditions (22) that any number ⟦ ⟧ is represented by its membership function ⟦ ⟧ ∈ [0,1] ℝ given by the identity Let symbol * denotes any arithmetic operation defined in ℝ. By symbol ⊛ we denote an extension of arithmetic operation * to . According to the Zadeh's Extension Principle, for any pair ( , ℒ) ∈ 2 represented respectively by their membership functions , ∈ [0,1] ℝ , Dubois and Prade [2] define the FN by means of its membership function ∈ [0,1] ℝ given by the identity: In line with the above, we can extend basic arithmetic operators to a fuzzy case in a following way:


For any pair ( , ℒ ) ∈ ℝ × , the "dot multiplication" represents the number a ∈ R, which is unoriented. All above facts imply that Let symbol * denote any arithmetic operation defined in R. By the symbol * we denote an extension of arithmetic operation * to K. In the revised theory, the arithmetic operations * are defined by where we haveǎ In this way, we obtain the revised arithmetic. Theorem 2 implies that It means that the revised arithmetic does not change proper results obtained with the use of Kosiński's arithmetic. In other cases, results obtained with the use of revised arithmetic are the best approximation of results obtained with the use of Kosinski's arithmetic [21].
Using (89), for any pair In an analogous way, for any pair β, where the functions L K and R M are given, respectively, by (52) and (62) and we have We note that the above determined arithmetic operations on OFNs and analogous operations on FN are quite different. Nevertheless, in an analogous way, we can determine following operators:

•
The unary minus operator "-" on R extended to the minus operator on K by the identity • The subtraction "-" on R extended to the subtraction on K by the identity The unary minus operator and the subtraction meet the condition (89-99) of the revised definition.
This together with the addition commutativity proves that □.
For the algebraic structure 〈 ,⊙, ⨁〉, we can determine following operators:


The unary minus operator "-" on ℝ extended to the minus operator ⊝ on by the identity


The subtraction "-" on ℝ extended to the subtraction ⊝ on by the identity The unary minus operator ⊖ and the subtraction ⊖ meet the condition (25) of Dubois and Prade definition.
Equations (25) and (29) are very difficult to apply. A great facilitation here is the fact that any FN can be equivalently defined as follows: is the additive identity in the algebraic structure K, , .
In this article, we will discuss an algebraic structure 〈 ,⊙, ⨁〉 understood as the space equipped with dot multiplication ⊙ and addition ⨁. From the identity (29) we get that the addition ⨁ is associative and commutative. Moreover, we have

Lemma 1. The number ⟦0⟧
is the additive identity (The additive identity is also sometimes called an additive neutral element) in the algebraic structure 〈 ,⊙, ⨁〉.
For the algebraic structure 〈 ,⊙, ⨁〉, we can determine following operators:


The unary minus operator "-" on ℝ extended to the minus operator ⊝ on by the identity


The subtraction "-" on ℝ extended to the subtraction ⊝ on by the identity The unary minus operator ⊖ and the subtraction ⊖ meet the condition (25) of Dubois and Prade definition.
Equations (25) and (29) are very difficult to apply. A great facilitation here is the fact that any FN can be equivalently defined as follows: is explicitly represented by its membership function given as follows where the starting function S 0 ∈ [0, 1] [0,0] and the ending function E 0 ∈ [0, 1] [0,0] are determined in the following way S 0 (0) = E 0 (0) = 1.
In this article, we will discuss an algebraic structure 〈 ,⊙, ⨁〉 understood as the space equipped with dot multiplication ⊙ and addition ⨁. From the identity (29) we get that the addition ⨁ is associative and commutative. Moreover, we have

Lemma 1. The number ⟦0⟧
is the additive identity (The additive identity is also sometimes called an additive neutral element) in the algebraic structure 〈 ,⊙, ⨁〉.


The unary minus operator "-" on ℝ extended to the minus operator ⊝ on by the identity


The subtraction "-" on ℝ extended to the subtraction ⊝ on by the identity The unary minus operator ⊖ and the subtraction ⊖ meet the condition (25) of Dubois and Prade definition.
Equations (25) and (29) are very difficult to apply. A great facilitation here is the fact that any FN can be equivalently defined as follows: and due to (108-111) we obtain .

Lemma 4.
The subtraction is an inverse operator to addition .

Proof of Lemma 4. Let us take into account any OFN
In this article, we will discuss an algebraic structure 〈 ,⊙, ⨁〉 understood as the space equipped with dot multiplication ⊙ and addition ⨁. From the identity (29) we get that the addition ⨁ is associative and commutative. Moreover, we have

Lemma 1. The number ⟦0⟧
is the additive identity (The additive identity is also sometimes called an additive neutral element) in the algebraic structure 〈 ,⊙, ⨁〉.


The unary minus operator "-" on ℝ extended to the minus operator ⊝ on by the identity


The subtraction "-" on ℝ extended to the subtraction ⊝ on by the identity Example 5 shows a very high level of formal complexity of any OFNs addition. Therefore, in many applications, researchers limit their considerations to the following kind of OFNs.
The symbol K Tr denotes the space of all TrOFNs. The space of all positively oriented TrOFNs is denoted by the symbol K + Tr . The space of all negatively oriented TrOFNs we denote by the symbol K − Tr . TrOFN ↔ Tr(a, a, a, a) = a varying degree. For this reason, an imprecise number is usually represented by a FN defined as a fuzzy subset of the family ℝ of all real numbers. The most general definition of FN was proposed by Dubois and Prade [1,2]. The fuzzy number (FN) is such a fuzzy subset ℒ ∈ ℱ(ℝ) represented by its upper semicontinuous membership function ∈ [0,1] ℝ satisfying the conditions: ∀ ( , , )∈ℝ 3 : The set of all FN we denote by the symbol . Each FN ℒ is interpreted as an imprecise number "about " for any ∈ ℴ ℯ(ℒ). Understanding the phrase "about " depends on the applied pragmatics of the natural language. Any real number ∈ ℝ is such FN ⟦ ⟧ ∈ that It implies that ℝ ⊂ . On the other hand, any FN fulfilling (22) is a real number. Moreover, we immediately obtain from conditions (22) that any number ⟦ ⟧ is represented by its membership function ⟦ ⟧ ∈ [0,1] ℝ given by the identity Let symbol * denotes any arithmetic operation defined in ℝ. By symbol ⊛ we denote an extension of arithmetic operation * to . According to the Zadeh's Extension Principle, for any pair ( , ℒ) ∈ 2 represented respectively by their membership functions , ∈ [0,1] ℝ , Dubois and Prade [2] define the FN by means of its membership function ∈ [0,1] ℝ given by the identity: In line with the above, we can extend basic arithmetic operators to a fuzzy case in a following way:


For any pair ( , ℒ ) ∈ ℝ × , the "dot multiplication" represents a crisp number a ∈ R, which is unoriented. Summing up, all above facts imply that For any pair ↔ Tr(a, b, c, d), Tr , the identities (101-111) imply that their sum is given as follows Tr(max{p, q}, q, r, min{r, s}), The identity (112)

Oriented Fuzzy Numbers Arithmetic vs. Fuzzy Numbers Arithmetic
Let us compare the algebraic structure F, , ⊕ determined by (27) and (29) with the algebraic structure K, , determined by (112) and (101). For the algebraic structure F, , ⊕ we can conclude as follows: • From the identity (29), we immediately get that that the addition ⊕ is associative and commutative, • Lemma 1 shows that the number In this article, we will discuss an algebraic structure 〈 ,⊙, ⨁〉 understood as the space equipped with dot multiplication ⊙ and addition ⨁. From the identity (29) we get that the addition ⨁ is associative and commutative. Moreover, we have Lemma 1. The number ⟦0⟧ is the additive identity (The additive identity is also sometimes called an additive neutral element) in the algebraic structure 〈 ,⊙, ⨁〉.
This together with the addition commutativity proves that □.
For the algebraic structure 〈 ,⊙, ⨁〉, we can determine following operators:


The unary minus operator "-" on ℝ extended to the minus operator ⊝ on by is the additive identity, • Lemma 2 shows that the subtraction is not an inverse operator to addition ⊕.
For the algebraic structure K, , we conclude: • From the identity (101), we immediately get that that the addition is commutative, • In [21] it is proved that the addition is not associative, • Lemma 3 shows that the number In this article, we will discuss an algebraic structure 〈 ,⊙, ⨁〉 understood as the space equipped with dot multiplication ⊙ and addition ⨁. From the identity (29) we get that the addition ⨁ is associative and commutative. Moreover, we have . The identities (23) and (29) imply that the sum is represented by its membership function given as follows This together with the addition commutativity proves that is the additive identity, • Lemma 4 shows that the subtraction is an inverse operator to addition .
The above comparison shows that algebraic structures F, , ⊕ and K, , are not isomorphic. Therefore, OFNs and FNs should be considered as a different models of imprecision numbers. For this reason, any theorem on FNs cannot be automatically applied to OFNs.
On the other hand, we have the following nice relationship between FNs and OFNs. Let us consider the mapping U : K → K given by identity For this mapping we have: We see that the mapping (147) is such axial symmetry on the space K that the symmetry axis is equal to the R. Moreover, Theorem 1 and Definition 9 imply that the space F and the space K + ∪ R are isomorphic. This fact together with (149) and (150) proves that the space K is a symmetry closure of the space F.

Evaluation of Imprecision for Oriented Fuzzy Numbers
After Klir [44] we understand information imprecision as a superposition of ambiguity and indistinctness of information. Ambiguity is understood as a lack of an explicit recommendation one alternative among various others. Indistinctness is interpreted as a lack of a clear distinction between recommended and not recommended alternatives.
The increase in an OFN ambiguity causes a higher number of recommended alternatives. This results in an increase in the risk of choosing an incorrect alternative from recommended ones. This may cause one to make a decision, which will result in the loss of chances ex post. The possibility of such an event is called the ambiguity risk. Therefore, the increase in the ambiguity of OFN implies the increase in ambiguity risk. The right tool for measuring the OFN ambiguity is an extension of energy measure d ∈ R + 0 F defined for FNs by de Luca and Termini [45].
For any FN L = L(a, b, c, d, L L , R L ) ∈ F, de Luca and Termini [45] propose to use energy measure determined as follows where µ L ∈ [0, 1] R is the membership function determining FN L.
In this paper, we propose to generalize the energy measure (151) to the ambiguity index a ∈ R K assessing the ambiguity of any OFN This fact implies the existence of isomorphism in the space F and the space K + ∪ R. For this reason, the mapping d ∈ R + 0 K given by the identity is an extension of energy measure d ∈ R + 0 F to the domain of all OFNs. We propose to use this extension as energy measure of any OFN. It all means that ambiguity index stores the information on energy measure and additionally also about the orientation of the assessed OFN. In decision analysis, we use the energy measure as a measure of the ambiguity risk. An increase in the indistinctness suggests that the distinction between recommended and not recommended alternatives is more difficult. This causes an increase in the indistinctness risk, that is, in a possibility of choosing a not recommended alternative. The proper tool for measuring the indistinctness of an OFN is the entropy measure e ∈ R + 0 F , also proposed by de Luca and Termini [46] and modified by Piasecki [47].
The most widely known kind of entropy measure is described by Kosko [48]. On the other hand, in [49,50], it is shown that Kosko's entropy measure is not convenient for portfolio analysis. Therefore, we propose to evaluate indistinctness of an arbitrary FN by Czogała-Gottwald-Pedrycz entropy measure introduced in [51].
For any FN L = L(a, b, c, d, L L , R L ) ∈ F, Czogała-Gottwald-Pedrycz entropy measure is determined as follows where µ L ∈ [0, 1] R is a membership function determining FN L.
In this paper, we propose to generalize the entropy measure (152)  where µ L ∈ [0, 1] R is the membership function determining OFN ↔ L . For any negatively oriented OFN, its indistinctness index is negative and for any positively oriented OFN its indistinctness index is positive. In an analogous way, as above presented, we can justify that the mapping e ∈ R + 0 K given by the identity is an extension of the entropy measure e ∈ R + 0 F to the domain of all OFNs. We propose to use this extension as the entropy measure of any OFN. It means that the indistinctness index stores the information on the entropy measure and additionally about the orientation of the assessed OFN. In decision analysis, we use the entropy measure as a measure of the indistictness risk. Imprecision risk consists of both ambiguity and indistinctness risk, combined. The notions of ambiguity index and of indistinctness one give new perspectives for imprecision management.

Imprecision Evaluation for Trapezoidal Oriented Fuzzy Numbers
Due to the high computational complexity, the practical applications of OFNs are limited in economics, finance, and decision analysis to the use of TrOFNs. Therefore, for greater clarity of exposition, we can confine our discussion to the case of TrOFNs. For any TrOFN ↔ Tr(a, b, c, d), its ambiguity index and indistinctness index are determined based on the following relations For any monotonic sequence (a, b, c, d) ⊂ R, we have It is very easy to check that for any pair (β, Moreover, for any pair ( , by using the identity (144) and (145) we can easily get In other cases, analogous imprecision assessments are a little more complicated. We have here: Tr(e, f , g, h)) ∈ (K + Tr ∪ R) × K − Tr . The addition is commutative. Therefore, we can restrict our considerations to the value d( Tr ∪ R then using (143) and (159) where We see that condition (169) is fulfilled for any sum ( Tr ∪ R, then then using (143) and (159) we obtain d( where We see that condition (169) is also met for any sum ( Tr(e, f , g, h)) ∈ (K + Tr ∪ R) × K − Tr . If ↔ K ↔ L ∈ K + Tr ∪ R then using (143) and (160) we get where the sequence (α, δ, ε, χ) is determined by (171). Moreover, here we have It implies that Therefore, we get e( We see that condition (174) is fulfilled for any sum ( where the sequence (α, δ, ε, χ) is determined by (173).
Moreover, then we have ( We see that condition (170) is also met for any sum (

Portfolio Diversification
All the results presented above may be presented in a form suitable for financial portfolio analysis. The relative benefit of the asset owning is defined as a function of the quotient of a benefit value to the asset value. The return rate and discount factor are examples of relative benefits. The value of a relative benefit of asset owning is shortly called asset benefit index.
We will consider two-assets portfolio without short positions. Then, the considered portfolio benefit index is equal to an average of the portfolio component benefit indexes. It is a typical financial model used to study the effects of portfolio diversification.
Let us consider a case when the portfolio component benefit indexes are imprecisely valued. We will consider two cases: The method of determining the parameter λ depends on the kind of considered relative benefits [50]. We can easily prove the following theorems: Proof of Theorem 6. Due to the existence of an isomorphism between F Tr and K + Tr ∪ R, the inequalities result immediately from (146), (183), and (184).
The Theorem 6 shows that when we use TrFNs, portfolio diversification only averages the imprecision risk assessments. This is illustrated by the results obtained in Example 6. Example 6. We consider TrFNs K = Tr(0, 4, 8, 12) and M = Tr (4,10,16,18). Let us take into account the value π(K, M, 0.5) = Tr (2,7,12,15). The obtained results confirm the pessimistic theses of the Theorem 6. The applied portfolio diversification only averaged the imprecision risk.
Theorem 6 shows that when we use TrOFNs, portfolio diversification may reduce the imprecision risk assessments. This is illustrated by the results obtained in Example 7. We see that if portfolio component benefit indexes have different orientation, then portfolio diversification significantly reduces the indistinctness measure of portfolio benefit index. This reduction is impossible if asset benefit indexes are valued by TrFN. It proves that in portfolio analysis, utilizing TrOFNs is more useful than utilizing TrFNs. On the other hand, if portfolio component benefit indexes have identical orientation, then portfolio diversification only averages the indistinctness measures of portfolio benefit indexes.

Final Conclusions
The aim of this work was to justify the expediency of developing the OFNs theory. In Section 6, we showed that algebraic structures F, , ⊕ and K, , are not isomorphic. For this reason, OFNs and FNs should be considered as a different models of imprecision numbers. In addition, in Section 9 we showed that for portfolio analysis TrOFNs are more useful than TrFNs. This demonstrates the need to replace TrFN by TrOFN in a portfolio analysis.
We can conclude that the OFNs theory meets the requirements of the postulates formulated by Klement and Mesiar [30]. Therefore, we recommend OFNs as:

•
Original research subject, • Useful tool for modelling a real-world problems.
In this work, four functionals were proposed to assess OFN imprecision: Ambiguity index, indistinctness index, energy measure, and entropy measure. The pair of ambiguity and indistinctness indexes is a very suitable formal tool, which may be applied for a formal analysis of properties of energy and entropy measures.
Imprecision risk is a possibility of negative consequences of taking actions under the influence of imprecise information. The pair of energy and entropy measures can be applied as two dimensional vector measure of imprecision. Section 9 gives an example of using this measure for management of financial assets portfolio. The results presented there can be directly applied to the financial portfolio model described in [10].
The results presented in the numerical example show the possibility of finding smaller dominant of a portfolio imprecision risk measure. In our opinion, the subject of further research should be a more accurate estimation of the portfolio ambiguity measure for the case when the portfolio component benefit indexes are differently oriented. Determining a more precise inequality will increase the effectiveness of ambiguity risk management.
In Section 9, it is proved that if asset benefit indexes have different orientation, then portfolio diversification reduces the imprecision ratings of portfolio benefit index. The example discussed there shows that this reduction may be significant. On the other hand, if asset benefit indexes have identical orientation, then portfolio diversification only averages the imprecision ratings of portfolio benefit index. Presumably, the differently oriented benefit indexes can play the same role in imprecision risk management, which negatively correlated return rates play in uncertainty risk management. The study of this phenomenon may constitute an interesting new research direction. It is purposeful to undertake research on the relations between diversified orientation and negative correlation of benefit indexes. Noticing such relationships should have a significant impact on risk management.