Fuzzy semi-numbers and a distance on them with a case study in medicine

In this paper, we present the novel concept of fuzzy semi-numbers. Then, a method for assigning distance between every pair of fuzzy semi-numbers is given. Moreover, it is shown that this distance is a metric on the set of all trapezoidal fuzzy semi-numbers with the same height and is a pseudo-metric on the set of all fuzzy semi-numbers. Also, by utilizing this distance, we propose an approximation of a fuzzy semi-number with given height and apply this approximation method in a medical case study.


Introduction
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 semi-number. Depending on some predefined conditions, the result of approximation can be either a fuzzy number or fuzzy semi-number. 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 semi-number 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 semi-numbers. Section ''Height source distance between fuzzy semi-numbers'' introduces a new distance namely Height Source Distance for fuzzy semi-numbers. In next section, the nearest trapezoidal fuzzy semi-number to an arbitrary fuzzy semi-number 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 FðRÞ be the set of all real fuzzy numbers (which are normal, upper semicontinuous, convex and compactly supported fuzzy sets).
The parametric form of a fuzzy number is denoted bỹ u ¼ ðu; uÞ, where functions u and u for each a 2 ½0; 1 satisfy the following requirements [18,19] Definition 1 A function s 2 C½0; 1 with the following properties is a source function [2,5] (regular reducing function in [20]) over all fuzzy numbers: Definition 2 The core of a fuzzy numberũ is defined as follows:

Fuzzy semi-numbers
Here, we present several novel definitions and lemmas which will be used throughout the paper. Proof For s h 2 we have the following relations: For a fuzzy semi-numberũ h 2 F h ðRÞ and a 2 ½0; h we define uðaÞ and uðaÞ as follows: For a trapezoidal fuzzy semi-number which is denoted bỹ u ¼ ða; b; c; d; hÞ, we have Lemma 2 Forũ h 2 F h ðRÞ and k 2 R, we have following properties: Proof straightforward. h Lemma 3 Forũ h 2 F h ðRÞ and k 2 R, we have following properties: Proof By definition of H-value and H-ambiguity we can write as follows:

Lemma 4
In FSðRÞ, for H-value and H-ambiguity we have following properties:

Definition 10
Let s be a source function, then the source number I s;h is defined on s as follows: Lemma 5 If s 1 and s 2 are equivalent source functions defined over ½0; h 1 and ½0; h 2 , respectively, then we have: Proof Using Definition 10 we have:

Lemma 6
For an arbitrary fuzzy semi-numberũ with height h, we have I s;h \ 1 2 h 3 . Proof By Mid-point Theorem, the proof is straightforward. h

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 semi-numbers which is invariant on translation only in cases where heights are the same.
Definition 11 Forũ h u ;ṽ h v 2 FSðRÞ with heights h u and h v , respectively, Height Source Distance HSD, is defined as follows: where d H is the Hausdorff meter, and ½w h is the h-cut of fuzzy setw.
In the following, we present some theorems of the proposed distance function and investigate its properties from the analytic geometry perspective.

Theorem 7
Forũ h u ;ṽ h v ;w h w 2 FSðRÞ and k 2 R, HSD satisfies the following properties: then the proof is trivial; otherwise, for each step we can write as follows: 1. Since d H is a meter we have: Since d H is a meter we have: By d H is a meter and triangle inequality we have: 4. By d H is a meter and Lemma 2 we have: with equal height h, HSD simplified as: Remark 8 Forũ;ṽ 2 FðRÞ, HSD, is the same as the source distance defined in [2].
In this case ifũ andṽ are two crisp real numbers, then HSDðũ;ṽÞ ¼ ja À bj: For the set of all fuzzy semi-numbers of the same height, we prove the following theorem.

Theorem 10
Forũ h ;ṽ h ;ũ 0 h ;ṽ 0 h 2 TF h ðRÞ and k 2 R, for a fixed height h and non-negative real number k, HSD satisfies the following properties: Thus, we investigate the proof in other cases. For the first part by d H is a meter and Lemma 3 we have: For the second part since d H is a meter by the triangle inequality and Lemma 4 we have: In definition of HSD, two source functions s 1 and s 2 are defined over ½0; h 1 and ½0; h 2 , respectively. If s 1 over ½0; h 1 is equivalent to s 2 over ½0; h 2 , (s 1 % s 2 ), then we denote this distance by HSD s with respect to the source function defined 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 semi-number. We start with some definitions and continue with presenting a theorem on the set of all fuzzy semi-numbers with equivalent source functions.
and hence which are equivalent to the following relations: Since Proof By Theorem 7 and Lemma 5 the proof is clear. h For an arbitrary fuzzy semi-numberũ h u 2 TF h u ðRÞ with height h u , the consistent nearest approximationṽ h v ¼ ða v ; b v ; c v ; d v ; h v Þ can be found by applying equation (18) for an arbitrary height h v Since the approximation depends on the height of known trapezoidal fuzzy semi-number, the trapezoidal approximation may not exist for a specific height. Example 16 explains this case.
Example 16 Letũ ¼ ð1; 2; 3; 4; 1 2 Þ and sðrÞ ¼ r. The consistent nearest approximation ofũ for height 1 3 , is v ¼ ð 63 8 ; 9 2 ; 27 4 ; 81 4 ; 1 3 Þ. Since a v ! b v ,ṽ is not a trapezoidal fuzzy semi-number. Therefore, in the following subsections, a range is specified for h associated with a non-negative or nonpositive trapezoidal fuzzy semi-number. This range must satisfy some constraints which will be explained in the following subsections.
A constraint on the nearest approximation' height of a non-negative fuzzy semi-number Letũ ¼ ða u ; b u ; c u ; d u ; h u Þ be an arbitrary non-negative trapezoidal fuzzy semi-number, for the consistent nearest approximation via HSD s , with height h v , wherẽ , the Lemmas 17 and 18 can be derived.

Lemma 17
If h v ! h u , the H-core of the consistent nearest approximated non-negative trapezoidal fuzzy seminumber,ṽ h v , tends to the left side and similarly if h v h u , the H-core of the consistent nearest approximated nonnegative trapezoidal fuzzy semi-number,ṽ h v , tends to the right side.
Proof When h v ! h u , from (20) and (21) we will have b v b u and c v c u , when h v h u , from (20) and (21) we

Lemma 18
Letũ be a non-negative trapezoidal fuzzy semi-number with fixed height h u . For each h v h u and the source function s h v which is equivalent to s h u , we have c v d v , similarly for each h v ! h u we have a v b v .
Proof Let I u ¼ I s hu ;h u and I v ¼ I s hv ;h v . Since s h u % s h v , with respect to Lemma 5 we have From Eqs. (21) and (22) we can write as follows: then by Eq. (23), since d u ! c u ! 0 and h u ! h v , we have d u h u À c u h v ! d u h u À c u h u and from that we have ð d u h u Àc u h v d u Àc u Þ ! h u . By multiply h 2 u to both sides and subtract 2I u from both side we'll get h 2 u ð d u h v Àc u h u d u Àc u Þ À 2I u ! h 3 u À 2I u . By Lemma 6 and d u À c u ! 0, Similarly for h v ! h u from Eqs. (20), (19), (23) and b u À a u ! 0, we have: h Theorem 19 Let s be a source function, for an arbitrary known non-negative trapezoidal fuzzy semi-numberũ ¼ ða u ; b u ; c u ; d u ; h u Þ with height h u . The consistent nearest approximation via HSD s is a trapezoidal fuzzy semi-number with height h v , if and only if h v satisfies one of the following conditions: Eqs. (20) and (21) it is clear that b v c v .
(i) From Lemma 17 and 18, whenever h v h u , it is sufficient that a v b v holds, therefore, by Eqs. (20) and (19) we write as follows: (ii) From Lemmas 17 and 18 ,whenever h v ! h u it is sufficient that c v d v holds; therefore, by Eqs. (21) and (22) we write as follows: Now, the Corollaries 20 and 21 can be derived from Theorem 19.
A constraint on the nearest approximation' height of a non-positive fuzzy semi-number Letũ ¼ ða u ; b u ; c u ; d u ; h u Þ be an arbitrary known non-positive trapezoidal fuzzy semi-number, for the consistent nearest approximation via HSD s , with height h v , whereṽ ¼

Lemma 22
If h v ! h u the H-core of the consistent nearest approximated non-positive trapezoidal fuzzy semi-number, v h v , tends to the right side and similarly if h v h u the H-core of the consistent nearest approximated non-positive trapezoidal fuzzy semi-number,ṽ h v , tends to the left side.

Lemma 23
Letũ h u 2 TF h u ðRÞ andũ h u be non-positive for a fixed h u . For each h v h u and the source function s h v which is equivalent to s h u , we have a v b v , similarly for each h v ! h u we have c v d v .
Theorem 24 Let s be a source function, for an arbitrary known non-positive trapezoidal fuzzy semi-numberũ ¼ ða u ; b u ; c u ; d u ; h u Þ with height h u . The consistent nearest approximation via HSD s is a trapezoidal fuzzy semi-number with height h v , if and only if h v satisfies one of the following conditions: Corollary 25 Letũ ¼ ða u ; b u ; c u ; d u ; hÞ be an arbitrary non-positive fuzzy semi-number with height h. When 1 a u h 3 À2a u I u þ2b u I u b u h 2 , the consistent nearest trapezoidal fuzzy number Corollary 26 Letũ ¼ ða u ; b u ; c u ; d u Þ be an arbitrary nonpositive trapezoidal fuzzy number. The consistent nearest fuzzy semi-number ofũ with height h for h ! a u À2a u I s;1 þ2b u I s; 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 ofa nonnegative fuzzy semi-number''). Alternatively, these proofs can be derived, assumingũ þ ¼ Àũ andṽ þ ¼ Àṽ.

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 sðrÞ ¼ r in our examples. Sinceũ is positive and h v ! h u ,ṽ tend to left side as it can be seen in Fig. 3 regarding Lemma 17, it was predictable and because h w h u ,w tend to right side as it can be seen in Fig. 4 according to Lemma 17 it could be foretold.
Example 29 Letũ ¼ ð1; 2; 3; 5; 1 2 Þ and sðrÞ ¼ r. The consistent nearest trapezoidal fuzzy semi-number ofũ with height 11 21  ; otherwise, the nearest approximation ofũ will not be a trapezoidal fuzzy semi-number. Sinceũ is a positive semi-number by Lemma 17 as it can be seen in Fig. 5 whenever the given height increases the nearest approximation ofũ tend to left and whenever the given height decreases the nearest approximation ofũ tend to right. As it can be seen in Fig. 5 Fig. 6. These can be written asũðrÞ ¼ ð4r; 5 À 2rÞ andṽðrÞ ¼ ð1 þ 4 3 r; 4 À 4 3 rÞ. By using sðrÞ ¼ r, Hvalue and H-ambiguity of them can be found as follows: The Height Source Distance between these fuzzy seminumbers is 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 Fig. 4 The nearest trapezoidal fuzzy semi-number with height 6 13 Fig. 5 The nearest fuzzy semi-number with height 7 8 Fig. 6 The nearest trapezoidal fuzzy semi-number with height 11 21 , 4 7 , 10 22 and 21 44 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 semi-number 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 semi-number we find the range of heights which indicates blood pressure variation. Then, we estimate a new fuzzy semi-number with given height related to the new blood pressure to help the anesthetist determine the dose of drugs through the anesthetic process.
To this end, we can estimate the range of height associated with the given fuzzy semi-number with (24) and (25) as follows: 0:548182 6 hB 6 0:804 This range shows how the blood pressure varies with the inflow concentration of Isoflurane by applying related defuzzification methods in medicine (Fig. 8).
Moreover, if we want to change blood pressure we approximate a new fuzzy semi-number,B Ã h B Ã , where h B Ã ¼ 0:8 is related to this new blood pressure by using (19) With this approximated fuzzy semi-number 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 well-known 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 non-normal 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 semi-numbers. Then, we proposed a distance to approximate an arbitrary fuzzy semi-number. Also, as we observed, depending on some predefined conditions, the result of approximation can be either a fuzzy number or a fuzzy semi-number. 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.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creative commons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.