Fuzzy Semi-Numbers and Their Elementary Arithmetic With a Medical Case Study

Sh. Yeganehmanesh; M. Amirfakhrian; P. Grzegorzewski

doi:10.2991/ijcis.11.1.75

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Volume 11, Issue 1, 2018, Pages 991 - 1004

Fuzzy Semi-Numbers and Their Elementary Arithmetic With a Medical Case Study

Authors

Sh. Yeganehmanesh¹^,shahab.yeganehmanesh@gmail.com, M. Amirfakhrian¹^{, *}^,amirfakhrian@iauctb.ac.ir, P. Grzegorzewski²^{, 3}^,pgrzeg@ibspan.waw.pl

¹Department of Mathematics, Central Tehran Branch, Islamic Azad University, Tehran, Iran.

²Systems Research Institute, Polish Academy of Sciences, Newelska 6, 01-447, Warsaw, Poland.

³Faculty of Mathematics and Information Science, Warsaw University of Technology, Koszykowa 75, 00-662, Warsaw, Poland.

^*Corresponding author.

Corresponding Author

M. Amirfakhrianamirfakhrian@iauctb.ac.ir

Received 30 July 2017, Accepted 17 April 2018, Available Online 2 May 2018.

DOI: 10.2991/ijcis.11.1.75 How to use a DOI?
Keywords: Fuzzy Number; Fuzzy Arithmetic; Fuzzy Semi-Number; Elevated Fuzzy Semi-Number
Abstract: A new methodology for processing non-normal fuzzy sets is proposed. To break the predominant constraint on normality of fuzzy numbers the concept of fuzzy semi-numbers is introduced Then it is shown how to generalize operations defined on fuzzy numbers onto a family of fuzzy semi-numbers with possibly different heights.
Copyright: © 2018, the Authors. Published by Atlantis Press.
Open Access: This is an open access article under the CC BY-NC license (http://creativecommons.org/licences/by-nc/4.0/).

1. Introduction

In most of the papers on fuzzy sets the authors consider only fuzzy numbers, i.e. fuzzy sets which are convex and normal (a fuzzy set is normal if its height is one, i.e. its membership function reaches value 1 at least in one point of its domain). Moreover, elementary arithmetic has been defined just on fuzzy numbers ^22,28 and, consequently, most of the contributions devoted to fuzzy ranking ^1,7,36, fuzzy approximation ^{2,3,4,8,13,16,37}, fuzzy differential equations ^18,27, fuzzy integral equations ^21,31 and systems of fuzzy equations ^11,33 are restricted to fuzzy numbers, especially to trapezoidal fuzzy numbers ^1,9,24,25. Fuzzy sets with height less than one appear have appeared in the literature rarely ^26,32. In particular, Chen considered the so-called Generalized Fuzzy Numbers (GFN) ¹⁵, i.e. fuzzy sets with triangular and trapezoidal membership functions but with height less than one.

To mitigate the possible information loss caused by applying normalization ^14,35,41 and minimum height methods ^10,12, we propose a new methodology for processing non-normal fuzzy sets. To eliminate the predominant constraint on normality we introduce the novel concept of fuzzy semi-numbers and suggest how to generalize operations defined on fuzzy numbers onto the broad family of fuzzy semi-numbers irrespective on their heigth.

The structure of this paper is as follows. In Section 2 we recall basic concepts related to fuzzy numbers. Next, in Section 3, we introduce fuzzy semi-numbers and define basic arithmetic operations on fuzzy semi-numbers of the same height, which further on are called equally high fuzzy semi-numbers. Then, in Section 4, we consider functions on equally high fuzzy semi-numbers. In Section 5 we introduce two useful operators: the elevator and fuzzy elevator which appear very helpful in generalizing arithmetic operations on fuzzy semi-numbers with arbitrary height. This way, in Section 6, we introduce a new method for performing calculations on arbitrary fuzzy semi-numbers. In Section 7 we provide a motivational medical case study showing some advantages of the proposed methodology. Finally, Section 8 concludes the contribution.

2. Preliminaries

Before delving into our contribution, let’s take a glance at some basic fuzzy theory definitions.

Let ũ denote a fuzzy set defined on the real line ℝ with its membership function μ_ũ : ℝ → [0,1].

Definition 2.1.

The support of a fuzzy set ũ is defined as follows ²⁰:

supp(u˜)={x|μu˜(x)>0}.

Definition 2.2.

The α-cut (where α ∈ [0,1]) of a fuzzy set ũ is defined by ⁴¹:

(2.1)[u˜]α={{x|μu˜(x)≥α}if α∈(0,1],supp(u˜)if α=0.

Definition 2.3.

A fuzzy set ũ is convex if ∀ x,y ∈ ℝ and ∀λ ∈ [0,1] we have ⁴¹:

μu˜(λx+(1−λ)y)≥min{μu˜(x),μu˜(y)}

Accordingly, if all α-cuts of ũ are convex, then ũ is a convex fuzzy set.

Definition 2.4.

The hight of a fuzzy set ũ is given by convex if ∀x,y∈S˜ and ∀ λ ∈ [0,1] we have ⁴¹:

hgt(u˜)=supx∈ℝμu˜(x).

Definition 2.5.

A fuzzy set ũ is called normal if given by hgt(ũ) = 1.

Definition 2.6.

The core of a normal fuzzy set ũ is defined as follows ²⁰:

core(u˜)={x|μu˜(x)=1}.

Let F(ℝ) be the set of all fuzzy numbers, (i.e. the set of all normal and convex fuzzy sets ^17,41) on the real line.

Definition 2.7.

A generalized LR fuzzy number ũ with the membership function µ_ũ(x), x ∈ ℝ can be defined as ^2,3,4,37:

μu˜(x)={lu˜(x),a≤x≤b,1,b≤x≤c,ru˜(x),c≤x≤d,0,otherwise,

where l_ũ is the left membership function and r_ũ is the right membership function. It is assumed that l_ũ is increasing in [a,b] and r_ũ is decreasing in [c,d], and that l_ũ(a) = r_ũ(d) = 0 and l_ũ(b) = r_ũ(c) = 1. In addition, if l_ũ and r_ũ are linear, then ũ is a trapezoidal fuzzy number, which is denoted by ũ = (a,b,c,d). Moreover, if b = c, we have a so-called triangular fuzzy number and we denote it by ũ = (a,c,d).

We say that a fuzzy number ũ is presented in its parametric form if ũ = (u,ū), where u and ū denote the left and right spread functions, respectively. Functions u(α) and ū(α) over [0,1], satisfy the following requirements ^6,29,30:

1.
u is increasing and left continuous,
2.
ū is decreasing and left continuous,
3.
u(α) ⩽ ū(α), 0 ⩽ α ⩽ 1.

Actually, given a family of α-cuts of a fuzzy number ũ, i.e. [ũ]^α for α ∈ [0,1], defined by (2.1), we obtain

(2.2)u_=inf[u˜]α,

(2.3)u¯=sup[u˜]α.

Note, that a crisp number a is a singleton set {a} with height 1. Hence u(α) = ū(α) = a, ∀ α ∈ [0,1], and we may denote it as a = {a}.

Let ũ= (u, ū), v˜=(v_,v¯)∈F(ℝ) and k ∈ ℝ. Basic arithmetic operations on fuzzy numbers ũ and ṽ (i.e. addition, subtraction and product by scalar) are defined as follows ^19,28:

•
k > 0 : kũ = (ku kū);
•
k < 0 : kũ = (kū, ku);
•
u˜+v˜=(u_+v_,u¯+v¯) ;
•
u˜−v˜=(u_−v¯,u¯−v_) .

A fuzzy number ũ is non-negative (non-positive) if for x < 0 (x > 0) we have µ_ũ(x) = 0. Similarly, a fuzzy number ṽ is positive (negative) if µ_ṽ(x) = 0 for x ⩽ 0 (x ⩾ 0).

Definition 2.8.

A fuzzy set with membership function μ is called a Generalized Fuzzy Number¹⁵ if the following conditions hold:

(i)
µ is a continuous mapping from ℝ to the closed interval [0, h], 0 < h ⩽ 1,
(ii)
µ(x) = 0 for x ∈ (−∞,a],
(iii)
µ is strictly increasing on [a, b],
(iv)
µ(x) = h for x ∈ [b, c],
(v)
µ is strictly decreasing on [c, d],
(vi)
µ(x) = 0 for x ∈ [d, +∞).

3. Fuzzy Semi-Numbers

Here we present several novel definitions and lemmas which will be used throughout the paper.

Definition 3.1.

A fuzzy set ũ_h is a generalized fuzzy LR semi-number^5,38, if there exist a positive number h ∈ (0,1] such that

μu˜h(x)={l(x),a≤x≤b,h,b≤x≤c,r(x),c≤x≤d,0,otherwise

where l(x) is nondecreasing on [a, b] and r(x) is non-increasing on [c, d] such that l(a) = r(d) = 0 and l(b) = r(c) = h.

We denote a fuzzy LR semi-number by (a, b, c, d;h)_LR. Obviously, if h = 1 then ũ is an LR fuzzy number.

A fuzzy semi-number is a (possibly) non-normal fuzzy set with the aforementioned conditions. It is clear that the set of all fuzzy semi-numbers is a subset of a family of all fuzzy sets and a superset of the family of of fuzzy numbers. Figure 1 shows a non-normal fuzzy set which is not a fuzzy semi-number because for lack of convexity.

The membership function of GFNs should be strictly increasing (decreasing) on left (right) spread but in fuzzy semi-numbers it should be only nondecreasing (nonincreasing). Therefore, the set of fuzzy semi-numbers is a superset of the family of the generalized fuzzy numbers. Figure 2 shows a fuzzy semi-number which is not a GFN because of the aforementioned reasons.

Moreover, a membership function of a GFN should be continuous, which is not required i the case of fuzzy semi-numbers. Figure 3 illustrates a fuzzy semi-number which is not a GFN.

Definition 3.2.

Fuzzy semi-numbers ũ and ṽ are called equally high if they have the same height, i.e.hgt(ũ) = hgt(ṽ).

We denote the set of all fuzzy semi-numbers of height h by F_h(ℝ). Therefore, the set of all fuzzy semi-numbers, denoted by FS(ℝ), satisfies

FS(ℝ)=∪h∈(0,1]Fh(ℝ).

It is clear that F(ℝ) is a proper subset of FS(ℝ).

If l(x) and r(x) are linear, then ũ_h is a trapezoidal fuzzy semi-number, which is denoted by (a,b,c,d;h). Moreover, if b = c, we obtain a triangular fuzzy semi-number and denote it by (a,b,d;h). Let TF(ℝ) = {(a, b, c, d) : a ⩽ b ⩽ c ⩽ d } and TF_h(ℝ) = {(a,b,c,d;h) : a ⩽ b ⩽ c ⩽ d, 0 < h ⩽ 1} denote the set of all trapezoidal fuzzy numbers and the set of all trapezoidal equally high fuzzy semi-numbers with height h, respectively. It TFS(ℝ) denote the set of all trapezoidal fuzzy semi-numbers then

TFS(ℝ)=∪h∈(0,1]TFh(ℝ).

Definition 3.3.

Suppose ũ_h ∈ F_h(ℝ). Then the H-core of a fuzzy semi-number ũ_h is defined as follows³⁸:

H-core(u˜h)={x|μu˜(x)=h}.

For any fuzzy semi-number ũ_h ∈ F_h(ℝ) we can consider its α-cuts defined by (2.1) for α ∈ [0,h]. Similarly, by (2.2)–(2.3) we define functions u(α) and ū(α) for α ∈ [0,h]. This way we obtain the parametric form of a fuzzy semi-number given by ũ_h = (u,ū;h), where functions u and ū satisfy the following requirements:

1.
u is increasing and left continuous on [0,h].
2.
ū is decreasing and left continuous on [0,h].
3.
u(α) ⩽ ū(α), 0 ⩽ α ⩽ h.

In particular, for a trapezoidal fuzzy semi-number ũ_h = (a,b,c,d;h) we obtain

u_(α)=a+b−ahα,u¯(α)=d−d−chα,

where 0 ⩽ α ⩽ h.

Since a fuzzy semi-number with height h = 1 is a fuzzy number, hence further on in such cases we’ll omit height in the notation, i.e. we denote ũ₁ = (u, ū;1) by ũ = (u, ū).

A crisp semi-number³⁸ ah° is a fuzzy singleton set with height h. Then u(α) = ū(α) = a, ∀ α ∈ [0, h], and we denote it as ah°={a;h} . We denote the set of all crisp semi-numbers of height h by ℝh° . Hence ℝ1°=ℝ .

4. Functions on equally high fuzzy semi-numbers

Suppose we want to generalize a mapping f : ℝⁿ → ℝ onto the framework of fuzzy semi-numbers. Thus we are interested in a mapping from Fhn(ℝ) into F_h(ℝ), where Fhn(ℝ)={u˜h=(u˜1,h,…,u˜n,h):u˜1,h,…,u˜n,h∈Fh(ℝ)} , such that

v˜h=f(u˜h).

By the Extension Principle, the membership function of ṽ_h ∈ F_h(ℝ) is given by

(4.1)μv˜(y)={sup(x1,…,xn)∈f−1(y)min{μ˜u˜1,h(x1),…,μu˜n,h(xn)}0,if f−1(y)≠∅,otherwise.

In particular, if f : F_h(ℝ) → F_h(ℝ) is a univariate function then

μv˜h(y)=supy=f(x)μu˜h(x),

which for an invertible function f reduces to

(4.2)μv˜h(y)=μu˜h(f−1(y)).

One can easily notice that for f : ℝ → ℝ and a fuzzy semi-number ũ_h = (u, ū;h) in F_h(ℝ) we obtain

f(u˜h)=(f(u_),f(u¯);h).

The proof of the following useful lemma is straightforward.

Lemma 4.1.

Let (ũ_h, ṽ_h) be a pair of fuzzy semi-numbers in Fh2(ℝ) and let f : ℝ² → ℝ be an increasing function with respect to both variables. Then, it is:

f(u˜h,v˜h)=(f(u_,v_),f(u¯,v¯);h).

Due to the aforementioned considerations we can define basic arithmetic operations on equally high fuzzy semi-numbers.

Definition 4.1.

Let ũ_h = (u, ū;h) and v˜h=(v_,v¯;h) be two equally high fuzzy semi-numbers with height h ∈ (0, 1]. Then

u˜h+v˜h=(u_+v_,u¯+v¯;h),u˜h⋅v˜h=(u⋅v_,u⋅v¯;h),

where

u⋅v_=min{u_⋅v_,u_⋅v¯,u¯⋅v_,u¯⋅v¯},u⋅v¯=max{u_⋅v_,u_⋅v¯,u¯⋅v_,u¯⋅v¯}.

Lemma 4.2.

If both of ũ_h and ṽ_h are two non-positive or non-negative fuzzy semi-numbers belonging to F_h(ℝ), then

u˜h⋅v˜h=(u_⋅v_,u¯⋅v¯;h).

Proof.

We know that (x, y) → xy is an increasing function over 𝒜² and ℬ² where 𝒜 = [0, +∞) and ℬ = (−∞, 0]. Therefore, according to Lemma 4.1 the proof is completed.

Lemma 4.3.

If ũ_h is a non-negative fuzzy semi-number and ṽ_h is a non-positive fuzzy semi-number belonging to F_h(ℝ), then

u˜h⋅v˜h=(u_⋅v¯,u¯⋅v_;h).

And if ũ_h is a non-positive fuzzy semi-number and ṽ_h is a non-negative fuzzy semi-number belonging to F_h(ℝ), then

u˜h⋅v˜h=(u¯⋅v_,u_⋅v¯;h).

Proof.

We know that (x, y) → xy is a decreasing function over 𝒜 × ℬ where 𝒜 = [0, +∞) and ℬ = (−∞, 0]. And similarly, we know that (x, y) → xy is a decreasing function over ℬ × 𝒜 where 𝒜 = [0, +∞) and ℬ = (−∞, 0]. Thus, according to Lemma 4.1 the proof is completed.

Note, that 0h°=(0;h) is the additive identity in F_h(ℝ), while 1h°=(1;h) is the multiplicative identity in F_h(ℝ). Indeed

u˜h+0h°=(u_+0,u¯+0;h)=(u_,u¯;h)=u˜h

and

u˜h⋅1h°=(u⋅1_,u⋅1¯;h)=(u_,u¯;h)=u˜h.

We can also define the scalar multiplication, subtraction ad division in F_h(ℝ).

Definition 4.2.

Let ũ_h = (u, ū;h) ∈ F_h(ℝ) and k∈ℝh° . Then

ku˜h={(ku_,ku¯;h)if kh°>0h°,(ku¯,ku_;h)if kh°<0h°,

Definition 4.3.

Let ũ_h = (u, ū;h) and v˜h=(v_,v¯;h) be two equally high fuzzy semi-numbers with height h ∈ (0,1]. Then

u˜h−v˜h=(u_−v¯,u¯−v_;h),u˜h÷v˜h=((uv)_,(uv)¯;h),

where

(uv)_=min{u_/v_,u_/v¯,u¯/v_,u¯/v¯},(uv)¯=max{u_/v_,u_/v¯,u¯/v_,u¯/v¯}.

Let ah° , bh°∈ℝh° be two equally high crisp semi-numbers. The proof of the following facts is straightforward:

ah°+bh°=(a+b)h°,ah°−bh°=(a−b)h°,ah°⋅bh°=(ab)h°,ah°bh°=(ab)h°, b≠0.

Let us consider the following examples.

Example 4.1.

Let u˜12 be a fuzzy semi-number with height 12 given by the following membership function

μu˜12(x)={x−12,1≤x<2,5−x6,2≤x<5.

Let v˜12=f(u˜12)=exp(u˜12) be a function on F_h(ℝ) to itself. Since f is a monotonic function hence by (4.2) we obtain μv˜12(y)=μu˜12(f−1(y))=μu˜12(ln(y)) given by

μv˜12(y)={ln(y)−12,e≤y<e2,5−ln(y)6,e2≤y<e5.

Membership functions of u˜12 and v˜12 are given in Figures 4 and 5, respectively.

Let us also consider w˜12=g(u˜12)=lnu˜12 . Since g is a monotonic function, we obtain μw˜12(y)=μu˜12(f−1(y))=μu˜12(exp(y)) , where

μw˜12(y)={exp(y)−12,0≤y<ln2;5−exp(y)6,ln2≤y<ln5.

The membership functions of w˜12 is given in Figure 6.

Example 4.2.

Let u˜13 be a fuzzy semi-number with height 13 with the following membership function:

μu˜13(x)={2πx,0≤x<π6,π2−xπ,π6≤x<π2.

Let v˜13=k(u˜13)=sin(u˜13) Since k is an one to ne function in [0,π2] hence we have μv˜13(y)=μu˜13(k−1(y))=μu˜13(arcsin(y))

μv˜13(y)={2πarcsin(y),0≤arcsin(y)<π6,π2−arcsin(y)π,π6≤arcsin(y)<π2={2πarcsin(y),0≤y<12,π2−arcsin(y)π,12≤y<1.

Membership functions of u˜12 and v˜12 are given in Figures 7 and 8, respectively.

5. Fuzzy elevator operator

In this section we introduce two operators: elevator and fuzzy elevator, which will come in handy in presentation of elementary arithmetic operations on fuzzy semi-numbers in general form.

Definition 5.1.

Let C_ML(A) be the set of all monotonic left continuous functions over the set A. Let f (α) ∈ C_ML([0, h]), where h ⩽ 1. An elevator operator is a mapping Ehh*:CML([0,h])→CML([0,h*]) , where h^* ∈ [h, 1], such that f*=Ehh*(f) , where f^* is given as follows⁵:

f*(α)={f(α),α∈(0,h],f(h),α∈(h,h*],0,α∉(0,h*].

It is clear that f* is an extension of f, i.e. f^*|_(0,_h_](α) = f(α) for any α ∈ (0,h]. It is also easily seen that an elevator is a linear operator, i.e. for any two monotonic left continuous functions f and g and a real number k we have

Ehh*(kf+g)=kEhh*(f)+Ehh*(g).

Definition 5.2.

A fuzzy elevator is an operator Ehh*:Fh(ℝ)→Fh*(ℝ) defined as follows⁵:

E˜hh*(u˜h)=E˜hh*(u_,u¯;h)=(Ehh*(u_),Ehh*(u¯);h*)=(u_*,u¯*;h*)=u˜h**,

where ũ_h = (u, ū; h) ∈ F_h(ℝ) and h* ∈ [h, 1]. A fuzzy semi-number u˜h**=(u_*,u¯*;h*)∈Fh*(ℝ) is called the elevated fuzzy semi-number.

By Definition 5.1 functions u^*(α) and ū^*(α) describing the elevated fuzzy semi-number u˜h**=(u_*,u¯*;h*) are obtained by applying an elevator operator to u and ū, respectively⁵. Hence we obtain

u_*(α)={u_(α),α∈(0,h],u_(h),α∈(h,h*],0,α∉(0,h*]

and

u¯*(α)={u¯(α),α∈(0,h],u¯(h),α∈(h,h*],0,α∉(0,h*].

We say that fuzzy semi-numbers ũ_h = (u, ū; h) ∈ F_h(ℝ) and u˜h**=(u_*,u¯*;h*)∈Fh*(ℝ) are associated.

The proofs of the following lemmas are straightforward.

Lemma 5.1.

Let ũ_h = (u, ū; h) ∈ F_h(ℝ) be an arbitrary fuzzy semi-number. The the spread functions u^*(α) and ū^*(α) of its associated elevated fuzzy semi-number u˜h**=(u_*,u¯*;h*)∈Fh*(ℝ) satisfy the following properties:

1.
u^* is an increasing left continuous function over [0, h^*],
2.
ū^* is a decreasing left continuous function over [0, h^*],
3.
u^*(α) ⩽ ū^*(α), 0 ⩽ α ⩽ h^*,
4.
u^*(α) = ū^*(α) = 0, α ∉ (0,h^*].

Lemma 5.2.

Fuzzy semi-numbers are closed under fuzzy elevator, i.e. the elevated fuzzy semi-number with respect to a given height h^* is a fuzzy semi-number with height h^*.

Obviously, the elevated of a crisp semi-number ah° with respect to a given height h^* is a crisp semi-number ah*° .

Example 5.1.

Figures 9 and 10 show the spread functions of a fuzzy semi-number u˜=(1,2,3,5;13) and its associated elevated fuzzy semi-number with respect to height 23 .

Similarly, Figures 11 and 12 present the spread functions of a fuzzy semi-number u˜=(1,2,2,5;13) and its associated elevated fuzzy semi-number with respect to height 23

6. Arithmetic operations on general fuzzy semi-numbers

To perform fuzzy arithmetic on non-normal fuzzy sets two main approaches have been used so far. Firstly, one may normalize the non-normal fuzzy sets by dividing them by their heights and then apply the desired operations well-defined for fuzzy numbers ^14,35,41. Alternatively, one may define an operation by the extension principle with respect to the minimum heights of non-normal fuzzy sets ^10,12. In this section, we propose another method for defining elementary arithmetic operations on fuzzy semi-number with different heights utilizing heir elevated forms with respect to the maximum of their heights.

Definition 6.1.

Let ũ_{h_u} = (u, ū; h_u) and v˜hv=(v_,v¯;hv) be any fuzzy semi-numbers and let h^* = max{h_u, h_v}. Then basic arithmetic operations on fuzzy semi-numbers are defined as follows:

•
addition
u˜hu+v˜hv=u˜h**+v˜h**=(u_*+v_*,u¯*+v¯*;h*),
•
subtraction
u˜hu−v˜hv=u˜h**−v˜h**=(u_*−v¯*,u¯*−v_*;h*).
•
multiplication
u˜hu⋅v˜hv=u˜h**⋅v˜h**=(u*⋅v*_,u*⋅v*¯;h*),
where:
u*⋅v*_=min{u_*⋅v_*,u_*⋅v¯*,u¯*⋅v_*,u¯*⋅v¯*},u*⋅v*¯=max{u_*⋅v_*,u_*⋅v¯*,u¯*⋅v_*,u¯*⋅v¯*}.
•
division
u˜hu÷v˜hv=u˜h**÷v˜h**=((u*v*)_,(u*v*)¯;h*),
where
(u*v*)_=min{u_*/v_*,u_*/v¯*,u¯*/v_*,u¯*/v¯*},(u*v*)¯=max{u_*/v_*,u_*/v¯*,u¯*/v_*,u¯*/v¯*}
and where v_hv and v¯hv are not zero.

As a straightforward conclusion of the above definition we obtain formulas for crisp semi-numbers. Let aha° and bhb° and let h_max = max{h_a, h_b}. Then we obtain

aha°+bhb°=(a+b)hmax°,aha°−bhb°=(a−b)hmax°,aha°⋅bhb°=(a⋅b)hmax°,aha°bhb°=(ab)hmax°,

where b ≠ 0.

One can easily prove the following lemma.

Lemma 6.1.

Trapezoidal equally high fuzzy semi-numbers are closed under addition and subtraction operations, i.e.

u˜h,v˜h∈TFh(ℝ)⇒(u˜h+v˜h),(u˜h−v˜h)∈TFh(ℝ).

Lemma 6.2.

If h^* ⩽ h then u˜h+0h*°=u˜h and u˜h⋅1h*°=u˜h .

Proof.

If h^* ⩽ h then u˜h+0h*°=(u_+0,u¯+0,h)=(u_,u¯;h)=u˜h , and u˜h⋅1h*°=(u⋅1_,u⋅1¯;h)=(u_,u¯;h)=u˜h .

In general, trapezoidal fuzzy semi-numbers are not closed under addition and subtraction operations, i.e. addition and subtraction of two arbitrary trapezoidal fuzzy semi-numbers with different heights are not trapezoidal fuzzy semi-numbers. Indeed, since the heights of the two given trapezoidal fuzzy semi-numbers can be different, the one with the smaller height must be elevated before addition or subtraction. Therefore, the elevated fuzzy semi-number can not be a trapezoidal fuzzy semi-number. Hence, their addition and subtraction can not be a trapezoidal fuzzy semi-number.

Similarly, trapezoidal fuzzy semi-numbers are not closed under multiplication and division operations, i.e. for two arbitrary trapezoidal fuzzy semi-numbers ũ_h, ṽ_h ∈ TFS(ℝ), their multiplication and division are not trapezoidal fuzzy semi-numbers. Indeed, since left and right spreads of a trapezoidal fuzzy semi-number are polynomials of degree one, their multiplication and division do not yield left and right spreads of degree one. Therefore, their multiplication and division are not trapezoidal fuzzy semi-numbers.

Lemma 6.3.

Elementary arithmetic operations between an arbitrary fuzzy semi-number and a crisp number gives a fuzzy number.

Proof.

Let ũ_h = (u, ū;h) be a fuzzy semi-number with height h ∈ (0, 1] and k ∈ ℝ. Since k = (k;1) and max{h, 1} = 1, we have the following results:

k+u˜h=(k+u_,k+u¯;1)=(k+u_,k+u¯),k⋅u˜h=(k⋅u_,k⋅u¯;1)=(k⋅u_,k⋅u¯).

It is also easily seen that for any fuzzy semi-number ũ_h ∈ FS(ℝ) the following properties hold

0+u˜h≠u˜h,1⋅u˜h≠u˜h.

To compute an algebraic expression of given fuzzy semi-numbers, the heights associated with these semi-numbers must be equalized. To this end, after selecting the maximum height of these semi-numbers, all the fuzzy semi-numbers are elevated to the maximum height by applying the fuzzy elevator operator. Therefore, now we have an algebraic expression consisting of equiheight fuzzy semi-numbers upon which the elementary arithmetic operators (defined in Section 5) can act. According to this, Arithmetic Mean of n fuzzy semi-numbers, ũ_i, i = 1, …, n denoted as AM is defined as follows:

(6.1)AM=1n∑i=1nu˜i.

Any arbitrary elementary arithmetic operation between two given fuzzy semi-numbers ũ_{h_u}, ṽ_{h_v} based on the Extension Principle can be computed by applying the aforementioned methods (introduced in this section) and then by obtaining the h^*-cut of the result where h_* = min{h_u, h_v}. i.e.

(6.2)u˜hu⊙v˜hv=[u˜h**⊙v˜h**]h*,

Here, ⊙ can be any elementary arithmetic operator.

Let us consider some numerical examples illustrating the proposed methodology.

Example 6.1.

Let u˜23=(−3,−1,2,3;23) and v˜13=(1,2,3,5;13) be two fuzzy semi-numbers. They are represented in the parametric form as follows

u˜=(u_,u¯;hu)=(−3+3α,3−32α;23),v˜=(v_,v¯;hv)=(1+3α,5−6α;13).

By applying a fuzzy elevator with respect to maximum height 23 we obtain the associated fuzzy semi-numbers u˜23* and v˜23* with the following spread functions

u_*(α)={−3+3α,0≤α≤13,−3+3α,13≤α≤23,0,otherwise,u¯*(α)={3−32α,0≤α≤13,3−32α,13≤α≤23,0,otherwise,

and

v_*(α)={1+3α,0≤α≤13,2,13≤α≤23,0,otherwise,v¯*(α)={5−6α,0≤α≤13,3,13≤α≤23,0,otherwise.

Now we are able to perform some arithmetic operators. In particular, we obtain the following sum

u˜23+v˜13=u˜23*+v˜23*=(u_*+v_*,u¯*+v¯*;23),

where

u_*(α)+v_*(α)={−2+6α,0≤α≤13,−1+3α,13≤α≤23,0,otherwise,u¯*(α)+v¯*(α)={8−152α,0≤α≤13,6−32α,13≤α≤23,0,otherwise.

The difference looks as follows

u˜23−v˜13=u˜23*−v˜23*=(u_*−v¯*,u¯*−v_*;23),

where

u_*(α)−v¯*(α)={−8+9α,0≤α≤13,−6+3α,13≤α≤23,0,otherwise,u¯*(α)−v_*(α)={2−92α,0≤α≤13,1−32α,13≤α≤23,0,otherwise.

Here is the product

u˜23⋅v˜13=u˜23*⋅v˜23*=(u*⋅v*_,u*⋅v*¯;23),

where

u*⋅v*_(α)={(3−3α)(−5+6α),0≤α≤13,−9+9α,13≤α≤23,0,otherwise,u*⋅v*¯(α)={(3−32α)(5−6α),0≤α≤13,9−92α,13≤α≤23,0,otherwise.

Finally, we obtain the following ratio

u˜23÷v˜13=u˜23*÷v˜23*=((u*v*)_,(u*v*)¯;23),

where

(u*v*)_(α)={3(−1+α)1+3α,0≤α≤13,32(−1+α),13≤α≤23,0,otherwise,(u*v*)¯(α)={6−3α2+6α,0≤α≤13,−34(−2+α),13≤α≤23,0,otherwise.

Example 6.2.

Let u˜=(1,2,3,5;12) and v˜=(1,3,4,5;13) be two fuzzy semi-numbers. The spread functions of ũ, ṽ and the associated elevated fuzzy semi-numbers with respect to the maximum of their heights are shown in Figures 13–16:

The spread functions of the results of addition, subtraction, multiplication and division of the elevated semi-numbers are shown in Figures 17–20

7. A medical case study

Mean Arterial Pressure (MAP), a common term in medicine, indicates the depth of patient’s anesthesia ^39,40. This feature helps the anesthetist to monitor patient’s fluid balance, ventilation (i.e. the supply of air to the lungs) and drug application. Inhaling anesthetic gases is one of the usual ways to take patients into a deep state of unconsciousness. Isoflurane, as one of these gases, is often deployed in conjunction with oxygen or nitrous oxide. In practice the MAP of a patient is measured by a fuzzy logic controller. As a typical example, a surgical team in an operating room uses a MAP controller to know the status of the blood pressure of a patient according to the amount of Isoflurane he has already inhaled. For greater accuracy, the patient blood pressure should be measured (in mm of Hg) at least twice with 2 hours interval. Suppose that the resulting outputs of blood pressure is obtained in terms of two fuzzy sets B˜1 , B˜2 with spread functions shown in Figures 21 and 22. These sets express the uncertainty in each blood pressure measurement.

An anesthetist needs these two outputs, namely B˜1 and B˜2 , to be somehow aggregated, so that she/he can make a more prudent and sensible decision about the anesthesic depth of a patient. The fundamental part of such aggregations is often done via normalization which is the most prevalent and somewhat even the sole way today. One approach to aggregating B˜1 and B˜2 is to take their arithmetic mean.

A traditional way for calculating the arithmetic mean of non-normal fuzzy sets goes through their normalization. However, in our case, B˜1 and B˜2 after normalization are indistinguishable (see Figures 23 and 24) co, consequently, the arithmetic mean of their normalized forms results in just the same fuzzy set.

However, by applying our elevation method for computing the arithmetic mean of B˜1 and B˜2 we get two equally high fuzzy sets while at the same time their differences are preserved. Figures 25, 26 and 27 illustrate these elevated values and their associated mean value, respectively.

As it has been shown, the normalization approach may lead to misleading results. Indeed, through the deformation of the shape of the input data caused by the normalization process we have lost some information on the initial differences between fuzzy observations. Fortunately, the proposed elevation method removes this weakness. It is obvious that such information loss may cause incorrect or false conclusions which sometimes may can result in serious and irrecoverable medical faults.

8. Conclusions

Fuzzy numbers of height one has formed the cornerstone of normal fuzzy sets. Based on this assumption, not only rudimentary arithmetic has been established but also, other topics e.g. fuzzy ranking, fuzzy approximation, fuzzy differential equations, fuzzy integral equations and systems of fuzzy equations have been developed.

Having gone against this prevailing thought of height one in fuzzy numbers, in this paper, we considered fuzzy sets not to be restricted by any predefined height. To achieve this goal we introduced the concept of fuzzy semi-number. Next we investigated elementary arithmetic on equally high fuzzy semi-numbers. Then a new method, based on elevation, for defining algebraic operations on general fuzzy semi-numbers in general form was introduced. As it was demonstrated in medical case study and some numerical examples, using the proposed methodology we avoid a loss of information in processing fuzzy data which may be caused by the traditional normalization method.

References

1.S Abbasbandy and B Asady, The nearest trapezoidal fuzzy number to a fuzzy quantity, Journal of Applied Mathematics and Computations, Vol. 156, 2004, pp. 381-386.

2.S Abbasbandy and M Amirfakhrian, The nearest approximation of a fuzzy quantity in parametric form, Journal of Applied Mathematics and Computations, Vol. 172, 2006, pp. 624-632.

3.S Abbasbandy and M Amirfakhrian, The nearest trapezoidal form of a generalized left right fuzzy number, Journal of Approximate Reasoning, Vol. 43, 2006, pp. 166-178.

4.M Amirfakhrian, Properties of parametric form approximation operator of fuzzy numbers, Analele Stiintifice ale Universitatii Ovidius Constanta, Vol. 18, 2010, pp. 23-34.

5.M Amirfakhrian, Sh Yeganehmanesh, and P Grzegorzewski, A new distance on fuzzy semi-numbers, Soft Computing, 2017. https://doi.org/10.1007/s00500-017-2902-2.

6.L Anzilli, G Facchinetti, and G Mastroleo, A parametric approach to evaluate fuzzy quantities, Fuzzy Sets and Systems, Vol. 250, 2014, pp. 110-133.

7.AI Ban and LC Coroianu, Simplifying the Search for Effective Ranking of Fuzzy Numbers, IEEE Transactions on Fuzzy Systems, Vol. 23, No. 2, 2015, pp. 327-339.

8.AI Ban, L Coroianu, and A Khastan, Conditioned weighted LR approximations of fuzzy numbers, Fuzzy Sets and Systems, Vol. 62, 2015, pp. 1-26.

9.AI Ban, L Coroianu, and P Grzegorzewski, Trapezoidal approximation and aggregation, Fuzzy Sets and Systems, Vol. 177, No. 1, 2011, pp. 45-59.

10.S Banerjee, Arithmetic Operations on Generalized Trapezoidal Fuzzy Number and its Applications, Turkish Journal of Fuzzy Systems, Vol. 3, No. 1, 2012, pp. 16-44.

11.D Behera and S Chakraverty, Solving fuzzy complex system of linear equations, Informations Science, Vol. 277, 2014, pp. 154-162.

12.D Chakraborty and D Guha, Addition of two Generalized Fuzzy Numbers, International Journal of Industrial Mathematics, Vol. 2, No. 1, 2010, pp. 9-20.

13.S Chanas, On the interval approximation of a fuzzy number, Fuzzy Sets and Systems, Vol. 122, 2001, pp. 353-356.

14.PT Chang and ES Lee, Ranking of fuzzy sets based on the concept of existence, Computers and Mathematics with Applications, Vol. 27, No. 10, 1994, pp. 1-21.

15.SH Chen, Ranking fuzzy numbers with maximizing set and minimizing set, Fuzzy Sets and Systems, Vol. 17, No. 2, 1985, pp. 113-129.

16.L Coroianu, M Gagolewski, and P Grzegorzewski, Nearest piecewise linear approximation of fuzzy numbers, Fuzzy Sets and Systems, Vol. 233, 2013, pp. 26-51.

17.M Delgado, MA Vila, and W Voxman, On a canonical representation of fuzzy numbers, Fuzzy Sets and Systems, Vol. 93, 1998, pp. 125-135.

18.SL Dong, H Vu, and DN Phu, The formulas of the solution for linear-order random fuzzy differential equations, Journal of Intelligent and Fuzzy Systems, Vol. 28, No. 2, 2015, pp. 795-807.

19.D Dubois and H Prade, Operations on fuzzy numbers, International Journal Systems Science, Vol. 9, 1978, pp. 613-626.

20.D Dubois and H Prade, Fuzzy Sets and Systems: Theory and Application, Academic Press, New York, 1980.

21.MA Fariborzi Araghi and Gh Kazemi Gelian, Solving fuzzy Fredholm linear integral equations using Sinc method and double exponential transformation, Soft Computing, Vol. 19, No. 4, 2015, pp. 1063-1070.

22.LS Gao, The fuzzy arithmetic mean, Fuzzy Sets and Systems, Vol. 107, No. 3, 1999, pp. 335-348.

23.P Grzegorzewski, Nearest interval approximation of a fuzzy number, Fuzzy Sets and Systems, Vol. 130, 2002, pp. 321-330.

24.P Grzegorzewski and E Mrówka, Trapezoidal approximations of fuzzy numbers, Fuzzy Sets and Systems, IFSA 2003, T Bilgic, B DeBaets, and O Kay nak (editors), Lecture Notes in Artificial Intelligence 2715, Springer, 2003, pp. 237-244.

25.P Grzegorzewski and E Mrówka, Trapezoidal approximations of fuzzy numbers, Fuzzy Sets and Systems, Vol. 153, 2005, pp. 115-135.

26.D Guha and D Chakraborty, A new approach to fuzzy distance measure and similarity measure between two generalized fuzzy numbers, Applied Soft Computing, Vol. 10, 2010, pp. 90-99.

27.A Khastan and R Rodriguez-López, On periodic solutions to first order linear fuzzy differential equations under differential inclusions’ approach, Information Science, Vol. 322, 2015, pp. 31-50.

28.M Ma, M Friedman, and A Kandel, A new fuzzy arithmetic, Fuzzy Sets and Systems, Vol. 108, 1999, pp. 83-90.

29.M Ma, A Kandel, and M Friedman, A new approach for defuzzification, Fuzzy Sets and Systems, Vol. 111, 2000, pp. 351-356.

30.M Ma, A Kandel, and M Friedman, Correction to “A new approach for defuzzification”, Fuzzy Sets and Systems, Vol. 128, 2002, pp. 133-134.

31.VH Ngo, Fuzzy fractional functional integral and differential equations, Fuzzy Sets and Systems, Vol. 280, 2015, pp. 58-90.

32.S Pushpinder, A novel method for ranking generalized fuzzy numbers, Information Science and Engineering, Vol. 31, 2015, pp. 1373-1385.

33.X Qu, F Sun, and T Wang, Matrix elementary transformations in solving systems of fuzzy relation equations, Applied Soft Computing, Vol. 31, 2015, pp. 25-29.

34.W Voxman, Some remarks on distance between fuzzy numbers, Fuzzy Sets and Systems, Vol. 100, 1998, pp. 353-365.

35.RR Yager, On the normalization of fuzzy belief structures, International Journal of Approximate Reasoning, Vol. 14, No. 2, 1996, pp. 127-153.

36.JS Yao and K Wu, Ranking fuzzy numbers based on decomposition principle and signed distance, Fuzzy Sets and Systems, Vol. 116, 2000, pp. 275-288.

37.Sh Yeganehmanesh and M Amirfakhrian, An Approximation of Fuzzy Numbers Based on Polynomial Form Fuzzy Numbers, International Journal of Analysis and Applications, Vol. 16, No. 2, 2018, pp. 290-305.

38.Sh Yeganehmanesh, M Amirfakhrian, and P Grzegorzewski , Fuzzy semi-numbers and a distance on them with a case study in medicine, Mathematical Sciences, Vol. 12, No. 1, 2018, pp. 4152.

39.H Ying and LC Sheppard, Regulating mean arterial pressure in postsurgical cardiac patients, IEEE Engineering in Medicine and Biology Magazine, Vol. 13, No. 5, 1994, pp. 671-677.

40.AM Zbinden, P Feigenwinter, S Petersen-felix, and S Hacisalizade, Arterial pressure control with isoflurane using fuzzy logic, British Journal of Anaesthesia, Vol. 74, 1995, pp. 66-72.

41.HJ Zimmermann, Fuzzy Set Theory and Its Applications, 2nd Edition, Kluwer Academic, Boston, 1991.

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 11 - 1
Pages: 991 - 1004
Publication Date: 2018/05/02
ISSN (Online): 1875-6883
ISSN (Print): 1875-6891
DOI: 10.2991/ijcis.11.1.75 How to use a DOI?
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TY  - JOUR
AU  - Sh. Yeganehmanesh
AU  - M. Amirfakhrian
AU  - P. Grzegorzewski
PY  - 2018
DA  - 2018/05/02
TI  - Fuzzy Semi-Numbers and Their Elementary Arithmetic With a Medical Case Study
JO  - International Journal of Computational Intelligence Systems
SP  - 991
EP  - 1004
VL  - 11
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.11.1.75
DO  - 10.2991/ijcis.11.1.75
ID  - Yeganehmanesh2018
ER  -

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