Fuzzy soft set theory: Application of classification rules in decision making during medical diagnosis*

Abstract

Present study is an interdisciplinary approach towards rapid and efficient medical diagnosis. The research articulated on data set of cross-sectional study of pregnant females dwelling rural area of Pakistan. The prognosis of gestational wellbeing followed through analyzing heterogenic medical information to develop a holistic picture of ongoing pregnancy. Therefore, for rapid medical diagnosis and precision in decision-making, Fuzzy Soft Set (denoted as FSS) theory selected to develop an algorithm. The algorithm constructed as single point, multipoint and cumulative diagnosis for predicting health status with respect of Hemoglobin, Body Mass Index and Random Glucose Concentration (Respectively denoted as Hb, BMI and RGC) of subjects under study. We successfully proposed novel approach for complex modeling and provision of algorithm for medical diagnosis. The algorithms successfully dealt with analyzing diversely attributed detailed medical tests/reports as input. The output of complex modeling effectively served efficient decision-making in predicting gestational wellbeing.

Keywords

Medical diagnosis fuzzy set soft set BMI maternal anemia

1 Introduction

Adverse gestational outcomes are leading contravene on human resource, particularly in world’s deprived regions [7 , 16]. Pakistan is a developing country. The country harbors population that constitutes larger segment as women of variable age clusters. Thus, women wellbeing secures mandate to nurture and nourish healthy progeny; consequently, resulting sustainable healthy community. However, health care provisions reported poor unanimously, while rural fragments of the country remained most unfortunate [23, 24]. An utmost importance revealed by scientists to realize window of time from pre-conception to delivery and up to second birthday of a newborn, [25] where, maternal health assures wellbeing of infant. But maternal health known as an assurance for healthy children by itself is a chronic health issue. Mothers lay down their lives during child. The impact of losses happened huge therefore, researches and global intellect envisaged estimates and terminologies. The estimates of maternal deaths during pregnancy are monitored globally and denoted as Maternal Mortality Rate (MMR). MMR specifies risk of death per pregnancy [4]. Rural areas of the country have higher burden of mortality than urban areas. Pakistan is placed at third number among developing nations with high MMR [33]. Therefore, maternal health monitoring needs to be prioritized.

Diagnostic procedure in terms of medical diagnosis means to identify health conditions that afflict people and manifest clinical findings. Diagnosis discloses a pro treatment procedure. In today’s World, information accumulation about health and disease has expanded to an extent where artificial intelligence and decision-making through various mathematical tools has become desirable. To predict health conditions by medical professionals is a routine; they group patients in accordance with disease. Their objective remains to render prognosis of health condition of patients or classification of laboratory specimens. However, observation of symptoms and assessing huge information received from laboratory tests reports took considerable time of medical personal to take decision. Interestingly, machine-based support for medical diagnosis and decision of treatment studied by scientists to assist health professionals [27]. Further, in coping challenge of limited human resource as, World Health Organization (WHO) estimated over four million medical personals required globally. The problem accelerates further, because medical professionals and facilities remain concentrated in cities leaving rural areas deprived from diagnosis and treatment [32]. Interestingly application of mathematical models in field of medicine [29] is picking momentum. In context of time and human resource constrains in medical diagnosis, it appears desirable to help medical professionals by developing multidisciplinary approach.

Several mathematical tools and fuzzy models have been developed to carryout analysis of big data. Big data based on multiple parameters studied rigorously by applying the tools and fuzzy models to achieve improvement in precision and decision-making [11,39,40,41,42,43 , 11,39,40,41,42,43]. Decision making and soft set theory have also been explored by researchers [13,2,14 , 13,2,14].

While doing mathematical working on big data containing enriched and diverse information, most crucial step seems situation of decision-making. Investigations revealed that fundamental cause of problem of uncertainty lies in classical logic and classical logic-based set theory. In 1999 Molodtsov [18] presented first ever theory of soft set, as a mathematical concept for handling with this limitation. The application of soft set theory and its significance also reported by [21,5, 21,5] and in dealing real life problems in field of business [34]. The applications of Fuzzy Soft Set (FSS) theory reported by [6] while, Kovkov in 2007 [12] introduced problems of optimization in soft set theory. Interesting applications of FSS in decision-making studied by [30] however, to go beyond traditional data handling and analysis most successful proposals remained fuzzy set theory by Zadeh [38] and concept of FSS [19].

Soft and fuzzy sets models recently used in medical diagnosis [37] of prostate cancer. A new model based on Dempster-Shafer generalized FSS developed by [35] and applied the method in medical problems. Langarizadeh and co-workers [17] used fuzzy expert system to distinguish between bacterial and aseptic meningitis. The fuzzy system showed a good agreement and high efficiency in terms of its application in medical diagnosis. Anjali and co-workers [1] developed fuzzy similarity measures for classifying gynecologists whether they reach at the unanimous diagnostic labels. They also explored relationship between symptoms and patients to know can patients be classified? indicating potential application of fussy methods in field of gynecology to cater pregnancy related health problems.

Challenges in health domain are tremendous. Hemoglobin (Hb) concentration in blood circulation is a predictor of anemia; one of major causal agent of internal bleeding and health impairments worldwide [31]. In context of mother and child health anemia is a chronic Public Health challenge. Maternal anemia has devastating impact on mother and fetus wellbeing. As studies reported, it served as a contributing factor in perinatal morbidity [9,15, 9,15] pregnancy outcomes, maternal and infant mortality [3, 22]. Gestational anemia reported; most prevalent nutritional deficiency problems inflicting pregnant women. According to WHO report, 52% of pregnant females in developing World happened anemic while, 23% in developed countries [31]. Maternal anemia described as most frequent gestational complication and needs to be monitored [9]. Gestational wellbeing is a multi-factor phenomenon in this context; Body Mass Index (BMI) is another important health parameter to be monitored during gestation [36]. In developing World, Asian countries female have lower BMI than in developed nations. In USA, 2% of pregnant women have BMI < 18.5 and more than 50% have BMI > 25 [26]. Random Glucose Concentration (RGC) another renounced health parameter studied for health monitoring. Research studies disclosed role of maternal glucose on birth weight and life of baby with life-long metabolic implications. Studies further revealed role of maternal glucose concentrations provides insight for gestational diabetes and fetal hyper-insulinemia [20]. Significance of maternal RGC has become worthwhile as International Association of Diabetes in Pregnancy Study Group proposed an increase in prevalence of gestational diabetes up to 18% [8].

1.1 Motivation of the study

Health domain by enlarge and gestational wellbeing of women encompass huge bulk of medical information which accounts real life problems. Limited health human resource constrains in well-timed medical diagnosis, complexity of diverse multiple attributed detailed medical tests/reports and urgency of rapid diagnosis involving precision in decision making served as motivation for present study to apply potential use of FSS theory. We selected Hb, BMI and RGC attributes to explore application of FSS in complex formulation in diagnosis of gestational wellbeing. The algorithm constructed as single point, multipoint and cumulative diagnosis for predicting health status with respect of Hb, BMI and RGC of subjects under study. The present paper articulated as introduction, motivation of the study, preliminaries, methodology, results, and conclusion, respectively.

2 Preliminaries

In this section, we carried out anatomization of soft set theory [18] fuzzy set theory [38] and fuzzy soft set [19].

Definition 1. Let’s begin by recalling definition of soft set from Molodtsov’s proposal reported in 1999 [18]. Let us consider the universal set U and E be consider as a set of parameters. If P (U) the power set of U and a non-empty subset A of E, then notion (F, A) for F is a mapping from A to P (U) is called a soft set over U.

Example 1. Let a soft set (F, A) over U. The universal set U = {c₁, c₂, c₃} where c₁, c₂ and c₃ are three cars to be selected by decision maker and properties of car are taken in parametric set as E = {costly, metallic color, cheap} = {e₁, e₂, e₃}. Let A = {e₁, e₂} where A is subset of parametric set E. (F, A) gave a detailed attractiveness of cars to make decision. Let $F (e_{1}) = {c_{1}, c_{2}}$ $F (e_{2}) = {c_{1}, c_{2}, c_{3},}$ The soft set $\begin{matrix} (F, A) = {(e_{1}, F (e_{1})), (e_{2}, F (e_{2}))} \\ {(costly, {c_{1}, c_{2},}), (metallic color, {c_{1}, c_{2} {, c}_{3}}} \end{matrix}$

Definition 2. By recalling Zadeh’s proposed model presented in 1965 [38]. Let us consider X be a non-empty set, a fuzzy set F on X defined as $μ F : X \to [0, 1]$

Where μ _F (x) is called membership of degree of element x ∈ X.

Definition 3. The Fuzzy Soft Set (FSS) defined as follows by recalling proposed method of Maji and Roy [19]. Let P (U) contains all fuzzy subsets of set U and E considered as set of parameters. The mapping F from E to P (U), for if e ∉ A, than F (e) =φ and A is a non-empty subset of E. The set (F, A) is a FSS over U. $(F, A) = {(e, F (e)) : e \in, A, F (e) \in P (U)}$

Example 2. Let (F, A) be a FSS over U. The set U denotes three cars under consideration of decision maker by U = {c₁, c₂, c₃} and parametric set as $E = {costly, metallic color, cheap} = {e_{1}, e_{2}, e_{3}}$

Let A = e₁, e₂ where A is subset of parametric set E. The FSS (F, A) describe outlook of cars to decision maker. The fuzzy Soft Set depicted in Table 1. Let

Table 1
Fuzzy Soft Set (FSS)

FSS (F, A)

(F,A) c₁ c₂ c₃

e₁ 0.6 0.4 0.3

e₂ 0.5 0.7 0.8

FSS (F, A)
e₁	0.6	0.4	0.3
e₂	0.5	0.7	0.8

$\begin{matrix} F (e_{1}) = {μ e_{1} (c_{1}), μ e_{1} (c_{2}), μ e_{1} (c_{3})} = \\ {0.6 / c_{1,} 0.4 / c_{2,} 0.3 / c_{3}} \\ F (e_{2}) = {μ e_{2} (c_{1}), μ e_{2} (c_{2}), μ e_{2} (c_{3})} = \\ {0.5 / c_{1,} 0.7 / c_{2,} 0.8 / c_{3}} \end{matrix}$

In present study data set is 341 pregnant females. Three parameters; level of hemoglobin (Hb g/dL) in maternal circulation, Body Mass Index (BMI kg/m²) and Random Glucose Concentration (RGC mg/dL) in blood were selected for assessment of general wellbeing of the study subjects. In the article the parametric set E = Hb, BMI,RGC and U be the set of 341 patients.

3 Methodology

In present paper, three attributes nominated from data set of cross-sectional study of pregnant females. The input data includes Hemoglobin (Hb), Body Mass Index (BMI) and Random Glucose Concentration (RGC) selected from matrix of health parameters of 341 pregnant females with informed consent who predominantly belonged to rural dwellings of Pakistan. The parameters selected for assessment of gestational wellbeing of study subjects.

The set criteria [31] used in present study for maternal anemia. However, we further assigned categories to levels of hemoglobin defined as; 1: in healthy range: 13.5–16.5 g/dL, 2: in normal range: 10.5–13.5 g/dL,3: at risk 7.5–10.5 g/dL, and 4: in critical range 4.5–7.5 g/dL, as represented in Table 2.

Table 2
Hemoglobin Level in Blood Circulation of Pregnant Females

Category Hb Present Study Criteria

1 Healthy 13.5–16.5 (g/dL)

2 Normal 10.5–13.5 (g/dL)

3 Risk 7.5–10.5 (g/dL)

4 Critical 4.5–7.5 (g/dL)

Category	Hb	Present Study Criteria
1	Healthy	13.5–16.5 (g/dL)
2	Normal	10.5–13.5 (g/dL)
3	Risk	7.5–10.5 (g/dL)
4	Critical	4.5–7.5 (g/dL)

The standard for Body Mass Index used as <18.5 kg/m² for underweight, range 18.5–24.9 kg/m² normal weight, range 25.0–29.9 kg/m² overweight, and ≥30 kg/m² for obese women respectively [26] Shown in Table 3.

Table 3

Body Mass Index of Pregnant Females

Category	BMI	Present Study Criteria
1	Underweight	<18.5 kg/m²
2	Normal weight	18.5–23 kg/m²
3	Over weight	23–27.5 kg/m²
4	Obese	>27.5 kg/m²

The health condition of pregnant females categorized in three groups on basis of levels of blood glucose concentration as recommended for therapeutic measures; ≤95 mg/dL for Fasting Blood Glucose <120 mg/dL and <140 mg/dL, at 1-2 hours postprandial spikes [42] represented in Table 4.

Table 4

Random Glucose Concentration of Pregnant Females

Category	RGC	Present Study Criteria
1	Critical	<80 (mg/dL)
2	Normal	80–120 (mg/dL)
3	Abnormal	>120 (mg/dL)

We developed algorithm by keeping in view the influence of confounders on plasma glucose levels in which important are length of overnight fast, BMI and glycosylated Hb. It is therefore, obvious to note significance of maternal blood glucose and its overlapping association with BMI and Hb levels. Research studies [28] reported relevance between blood glucose and glycosylated hemoglobin variability during gestation [8]. The algorithm constructed as single point, multipoint and cumulative diagnosis for predicting gestational wellbeing.

4 Results

4.1 Structure criteria for evaluation

Our proposed model of algorithm revolves around the concerns of decision-making process, ranking attributes, and simultaneous computation of three different health attributes for medical diagnosis. As consequence, it reduces time of analysis of these medical problems by medical personals hence helps to manage large number of patients even with limited human resource.

We proceeded by defining algorithm for decision-making criterion. After performing a series of experiments on decision-making algorithm to diagnose patient health condition, we found developed criteria seem workable. The fuzzy decision-making method is to save cognitive effort by only processing information according to decision related steps. The algorithm for selection of appropriate choice is given as;

4.2 Input data

We selected set standard for input data that established by WHO. The complexity lies in these set standards is existence of various ranges of attributes. Therefore, we structured our criteria by process of fuzzification. The Hb, (Table 2) BMI (Table 3) and RGC (Table 4) values are defined.

4.3 Fuzzification

The input data used as reference values while different ranges of each parameter are further assigned with fuzzified values as Membership Function (MF).

In example, a sample of twenty pregnant women (Table 5) used to illustrate the use of proposed method. The samples of twenty patients chosen as the evaluation object.

Table 5
The input values of 20 subjects of present study

Subjects Age Hb BMI RGC

_s1 22 9.1 14.03 76

_s2 27 13 19.56 97

_s3 25 8.3 14.76 91

_s4 33 13.1 17.24 77

_s5 23 10.2 21.89 102

_s6 26 13.1 27.61 114

_s7 27 9.2 16.2 90

_s8 29 8.7 23.4 82

_s9 31 12.8 21.47 107

_s10 33 11.4 30.72 106

_s11 34 10 21.85 96

_s12 35 13.2 19.36 86

_s13 21 12.1 16.96 71

_s14 33 10.9 17.02 89

_s15 24 11.7 20.85 112

_s16 24 8 20.41 125

_s17 26 13.2 21.23 121

_s18 28 10 22.61 101

_s19 33 12.7 21.42 83

_s20 41 4.5 18 111

Subjects	Age	Hb	BMI	RGC
_s1	22	9.1	14.03	76
_s2	27	13	19.56	97
_s3	25	8.3	14.76	91
_s4	33	13.1	17.24	77
_s5	23	10.2	21.89	102
_s6	26	13.1	27.61	114
_s7	27	9.2	16.2	90
_s8	29	8.7	23.4	82
_s9	31	12.8	21.47	107
_s10	33	11.4	30.72	106
_s11	34	10	21.85	96
_s12	35	13.2	19.36	86
_s13	21	12.1	16.96	71
_s14	33	10.9	17.02	89
_s15	24	11.7	20.85	112
_s16	24	8	20.41	125
_s17	26	13.2	21.23	121
_s18	28	10	22.61	101
_s19	33	12.7	21.42	83
_s20	41	4.5	18	111

$X = {s_{1}, s_{2}, \dots, s_{20}}$ and evaluate based on three parameters Hb, BMI and RGC.

Example 3. The data of pregnant women tabulated in Table 5.

4.4 Fuzzification of the parameters

Fuzzification of the parameters envisaged through Membership Function. Here we have three parameters. We use e₁, e₂, e₃ for Hemoglobin (Hb), Body Mass Index (BMI), Random Glucose Concentration (RGC) respectively.

4.4.1 Membership function for parameter e₁

The Membership Function for The Membership Function for hemoglobin e₁ is denoted by μ_{s
_i} (e₁) and is defined as; $μ_{si} (e_{1}) = {\begin{matrix} \frac{1}{e_{1} (e_{1} + 1)} & if & 4.5 < e_{1} < 7.5 \\ \frac{1}{e_{1}} & if & 7.5 ⩽ e_{1} < 10.5 \\ \frac{e_{1} - 1}{e_{1} + 1} & if & 10.5 ⩽ e_{1} < 13.5 \\ \frac{1}{e_{1} (e_{1} + 1)} & if & 13.5 ⩽ e_{1} < 16.5 \end{matrix}$

Where s_i denotes the patients/subject numbering.

4.4.2 Membership function for parameter e₂

The Membership Function for BMI is denoted by μ_{s
_i} (e₂) where s_i represents the patients/subject numbering. It is defined as follows. $μ_{s_{i}} (e_{2}) = {\begin{matrix} \frac{e_{2}}{1 + e_{2}} & if & e_{2} < 18 . 5 \\ 1 - \frac{1}{e_{2} - 1} & if & 18.5 ⩽ e_{2} ⩽ 23 \\ \frac{1}{e_{2}} + \frac{1}{e_{2} + 1} & if & 23 ⩽ e_{2} ⩽ 27.5 \end{matrix}$

4.4.3 Membership function for RGC

The Membership Function of RGC denoted by μ_{s
_i} (e₃). It is defined as follows. $μ_{s_{i}} (e_{3}) = {\begin{matrix} \frac{e_{3} - 1}{2 e_{3}} & if & e_{3} < 80 \\ \frac{e_{3} - 1}{e_{3} + 1} & if & 80 ⩽ e_{3} ⩽ 120 \\ \frac{1}{e_{3} - 1} & if & e_{3} > 120 \end{matrix}$

4.5 Fuzzy soft Set

The fuzzy soft set is defined in the following way. $ψ (s_{i}) = {(e_{1}, μ_{s_{i}} (e_{1})), (e_{2}, μ_{s_{i}} (e_{2})), (e_{3}, μ_{s_{i}} (e_{3}))}$

Where e₁, e₂, e₃ ∈ Eare the set of parameters HB, BMI and RGC respectively and μ_{s
_i} (e₁), μ_{s
_i} (e₂) and μ_{s
_i} (e₃) are respective fuzzy functions defined above.

Example 4. In continuation of example and Table 5, the fuzzification of parameters are given here in Table 6.

Table 6
Membership Function for Hb, BMI, RGC

Subject Hb BMI RGC

μ_{s
_i} (e₁) μ_{s
_i} (e₂) μ_{s
_i} (e₃)

_s1 0.10989 0.933466 0.493421

_s2 0.857143 0.946121 0.887755

_s3 0.120482 0.936548 0.880435

_s4 0.858156 0.945175 0.493506

_s5 0.098039 0.95213 0.893204

_s6 0.858156 0.034953 0.904348

_s7 0.108696 0.94186 0.879121

_s8 0.114943 0.083719 0.86747

_s9 0.855072 0.951148 0.898148

_s10 0.83871 0.031526 0.897196

_s11 0.1 0.952038 0.886598

_s12 0.859155 0.945534 0.873563

_s13 0.847328 0.944321 0.492958

_s14 0.831933 0.944506 0.877778

_s15 0.84252 0.949622 0.902655

_s16 0.125 0.94848 0.008065

_s17 0.994964 0.950568 0.008333

_s18 0.1 0.953725 0.892157

_s19 0.854015 0.951028 0.869048

_s20 0.040404 0.947368 0.901786

Subject	Hb	BMI	RGC
_s1	0.10989	0.933466	0.493421
_s2	0.857143	0.946121	0.887755
_s3	0.120482	0.936548	0.880435
_s4	0.858156	0.945175	0.493506
_s5	0.098039	0.95213	0.893204
_s6	0.858156	0.034953	0.904348
_s7	0.108696	0.94186	0.879121
_s8	0.114943	0.083719	0.86747
_s9	0.855072	0.951148	0.898148
_s10	0.83871	0.031526	0.897196
_s11	0.1	0.952038	0.886598
_s12	0.859155	0.945534	0.873563
_s13	0.847328	0.944321	0.492958
_s14	0.831933	0.944506	0.877778
_s15	0.84252	0.949622	0.902655
_s16	0.125	0.94848	0.008065
_s17	0.994964	0.950568	0.008333
_s18	0.1	0.953725	0.892157
_s19	0.854015	0.951028	0.869048
_s20	0.040404	0.947368	0.901786

where e₁, e₂, e₃ ∈ Eare the set of parameters Hb, BMI and RGC respectively and μ_{s
_i} (e₁), μ_{s
_i} (e₂) and μ_{s
_i} (e₃) are respective fuzzy functions defined above.

4.6 Output for Hb and BMI

To generate output from our input data and fuzzified attributes we developed a dynamic synthetic model. Initially we applied it to our two parameters Hb and BMI for decision-making and envisaged ranking values.

4.6.1 Synthetic dynamic model and outcomes

$θ (s_{i}) = μ_{s_{i}} (e_{1}) + μ_{s_{i}} (e_{2})$

θ (s_i) denotes picture of two different medical problems of a subjects, that is, Hb and BMI.

We constructed FSS Table 6 recalling previously defined values of Hb Table 2 and BMI Table 3. According to value of θ (s_i) each individual subject received different ranking score.

In Table 7, our proposed synthetic dynamic model provides more than one interval for critical and risk values, while single interval for normal and healthy condition. Therefore, it is assumed that when we apply our proposed model on multiple as well as different medical conditions or diseases simultaneously. It is predisposed to generate more than one interval for our defined health conditions.

Table 7
Ranking of Hb and BMI developed for subjects under study

Ranking Intervals of Hb and BMI

Healthy Values of Subject i = 1.3455< θ (s_i) ≤1.9511

Normal Values of Subject i = 0.8898< θ (s_i) ≤0.9335

i = 0< θ (s_i) ≤0.0871

Critical Values of Subject ii = 0.1667< θ (s_i) ≤0.4442

iii = 0.9335< θ (s_i) ≤0.9702

i = 0.0871< θ (s_i) ≤0.1667

Risk Values of Subject ii = 0.4442< θ (s_i) ≤0.8898

iii = 0.9702< θ (s_i) ≤1.3455

Ranking Intervals of Hb and BMI
Healthy Values of Subject	i = 1.3455< θ (s_i) ≤1.9511
Normal Values of Subject	i = 0.8898< θ (s_i) ≤0.9335
	i = 0< θ (s_i) ≤0.0871
Critical Values of Subject	ii = 0.1667< θ (s_i) ≤0.4442
	iii = 0.9335< θ (s_i) ≤0.9702
	i = 0.0871< θ (s_i) ≤0.1667
Risk Values of Subject	ii = 0.4442< θ (s_i) ≤0.8898
	iii = 0.9702< θ (s_i) ≤1.3455

Example 5. In continuation of example (Table 5) and fuzzy membership functions (Table-) and ranking intervals of parameters (Table 7), the output expressed in Table 8.

Table 8

Cumulative Output of Hb and BMI developed for subjects under study

Subject	μ_{s _i} (e₁)	μ_{s _i} (e₂)	μ_{s _i} (e₃)	Output
_s1	0.10989	0.933466	1.043357	Risk
_s2	0.857143	0.946121	1.803264	Healthy
_s3	0.120482	0.936548	1.05703	Risk
_s4	0.858156	0.945175	1.803331	Healthy
_s5	0.098039	0.95213	1.050169	Risk
_s6	0.858156	0.034953	0.893109	Normal
_s7	0.108696	0.94186	1.050556	Risk
_s8	0.114943	0.083719	0.198661	Critical
_s9	0.855072	0.951148	1.80622	Risk
_s10	0.83871	0.031526	0.870236	Risk
_s11	0.1	0.952038	1.052038	Risk
_s12	0.859155	0.945534	1.804689	Healthy
_s13	0.847328	0.944321	1.791649	Risk
_s14	0.831933	0.944506	1.776439	Risk
_s15	0.84252	0.949622	1.792142	Risk
_s16	0.125	0.94848	1.07348	Risk
_s17	0.994964	0.950568	1.945532	Risk
_s18	0.1	0.953725	1.053725	Risk
_s19	0.854015	0.951028	1.805043	Risk
_s20	0.040404	0.947368	0.987772	Risk

4.7 Output for multiple attributes

After successful functioning of our proposed model with input data of two parameters we added a third variable in existing model to deduce weather it is workable for multiple parameters. Here we added RGC as third attribute and adopted approach for complex modeling.

4.7.1 Approach for complex modeling

Keeping in view handling of multiple health conditions, here we opted complex modeling approach third health condition that is RGC. This parameter has three different ranges for evaluation of blood glucose concentration of subject (Table 4). This approach helped us to testify application of classification rule for multiple, diverse medical problems. After further calculation (Table 8), Table 8.: Interval values of Hb, BMI and RGC developed for subjects under study.(Table 9), results were established for overall score of each subject for fuzzified values of Hb, BMI and RGC. $φ (s_{i}) = θ (s_{i}) + μ_{s_{i}} (e_{3})$

Table 9
Interval Values of Hb, BMI and RGC developed for subjects under study.

Interval Values

i = 1.3836< φ (s_i)≤1.4273

Interval Values of Healthy Subject ii = 1.8733< φ (s_i)≤1.9170

iii = 2.4448< φ (s_i)≤2.9346

i = 0.8932< φ (s_i)≤0.9369

ii = 1.3489< φ (s_i)≤1.3836

Interval Values of Normal Subject iii = 1.7048< φ (s_i)≤1.8200

iv = 1.8393< φ (s_i)≤1.8733

v = 1.9544< φ (s_i)≤2.4448

i = 0< φ (s_i)≤0.1264

ii = 0.1700< φ (s_i)≤0.5809

iii = 0.6604< φ (s_i)≤0.8932

Interval Values of Critical Subject iv = 0.9369< φ (s_i)≤1.0277

v = 1.0531< φ (s_i)≤1.0706

vi = 1.1502< φ (s_i)≤1.3482

i = 0.1264< φ (s_i)≤0.1700

ii = 0.5809< φ (s_i)≤0.6604

iii = 1.0277< φ (s_i)≤1.0531

Interval Values of at Risk Subject iv = 1.0706< φ (s_i)≤1.0713

v = 1.3482< φ (s_i)≤1.3489

vi = 1.4640< φ (s_i)≤1.5617

vii = 1.8200< φ (s_i)≤1.8393

viii = 1.9537< φ (s_i)≤1.9544

Interval Values
	i = 1.3836< φ (s_i)≤1.4273
Interval Values of Healthy Subject	ii = 1.8733< φ (s_i)≤1.9170
	iii = 2.4448< φ (s_i)≤2.9346
	i = 0.8932< φ (s_i)≤0.9369
	ii = 1.3489< φ (s_i)≤1.3836
Interval Values of Normal Subject	iii = 1.7048< φ (s_i)≤1.8200
	iv = 1.8393< φ (s_i)≤1.8733
	v = 1.9544< φ (s_i)≤2.4448
	i = 0< φ (s_i)≤0.1264
	ii = 0.1700< φ (s_i)≤0.5809
	iii = 0.6604< φ (s_i)≤0.8932
Interval Values of Critical Subject	iv = 0.9369< φ (s_i)≤1.0277
	v = 1.0531< φ (s_i)≤1.0706
	vi = 1.1502< φ (s_i)≤1.3482
	i = 0.1264< φ (s_i)≤0.1700
	ii = 0.5809< φ (s_i)≤0.6604
	iii = 1.0277< φ (s_i)≤1.0531
Interval Values of at Risk Subject	iv = 1.0706< φ (s_i)≤1.0713
	v = 1.3482< φ (s_i)≤1.3489
	vi = 1.4640< φ (s_i)≤1.5617
	vii = 1.8200< φ (s_i)≤1.8393
	viii = 1.9537< φ (s_i)≤1.9544

By adding another parameter, we found increasing trends in number of intervals in our defined categories. The results were shown in Table 9

In our proposed model by applying complex approach we found that when we need to handle multiple parameters and their intervals are also multiple then outcomes would have more than one value. We also found this novel approach is equally applicable for a large data set as we proved in present study. The study is articulated as pictorial expression in Fig. 1.

Fig. 1

The conceptualized flowchart of present study.

Figure represents input and output of medical information. Parameters; Hemoglobin (Hb), Body Mass Index (BMI) and Random Glucose Concentration (RGC) selected as Input Data Sets further Member Function (MF) and Synthetic Dynamic Model developed for an approach of Complex Modeling, through fuzzy soft set theory and classification rules to obtain output as Decision Making for medical diagnosis.

Example 6. The final decision from the previous example incorporated and finalized by selecting the parameter e₃. The decision for medical diagnosis as whether the individual subject is critical, risk, normal or healthy with respect of gestational wellbeing (Table 10)

Table 10

Complex Molding and Medical Diagnosis of Subjects as Critical, Normal or Healthy with Respect of Gestational Wellbeing

Subject	θ (s_i)	μ (s_i)e₃	φ (s_i)	Decision
_s1	1.043357	0.493421	1.536777563	Risk
_s2	1.803264	0.887755	2.691018649	Healthy
_s3	1.05703	0.880435	1.937464934	Risk
_s4	1.803331	0.493506	2.29683796	Healthy
_s5	1.050169	0.893204	1.943373305	Risk
_s6	0.893109	0.904348	1.797456668	Normal
_s7	1.050556	0.879121	1.929676996	Risk
_s8	0.198661	0.86747	1.066131058	Critical
_s9	1.80622	0.898148	2.704368633	Healthy
_s10	0.870236	0.897196	1.76743179	Normal
_s11	1.052038	0.886598	1.938636307	Risk
_s12	1.804689	0.873563	2.678251917	Healthy
_s13	1.791649	0.492958	2.284606703	Normal
_s14	1.776439	0.877778	2.654216655	Healthy
_s15	1.792142	0.902655	2.694796719	Healthy
_s16	1.07348	0.008065	1.081544681	Risk
_s17	1.945532	0.008333	1.953865535	Risk
_s18	1.053725	0.892157	1.94588199	Risk
_s19	1.805043	0.869048	2.674090621	Healthy
_s20	0.987772	0.901786	1.889558176	Risk

5 Conclusion

Present study designed as interdisciplinary approach towards rapid and efficient medical diagnosis to assist medical personals who have to manage a large number of patients with huge bulk of diverse medical information, thus face time constraints, and need methods helpful in prognosis and diagnosis in lesser time with precision in decisions.

This paper proposed a novel approach and successfully applied FSS based on ranking technique to rank health condition by the recruitment of heterogeneous attributes (Hb, BMI and RGC). The algorithm was constructed as single point, multipoint and cumulative diagnosis for predicting gestational wellbeing.

Fuzzy and soft sets hold potential for complex modeling and provision of algorithm for decision-making involving multiple health attributes detailed as medical tests/reports. Countries suffering from public health crises are at most demanding end for innovative approaches. Our novel algorithm is a unique effort to study health parameters for holistic insight of ongoing pregnancy. The algorithm also has equally parallel significance in time reduction by medical personals for diagnosis of anemia, hypertension, and diabetes during ongoing gestation pregnancy.

By applying our complex approach, we became able to compute multiple conditions for holistic insight about health of pregnant females of ruler area of Pakistan. We also assume that our proposed method is applicable for a large data set as we successfully applied it for a data set of 341 subjects. The future prospective of our proposed model is to refine model for handling the increasing trend of intervals.

References

Anjali

, Vilas

and Pradip

, Efficacy of Fuzzy-Stat Modeling in Classification of Gynecologists and Patients, Journal of Intelligent Systems (2015). Doi: 10.1515/jisys-2015-0001

Aygüna

and Kamaci

, Some Generalized Operations in Soft Set Theory and their Role in Similarity and Decision Making, Journal of Intelligent & Fuzzy Systems 36 (2019), 6537–6547

10.3233/JIFS-182924

Brabin

B.J.

, Hakimi

and Pelletier

, An Analysis of Anemia and Pregnancy-related Maternal Mortality, Nutrition J 131 (2001), 604S–615S.

Bradshaw

, Chopra

, Kerber

, Lawn

J.E.

, Bamford

, Moodley

, Pattinson

, Patrick

, Stephen

and Velaphi

, Every Death Counts: Use of Mortality Audit Data for Decision Making to Save the Lives of Mothers, Babies, and Children in South Africa, Lancet 12;371(9620) (2008), 1294–1304. doi: 10.1016/S01406736(08)60564-4

Chen

, Tsang

E.C.C.

, Yeung

D.S.

and Wang

, The Parameterization Reduction of Soft Sets and its Applications, Computers and Mathematics with Applications (2005), 757–763. https://doi.org/10.1016/j.camwa.2004.10.036

Çağman

, Enginoğlu

and Çitak

, Fuzzy Soft Set Theory and Its Applications, Iranian Journal of Fuzzy Systems 8(3) (2011), 137–147.

Chatterji

, Lahiri

and Kim

, Fetal Growth and Neurobehavioral Outcomes in Childhood, Econ Hum Biol 15 (2014), 187–200.

Gillmer

M.D.

, Beard

R.W.

, Rooke

F.M.

and Oakley

N.W.

, Carbohydrate Metabolism in Pregnancy, Part I. Diurnal Plasma Glucose Profile in Normal and Diabetic Women, BMJ 3(1975), 399–402. doi: 10.1136/bmj.3.5980.399

Godfrey

K.M.

, Redman

C.W.G.

, Barker

D.J.P.

and Osmond

, The Effect of Maternal Anaemia and Iron Deficiency on the Ratio of Fetal Weight to Placental Weight, British J Obstetrics and Gynecology 98 (1991), 886–891. Doi: 10.1111/j.1471-0528.1991.tb13510.x

10.

Johnson

and Schoeni

, The Influence of Early-Life Events on Human Capital, Health Status, and Labor Market Outcomes Over the Life Course, The B.E. Journal of Economic Analysis & Policy 154(11) (2014), 2262–2265.

11.

Jiang

, Zhan

and Chen

, Rough Sets with Applications to Multi Attribute, IEEE Transactions on Fuzzy Systems 27(8) (2019), 1558–1572.

12.

Kovkov

D.V.

, Kolbanov

V.M.

and Molodtsov

D.A.

, Soft Sets Theory-based Optimization, J Comp Sys Sc Int 46 (2007), 872–880.

13.

Kamaci

, Atagüna

A.O.

and Toktas

, Bijective Soft Matrix Theory and Multi-Bijective Linguistic Soft Decision System, Filomat (2018), 3799–3814. https://doi.org/10.2298/FIL1811799K.

14.

Kamaci

, Selectivity analysis of Parameters in Soft Set and its Effect on Decision Making, International Journal of Machine Learning and Cybernetics 11(2020), 313–324.

15.

Lone

F.W.

, Qureshi

R.N.

and Emanuel

, Maternal Anemia and its Impact on Perinatal Outcome, Tropical Medicine and International Health 9 (2004), 486–490.

16.

Lavy

, Schlosser

and Shany

, Out of Africa: Human Capital Consequences of In Utero and Prenatal Conditions, National Bureau of Economic Research (2014), 1–41.

17.

Langarizadeh

, Khajehpour

, Farokhnia

and Eftekhari

, A Fuzzy Expert System for Distinguishing between Bacterial and Aseptic Meningitis, Iranian Journal of Medical Physics 12 (2015), 1–5. Doi: 10.22038/IJMP.2015.4322

18.

Molodtsov

, Soft Set Theory First Results, Comp and Math With Appl 37 (1999), 19–31.

19.

Maji

P.K.

, Biswas

and Roy

A.R.

, Fuzzy Soft Sets, J Fuzzy Math 9 (2001), 589–602.

20.

Metzger

B.E.

, Lowe

L.P.

, Dyer

A.R.

, Trimble

E.R.

, Chaovarindr

, Coustan

D.R.

, Hadden

D.R.

, McCance

D.R.

, Hod

, McIntyre

H.D.

, Oats

J.J.N.

, Persson

, Rogers

M.S.

and Sacks

D.A.

, Hyperglycemia and Adverse Pregnancy Outcomes, N Engl J Med 358 (2002), 1991–2002. Doi: 10.1056/NEJMoa0707943

21.

Maji

P.K.

, Roy

A.R.

and Biswas

, An Application of Soft Sets in a Decision Making problem, Computers and Mathematics with Applications 44 (2002), 1077–1083. https://doi.org/10.1016/S0898-1221(02)00216-X.

22.

Marchant

, Schellenberg

J.A.

, Nathan

, Abdulla

, Mukasa

, Mshinda

and Lengeler

, Anemia in Pregnancy and Infant Mortality in Tanzania, Tropical Medicine and International Health 9 (2004), 262–266.

23.

Mahmud

, Zaman

, Jafarey

, Khan

R.L.

, Sohail

and Fatima

, Achieving Millennium Development Goals 4 and 5 in Pakistan, An International J Obstetrics and Gynecology 118 (2011), 69–77.

24.

Mansuri

, Situation Analysis of Millennium Development Goals 4 & 5: Pakistan’s Perspective, Annals of Abbasi Shaheed Hospital & Karachi Medical & Dental College 19(2) (2014), 91–96.

25.

Moore

, Arefadib

, Deery

and West

, Days, The First Thousand Days: An Evidence Paper-Summary, Centre for Community Child Health’ (2017), 1–93.

26.

Ota

, Haruna

and Suzuki

, Maternal Body Mass Index and Gestational Weight Gain and their Association with Perinatal Outcomes in Viet Nam, Bull World Health Organ 89 (2011), 127–136.

27.

Oguntimilehin

, Adetun

A.O.

and Olatunji

K.A.

, A Machine Learning Based Clinical Decision Support System for Diagnosis and Treatment of Typhoid Fever, IJARCSSE 4(6) (2014), 961–969.

28.

Phelps

R.L.

, Honig

and Green

, Biphasic Changes in Hemoglobin Concentrations During Normal Human Pregnancy, Am J Obstet Gynecol 147 (1983), 651–653.

29.

Phuong

N.H.

and Kreinovich

, Fuzzy Logic and its Applications in Medicine, Int J Med Inform 62 (2001), 165–173. DOI: 10.1016/s1386-5056(01)00160-5

30.

Roy

A.R.

and Maji

P.K.

, A Fuzzy Soft Set Theoretic Approach to Decision Making Problems, Journal of Computational and Applied Mathematics 203 (2007), 412–418.

31.

World Health Organization, Iron Deficiency Anemia: Assessment, Prevention, and Control, World Health Organization Geneva Switzerland (2001).

32.

World Health Organization, Taking Stock: Health Workers Shortages and the Response to AIDS, (2007). www.who.int/healthsystems.

33.

World Health Organization. Every Woman, Every Child, and Every Adolescent: Achievements and Prospects, IERG (2015).

34.

Xiao

, li

, Zhong

and Yang

, Research on Synthetically Evaluating Method for Business Competitive Capacity Based on Soft Set, Stat Methods Med Res (2003), 52–54.

35.

Xiao

, Yang

and Niu

, A New Evaluation Method Based on D-S Generalized Fuzzy Soft Sets and its Application in Medical Diagnosis Problem, Appl Math Model 10 (2012), 4592–4604.

36.

Yazdani

, Yosofniyapasha

, Nasab

, Mojaveri

and Bouzari

, Effect of Maternal Body Mass Index on Pregnancy Outcome and Newborn Weight, BMC Research Notes 5 (2012), 34. Doi: 10.1186/1756-0500-5-34

37.

Yuksel

, Dizmen

, Yildizdan

and Sert

, Application of Soft Sets to Diagnose the Prostate Cancer Risk, J Inequalities and Applications 22 (2013).

38.

Zadeh

L.A.

, Fuzzy Sets, Information Control 8 (1965), 338–353.

39.

Zhan

and Alcantud

J.C.R.

, A Novel Type of Soft Rough Covering and its Application to Multicriteria Group Decision Making. Artificial Intelligence. (2018), Review. https://doi.org/10.1007/s10462-018-9617-3

40.

Zhan

and Alcantud

J.C.R.

, A Survey of Parameter Reduction of Soft Sets and Corresponding Algorithms, Artificial Intelligence 52(3) (2019), 1839–1872. https://doi.org/10.1007/s10462-017-9592-0

41.

Zhan

, Sun

and Alcantud

J.C.R.

, Covering Based Multigranulation (I T)-Fuzzy Rough Set Models and Applications in Multi-attribute Group Decision-making, Information Sciences 476 (2019), 290–318. https://doi.org/10.1016/j.ins.2018.10.016

42.

Zhang

, Zhan

, Wu

and Alcantud

J.C.R.

, Fuzzy β-covering based (I T)-Fuzzy Rough Set Models and Applications to Multi-attribute Decision-making, Computers & Industrial Engineering 128 (2019), 605–621. https://doi.org/10.1016/j.cie.2019.01.004

43.

Zhang

, Zhan

and Xu

, Covering-based generalized, IF rough sets with applications, Information Sciences 478 (2019), 275–302.

Fuzzy soft set theory: Application of classification rules in decision making during medical diagnosis*

Abstract

Keywords

1 Introduction

1.1 Motivation of the study

2 Preliminaries

Table 1 Fuzzy Soft Set (FSS) FSS (F, A) (F,A) c1 c2 c3 e1 0.6 0.4 0.3 e2 0.5 0.7 0.8

Table 2 Hemoglobin Level in Blood Circulation of Pregnant Females Category Hb Present Study Criteria 1 Healthy 13.5–16.5 (g/dL) 2 Normal 10.5–13.5 (g/dL) 3 Risk 7.5–10.5 (g/dL) 4 Critical 4.5–7.5 (g/dL)

4.1 Structure criteria for evaluation

4.2 Input data

4.3 Fuzzification

4.4.1 Membership function for parameter e1

4.4.2 Membership function for parameter e2

4.4.3 Membership function for RGC

4.5 Fuzzy soft Set

4.6.1 Synthetic dynamic model and outcomes

4.7.1 Approach for complex modeling

References

Table 1
Fuzzy Soft Set (FSS)

FSS (F, A)

(F,A) c₁ c₂ c₃

e₁ 0.6 0.4 0.3

e₂ 0.5 0.7 0.8

Table 2
Hemoglobin Level in Blood Circulation of Pregnant Females

Category Hb Present Study Criteria

1 Healthy 13.5–16.5 (g/dL)

2 Normal 10.5–13.5 (g/dL)

3 Risk 7.5–10.5 (g/dL)

4 Critical 4.5–7.5 (g/dL)

4.4.1 Membership function for parameter e₁

4.4.2 Membership function for parameter e₂