Fuzzy seminumbers and a distance on them with a case study in medicine
Abstract
In this paper, we present the novel concept of fuzzy seminumbers. Then, a method for assigning distance between every pair of fuzzy seminumbers is given. Moreover, it is shown that this distance is a metric on the set of all trapezoidal fuzzy seminumbers with the same height and is a pseudometric on the set of all fuzzy seminumbers. Also, by utilizing this distance, we propose an approximation of a fuzzy seminumber with given height and apply this approximation method in a medical case study.
Keywords
Fuzzy sets Fuzzy numbers Fuzzy seminumbers Generalized LR fuzzy seminumbers Hvalue HambiguityIntroduction
Conventional works on fuzzy logic assume that fuzzy sets satisfy the conditions of convexity and normality (the height is one) which are called fuzzy numbers. Many of the researches in fuzzy logic have contributed to the approximation methods of a fuzzy number. Such an approximation has been done in several ways. Some authors have assigned a crisp number to a fuzzy number as a ranking method. Such methods suffer from a great amount of data loss. Some other methods [10, 14] have defined an interval as an approximation of a fuzzy number. But, in this case, the modal value (the core with height 1) of the fuzzy numbers is lost. In some works such as [1, 3, 13, 15, 16], the authors have tried to solve an optimization problem to obtain a trapezoidal fuzzy number as a nearest approximation. Recently, some works have been done on approximation of a fuzzy number [4, 6, 7, 8, 9, 11] by defining distance functions.
All the aforementioned methods destroy much of the useful data associated with the fuzzy number and as a result imprecision of the approximation increases. Such loss is not negligible specially in high critical domains such as medicine where most of the clinical data are intrinsically inexact and ambiguous. To mitigate this, most often applying fuzzy mathematical modeling is favorable approach. However, these data are not necessarily normal fuzzy sets. To overcome these problems, this paper works on fuzzy sets in general form and without regard to their height. At first, we present the novel concept of fuzzy seminumber. Then, we propose a distance to approximate an arbitrary fuzzy seminumber. Depending on some predefined conditions, the result of approximation can be either a fuzzy number or fuzzy seminumber. To this end, in this paper an approximation for a given fuzzy set without considering their height is proposed. Also, for more clarification, a real application of fuzzy seminumber will be studied in terms of a case study.
The structure of the this paper is as follows. In “Preliminaries”, the basic concepts of our work are introduced; then in next section, we review fuzzy seminumbers. Section “Height source distance between fuzzy seminumbers” introduces a new distance namely Height Source Distance for fuzzy seminumbers. In next section, the nearest trapezoidal fuzzy seminumber to an arbitrary fuzzy seminumber is introduced and a simple method for its computation is presented. Next section contains some numerical examples. Section “A medical case study” presents a case study in medicine and the final section concludes the article.
Preliminaries
Before delving in to our contributions, let us take a glance at some basic fuzzy theory definitions. Let \(F({\mathbb{R}})\) be the set of all real fuzzy numbers (which are normal, upper semicontinuous, convex and compactly supported fuzzy sets).
 1.
\(\underline{u}\) is a monotonically increasing left continuous function.
 2.
\(\overline{u}\) is a monotonically decreasing left continuous function.
 3.
\(\underline{u}(\alpha )\le \overline{u}(\alpha ) ,\quad 0\le \alpha \le 1\).

\(x > 0\) : \(x\tilde{u} =(x\underline{u},x\overline{u})\);

\(x < 0\) : \(x\tilde{u} =(x\overline{u},x\underline{u})\);

\(\tilde{u}+\tilde{v} = (\underline{u}+\underline{v},\overline{u}+\overline{v})\);

\(\tilde{u}\tilde{v} = (\underline{u}\overline{v},\overline{u}\underline{v})\).
Definition 1
Definition 2
Definition 3
 1.
\(\text{ V }(\tilde{u})=\int _0^1{s(\alpha )[\overline{u}(\alpha )+\underline{u}(\alpha )]\mathrm{d}\alpha }\),
 2.
\(\text{ A }(\tilde{u})=\int _0^1{s(\alpha )[\overline{u}(\alpha )\underline{u}(\alpha )]\mathrm{d}\alpha }\).
Fuzzy seminumbers
Here, we present several novel definitions and lemmas which will be used throughout the paper.
Definition 4
A fuzzy seminumber \(\tilde{v}_h\) is nonnegative (nonpositive) if for \(x<0\ (x>0)\), we have \(\mu _{\tilde{v}}(x)=0\), equivalently if on [0, h] we have \(\underline{v}\ge 0\ (\overline{v}\le 0)\). Also a fuzzy seminumber \(\tilde{v}_h\) is positive (negative) if for \(x\le 0\ (x\ge 0)\), we have \(\mu _{\tilde{v}}(x)=0\), equivalently if on [0, h] we have \(\underline{v}> 0\ (\overline{v}< 0)\).
Definition 5
Definition 6
 1.
\(s_h(\alpha )\ge 0 ,\quad \alpha \in [0,h]\)
 2.
\(s_h(0)=0\),
 3.
\(s_h(h)=h\),
 4.
\(\int _0^h{s_h(\alpha )\mathrm{d}\alpha =\frac{1}{2}h^2}\).
Definition 7
Let \(s_{h_1}\) be a source function over \([0,h_1]\). \(s_{h_2}\) is equivalent to \(s_{h_1}\) if for a given height \(h_2\), \(s_{h_2}\) is defined as \(s_{h_2}(\alpha )=\frac{h_2}{h_1}s_{h_1}(\frac{h_1}{h_2}\alpha )\). This equivalence is denoted by \(s_{h_1}\approx s_{h_2}\).
Lemma 1
Let \(s_{h_1}\) be a source function over \([0,h_1]\). If \(s_{h_2}\approx s_{h_1}\), then \(s_{h_2}\) is a source function over \([0,h_2]\).
Proof
 1.
\(s_{h_2}(\alpha )=\frac{h_2}{h_1}s_{h_1}(\frac{h_1}{h_2}\alpha )\ge 0 ,\quad \alpha \in [0,h_2]\)
 2.
\(s_{h_2}(0)=\frac{h_2}{h_1}s_{h_1}(0)=0\),
 3.
\(s_{h_2}(h_2)=\frac{h_2}{h_1}s_{h_1}(\frac{h_1}{h_2}h_2)=h_2\),
 4.
\(\int _0^{h_2}{s_{h_2}(\alpha )\mathrm{d}\alpha = \frac{h_2}{h_1} \int _0^{h_2}s_{h_1}(\frac{h_1}{h_2}\alpha )\mathrm{d}\alpha = \frac{h_2}{h_1} \frac{h_2}{h_1}\int _0^{h_1}s_{h_1}(\beta )\mathrm{d}\beta =\frac{1}{2}h_2^2}\),
Hence \(s_{h_2}\) is a source function over \([0,h_2]\). \(\square\)
 1.
\(\underline{u}\) is a monotonically increasing left continuous function over [0, h].
 2.
\(\overline{u}\) is a monotonically decreasing left continuous function over [0, h].
 3.
\(\underline{u}(\alpha )\le \overline{u}(\alpha ) ,\quad 0\le \alpha \le h\).
 4.
\(\underline{u}(\alpha )= \overline{u}(\alpha )=0\), for \(\alpha \not \in [0,h]\).
Definition 8
Crisp seminumber \(a^\circ _h\), is a fuzzy singleton set with height h, then \(\underline{u}(\alpha )= \overline{u}(\alpha )=a\), for \(\forall \alpha \in [0,h]\) and we denote \(a^\circ _h= (a;h)\). We denote the set of all crisp seminumbers of height h by \({\mathbb{R}}^\circ _h\) and we have \({\mathbb{R}}^\circ _1 = {\mathbb{R}}\). The additive identity on \(F_h({\mathbb{R}})\) is \(0^\circ _h = (0;h)\).
Definition 9
 1.
\(\text{ HV }(\tilde{u}_h)=\int _0^h{s_h(\alpha )[\overline{u}(\alpha )+\underline{u}(\alpha )]\mathrm{d}\alpha }\),
 2.
\(\text{ HA }(\tilde{u}_h)=\int _0^h{s_h(\alpha )[\overline{u}(\alpha )\underline{u}(\alpha )]\mathrm{d}\alpha }\).
Lemma 2
 1.
\(\text{ HV }(k\tilde{u}_h)=k\text{ HV }(\tilde{u}_h)\),
 2.
\(\text{ HA }(k\tilde{u}_h)=k\text{ HA }(\tilde{u}_h)\).
Proof
straightforward. \(\square\)
Lemma 3
 1.
\(\text{ HV }(\tilde{u}_h+k)=\text{ HV }(\tilde{u}_h)+h^2k\),
 2.
\(\text{ HA }(\tilde{u}_h+k)=\text{ HA }(\tilde{u}_h)\).
Proof
Lemma 4
 1.
\(\text{ HV }(\tilde{u}_h\pm \tilde{v}_h)=\text{ HV }(\tilde{u}_h)\pm \text{ HV }(\tilde{v}_h)\),
 2.
\(\text{ HA }(\tilde{u}_h\pm \tilde{v}_h)=\text{ HA }(\tilde{u}_h)+\text{ HA }(\tilde{v}_h)\).
Proof
straightforward. \(\square\)
Definition 10
Lemma 5
Proof
Lemma 6
For an arbitrary fuzzy seminumber \(\tilde{u}\) with height h, we have \(I_{s,h}<\frac{1}{2}h^3\).
Proof
By Midpoint Theorem, the proof is straightforward. \(\square\)
Height source distance between fuzzy seminumbers
In this section, we introduce a new distance function namely Height Source Distance to measure the distance between fuzzy seminumbers which is invariant on translation only in cases where heights are the same.
Definition 11
In the following, we present some theorems of the proposed distance function and investigate its properties from the analytic geometry perspective.
Theorem 7
 1.
\(\hbox{HSD}(\tilde{u}_{h_u},\tilde{u}_{h_u})=0\),
 2.
\(\hbox{HSD}(\tilde{u}_{h_u},\tilde{v}_{h_v})=\hbox{HSD}(\tilde{v}_{h_v},\tilde{u}_{h_u})\),
 3.
\(\hbox{HSD}(\tilde{u}_{h_u},\tilde{w}_{h_w})\le \hbox{HSD}(\tilde{u}_{h_u},\tilde{v}_{h_v})+\hbox{HSD}(\tilde{v}_{h_v},\tilde{w}_{h_w})\),
 4.
\(\hbox{HSD}(k\tilde{u}_{h_u},k\tilde{v}_{h_v})=\left k\right \hbox{HSD}(\tilde{u}_{h_u},\tilde{v}_{h_v})\).
Proof
 1.Since \(d_H\) is a meter we have:$$\begin{aligned} \hbox{HSD}(\tilde{u}_{h_u},\tilde{u}_{h_u}) &= \frac{1}{2}\left\{ \left h_u\text{ HV }(\tilde{u}_{h_u})  h_u\text{ HV }(\tilde{u}_{h_u})\right + \left h_u\text{ HA }(\tilde{u}_{h_u})  h_u\text{ HA }(\tilde{u}_{h_u})\right \right. \\ &\quad \left. +d_H(h_u^2[\tilde{u}_{h_u}]^{h_u},h_u^2[\tilde{u}_{h_u}]^{h_u})\right\} \\ &=0 \end{aligned}$$
 2.Since \(d_H\) is a meter we have:$$\begin{aligned} \hbox {HSD}(\tilde{u}_{h_u},\tilde{u}_{h_u})&= \frac{1}{2}\left\{ \left h_u\text{ HV }(\tilde{u}_{h_u})  h_u\text{ HV }(\tilde{u}_{h_u})\right + \left h_u\text{ HA }(\tilde{u}_{h_u})  h_u\text{ HA }(\tilde{u}_{h_u})\right \right. \\&\quad \left. +d_H(h_u^2[\tilde{u}_{h_u}]^{h_u},h_u^2[\tilde{u}_{h_u}]^{h_u})\right\} \\&=0 \end{aligned}$$
 3.By \(d_H\) is a meter and triangle inequality we have:$$\begin{aligned} \hbox{HSD}(\tilde{u}_{h_u},\tilde{w}_{h_w}) &= \frac{1}{2}\left\{ \left h_u\text{ HV }(\tilde{u}_{h_u})  h_w\text{ HV }(\tilde{w}_{h_w})\right + \left h_u\text{ HA }(\tilde{u}_{h_u})  h_w\text{ HA }(\tilde{w}_{h_w})\right \right. \\ &\quad \left. +d_H(h_u^2[\tilde{u}_{h_u}]^{h_u},h_w^2[\tilde{w}_{h_w}]^{h_w}) \right\} \\ &=\frac{1}{2}\left\{ \left h_u\text{ HV }(\tilde{u}_{h_u})  h_v\text{ HV }(\tilde{v}_{h_v})+h_v\text{ HV }(\tilde{v}_{h_v})h_w\text{ HV }(\tilde{w}_{h_w})\right \right. \\ &\quad +\left h_u\text{ HA }(\tilde{u}_{h_u})  h_v\text{ HA }(\tilde{v}_{h_v})+h_v\text{ HA }(\tilde{v}_{h_v})h_w\text{ HA }(\tilde{w}_{h_w})\right \\ &\quad \left. +d_H(h_u^2[\tilde{u}_{h_u}]^{h_u},h_w^2[\tilde{w}_{h_w}]^{h_w}) \right\} \\ &\le \frac{1}{2}\left\{ h_u\left \text{ HV }(\tilde{u}_{h_u})  h_v\text{ HV }(\tilde{v}_{h_v})\right +\left h_v\text{ HV }(\tilde{v}_{h_v})h_w\text{ HV }(\tilde{w}_{h_w})\right \right. \\ &\quad +\left h_u\text{ HA }(\tilde{u}_{h_u})  h_v\text{ HA }(\tilde{v}_{h_v})\right +\left h_v\text{ HA }(\tilde{v}_{h_v})h_w\text{ HA }(\tilde{w}_{h_w})\right \\ &\quad \left. +d_H(h_u^2[\tilde{u}_{h_u}]^{h_u},h_v^2[\tilde{v}_{h_v}]^{h_v})+d_H(h_v^2[\tilde{v}_{h_v}]^{h_v},h_w^2[\tilde{w}_{h_w}]^{h_w}) \right\} \\ &=\frac{1}{2}\left\{ \left h_u\text{ HV }(\tilde{u}_{h_u})  h_v\text{ HV }(\tilde{v}_{h_v})\right +\left h_u\text{ HA }(\tilde{u}_{h_u})  h_v\text{ HA }(\tilde{v}_{h_v})\right \right. \\ &\quad \left. +d_H(h_u^2[\tilde{u}_{h_u}]^{h_u},h_v^2[\tilde{v}_{h_v}]^{h_v}) \right\} +\frac{1}{2}\left\{ \left h_v\text{ HV }(\tilde{v}_{h_v})h_w\text{ HV }(\tilde{w}_{h_w})\right \right. \\ &\quad \left. +\left h_v\text{ HA }(\tilde{v}_{h_v})h_w\text{ HA }(\tilde{w}_{h_w})\right d_H(h_v^2[\tilde{v}_{h_v}]^{h_v},h_w^2[\tilde{w}_{h_w}]^{h_w}) \right\} \\ &= \hbox{HSD}(\tilde{u}_{h_u},\tilde{v}_{h_v})+ \hbox{HSD}(\tilde{v}_{h_v},\tilde{w}_{h_w}). \end{aligned}$$
 4.By \(d_H\) is a meter and Lemma 2 we have:$$\begin{aligned} \hbox{HSD}(k\tilde{u}_{h_u},k\tilde{v}_{h_v}) = \frac{1}{2}\left\{ \left h_u\text{ HV }(k\tilde{u}_{h_u})  h_v\text{ HV }(k\tilde{v}_{h_v})\right + \left h_u\text{ HA }(k\tilde{u}_{h_u})  h_v\text{ HA }(k\tilde{v}_{h_v})\right \right. \\ \qquad \left. +d_H(h_u^2[k\tilde{u}_{h_u}]^{h_u},h_v^2[k\tilde{v}_{h_v}]^{h_v})\right\} \\ =\frac{1}{2}\left\{ \left kh_u\text{ HV }(\tilde{u}_{h_u})  kh_v\text{ HV }(\tilde{v}_{h_v})\right + \left kh_u\text{ HA }(\tilde{u}_{h_u})  kh_v\text{ HA }(\tilde{v}_{h_v})\right \right. \\ {}\qquad \left. +d_H(h_u^2[k\tilde{u}_{h_u}]^{h_u},h_v^2[k\tilde{v}_{h_v}]^{h_v}) \right\} \\ =\frac{1}{2}\left\{ \left k\right \left h_u\text{ HV }_{h_u}(\tilde{u})  h_v\text{ HV }_{h_v}(\tilde{v})\right + \left k\right \left h_u\text{ HA }_{h_u}(\tilde{u})  h_v\text{ HA }_{h_v}(\tilde{v})\right \right. \\ {} \left. \qquad +\left k\right d_H(h_u^2[\tilde{u}]^{h_u},h_v^2[\tilde{v}]^{h_v}) \right\} \\ =\frac{1}{2}\left k\right \left\{ \left \text{ HV }(\tilde{u}_{h_u})  \text{ HV }(\tilde{v}_{h_v})\right + \left \text{ HA }(\tilde{u}_{h_u})  \text{ HA }(\tilde{v}_{h_v})\right \right. \\ {} \qquad \left. +d_H(h_u^2[\tilde{u}_{h_u}]^{h_u},h_v^2[\tilde{v}_{h_v}]^{h_v}) \right\} \\ =\left k\right \hbox{HSD}(\tilde{u}_{h_u},\tilde{v}_{h_v}) \end{aligned}$$
Remark 8
For \(\tilde{u},\tilde{v} \in F({\mathbb{R}})\), HSD, is the same as the source distance defined in [2].
Example 9
For the set of all fuzzy seminumbers of the same height, we prove the following theorem.
Theorem 10
 1.
\(\hbox{HSD}(\tilde{u}_{h}+k,\tilde{v}_{h}+k)= \hbox{HSD}(\tilde{u}_{h},\tilde{v}_{h})\),
 2.
\(\hbox{HSD}(\tilde{u}_{h}+\tilde{v}_{h},\tilde{u}'_{h}+\tilde{v}'_{h})\le \hbox{HSD}(\tilde{u}_{h},\tilde{u}'_{h})+ \hbox{HSD}(\tilde{v}_{h},\tilde{v}'_{h}).\)
Proof
In definition of HSD, two source functions \(s_1\) and \(s_2\) are defined over \([0,h_1]\) and \([0,h_2]\), respectively. If \(s_1\) over \([0,h_1]\) is equivalent to \(s_2\) over \([0,h_2]\), (\(s_1\approx s_2\)), then we denote this distance by \(\hbox{HSD}_s\) with respect to the source function defined over [0, 1].
Example 11
In Fig. 1, functions \(s_1(\alpha ) = \frac{1}{2}s(2\alpha )\) over \([0,\frac{1}{2}]\), \(s_2(\alpha ) = \frac{1}{5}s(5\alpha )\) over \([0,\frac{1}{5}]\) and \(s_h(\alpha ) = \frac{1}{h}s(h\alpha )\) over \([0,\frac{1}{h}]\) (where \(h\ge 1\)) are all equivalent to the trivial source function \(s(\alpha ) = \alpha\) over [0, 1].
Example 12
In Fig. 2, functions \(s_1(\alpha ) = \frac{1}{2}s(2\alpha )\) over \([0,\frac{1}{2}]\) and \(s_2(\alpha ) = \frac{1}{5}s(5\alpha )\) over \([0,\frac{1}{5}]\) are both equivalent to a nontrivial source function \(s(\alpha ) = 2\alpha ^3  3\alpha ^2+2\alpha\) over [0, 1].
The nearest approximation of a fuzzy seminumber
In this section, we use HSD to find the nearest approximation of an arbitrary fuzzy seminumber. We start with some definitions and continue with presenting a theorem on the set of all fuzzy seminumbers with equivalent source functions.
Definition 12
Definition 13
Theorem 13
Let \(\tilde{u}_{h_u},\tilde{v}_{h_v}\in TF_h({\mathbb{R}})\), then \(\hbox{HSD}_s(\tilde{u}_{h_u},\tilde{v}_{h_v})=0\), if and only if \(\tilde{u}_{h_u}=\tilde{v}_{h_v}\).
Proof
If \(\tilde{u}_{h_u}=\tilde{v}_{h_v}\), from Theorem 7 we have \(\hbox{HSD}_s(\tilde{u}_{h_u},\tilde{v}_{h_v})=0\).
From Theorem 7, Corollaries 4.2 and 4.3 can be immediately derived.
Corollary 14
\(\hbox{HSD}_s\), is a metric on \(TF_h({\mathbb{R}})\), for a fixed height h.
Corollary 15
Example 16
Let \(\tilde{u}=(1,2,3,4;\frac{1}{2})\) and \(s(r)=r\). The consistent nearest approximation of \(\tilde{u}\) for height \(\frac{1}{3}\), is \(\tilde{v}=(\frac{63}{8},\frac{9}{2},\frac{27}{4},\frac{81}{4};\frac{1}{3})\). Since \(a_v\ge b_v\), \(\tilde{v}\) is not a trapezoidal fuzzy seminumber.
Therefore, in the following subsections, a range is specified for h associated with a nonnegative or nonpositive trapezoidal fuzzy seminumber. This range must satisfy some constraints which will be explained in the following subsections.
A constraint on the nearest approximation’ height of a nonnegative fuzzy seminumber
Let \(\tilde{u}=(a_u,b_u,c_u,d_u;h_u)\) be an arbitrary nonnegative trapezoidal fuzzy seminumber, for the consistent nearest approximation via \(\hbox{HSD}_s\), with height \(h_v\), where \(\tilde{v}=(a_v,b_v,c_v,d_v;h_v) \in TF_{h_v}({\mathbb{R}})\), the Lemmas 17 and 18 can be derived.
Lemma 17
If \(h_v\ge h_u\), the Hcore of the consistent nearest approximated nonnegative trapezoidal fuzzy seminumber, \(\tilde{v}_{h_v}\), tends to the left side and similarly if \(h_v\le h_u\), the Hcore of the consistent nearest approximated nonnegative trapezoidal fuzzy seminumber, \(\tilde{v}_{h_v}\), tends to the right side.
Proof
When \(h_v\ge h_u\), from (20) and (21) we will have \(b_v\le b_u\) and \(c_v\le c_u\), when \(h_v\le h_u\), from (20) and (21) we will have \(b_v\ge b_u\) and \(c_v\ge c_u\), since \(a_u\ge b_u\ge c_u\ge d_u\ge 0\), the proof of the lemma is complete. \(\square\)
Lemma 18
Let \(\tilde{u}\) be a nonnegative trapezoidal fuzzy seminumber with fixed height \(h_{u}\). For each \(h_v\le h_u\) and the source function \(s_{h_{v}}\) which is equivalent to \(s_{h_{u}}\), we have \(c_v\le d_v\), similarly for each \(h_v\ge h_u\) we have \(a_v\le b_v\).
Proof
Theorem 19
 (i)If \(h_v \le h_u\), then$$h_v\ge \frac{a_uh_u^32a_u I_{s,h_u}+2b_u I_{s,h_u}}{b_uh_u^2},$$(24)
 (ii)If \(h_v \ge h_u\), then$$h_v \le \frac{d_uh_u^32d_u I_{s,h_u}+2c_u I_{s,h_u}}{c_uh_u^2}.$$(25)
Proof
 (i)From Lemma 17 and 18, whenever \(h_v\le h_u\), it is sufficient that \(a_v \le b_v\) holds, therefore, by Eqs. (20) and (19) we write as follows:From Lemma 6, \(b_v \ge a_v\) implies that$$\begin{aligned} b_va_v = \left( \frac{h_u}{h_v}\right) ^2 b_u  \frac{\ h_u^32I_u}{\ h_v^32I_v}a_u\frac{2 h_v^2I_u2h_u^2I_v}{\ h_v^52h_v^2I_v}b_u,\\ = \left( \frac{h_u}{h_v}\right) ^3\left( \frac{a_uh_u^3+b_uh_u^2h_v+2a_uI_u2b_uI_u}{h_u^3  2 I_u}\right) , \end{aligned}$$since \(b_u\ge a_u\ge 0\) hence$$a_uh_u^3+b_uh_u^2h_v+2a_uI_u2b_uI_u \ge 0,$$$$h_v\ge \frac{a_uh_u^32a_uI_u+2b_uI_u}{b_uh_u^2}.$$
 (ii)From Lemmas 17 and 18 ,whenever \(h_v\ge h_u\) it is sufficient that \(c_v \le d_v\) holds; therefore, by Eqs. (21) and (22) we write as follows:From Lemma 6, \(d_v \ge c_v\) implies that$$d_vc_v = \left( \frac{h_u}{h_v}\right) ^3\frac{d_uh_u^3c_uh_u^2h_v2d_uI_u+2c_uI_u}{h_u^3  2 I_u},$$since \(d_u\ge c_u\ge 0\), then$$d_uh_u^3c_uh_u^2h_v2d_uI_u+2c_uI_u \ge 0,$$$$h_v \le \frac{d_uh_u^32d_uI_u+2c_uI_u}{c_uh_u^2}.$$
Now, the Corollaries 20 and 21 can be derived from Theorem 19.
Corollary 20
Proof
Straightforward. \(\square\)
Corollary 21
Proof
Straightforward. \(\square\)
A constraint on the nearest approximation’ height of a nonpositive fuzzy seminumber
Let \(\tilde{u}=(a_u,b_u,c_u,d_u;h_u)\) be an arbitrary known nonpositive trapezoidal fuzzy seminumber, for the consistent nearest approximation via \(\hbox{HSD}_s\), with height \(h_v\), where \(\tilde{v}=(a_v,b_v,c_v,d_v;h_v) \in F_{h_v}({\mathbb{R}})\), Lemmas 22 and 23 are obtained.
Lemma 22
If \(h_v\ge h_u\) the Hcore of the consistent nearest approximated nonpositive trapezoidal fuzzy seminumber, \(\tilde{v}_{h_v}\), tends to the right side and similarly if \(h_v\le h_u\) the Hcore of the consistent nearest approximated nonpositive trapezoidal fuzzy seminumber, \(\tilde{v}_{h_v}\), tends to the left side.
Lemma 23
Let \(\tilde{u}_{h_u}\in TF_{h_{u}}({\mathbb{R}})\) and \(\tilde{u}_{h_u}\) be nonpositive for a fixed \(h_{u}\). For each \(h_v\le h_u\) and the source function \(s_{h_{v}}\) which is equivalent to \(s_{h_{u}}\), we have \(a_v\le b_v\), similarly for each \(h_v\ge h_u\) we have \(c_v\le d_v\).
Theorem 24
 (i)If \(h_v \le h_u\), then$$h_v\ge \frac{d_uh_u^32d_uI_u+2c_uI_u}{c_uh_u^2},$$(26)
 (ii)If \(h_v \ge h_u\), then$$h_v \le \frac{a_uh_u^32a_uI_u+2b_uI_u}{b_uh_u^2}.$$(27)
Corollary 25
Corollary 26
The proofs of the lemmas, theorem and corollaries in this subsection are the same as the previous subsection (“A constraint on the nearest approximation' height of a nonnegative fuzzy seminumber”). Alternatively, these proofs can be derived, assuming \(\tilde{u}^+ = \tilde{u}\) and \(\tilde{v}^+=\tilde{v}\).
Numerical examples
In this section, we present some numerical examples using the proposed method during paper. The simplest source function is identity function and because of computational complexity we use \(s(r)=r\) in our examples.
Example 27
Let \(\tilde{u}=(1,2,3,4;\frac{1}{2})\) and \(s(r)=r\). The consistent nearest trapezoidal fuzzy seminumber of \(\tilde{u}\) with height \(\frac{7}{13}\), is \(\tilde{v}=(\frac{1521}{2744},\frac{169}{98},\frac{507}{196},\frac{3887}{1372};\frac{7}{13})\). In Fig. 3, \(\tilde{u}\) and \(\tilde{v}\) are shown by solid and dashed lines, respectively.
If \(h_v \ge \frac{1}{2}\), then from Theorem 19 we should have \(h_v \le \frac{5}{9}\) and if \(h_v \le \frac{1}{2}\), then from Theorem 19 we should have \(h_v \ge \frac{5}{12}\). Since \(\tilde{u}\) is positive and \(h_v \ge h_u\), \(\tilde{v}\) tend to left side as it can be seen in Fig. 3 regarding Lemma 17, it was predictable and because \(h_w \le h_u\), \(\tilde{w}\) tend to right side as it can be seen in Fig. 4 according to Lemma 17 it could be foretold.
Example 28
Let \(\tilde{u}=(5,3,2,1,\frac{3}{4})\) and \(s(r)=r\). The consistent nearest fuzzy seminumber of \(\tilde{u}\) with height \(\frac{7}{8}\), is \(\tilde{v}=(\frac{864}{343},\frac{108}{49},\frac{72}{49},\frac{72}{343},\frac{7}{8})\). In Fig. 4, \(\tilde{u}\) and \(\tilde{v}\) are shown by solid and dashed lines, respectively.
The nearest fuzzy seminumber of \(\tilde{u}\) with height \(\frac{2}{3}\), is \(\tilde{w}=(\frac{4131}{512},\frac{243}{64},\frac{81}{32},\frac{1053}{512},\frac{2}{3})\). In Fig. 5, \(\tilde{u}\) and \(\tilde{w}\) are shown by solid and dashed lines, respectively.
Example 29
If \(h_v \ge \frac{1}{2}\), then from Theorem 19 we should have \(h_v \le \frac{3}{5}\) and if \(h_v \le \frac{1}{2}\), then from Theorem 19 we should have \(h_v \ge \frac{9}{22}\). Hence, when \(h_u= \frac{1}{2}\), given height should belong to \([\frac{9}{22},\frac{3}{5}]\); otherwise, the nearest approximation of \(\tilde{u}\) will not be a trapezoidal fuzzy seminumber. Since \(\tilde{u}\) is a positive seminumber by Lemma 17 as it can be seen in Fig. 5 whenever the given height increases the nearest approximation of \(\tilde{u}\) tend to left and whenever the given height decreases the nearest approximation of \(\tilde{u}\) tend to right. As it can be seen in Fig. 5, according to Lemma 18 when the given height increases we do not have to worry about left spread and when the given height decreases we do not have to worry about right spread.
Example 30
A medical case study
Without a doubt, controlling the depth of anesthesia is the main task of an anesthetist during surgery. According to the report in [17], annually, anesthesia is the direct responsible for the death of approximately 34 patients. Also it plays an indirect role in another 281 deaths especially in the elderly. Measuring the depth of anesthesia is inherently not straightforward. Thus, anesthetists solve the matter via measuring some other available factors such as blood pressure, heart rate and so on.
In [12], Cullen showed that there is a plausible correlation between blood pressure and anesthetic dose. In other words, gaining a sufficient depth of anesthesia is feasible through controlling the blood pressure related factors such as the MAP (Mean Arterial Pressure which is measured in mmHg) so that it falls within a predetermined range. This approach has been one of the most prevalent ones in determining the necessary anesthetic dose for decades.
Surgical operations are so risky that most often a small oversight may lead to serious losses. Thus, it is reasonable to automate such tasks as far as possible. In reality, controlling of depth of anesthesia automatically is of highly importance among these tasks, and it releases the surgery staff to pay much more attention to the other activities in the operating room.
There are actually two ways to anesthetize a patient. This is performed either by intravenous injection of drugs or inhaling gasses which are most often a mixture of Isoflurane in oxygen and potentially nitrous oxide. The concentration of Isoflurane used in the process is determined and tuned regarding the type of surgery and the patients psychological condition.
As a case study, in this paper, a fuzzy logic controller is deployed to measure the MAP [21, 22]. This controller simulates the relationship between the inflow concentration of Isoflurane, and the blood pressure. Assume that we are given a fuzzy seminumber which is the average of patient’s blood pressure. Due to some disturbances, the blood pressure measurement is an ambiguous process; thus, the given data should be a fuzzy set. Actually, there are different sorts of disturbances such as surgical disturbances to the patient and measurement noise which cause the ambiguity of blood pressure measurement to increase. For this given fuzzy seminumber we find the range of heights which indicates blood pressure variation. Then, we estimate a new fuzzy seminumber with given height related to the new blood pressure to help the anesthetist determine the dose of drugs through the anesthetic process.
The given fuzzy seminumber \(\tilde{B}_{h_B}=(1,2,3,5;0.67)\), shown in Fig. 7, represents the uncertainty in each blood pressure measurement. Here, the height of given seminumber, \(h_B=0.67\), is proportional to blood pressure and its support [1, 5] shows the anesthesia duration.
With this approximated fuzzy seminumber we will allow the anesthetist to determine the next drug application for the patient.
Conclusion
In most of the works on fuzzy sets, the fuzzy sets are convex and normal (the height is one) which are called fuzzy numbers. Approximation of a fuzzy set has been done in several ways. Assigning a single crisp number to a fuzzy set, defining an interval as an approximation of a fuzzy set, defining distance function and solving an optimization problem in order to obtain a trapezoidal fuzzy set as a nearest approximation are among the most wellknown ones. All of these methods suffer from some deficiencies such as precision loss. Moreover, none of them have addressed fuzzy sets with heights less than one. Since too many subjects work on nonnormal fuzzy sets, in this paper we worked on fuzzy sets in general, without regard to their height. At first, we reviewed the novel concept of fuzzy seminumbers. Then, we proposed a distance to approximate an arbitrary fuzzy seminumber. Also, as we observed, depending on some predefined conditions, the result of approximation can be either a fuzzy number or a fuzzy seminumber. We finished the paper with some numerical examples and presented a critical medicine case study which improved anesthetist decision for prescribing dose of drugs and controlling the depth of anesthesia with our approximation method.
Notes
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