J. Nig. Soc. Phys. Sci. 4 (2022) 174–182 Journal of the Nigerian Society of Physical Sciences Modified Szmidt and Kacprzyk’s Intuitionistic Fuzzy Distances and their Applications in Decision-making P. A. Ejegwaa,∗, I. C. Onyekeb, B. T. Terhemena, M. P. Onojaa, A. Ogijia, C. U. Opeha aDepartment of Mathematics, University of Agriculture, P.M.B. 2373, Makurdi, Nigeria bDepartment of Computer Science, University of Agriculture, P.M.B. 2373, Makurdi, Nigeria Abstract Intuitionistic fuzzy models are significant in resolving decision-making. Distance measures under intuitionistic fuzzy environment are reliable techniques deployed to express the application of IFSs. Some approaches of estimating distances between IFSs have been explored by Szmidt and Kacprzyk, where the complete parameters of IFSs are considered. Albeit, the distance operators lack reliability because of certain setbacks. In this paper, we modified Szmidt and Kacprzyk’s distance operators between IFSs to enhance reliability in terms of applications. Some theorems are given to substantiate the validity of the modified intuitionistic fuzzy distance operators. Futhermore, decision-making cases of pattern recognition and disease identification are discussed using the Szmidt and Kacprzyk’s distances and their improved versions where information are represented in intuitionistic fuzzy pairs. From the study, it is observed that the modified Szmidt and Kacprzyk’s distance operators between IFSs yield better results compare to the Szmidt and Kacprzyk’s distance operators between IFSs. DOI:10.46481/jnsps.2022.530 Keywords: Decision-making, Distance measure, Intuitionistic fuzzy set, Pattern recognition, Medical diagnosis Article History : Received: 12 December 2021 Received in revised form: 21 April 2022 Accepted for publication: 24 April 2022 Published: 29 May 2022 c©2022 Journal of the Nigerian Society of Physical Sciences. All rights reserved. Communicated by: T. Latunde 1. Introduction Decision-making is a critical task enmeshed with vague- ness. With the introduction of fuzzy sets by Zadeh [1], the problem of vagueness has be considerably tackled to enhance the solution of many decision-making problems including med- ical diagnosis, career determination, pattern recognition among others. Fuzzy set theory although relevant has a setback in the ∗Corresponding author tel. no: +2347062583323 Email addresses: ejegwa.augustine@uam.edu.ng (P. A. Ejegwa ), onyeke.idoko@uam.edu.ng (I. C. Onyeke), terhementarfabenjamin@gmail.com (B. T. Terhemen), onojaexcel1234@gmail.com (M. P. Onoja), andrewogiji@yahoo.com (A. Ogiji), christophernathaniel360@gmail.com (C. U. Opeh) sense that it considers only the membership degree υ (MD) of the case under consideration. Because of this setback, Atanassov [2] proposed a generalized fuzzy set called intuitionistic fuzzy set (IFS). IFS is described by membership degree υ, nonmem- bership degree ν and intuitionistic fuzzy index $ with the prop- erty that their sum is unity. IFSs have been applied in sundry cases [3, 4, 5]. Chen et al. [6] discussed fuzzy queries process based on intuitionistic fuzzy social networks, De et al. [7] applied IFSs to medical decision via composite relation, Ejegwa and Onasanya [8] im- proved the intuitionistic fuzzy composite relation in [7] with ap- plication to medical diagnosis, and Liu and Chen [9] presented a group decision-making based on Heronian aggregation opera- tors of IFSs. Several information measures have been studied to 174 Ejegwa et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 174–182 175 enhance the application of IFSs in real-life problems. Ejegwa [10] proposed a new correlation coefficient of IFSs and applied the measure to solve multi-criteria decision-making (MCDM) problems. Many correlation coefficients of IFSs have been pro- posed and applied to several decision-making problems [11, 12, 13, 14, 15]. Medical diagnostic problems were solved based on intuitionistic fuzzy correlation coefficient as seen in [16, 17, 18]. Garg [19] presented a correlation coefficient under intu- itionistic multiplicative environment with application in decision- making process, and a new correlation coefficient of IFSs based on the connection number of set pair analysis was studied with application [20]. A robust technique of computing the corre- lation coefficients of complex IFSs and their applications in decision-making were discussed in [21]. TOPSIS method based on correlation coefficient under intuitionistic fuzzy soft sets was discussed and with application [22]. The idea of aggregation operators has been applied in many cases of decision-making [23, 24, 25, 26], and fuzzy soft set-valued maps has been intro- duced and applied to homotopy [27]. Similarly, the concepts of similarity and distance measures under intuitionistic fuzzy context have been discussed as re- liable information measures. Boran and Akay [28] proposed a biparametric similarity measure and applied the measure to pattern recognition. A similarity measure of IFSs based on transformation technique has been studied and applied to pat- tern recognition [29]. In [30, 31], some similarity measures of IFSs were introduced and applied to medical diagnostic rea- soning. In addition, some similarity measures based on dice and Jaccard approaches have been studied on expected intervals of trapezoidal neutrosophic fuzzy numbers with application to MCDM [32]. Burillo and Bustince [33] initiated the concept of distances for IFSs and interval-valued fuzzy sets. Szmidt and Kacprzyk [34] modified the distances in [33], and showed that all the three parameters describing IFSs should be taken into ac- count while calculating distances between IFSs. Hatzimichai- lidis et al. [35] introduced a novel distance measure between IFSs with application to cases of pattern recognition. Wang and Xin [36] proposed a novel distance measure between IFSs and its weighted version with application to the solution of pat- tern recognition problem, Davvaz and Sadrabadi [37] revised some existing distance measures and applied them to medical diagnostic process, and other applications of distance measures between IFSs have been studied [38, 39, 40, 41, 42]. Among the distances between IFSs studied in literature, dis- tances in [34] are prominent for their reliable interpretations. Albeit, these distances show some limitations which needed to be strengthened to enhance reliable outputs. Although Szmidt and Kacprzyk [34] modified the intuitionistic fuzzy distance measures in [33] with better rating, they do not take account the number of the considered parameters, but just added the hesitation margins to the methods introduced in [33]. This set- back adversely influence the performance rating of Szmidt and Kacprzyk’s distances [34]. The motivation for this work is to propose modified Szmidt and Kacprzyk’s distances which have better performance indexes compare to Szmidt and Kacprzyk’s distances taking into account the number of the considered pa- rameters to forestall inaccurate outputs. Specifically, the objec- tives of this paper are to (i) revisit Szmidt and Kacprzyk’s dis- tances between IFSs, (ii) propose modified versions of the dis- tances between IFSs in [34], (iii) apply the modified distances between IFSs to determine pattern recognition and disease di- agnosis, and (iv) present comparison analyses of the modified distances with the Szmidt and Kacprzyk’s distances in intuition- istic fuzzy domain. The paper is outlined as follow; Section 2 presents the concept of IFSs and discusses the Szmidt and Kacprzyk’s distances between IFSs [34], Section 3 introduces the modified Szmidt and Kacprzyk’s distances between IFSs, Section 4 discusses the applications of Szmidt and Kacprzyk’s distances and their modified versions in pattern recognition and disease diagnosis, and Section 5 concludes the paper with rec- ommendations for future studies. 2. Preliminaries This section presents the concept of IFSs and discusses the Szmidt and Kacprzyk’s distances between IFSs. 2.1. Intuitionistic fuzzy sets Numerous works on IFSs have been carried out [2, 43, 3]. Here, some basic concepts of IFSs are presented. Let us assume that Y is a non-empty set throughout this paper. Definition 2.1. [43] An intuitionistic fuzzy set C of Y is de- fined by C = {〈y,υC (y),νC (y)〉 : y ∈ Y}, where the functions υC, νC : Y → [0, 1] are the membership and non-membership degrees of y ∈ Y , and 0 ≤ υC (y) + νC (y) ≤ 1. For a IFS C in Y , $C (y) ∈ [0, 1] = 1 −υC (y) −νC (y) is the intuitionistic fuzzy index or hesitation margin of C. Definition 2.2. [3] Suppose C and D are IFSs in Y , then for all y ∈ Y we have (i) C = D iff υC (y) = υD(y), νC (y) = νD(y). (ii) C ⊆ D iff υC (y) ≤ υD(y), νC (y) ≥ νD(y). (iii) C = {〈y,νC (y),υC (y)〉 : y ∈ Y}. (iv) C ∪ D = {〈y, max(υC (y),υD(y)), min(νC (y),νD(y))〉 : y ∈ Y}. (v) C ∩ D = {〈y, min(υC (y),υD(y)), max(νC (y),νD(y))〉 : y ∈ Y}. Definition 2.3. [3] Intuitionistic fuzzy pair (IFP) is character- ized by the form 〈c, d〉 such that c + d ≤ 1 where c, d ∈ [0, 1]. IFP evaluate the IFS for which the components (c and d) are interpreted as membership and non-membership degrees. 2.2. Distances between intuitionistic fuzzy sets Distance measure is a soft computing tool use in the appli- cations of IFSs. The definition of distance measure between IFSs is a follows. Definition 2.4. [34] If C and D are IFSs of Y , then the distance between C and D denoted by φ(C, D) is a function φ: I FS × I FS → [0, 1] which satisfies (i) 0 ≤ φ(C, D) ≤ 1 175 Ejegwa et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 174–182 176 (ii) φ(C, D) = 0 iff C = D (iii) φ(C, D) = φ(D, C) (iv) φ(C, E) ≤ φ(C, D) + φ(D, E), where E is also an IFS of Y . When φ(C, D) reaches 0, it shows that C and D are more close or related. Again, if φ(C, D) reaches 1 then C and D are not related or close. For any two IFSs C and D in Y = {y1, · · · , yn}, we present the following distances between them. 2.2.1. Burillo and Bustince’s distances between IFSs By extending the distances between fuzzy sets as presented in [44], Burillo and Bustince [33] proposed the following dis- tances under intuitionistic fuzzy environment: φ1(C, D) = 1 2 Σ n i=1 ( |υC (yi) −υD(yi)| + |νC (yi) −νD(yi)| ) (1) φ2(C, D) = (1 2 Σ n i=1 ( (υC (yi)−υD(yi)) 2 +(νC (yi)−νD(yi)) 2))12 (2) φ3(C, D) = 1 2n Σ n i=1 ( |υC (yi)−υD(yi)|+ |νC (yi)−νD(yi)| ) (3) φ4(C, D) = ( 1 2n Σ n i=1 ( (υC (yi)−υD(yi)) 2 +(νC (yi)−νD(yi)) 2))12 (4) The denominator in Eqs. (1-4) depicts the number of parame- ters of IFSs considered similar to the approach in [44] where the denominator is unity because fuzzy set considers only member- ship function. The limitation of these approaches [33] is that the hesitation margin is not considered in the computations. 2.2.2. Szmidt and Kacprzyk’s distances between IFSs Because of the limitation in [33], Szmidt and Kacprzyk [34] proposed the some distances by incorporating hesitation margin of the considered IFSs. For simplicity sake, let υC (yi) = υC , νC (yi) = νC , $C (yi) = $C , υD(yi) = υD, νD(yi) = νD, $D(yi) = $D. The distances are as follow: φ̂1(C, D) = 1 2 Σ n i=1 ( |υC −υD| + |νC −νD| + |$C −$D| ) (5) φ̂2(C, D) = (1 2 Σ n i=1 ( (υC−υD) 2 +(νC−νD) 2 +($C−$D) 2))12 (6) φ̂3(C, D) = 1 2n Σ n i=1 ( |υC −υD| + |νC −νD| + |$C −$D| ) (7) φ̂4(C, D) = ( 1 2n Σ n i=1 ( (υC−υD) 2 +(νC−νD) 2 +($C−$D) 2))12 (8) Though the distances in [34] captured the three parameters of IFSs, the denominators in each equations do not depict the num- ber of parameters of the considered IFSs. These omissions will lead to information loss and thus, affect the interpretations. 3. Modified Szmidt and Kacprzyk’s Distances between IFSs To enhance reliable outputs and avoid information loss, we modified the Szmidt and Kacprzyk’s distances between IFSs [34] using the same hypotheses in Subsubsection 2.2.2 as fol- low: φ̃∗(C, D) = (1 3 Σ n i=1 ( |υC−υD| r +|νC−νD| r +|$C−$D| r))1r (9) φ̃∗∗(C, D) = ( 1 3n Σ n i=1 ( |υC−υD| r +|νC−νD| r +|$C−$D| r))1r (10) for r ≤ 2. If r = 1, we get φ̃1(C, D) = 1 3 Σ n i=1 ( |υC −υD| + |νC −νD| + |$C −$D| ) (11) φ̃2(C, D) = 1 3n Σ n i=1 ( |υC −υD|+ |νC −νD|+ |$C −$D| ) (12) If r = 2, we get φ̃3(C, D) = (1 3 Σ n i=1 ( (υC−υD) 2 +(νC−νD) 2 +($C−$D) 2))12 (13) φ̃4(C, D) = ( 1 3n Σ n i=1 ( (υC−υD) 2 +(νC−νD) 2 +($C−$D) 2))12 (14) Proposition 3.1. Suppose C and D are IFSs of Y = {y1, . . . , yn}, then we have (i) φ̃1(C, D) = nφ̃2(C, D) (ii) φ̃3(C, D) = √ nφ̃4(C, D). Proof. Given that φ̃2(C, D) = 1 3n Σni=1 ( |υC − υD| + |νC − νD| + |$C −$D| ) . Then it implies that φ̃2(C, D) = 1 3 Σni=1 ( |υC −υD| + |νC −νD| + |$C −$D| ) n = φ̃1(C, D) n . Hence (i) holds. Similarly, φ̃4(C, D) = ( 1 3n Σ n i=1 ( (υC −υD) 2 + (νC −νD) 2 + ($C −$D) 2))12 = (1 3 Σni=1 ( (υC −υD)2 + (νC −νD)2 + ($C −$D)2 ))12 √ n = φ̃3(C, D) √ n , and so (ii) holds. Proposition 3.2. If C and D are IFSs in Y , then the following hold: (i) φ̃∗(C, D) = φ̃∗(D, C) (ii) φ̃∗(C, D) = φ̃∗(C, D). 176 Ejegwa et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 174–182 177 Proof. For the proof of (i), we have φ̃∗(C, D) = (1 3 Σ n i=1 ( |υC −υD| + |νC −νD| + |$C −$D| ))1 r = (1 3 Σ n i=1 ( |υD −υC| + |νD −νC| + |$D −$C| ))1 r = φ̃∗(D, C). Hence (i) holds. The proof of (ii) is similar. Similarly, we have the following proposition. Proposition 3.3. Suppose C and D are IFSs in Y , then the fol- lowing hold: (i) φ̃∗∗(C, D) = φ̃∗∗(D, C) (ii) φ̃∗∗(C, D) = φ̃∗∗(C, D). Theorem 3.4. Suppose C, D and E are IFSs in Y , then the function φ̃∗(C, D) satisfies (i) 0 ≤ φ̃∗(C, D) ≤ 1 (ii) φ̃∗(C, D) = 0 iff C = D (iii) φ̃∗(C, D) = φ̃∗(D, C). Proof. The proof of (i) is straightforward. Recall that φ̃∗(C, D) = (1 3 Σ n i=1 ( |υC −υD| r + |νC −νD| r + |$C −$D| r))1r . To proof (ii), suppose that φ̃∗(C, D) = 0. Then |υC −υD| r = 0, |νC −νD| r = 0, and |$C −$D| r = 0, and thus, υC = υD, νC = νD and $C = $D, hence C = D. The converse is easy to see, so omitted. The proof of (iii) is the same as (i) of Proposition 3.2. From Theorem 3.4, we have the following result. Theorem 3.5. If C, D and E are IFSs in Y , then the function φ̃∗∗(C, D) satisfies (i) 0 ≤ φ̃∗∗(C, D) ≤ 1 (ii) φ̃∗∗(C, D) = 0 iff C = D (iii) φ̃∗∗(C, D) = φ̃∗∗(D, C). Theorem 3.6. Suppose C, D and E are IFSs in Y and C ⊆ D ⊆ E. Then we have (i) φ̃∗(C, E) ≥ φ̃∗(C, D), (ii) φ̃∗(C, E) ≥ φ̃∗(D, E), (iii) φ̃∗(C, E) ≥ max[φ̃∗(C, D), φ̃∗(D, E)]. Proof. If C ⊆ D ⊆ E, then |υC −υE|r ≥ |υC −υD|r , |νC −νE|r ≥ |νC −νD| r and |$C −$E|r ≥ |$C −$D|r . Thus |υC −υE| r + |νC −νE| r + |$C −$E| r ≥ |υC −υD| r + |νC −νD| r + |$C −$D| r. So, φ̃∗(C, E) ≥ φ̃∗(C, D), which proves (i). By the same logic we see that φ̃∗(C, E) ≥ φ̃∗(D, E), so (ii) holds. Combining (i) and (ii), it follows that φ̃∗(C, E) ≥ max[φ̃∗(C, D), φ̃∗(D, E)], which proves (iii). By consequence, we have Theorem 3.7. Theorem 3.7. Suppose C, D and E are IFSs in Y and C ⊆ D ⊆ E. Then we have (i) φ̃∗∗(C, E) ≥ φ̃∗∗(C, D), (ii) φ̃∗∗(C, E) ≥ φ̃∗∗(D, E), (iii) φ̃∗∗(C, E) ≥ max[φ̃∗∗(C, D), φ̃∗∗(D, E)]. 4. Experimental Examples In this section, we apply both the Szmidt and Kacprzyk’s distances and their modified versions to problems of pattern recognition and medical diagnosis to determine which of the approaches are better in terms of performance indexes. 4.1. Case I Pattern recognition is the process of identifying patterns by using machine learning method. The idea of pattern recognition is important because of its application potential in diverse areas. Assume there are three patterns P1, P2, P3 denoted with IFPs in Y = {y1, y2, y3}. If there is an unknown pattern Q denoted with IFP in the same feature space Y . The intuitionistic fuzzy representations of these patterns are in Table 1. Table 1: Intuitionistic fuzzy representations of patterns Feature space IFPs y1 y2 y3 υP1 νP1 $P1 0.1000 0.1000 0.8000 0.5000 0.1000 0.4000 0.1000 0.9000 0.0000 υP2 νP2 $P2 0.5000 0.5000 0.0000 0.7000 0.3000 0.0000 0.0000 0.8000 0.2000 υP3 νP3 $P3 0.7000 0.2000 0.1000 0.1000 0.8000 0.1000 0.4000 0.4000 0.2000 υQ νQ $Q 0.4000 0.4000 0.2000 0.6000 0.2000 0.2000 0.0000 0.8000 0.2000 Take SKD as Szmidt and Kacprzyk’s distance and MSK as Modified Szmidt and Kacprzyk’s distance. The task is to determine which of the patterns can the sam- ple Q be associated with. Table 2 contains the distances using the Szmidt and Kacprzyk’s distances while Table 3 contains the distances based on the modified Szmidt and Kacprzyk’s dis- tances. From Tables 2 and 3, the sample Q can be classified with pattern P2 since the distances between them are the smallest. Tables 2 and 3 can be represented by the following figures to show the superiority of the modified Szmidt and Kacprzyk’s distances over the Szmidt and Kacprzyk’s distances. 177 Ejegwa et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 174–182 178 Table 2: Results using Szmidt and Kacprzyk’s distances Distances (P1, Q) (P2, Q) (P3, Q) φ1 1.0000 0.4000 1.3000 φ2 0.3333 0.1333 0.4333 φ3 0.5745 0.2449 0.7348 φ4 0.3317 0.1414 0.4243 Table 3: Results using modified Szmidt and Kacprzyk’s distances Distances (P1, Q) (P2, Q) (P3, Q) φ̃1 0.6667 0.2667 0.8667 φ̃2 0.2222 0.0889 0.2889 φ̃3 0.4690 0.2000 0.6000 φ̃4 0.2708 0.1155 0.3464 Table 4: Intuitionistic fuzzy representations of diagnostic process Clinical manifestations IFSs s1 s2 s3 s4 s5 υV νV $V 0.4000 0.0000 0.6000 0.3000 0.5000 0.2000 0.1000 0.7000 0.2000 0.4000 0.3000 0.3000 0.1000 0.7000 0.2000 υM νM $M 0.7000 0.0000 0.3000 0.2000 0.6000 0.2000 0.0000 0.9000 0.1000 0.7000 0.0000 0.3000 0.1000 0.8000 0.1000 υT νT $T 0.3000 0.3000 0.4000 0.6000 0.1000 0.3000 0.2000 0.7000 0.1000 0.2000 0.6000 0.2000 0.1000 0.9000 0.0000 υS νS $S 0.1000 0.7000 0.2000 0.2000 0.4000 0.4000 0.8000 0.0000 0.2000 0.2000 0.7000 0.1000 0.2000 0.7000 0.1000 υC νC $C 0.1000 0.8000 0.1000 0.0000 0.8000 0.2000 0.2000 0.8000 0.0000 0.2000 0.8000 0.0000 0.8000 0.1000 0.1000 υP νP $P 0.6000 0.1000 0.3000 0.5000 0.4000 0.1000 0.3000 0.4000 0.3000 0.7000 0.2000 0.1000 0.3000 0.4000 0.3000 Table 5: Results using Szmidt and Kacprzyk’s distances Distances (V, P) (M, P) (T, P) (S, P) (C, P) φ1 1.4000 1.5000 1.9000 2.2000 2.7000 φ2 0.5568 0.6557 0.7810 0.9644 1.1091 φ3 0.2800 0.3000 0.3800 0.4400 0.5400 φ4 0.2490 0.2933 0.3493 0.4313 0.4960 4.2. Case II Diagnosis of diseases is challenging due to embedded fuzzi- ness in the processes. Here, we present a scenario of mathemat- ical approach of diagnosing a patient medical status using the Szmidt and Kacprzyk’s distance and the modified Szmidt and Table 6: Results using modified Szmidt and Kacprzyk’s distances Distances (V, P) (M, P) (T, P) (S, P) (C, P) φ1 0.9333 1.0000 1.2667 1.4667 1.8000 φ2 0.4546 0.5354 0.6377 0.7874 0.8200 φ3 0.1867 0.2000 0.2533 0.2933 0.3600 φ4 0.2033 0.2394 0.2852 0.3521 0.4050 Kacprzyk’s distance, where the symptoms or clinical manifesta- tions of the diseases are represented as IFPs using hypothetical case. Suppose we have a set of diseases namely; viral fever (V), malaria (M), typhoid fever (T), Stomach ulcer (S) and chest 178 Ejegwa et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 174–182 179 Figure 1: Distances using Eqs. (5) and (11) (P1 Q) (P2 Q) (P3 Q) 0.2 0.4 0.6 0.8 1 1.2 1.4 Classifications D is ta nc es SKD MSKD Figure 2: Distances using Eqs. (6) and (13) (P1 Q) (P2 Q) (P3 Q) 0.2 0.4 0.6 Classifications D is ta nc es SKD MSKD problem (C) represented by IFPs, and a set of symptoms S = {s1, s2, s3, s4, s5} where s1 = temperature, s2 = headache, s3 = stomach pain, s4 = cough, s5 = chest pain. These symptoms are the clinical manifestations of the mentioned diseases. Assume a patient P manifests symptoms as mentioned above, represented by IFPs. Table 4 contains intuitionistic fuzzy information of the diseases and patient P with respect to the symptoms. Now, we find which of the diseases has the smallest distance with the patient with respect to the symptoms by deploying the Szmidt and Kacprzyk’s distances and their modifications. Ta- Figure 3: Distances using Eqs. (7) and (12) (P1 Q) (P2 Q) (P3 Q) 0.1 0.2 0.3 0.4 Classifications D is ta nc es SKD MSKD Figure 4: Distances using Eqs. (8) and (14) (P1 Q) (P2 Q) (P3 Q) 0.1 0.2 0.3 0.4 Classifications D is ta nc es SKD MSKD bles 5 and 6 contain the results. From Tables 5 and 6, it is inferred that the patient is suffering from viral fever since the distance between the patient and viral fever is the smallest. Tables 5 and 6 can be represented by the following figures to show the supremacy of the modified Szmidt and Kacprzyk’s distances over the Szmidt and Kacprzyk’s dis- tances. From Figs. 1–8, it is observed that the modified Szmidt and Kacprzyk’s distances outperformed the Szmidt and Kacprzyk’s distances in terms of accuracy because while the modified Szmidt 179 Ejegwa et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 174–182 180 Figure 5: Distances using Eqs. (5) and (11) (V P) (MP) (T P) (S P) (CP) 1 1.5 2 2.5 Classifications D is ta nc es SKD MSKD Figure 6: Distances using Eqs. (6) and (13) (V P) (MP) (T P) (S P) (CP) 0.4 0.6 0.8 1 Classifications D is ta nc es SKD MSKD and Kacprzyk’s distances take cognizance of the average of the differences among the three parameters of IFSs, the Szmidt and Kacprzyk’s distances do not consider the average of the differ- ences. Since distance measure lies between 0 and 1, we con- clude that φ1, φ2, φ̃1 and φ̃2 are not reliable distances. 5. Conclusion In this paper, we have studied the Szmidt and Kacprzyk’s distances between IFSs and noticed a setback with the distance Figure 7: Distances using Eqs. (7) and (12) (V P) (MP) (T P) (S P) (CP) 0.2 0.3 0.4 0.5 Classifications D is ta nc es SKD MSKD Figure 8: Distances using Eqs. (8) and (14) (V P) (MP) (T P) (S P) (CP) 0.2 0.3 0.4 0.5 Classifications D is ta nc es SKD MSKD measures. Because of this setback, modifications of the Szmidt and Kacprzyk’s distances between IFSs were proposed to en- hance accuracy of measure. It was verified mathematically that the modified Szmidt and Kacprzyk’s distances between IFSs satisfied the conditions for distance measure. Both the Szmidt and Kacprzyk’s distances and their modified versions were ap- plied to determine pattern recognition and diagnostic medical reasoning where information were represented as IFPs. From the work, it is observed that the modified Szmidt and Kacprzyk’s distances outperformed the Szmidt and Kacprzyk’s distances 180 Ejegwa et al. / J. Nig. Soc. Phys. Sci. 4 (2022) 174–182 181 in terms of accuracy because while the modified Szmidt and Kacprzyk’s distances take account of the number of the con- sidered parameters, the Szmidt and Kacprzyk’s distances just added the hesitation margins to the methods introduced in [33]. 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