Abstract

One of the primary aspects of utilization of any database model lies in its potential in processing information and queries accurately. In the present work, the authors intend to make a comparative analysis on the capability of fuzzy and vague relational database models in treating uncertain queries. A new algorithm is proposed and query testing related to a real life example is performed. The investigation demonstrates that a relational data model based on vague set theory produces more refined decisions than a fuzzy data model. It may thus be asserted that a relational database management system (RDBMS) using vague theoretic concept might lead to better software fabrication than the presently accessible ones.

Keywords: Fuzzy set, vague set, database model, similarity measure, SQL, fuzzy SQL, vague SQL

25.1 Introduction

In real life, information is very often imprecise or incomplete. The conventional relational database system fails to treat such uncertain data. The theory of fuzzy sets, as formalized by Zadeh [20] in 1965, is extensively applied to handle such inexact or imprecise data. Several authors [1, 6, 10, 11, 14, 15, 17–19] have also worked on formulation of a query language for a database representation based on fuzzy sets. Bosc et al. [1] and Nakajima et al. [15] have outstretched the familiar SQL language in the framework of fuzzy set theory, namely, SQLF. More recently, Moreau et al. [14] have discussed a procedure wherein one may fetch data with no prior information about the database constitution or precise query language. The potential benefits and efficacy of the use of fuzzy query in a traditional data model have been thoroughly explained in [17]. An intelligent approach to extend SQL language to treat flexible conditions in queries was proposed by Mama et al. [11] in 2021.

Gau and Buehrer [4] initiated the theory of vague sets in 1993 as an abstraction of a fuzzy set theoretic approach. It is believed that a vague set that employs interval-based membership values may process uncertain information in a more efficient manner compared to a fuzzy set. The vague theoretic concept was further embodied in relations by Lu and Ng [8, 9] and a new query language called VSQL evolved. A vague database model was designed by Zhao and Ma [21] and vague querying strategies with SQL were also investigated. In [3], Dutta et al. have used vague sets to rejuvenate “vague search”, a new method of intelligent search that is capable of answering any uncertain query put forth by the user. In [12], the detailed design aspects of a vague database model have been thoroughly discussed. An architecture for processing of hesitant queries has been designed by Mishra et al. [13] along with a comparison of fuzzy and vague sets in handling imprecise queries.

In the present work, the authors also aspire to analyze the ability of fuzzy and vague database models in relation to uncertain query processing. It has been noticed that the algorithms used in literature (see refs. [10, 13, 16]) to obtain membership values possess certain drawbacks. In the current investigation, the authors present a novel algorithm which is devoid of such deficiencies. The proposed algorithm generates membership functions for fuzzy or vague sets for the calculation of membership values that are independent of the attribute. However, it is worthwhile to mention that the membership functions employed earlier in literature [10, 16] depend on the attribute type.

The framework of the chapter is as follows. The definitions of fuzzy and vague sets as well as related fundamental concepts are presented in Sections 25.2.1 and 25.2.2. The similarity measure formulae used in this study appear in Section 25.2.3. The algorithm designed in the present analysis to generate membership values is proposed in Section 25.3. Real life examples are demonstrated in Section 25.4 to observe that a vague theoretic approach is better suited in processing queries that are not precise. The concluding remarks are reported in Section 25.5.

25.2 Basic Definitions

Let X denote the universe of discourse and x represent an element of X.

25.2.1 Fuzzy Set

Definition 25.2.1.1: A fuzzy set S, defined in the universe of discourse X, is a set of ordered pairs where µ_S: X → [0,1] denotes the grade of membership of x in S.

It may be easily observed that an ordinary subset A of X may be treated as a fuzzy set with membership function µ_A that takes binary values, i.e.,

25.2.3 Similarity Measure

The concept of similarity of vague sets has been studied by several researchers [2, 5, 7, 8]. The similarity measure proposed by Lu et al. in [8] was shown to be more effective in general cases. This measure has been used in this work, which is presented below:

Definition 25.2.3.1: Similarity Measure for Vague Values

Let u and v be vague values such that u = [t₁, 1 − f₁], and v = [t₁, 1 − f₂], where 0 ≤ t₁ ≤ 1 – f₁ ≤ 1 and 0 ≤ t₂ ≤ 1 – f₂ ≤ 1. If SM(u, v) denotes the similarity measure between u and v, then

Definition 25.2.3.2: Similarity Measure for Vague Sets

Let X = {x₁,x₂,x₃,...,x_n} be the universe. Let S₁ and S₂ be two vague sets of X, such that

where

and

where

Now, the similarity measure between S₁ and S₂, denoted by SM(S₁, S₂), is defined as:

25.3 Algorithm to Generate Membership Values

In the literature, various membership functions have been employed by different authors for finding the membership values for different fuzzy attributes.

For an EMPLOYEE database, the membership function deployed by Raju et al. [16] corresponding to a fuzzy set ‘close to u’ and fuzzy attribute Salary is the following:

However, for the Experience attribute, Raju et al. defined the membership function as

On the contrary, Ma et al. [10] used the following membership functions for the same fuzzy attributes, Salary and Experience:

It is obvious that the above membership functions depend on the specific attribute under investigation.

In this study, the authors have made an effort to devise an algorithm for calculation of membership values for different fuzzy/vague attributes. It may be pointed out that the membership function generated by said algorithm is attributed as independent for a fuzzy/vague set. The membership values generated by the proposed algorithm also compare well with that of others obtained in the literature.

A similar algorithm was designed by Mishra et al. [13] for the purpose of estimation of membership values. It is observed that the same algorithm has been utilized by Yadav in [19] for SQL query processing. But, the formula put forth in [13] is found to possess the following deficiencies:

1. Negative membership value is generated in certain instances.
2. Addition of new records in the given data leads to change of membership values for the existing data that is not sensible.

The algorithm devised in this study is presented below. It may be noted that it is devoid of such anomalies.

25.4 Real Life Applications

We now use the following EMPLOYEE relation (Table 25.1) as a real life example and process certain imprecise queries with one or more attributes. A comparative analysis is presented in this section for each of the queries on the framework of fuzzy and vague models.

Query 1: “To fetch the employees with age close to 53 years”.

Query 2: “To retrieve the data of the employees whose age is more or less than 53 years with work experience close to 22 years.”

The fuzzy and vague database models were used to execute the above queries with one or more attributes. In each case, the vague database model produced better results than the fuzzy data model, as may be observed from the analysis presented herein.

Query 1: “To fetch the employees with age close to 53 years”.

Solution with Fuzzy Model:

The fuzzy characteristic here is Age and the fuzzy data is close to 53 years.

We employ the algorithm discussed in Section 25.3 to find the membership value for each domain value of the Age attribute.

Here,

Domain of Age attribute = {30, 31, 34, 51, 57, 49, 50, 41, 39, 53, 52, 56, 54, 55, 62}

f_data = 53

Range = 62 – 30 = 32

We now deploy the formula provided in Algorithm 25.3.1 to determine the membership values for every domain value of Age attribute as follows:

1^st tuple: Membership Value = 1 - (|53–30| / 32) = 0.28125

2^nd tuple: Membership Value = 1 - (|53–31| / 32) = 0.3125

The complete record of membership values for each tuple is shown in Table 25.2.

Table 25.3 displays the complete representation of this relation corresponding to Query 1. For the fuzzy attribute Age, the fuzzy representation appears in column 3 and its equivalent vague formulation occurs in column 4. This vague formulation is now utilized to compute the similarity measures (SM) with the fuzzy data ‘close to 53’, which may be represented as <53, [1, 1]> in vague notation. The formula for similarity measure, as introduced in Definition 25.2.3.1, is then applied.

If one considers the vague data u = <53,[1,1]> and v = <30,[0.28125, 0.28125]>, then

Name	Age (years)	Fuzzy representation of age	Vague representation of age	SM of age with <53,[1,1]>	Experience (years)	Remuneration	SM (tuple)
Mr. Roy	30	<30, 0.28125>	<30, [0.28125, 0.28125]>	0.53033	4	32000	0.53033
Mr. Roychowdhury	31	<31, 0.3125>	<31, [0.3125, 0.3125]>	0.55902	6	32500	0.55902
Mr. Pramanik	34	<34, 0.40625>	<34, [0.40625, 0.40625]>	0.63738	8	34000	0.63738
Mr. Barik	51	<51, 0.9375>	<51, [0.9375, 0.9375]>	0.96825	18	58000	0.96825
Mr. Bose	57	<57, 0.875>	<57, [0.875, 0.875]>	0.93541	22	77900	0.93541
Mr. Ghosh	49	<49, 0.87875>	<49, [0.875, 0.875]>	0.93541	17	57400	0.93541
Mr. Samanta	50	<50, 0.9090625>	<50, [0.90625, 0.90625]>	0.95197	16	56800	0.95197
Mr. Bhowmick	41	<41, 0.62625>	<41, [0.625,. 0.625]>	0.79057	13	46500	0.79057
Mr. Banerjee	39	<39, 0.5625>	<39, [0.5625, 0.5625]>	0.75000	12	42000	0.75000
Mr. Chattopadhyay	53	<53, 1>	<53, [1,1]>	1	20	80900	1
Mr. Ganguly	52	<52, 0.9696875>	<52, [0.96875, 0.96875]>	0.98425	19	80200	0.98425
Mr. Kher	56	<56, 0.90625>	<56, [0.90625, 0.90625]>	0.95197	21	81100	0.95197
Mr. Mukhopadhyay	54	<54, 0.96875>	<54, [0.96875, 0.96875]>	0.9842	23	82200	0.9842
Mr. Patnaik	55	<55, 0.939375>	<55, [0.9375, 0.9375]>	0.96825	27	82000	0.96825
Mr. Sahoo	62	<62, 0.7171875>	<62, [0.71875, 0.71875]>	0.84779	35	122000	0.84779

and thus,

Again, for u = <53,[1,1]> and v = <31,[0.3125,0.3125]>,

Then,

and so on. The complete results are shown in Table 25.3.

Now, the following SQL statement is generated to execute the given query at an α-cut or threshold value 0.95, provided by the decision maker:

SELECT^⋆ FROM EMPLOYEE WHERE SM(tuple) ≥ 0.95.

Table 25.4 now presents the resultant tuples retrieved from the EMPLOYEE database.

Name	Age (years)	Experience (years)	Remuneration ()
Mr. Barik	51	18	58000
Mr. Samanta	50	16	56800
Mr. Chattopadhyay	53	20	80900
Mr. Ganguly	52	19	80200
Mr. Kher	56	21	81100
Mr. Mukhopadhyay	54	23	82200
Mr. Patnaik	55	27	82000

Name	Age (years)	Vague representation of age	SM of age with <53,[1,1]>	Experience (years)	Remuneration	SM (tuple)
Mr. Roy	30	<30, [0.28125, 0.440416]	0.537289	4	32000	0.537289
Mr. Roychowdhury	31	<31, [0.3125, 0.485216]>	0.574358	6	32500	0.574358
Mr. Pramanik	34	<34, [0.40625, 0.516927]>	0.640691	8	34000	0.640691
Mr. Barik	51	<51, [0.9375, 0.953667]>	0.964182	18	58000	0.964182
Mr. Bose	57	<57, [0.875, 0.878291]>	0.929415	22	77900	0.929415
Mr. Ghosh	49	<49, [0.875, 0.875770]>	0.930180	17	57400	0.930180
Mr. Samanta	50	<50, [0.90625,0.921369]>	0.951472	16	56800	0.951472
Mr. Bhowmick	41	<41, [0.625, 0.696428]>	0.785632	13	46500	0.785632
Mr. Banerjee	39	<39, [0.5625, 0.681963]>	0.743212	12	42000	0.743212
Mr. Chattopadhyay	53	<53, [1.000000,1.00000]>	1.000000	20	80900	1.000000
Mr. Ganguly	52	<52, [0.96875, 0.977372]>	0.982317	19	80200	0.982317
Mr. Kher	56	<56, [0.90625, 0.914506]>	0.946656	21	81100	0.946656
Mr. Mukhopadhyay	54	<54, [0.96875,0.976538]>	0.982701	23	82200	0.982701
Mr. Patnaik	55	<55, [0.9375, 0.943755]>	0.966286	27	82000	0.966286
Mr. Sahoo	62	<62, [0.71875, 0.783282]>	0.835870	35	122000	0.835870

Solution with Vague Model:

We now analyze the same query using a vague data model. Age is the vague attribute and vague data is ‘close to 53’.

To determine the vague representation of the Age attribute, Algorithm 25.3.1 is now utilized to obtain the truth membership values. On the other hand, the decision maker provides the false membership values randomly along with the constraint that the sum of the two membership values cannot exceed 1. Next, the similarity measures are computed and are displayed in Table 25.5.

The tuples recovered from the EMPLOYEE relation at this α-cut value 0.95 are shown in Table 25.6.

It may be clearly observed from Tables 25.4 and 25.6 that the vague model is better off than its fuzzy version. The tuple of Mr. Kher, aged 56, was not retrieved by the SQL statement with vague query although it was fetched as fuzzy query. It is worthwhile to note that compared to the other values returned by the SQL statement, 56 is less close to 53.

We now look at another imprecise question that has two fuzzy or vague characteristics.

Query 2: “To retrieve the data of the employees whose age is more or less than 53 years with work experience close to 22 years.”

Solution with Fuzzy Model:

The current query has two attributes: Age and Experience. Table 25.7 shows the fuzzy representation of the EMPLOYEE relationship produced using Definition 25.2.3.1 and Algorithm 25.3.1.

Name	Age (years)	Experience (years)	Remuneration
Mr. Barik	51	18	58000
Mr. Samanta	50	16	56800
Mr. Chattopadhyay	53	20	80900
Mr. Ganguly	52	19	80200
Mr. Mukhopadhyay	54	23	82200
Mr. Patnaik	55	27	82000

Name	Age (years)	Fuzzy representation of age	Vague representation of age	SM of age with <53, [1,1]>	Experience (years)	Fuzzy representation of experience	Vague representation of experience	SM of experience with <22,1,1]>	Remuneration	SM (tuple)
Mr. Roy	30	<30, 0.28125>	<30, [0.28125, 0.28125]>	0.53033	4	<4,0.419355>	<4,[0.4193550 .419355]>	0.647576	32000	0.53033
Mr. Roychowdhury	31	<31, 0.3125>	<31, [0.3125, 0.3125]>	0.55902	6	<6,0.483871>	<6,[0.483871, 0.483871]>	0.695608	32500	0.55902
Mr. Pramanik	34	<34, 0.40625>	<34, [0.40625, 0.40625]>	0.63738	8	<8, 0.548387>	<8, [0.548387, 0.548387]>	0.740532	34000	0.63738
Mr. Barik	51	<51, 0.9375>	<51, [0.9375, 0.9375]>	0.96825	18	<18,.0.870968>	<18,[0.870968, 0.870968]>	0.933257	58000	0.933257
Mr. Bose	57	<57, 0.875>	<57, [0.875, 0.875]>	0.93541	22	<22,1>	<22,[1,1]>	1.000000	77900	0.93541
Mr. Ghosh	49	<49, 0.875>	<49, [0.875,. 0.875]>	0.93541	17	<17,0.838710>	<17,[0.838710, 0.838710]>	0.915811	57400	0.915811
Mr. Samanta	50	<50, 0.90625>	<50, [0.90625, 0.90625]>	0.95197	16	<16,0.806452>	<16,[0.806452, 0.806452]>	0.898027	56800	0.898027
Mr. Bhowmick	41	<41, 0.625>	<41, [0.625,. 0.625]>	0.79057	13	<13,0.709677>	<13,[0.709677, 0.709677]>	0.842424	46500	0.79057
Mr. Banerjee	39	<39, 0.5625>	<39, [0.5625, 0.5625]>	0.75000	12	<12,0.677419>	<12,[0.677419, 0.677419]>	0.823055	42000	0.75000
Mr. Chattopadhyay	53	<53, 1>	<53, [1,1]>	1	20	<20,0.935484>	<20, [0.935484, 0.935484]>	0.967204	80900	0.967204
Mr. Ganguly	52	<52, 0.96875>	<52, [0.96875, 0.96875]>	0.98425	19	<19,.0.903226>	<19,[.0.903226, .0.903226>	0.950382	80200	0.950382
Mr. Kher	56	<56, 0.90625>	<56, [0.90625, 0.90625]>	0.95197	21	<21,0.967742>	<21,[0.967742, 0.967742]>	0.983739	81100	0.95197
Mr. Mukhopadhyay	54	<54, 0.96875>	<54, [0.96875, 0.96875]>	0.9842	23	<23,0.967742>	<23,[0.967742, 0.967742]>	0.983739	82200	0.983739
Mr. Patnaik	55	<55, 0.9375>	<55, [0.9375, 0.9375]>	0.96825	27	<27,0.838710>	<27,[0.838710, 0.838710]>	0.915811	82000	0.915811
Mr. Sahoo	62	<62, 0.71875>	<62, [0.71875, 0.71875]>	0.84779	35	<35,0.580645>	<35, [0.580645, 0.580645]>	0.762001	122000	0.762001

Name	Age (years)	Experience (years)	Remuneration
Mr. Chattopadhyay	53	20	80900
Mr. Ganguly	52	19	80200
Mr. Kher	56	21	81100
Mr. Mukhopadhyay	54	23	82200

Name	Age (years)	Experience (years)	Remuneration
Mr. Barik	51	18	58000
Mr. Bose	57	22	77900
Mr. Ghosh	49	17	57400
Mr. Chattopadhyay	53	20	80900
Mr. Ganguly	52	19	80200
Mr. Kher	56	21	81100
Mr. Mukhopadhyay	54	23	82200
Mr. Patnaik	55	27	82000

It may be noted that as Query 2 involves two different attributes, the intersection of the similarity measures of the two attributes give the similarity measure of the corresponding tuple. The given query is examined at different threshold values provided by the decision maker. The results generated at the α-cut values 0.95 and 0.91 are demonstrated in Tables 25.8 and 25.9, respectively.

Next, we present the solution using a vague set for the same query.

Solution with Vague Model:

Once again, Algorithm 25.3.1 and Definition 25.2.3.1 are employed to obtain the vague representation of the EMPLOYEE relation corresponding to the Query 2, as presented in Table 25.10.

Name	Age (years)	Vague representation of age	SM of age with <53, [1,1]>	Experience (years)	Vague representation of experience	SM of experience with <22, [1,1]>	Remuneration	SM (tuple)
Mr. Roy	30	<30, [0.28125, 0.440416]>	0.537289	4	<4,[0.419355, 0.544234]>	0.647989	32000	0.537289
Mr. Roychowdhury	31	<31, [0.3125, 0.485216]>	0.574358	6	<6,[0.483871, 0.560575]>	0.695474	32500	0.574358
Mr. Pramanik	34	<34, [0.40625, 0.516927]>	0.640691	8	<8, [0.548387, 0.603663]>	0.739961	34000	0.640691
Mr. Barik	51	<51, [0.9375, 0.953667]>	0.964182	18	<18, [0.870968, 0.888225]]>	0.931279	58000	0.931279
Mr. Bose	57	<57, [0.875, 0.878291]>	0.929415	22	<22,[1,1]>	1.000000	77900	0.929415
Mr. Ghosh	49	<49, [0.875, 0.875770]>	0.930180	17	<17,[0.83871, 0.884063]>	0.910255	57400	0.910255
Mr. Samanta	50	<50, [0.90625, 0.921369]>	0.951472	16	<16, [0.80645, 0.839616]>	0.888711	56800	0.888711
Mr. Bhowmick	41	<41, [0.625, 0.696428]>	0.785632	13	<13, [0.70967, 0.751489]>	0.841028	46500	0.785632
Mr. Banerjee	39	<39, [0.5625, 0.681963]>	0.743212	12	<12,[0.67741, 0.704832]>	0.820811	42000	0.743212
Mr. Chattopadhyay	53	<53, [1.000000, 1.00000]>	1.000000	20	<20, [0.935484, 0.942255]>	0.963133	80900	0.963133
Mr. Ganguly	52	<52, [0.96875, 0.977372]>	0.982317	19	<19, [.0.90326, 0.914820>	0.945563	80200	0.945563
Mr. Kher	56	<56, [0.90625, 0.914506]>	0.946656	21	<21,[0.967742, 0.975348]>	0.983458	81100	0.946656
Mr. Mukhopadhyay	54	<54, [0.96875, 0.976538]>	0.982701	23	<23, [0.96774, 0.971200]>	0.982917	82200	0.982701
Mr. Patnaik	55	<55, [0.9375, 0.943755]>	0.966286	27	<27,[0.83871, 0.873021]>	0.908944	82000	0.908944
Mr. Sahoo	62	<62, [0.71875, 0.783282]>	0.835870	35	<35, [0.58064, 0.667873]>	0.756300	122000	0.756300

The resultant tuples generated after processing this imprecise query at the same threshold or α-cut values are presented below in Tables 25.11 and 25.12.

Tables 25.8 and 25.11 confirm that when the threshold value is set to 0.95, the vague model produces finer results. The tuples of Mr. Ganguly and Mr. Kher are not returned by the vague SQL, as it may be noticed that their experience or age are not close enough to the said query. The results shown in Tables 25.9 and 25.12 for the α-cut value 0.91 re-establish the fact that the vague SQL performs more effectively than its fuzzy representation. It may be noted that the tuple of Mr. Patnaik, who is 55 years old and has 27 years of experience, was not retrieved utilizing the vague framework.

Name	Age (years)	Experience (years)	Remuneration
Mr. Chattopadhyay	53	20	80900
Mr. Mukhopadhyay	54	23	82200

25.5 Conclusion

An algorithm has been designed in this research work to generate the fuzzy or vague representation of attributes for an imprecise query. The suggested technique produces a membership function for calculating membership values that do not depend on the type of the attribute and are free of various drawbacks identified in existing membership functions studied in the literature. Using an Employee database, a comparison of fuzzy and vague sets for handling imprecise queries has been performed. The results of this study confirms that a database model based on vague sets can process uncertain queries more efficiently than a fuzzy model.

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