Despite the fact that several technologies have been developed to assist healthcare workers in reducing errors and improving accuracy in illness diagnosis, there is still substantial ambiguity regarding the accurate disease diagnosis based on symptoms. The goal of this work is to establish a group decision-making problem in an uncertain situation to assist medical practitioners in generating accurate illness predictions based on symptoms. This study proposes a novel distance measure for Pythagorean fuzzy sets that incorporates the inherent uncertainty of complex, uncertain data by incorporating indeterminacy in the computation. First, we establish the proposed Mabala distance measure by describing it’s properties. Then, the suggested distance measure is applied to solve group decision-making problems in uncertain situations. A case study of disease analysis based on symptoms is presented to illustrate the decision-making procedure involving four medical professionals, five symptoms, and five probable diseases. Furthermore, We have presented two cases of disease analysis using non-standard and standard Pythagorean fuzzy soft matrices. The results suggest that the proposed Mabala’s distance measure has great potential for improving disease analysis. The proposed Mabala distance measure is compared to five existing distance measures using an identical data set of prospective disease symptoms. The comparative analysis indicates that the suggested Mabala distance measure’s result almost coincides with the results of the other distance measurements. A set of sensitivity analysis is provided to analyze the durability and consistency of the proposed distance measurements across different input scenarios.
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Introduction
The problem of medical diagnosis involves identifying and determining the underlying health condition or disease affecting an individual based on a comprehensive analysis of symptoms, medical history, diagnostic tests, expertise, clinical judgment, and evidence-based guidelines [2, 18, 23]. It requires healthcare professionals to make accurate and informed judgments through meticulous analysis of patient symptoms, medical history, and advanced diagnostic tests to provide appropriate treatment and care. The integration of cutting-edge technology, such as medical imaging and genetic profiling, contributes to accurate disease identification. Collaborative consultations among multidisciplinary medical teams and adherence to evidence-based guidelines further refine the precision of medical diagnoses. Wrong or partially wrong medical diagnoses arise from incomplete or inaccurate patient information, inadequate communication between healthcare providers, and misinterpretation of test results. Inaccuracies can also happen because of cognitive biases, the fact that a condition may not show up in the usual way or very often, or because of a lack of time during the diagnosis [47]. Lack of access to advanced diagnostic tools [46], suboptimal follow-up, and insufficient experience among healthcare professionals may also play a crucial role. To cut down on diagnostic mistakes and improve patient outcomes, these many factors need to be dealt with through better communication, thorough training, better use of technology, and systemic safeguards.
Effective decision-making should account for ethical and legal considerations, human cognitive constraints, and changing circumstances. In the field of decision-making methods, several challenges and limitations are associated with the existing methods, such as subjectivity and biases. These challenges and limitations can affect the effectiveness and efficiency of decision-making processes in various domains, such as product development, project management, and policy evaluation. Addressing them can be challenging due to limited data quality and availability. Efficient data collection and analysis can overcome limitations in decision-making. However, multiple-variable problems are still challenging to handle with existing methods. Effective models for such complexity remain an ongoing challenge due to their limited scalability to handle a vast number of options or variables. An algorithmic approach, such as artificial intelligence and machine learning, may be an option to make illness prediction systems more accurate and dependable [16, 35]. Uncertainty developments have improved the accuracy of disease diagnosis by assessing and predicting the correct disease using real-time data. Uncertainty and risk need to be dealt with using probabilistic and risk assessment methods, but cognitive limitations can make decisions less effective [23]. Researchers and practitioners continuously develop and refine techniques, leverage advanced technologies, and emphasize interdisciplinary collaboration to address these challenges.
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Pythagorean fuzzy sets (PFSs) are a modification of fuzzy sets that permit the representation of ambiguity and imprecision in medical data, making disease diagnosis flexible and comprehensive. However, the use of PFSs in disease analysis requires an appropriate distance measure that can capture the complex relationships between medical variables. This study introduces a new distance measure for PFSs that considers the membership (MS), non-membership (NMS) and hesitancy grades of each variable. We have presented two cases of disease analysis algorithms on Pythagorean fuzzy soft matrices (PFSMs). The use of fuzzy logic in disease diagnosis can improve accuracy by accommodating uncertainty and imprecision. Fuzzy inference systems can integrate ambiguous medical data, patient histories, and expert knowledge to generate more nuanced diagnostic outcomes. The MS function of PFS can represent varying degrees of severity of symptoms, aiding in better classification. Fuzzy clustering techniques can identify patterns in complex patient data, improving disease categorization [56]. Fuzzy decision support systems can assist healthcare professionals in making well-informed diagnostic choices, considering multiple factors. Using fuzzy logic’s adaptability makes the diagnostic process stronger and better able to handle uncertain medical situations. This makes disease diagnosis more accurate [10]. The utilization of fuzzy logic through an expert system-based approach has proven to effectively address the challenges posed by uncertainty and imprecision in medical data [33].
The multi-criteria decision-making (MCDM) techniques systematically prioritize diagnostic criteria based on relative importance, aiding structured decision-making. Multi-criteria group decision-making (MCGDM) promotes collaborative decisions [12], lessening the impact of individual imprecision. Crisp MCDM approaches fall short of adequately addressing this issue due to their inability to handle the inherent uncertainty and imprecision present in medical diagnosis. Medical data often encompasses intricate relationships and varying degrees of relevance, which crisp methods struggle to capture effectively. The deterministic MCDM [28] lacks the flexibility to accommodate ambiguous or conflicting information, leading to oversimplified and potentially inaccurate results. On the contrary, the complex nature of disease diagnosis requires a more nuanced approach that can handle fuzzy and uncertain inputs [15], making fuzzy MCDM a more suitable choice to improve accuracy and decision-making in this context. Therefore, it is necessary to incorporate fuzzy MCDM or MCGDM processes into medical data analysis to improve the accuracy of treatment. In this regard, we proposed a novel distance measure to analyze medical data using the group decision-making (GDM) process.
To facilitate tracking, this study is structured into the following divisions: “Literature survey”, we provide an updated overview through a literature study. “Materials and developments” delves into fuzzy sets and fuzzy logic concepts, exploring recent extensions like PFSs and their novel operations, along with the introduction of a novel distance function for the Pythagorean fuzzy framework. “Disease analysis algorithm” introduces a new MCGDM algorithm, leveraging standardization for analyzing medical diagnostic reports. Approaches for solving MCGDM algorithms, along with a case study applying the proposed algorithm, are discussed in “Medical case study: disease analysis”. “Distance function comparison” elaborates on various solutions for existing distance functions, examining the proposed distance function’s sensitivity as well as solutions for existing and proposed MCGDM techniques. Finally, “Conclusion” concludes this work with some future research directions. The general outline of this research has been figured out in Fig. 1.
Fig. 1
Schematic representation of a patient’s symptom-based decision analysis
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Literature survey
This section presents an updated review that aims to identify the research gap in the current literature. In light of this, this work’s contribution is provided.
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Fuzzy MCDM approach in medical diagnosis
The idea behind fuzzy sets [69] is to allow for a more nuanced representation of reality where objects or concepts can have varying MS degrees in a set. Atanossov [5, 6] introduced and investigated intuitionistic fuzzy sets (IFSs), which take into account their MS (\(\tilde{\eta }\)) and NMS (\({\tilde{\Upsilon }}\)) degrees, with the sum of these two numbers being less than or equal to one. Subsequently, Yager introduced the PFSs [65], which was a significant step forward in this regard. The PFSs offer a broader range of possibilities as they allow for an increased admissible area. Specifically, the constraint that the total of the squares is restricted to one provides greater flexibility in modeling real-world situations and allows for more nuanced analysis.
In medical data analysis, it is frequently encountered that the classification of disease becomes a complex task due to the inherent imprecision in symptoms, test results, or patient information. Fuzzy sets may improve medical diagnosis [13, 14, 34, 68, 75] by examining various symptoms and their degrees of MS in distinct illness categories, allowing for a more complete examination of a patient’s health when dealing with overlapping symptoms. Fuzzy logic has been used to classify the risk factors for contracting a certain illness, evaluate the efficacy of treatments with confusing outcomes, anticipate the progress of an illness, and assess the probability of recovery or further deterioration. Here, we have reviewed some related applications of fuzzy decision-making in medical diagnosis. Chen et al. [11] applied interval type-2 fuzzy numbers to an MCDM technique to address a medical decision-making issue involving acute inflammatory demyelinating disease. Lin et al. [41] developed a directional correlation coefficient between PFSs for medical diagnosis. Collaborative decision-making [67] within the fuzzy MCDM framework ensures well-informed and consensus-driven criteria rankings. By employing fuzzy sets and rules, this approach enhances the accuracy of diagnosis and treatment decisions. Fuzzy clustering techniques for medical image analysis are suggested by Huang et al. [26] in an intuitionistic fuzzy environment. Kumar et al. [66] presented fuzzy decision support systems in medical diagnosis and highlighted the benefits of fuzzy logic-based approaches in handling uncertainty and vagueness in medical data, facilitating more accurate and personalized disease analysis. The fuzzy data mining technique by Zheng et al. [74] is also one decision-making technique to analyze the data and make decisions in different areas, especially for disease analysis. The flexibility offered by fuzzy MCDM in assigning weights to various diagnostic criteria [49], allows for a comprehensive evaluation of the importance of criteria, refining the ranking process.
Recent advancements in fuzzy methods have emerged as effective strategies to address the challenges inherent in medical diagnoses. Fuzzy clustering techniques have gained prominence, with cutting-edge algorithms like fuzzy C-means and Possibilistic C-means [3] being applied to meticulously uncover subtle patterns and relationships concealed within imprecise medical data. By discerning these intricate connections, these algorithms contribute significantly to the precise categorization of diseases, thereby enhancing diagnostic accuracy. Modern medical diagnostics are benefiting from the integration of advanced fuzzy inference systems, including the adoption of type-2 fuzzy logic systems [29]. The fusion of fuzzy logic into deep learning frameworks empowers us to adeptly handle the challenges posed by uncertain and vague medical data [27]. By imbuing deep learning with the capability to navigate such data intricacies, disease prediction and diagnosis are significantly improved.
Distance function approach in decision-making
Distance measures provide a method for quantifying the similarity or dissimilarity between fuzzy sets or fuzzy numbers [20]. Several fuzzy distance measures and score functions continue to be employed to address diagnostic concerns and enhance accuracy. Despite the fact that medical data is not always accurate, fuzzy similarity measures like fuzzy Euclidean distance, fuzzy Hamming distance, and fuzzy Jaccard similarity make it possible to quantify the similarity between patient profiles and disease patterns. The utilization of these distance measures has garnered significant attention from scholars owing to their extensive applications in diverse domains, including decision-making [9, 73], medical diagnosis [17, 32, 52, 57, 59], pattern recognition [44, 63], clustering and data analysis [22, 54], and control systems [8].
Pythagorean MCDM and MCGDM approach in medical diagnosis
The PFSs have been used in a variety of fields, leading to improved outcomes in decision-making for medical diagnosis. Table 1 provides an overview of recent Pythagorean MCDM and MCGDM techniques employed in various domains, with a specific emphasis on medical diagnostics.
Table 1
Recent implementation of Pythagorean multi-criteria decision-making approaches
The literature review reveals that most distance functions for PFSs are derived from their IFS counterparts. Only a few of these functions have undergone normalization, allowing them to be reintroduced with appropriate adjustments. The GDM is tough in the face of current and forthcoming problems. Table 2 shows the abbreviations and symbols used in this study, which will be helpful in understanding the work.
Table 2
List of abbreviations and symbols with their expansions
Abbreviations
Symbols
Notation
Expansion
\(\tilde{\pi }_{\wp }(r)\)
Indeterminacy value of an element r in the PFS \({\tilde{\wp }}\)
NMS scalar cardinality of parameter e in PFS \(\rho _A\)
\(\textrm{HD}_2\)
Hamming distance 2
\({\textbf {D}}(\tilde{\wp _1}, \tilde{\wp _2})\)
Proposed Mabala’s distance between \(\tilde{\wp _{1}}\) and \(\tilde{\wp _{2}}\)
\(\textrm{ED}_1\)
Euclidean distance 1
\(\sigma _{\eta ,j}\)
MS variance of PFSM of order \(m \times n\) corresponding to a parameter \(e_j\)
\(\textrm{ED}_2\)
Euclidean distance 2
\({\sigma _{\Upsilon ,j}}\)
NMS variance of PFSM of order \(m \times n\) corresponding to a parameter \(e_j\)
\({{\tilde{\varrho }}}_{\varsigma }\)
Standard form of a PFSM \({{\tilde{\varrho }}}\)
G
Universe of possible diagnoses for health problems
Symbols
S
Set of symptoms (parameters)
Notation
Expansion
H
Set of physicians (DMs)
\(\mathfrak {R}\)
Universe of discourse
\(\tilde{\varrho _i}\)
Opinion PFSM of Physician \(p_i\)
\({\tilde{\wp }}\)
PFS
\(\textrm{CM}p_i\)
Choice matrix of physician \(p_i\)
\( \tilde{\eta _{\wp }}(r)\)
MS value of an element r in the PFS \({\tilde{\wp }}\)
\(W(d_i)\)
MS weights of disease \(d_i\)
\( \tilde{\upsilon _{\wp }}(r)\)
NMS value of an element r in the PFS \({\tilde{\wp }}\)
\({{\tilde{\varrho }}}_{{i}(\varsigma )}\)
Standardized opinion PFSM of physician \(p_i\)
Research gap and contributions
However, the complex mathematical formulations of certain distance measures impede their practicality and hinder their adoption as a convenient mathematical technique. Furthermore, the existing measures lack the ability to distinguish highly uncertain PFSs, which exhibit extremely low MS and NMS grades. These PFSs often occur in situations where there is limited knowledge or information about a system. Consequently, there exists a pressing need for a suitable distance measure specifically designed to address these types of PFSs. To bridge these gaps, we developed a novel distance function for PFSs, that boasts a simplified mathematical form. This novel function demonstrates effectiveness in handling highly uncertain PFSs while remaining equally efficient in other scenarios. This article makes the following contributions in light of the identified research gaps:
A novel distance measure is proposed for determining the distance between two PFSs.
The concept of standardizing PFSMs is introduced as an alternative to the conventional normalization approach.
A Pythagorean fuzzy GDM framework is developed to present an algorithm aimed at effectively evaluating disease symptoms.
A case study has been selected to illustrate the practical use of the proposed distance function.
The proposed standardization-based algorithm decisions are compared with normalization-based algorithm decisions.
Sensitivity analyses were carried out on both proposed and existing distance measures by systematically varying parameters.
Materials and developments
The base materials for constructing this article are defined in this section.
Definition 1
Pythagorean fuzzy set (PFS): [65] The PFS \(\tilde{\wp }\) describes in \(\mathfrak {R} \ne \emptyset \) as a structure
within which the capabilities of the functions \({\tilde{\eta }}_{\wp }(r):\mathfrak {R} \rightarrow [0,~1]\) and \({\tilde{\Upsilon }}_{\wp }(r):\mathfrak {R} \rightarrow [0,1]\), indicate the level of MS and NMS for every component \(r \in \mathfrak {R}\) to the set \(\wp \) respectively, and \(0 \le (\tilde{\eta }_{\wp }(r))^2+({\tilde{\Upsilon }}_{\wp }(r))^2 \le 1\), for all \(r \in X\). For any PFS \(\wp \) and \(r \in \mathfrak {R}\),
is can explaining a indeterminacy of r to \(\wp \).
Let the discourse universe and the set of parameters be \(\mathfrak {R}=\{r_{1},~r_{2},~...,~r_{m}\}\) and \(E=\{e_{1},~e_{2},~...,~e_{n}\}\), respectively. Let \(A \subseteqq E\) then, soft set, fuzzy soft set (FSS), and Pythagorean fuzzy soft sets (PFSS) are defined as follows:
Definition 2
Soft set: [45] Let \(C_{A}\) be a mapping defined by \(C_{A}: A \rightarrow P(\mathfrak {R})\), where \(P(\mathfrak {R})\) denotes the power set of X. Then the pair \((C_{A},~E)\) is called a soft set over \(\mathfrak {R}\).
Definition 3
Fuzzy soft set (FSS): Let \(F_{A}\) be a mapping defined by \(F_{A}: A \rightarrow F(\mathfrak {R})\), where \(F(\mathfrak {R})\) denotes the set of all fuzzy sets over \(\mathfrak {R}\). Then, the pair \((F_{A},~E)\) is called a FSS over \(\mathfrak {R}\).
Definition 4
Pythagorean fuzzy soft set (PFSS): [50] Let \(\rho _{A}\) be a mapping defined by \(\rho _{A}: A \rightarrow \textrm{PFS}(\mathfrak {R})\) where \(\textrm{PFS}(\mathfrak {R})\) denotes the set of all PFSs over \(\mathfrak {R}\). Then, the pair \((\rho _{A},~E)\) is called a PFSS over \(\mathfrak {R}\).
Definition 5
Cardinalities of a PFSS: [32] For all \(e \in E\), the scalar cardinalities \({({\tilde{\eta }})}_{\textrm{SC} \rho _A}(e)\) and \({(\tilde{\Upsilon })}_{\textrm{SC} \rho _A} (e)\) of a PFSS \((\rho _A,~E)\) is defined as,
Scalar multiplication of PFSS: [24] Let \({\tilde{\varrho }}\) = \(\left[ {{\varrho }}(\tilde{\eta }_{ij}),~{{\varrho }}({\tilde{\Upsilon }}_{ij})\right] _{m \times n}\) be a PFSS. The scalar multiplication is defined as
Pythagorean fuzzy soft matrix (PFSM): Let \((\rho _{A},~E)\) be a PFSS over \(\mathfrak {R}\). Let \(M_A\) be a set defined by \(M_{A}=\{(r_{i},~e_{j}): r_{i} \in \mathfrak {R},~e_{j} \in A\}\). \(M_{A}\) can be characterized by its MS \({\tilde{\eta }}_{M_{A}}:~\mathfrak {R} \times E \rightarrow [0,~1]\) and NMS \({\tilde{\Upsilon }}_{M_A}: \mathfrak {R} \times E \rightarrow [0,~1]\) defined respectively by
If \({\tilde{\varrho }}(\eta _{ij})={\tilde{\eta }}_{M_A} (r_{i},~e_{j})\) and \({\tilde{\varrho }}({\tilde{\Upsilon }}_{ij})={\tilde{\Upsilon }}_{M_A} (r_{i},~e_{j})\) then, \({\tilde{\varrho }}=[{\tilde{\varrho }}_{ij}]=\left[ {{\varrho }}({\tilde{\eta }}_{ij}),~{{\varrho }} ({\tilde{\Upsilon }}_{ij})\right] _{m \times n}\) is called the PFSM over \(\mathfrak {R}\).
Definition 8
Addition of PFSMs: [32] Let \(\tilde{\varrho _1}\) = \(\left[ {{\varrho _1}}(\tilde{\eta }_{ij}),~{{\varrho _1}}({\tilde{\Upsilon }}_{ij})\right] _{m \times n}\) and \(\tilde{\varrho _2}\) = \(\left[ {{\varrho _2}}(\tilde{\eta }_{ij}),~{{\varrho _2}}({\tilde{\Upsilon }}_{ij})\right] _{m \times n}\) be two PFSMs over \(\mathfrak {R}\). Then the sum of \(\tilde{\varrho _1}\) and \(\tilde{\varrho _2}\) is defined as
Equality of PFSMs: [24] Let \(\tilde{\varrho _1}\) = \(\left[ {{\varrho _1}}(\tilde{\eta }_{ij}),~{{\varrho _1}}({\tilde{\Upsilon }}_{ij})\right] _{m \times n}\) and \(\tilde{\varrho _2}\) = \(\left[ {{\varrho _2}}(\tilde{\eta }_{ij}),~{{\varrho _2}}({\tilde{\Upsilon }}_{ij})\right] _{m \times n}\) be two PFSMs over \(\mathfrak {R}\). Then, \(\tilde{\varrho _1}\) and \(\tilde{\varrho _2}\) are said to be equal if \({{\varrho _1}}(\tilde{\eta }_{ij})={{\varrho _2}}({\tilde{\eta }}_{ij})\) and \({{\varrho _2}}({\tilde{\Upsilon }}_{ij})={{\varrho _2}}(\tilde{\Upsilon }_{ij})~~\forall \) i and j.
Definition 10
Max-min product of PFSMs: [32] Let \({\tilde{\varrho _1}}=\left[ {{\varrho _1}}(\tilde{\eta }_{ij}),~{\varrho _1}({\tilde{\Upsilon }}_{ij})\right] _{m \times n}\) and \({\tilde{\varrho _2}}\) = \(\left[ {{\varrho _2}}({\tilde{\eta }}_{ij}),~{\varrho _2}(\tilde{\Upsilon }_{ij})\right] _{m \times n}\) be two PFSMs over \(\mathfrak {R}\). Then the Max-min product of \({\tilde{\varrho _1}}\) and \({\tilde{\varrho _2}}\) is defined as
Choice matrix: [32] A choice matrix (CM) is a square matrix in which both rows and columns represent the parameter set’s elements.
The \((i,j)\textrm{th}\) element of \({CM}= {\left\{ \begin{array}{ll} (1,~0),~\text {if}~i\textrm{th}~\text {and}~j\textrm{th} \text { are choice parameters for DM} \\ (0,~1),~\text {if}~i\textrm{th}~\text {or}~j\textrm{th} \text { parameters be a choice of DM} \end{array}\right. }\).
A combined CM for GDM process differs from the CM if the elements of the rows in the choice parameter set (CPS) represent the concerned DM and the elements of the columns represent the remaining DMs.
Definition 12
Hamming distance 1 (\(\textrm{HD}_{1}\)): [64] Let \(\mathfrak {R}=\{r_{1},~r_{2},~...,~r_{m}\}\) be a given universe of discourse. The \(\textrm{HD}_{1}\) between any two PFSs \(\tilde{\wp _1}\) and \(\tilde{\wp _2}\) on \(\mathfrak {R}\) is defined as,
Hamming distance 2 (\(\textrm{HD}_{2}\)): [19] Let \(\mathfrak {R}=\{r_{1},~r_{2},~...,~r_{m}\}\) be a given universe of discourse. The \(\textrm{HD}_{2}\) between any two PFSs \(\tilde{\wp _{1}}\) and \(\tilde{\wp _{2}}\) on \(\mathfrak {R}\) is defined as,
Euclidean distance 1 (\(\textrm{ED}_{1}\)): [64] Let \(\mathfrak {R}=\{r_{1},~r_{2},~...,~r_{m}\}\) be a given universe of discourse. The \(\textrm{ED}_{1}\) between any two PFSs \(\tilde{\wp _1}\) and \(\tilde{\wp _2}\) on \(\mathfrak {R}\) is defined as,
Euclidean distance 2 (\(\textrm{ED}_{2}\)): [19] Let \(\mathfrak {R}=\{r_{1},~r_{2},~...,~r_{m}\}\) be a given universe of discourse. The \(\textrm{ED}_{2}\) between any two PFSs \(\tilde{\wp _1}\) and \(\tilde{\wp _2}\) on \(\mathfrak {R}\) is defined as,
Juthika’s distance: [44] Let \(\mathfrak {R}=\{r_{1},~r_{2},~..., r_{m}\}\) be a given universe of discourse. The Juthika’s distance between any two PFSs \(\tilde{\wp _1}\) and \(\tilde{\wp _2}\) on \(\mathfrak {R}\) is defined as,
Introducing a novel distance metric within a Pythagorean fuzzy framework unveils a fresh perspective on the assessment of similarity. This innovative approach incorporates the inherent uncertainty and ambiguity in data, enabling a more accurate representation of complex relationships. By harnessing the principles of PFSs, the proposed distance measure offers a refined tool for quantifying intricate nuances between elements. This advancement promises to enhance decision-making processes and refine pattern recognition tasks in real-world applications. The prior distance functions [44] only included the MS and NMS grades; however, our proposed distance function included an indeterminacy function as well. The following is a description of the proposed distance function:
Definition 17
Let \(\mathfrak {R}=\{r_{1},~r_{2},~...,~r_{m}\}\) be a given universe of discourse. The proposed Mabala’s distance function to find the distance of any two PFSs \(\tilde{\wp _{1}}\) and \(\tilde{\wp _{2}}\) on \(\mathfrak {R}\) is defined as
Let \(\tilde{\wp _l},~\tilde{\wp _m},~\tilde{\wp _n}\) be three PFSs describing the set \(\Re \ne \emptyset \). The proposed Mabala’s distance satisfies the following properties.
(i)
Non-negativity and being in [0, 1]: \(0 \le D(\tilde{\wp _l},~\tilde{\wp _m}) \le 1~\forall ~\tilde{\wp _l},~\tilde{\wp _m}\)
(ii)
Necessity and sufficiency for equality to 0 and separability: \(D(\tilde{\wp _l},~\tilde{\wp _m})=0~\text {iff}~\tilde{\wp _l}=\tilde{\wp _m}\)
Thus, \(D(\tilde{\wp _l},~\tilde{\wp _m})\) is a non-negative quantity. Therefore, \(0 \le D(\tilde{\wp _l},~\tilde{\wp _m}) \le 1\). Since, \(\tilde{\wp _l}\) and \(\tilde{\wp _m}\) are arbitrary, therefore the above equality is true for all pairs of PFSs.
(ii) Let \( \tilde{\wp _l}=\tilde{\wp _m}\), then the result is trivial.
Conversely, let’s assume that \(D(\tilde{\wp _l},~\tilde{\wp _m})=0\)
The standardization process makes it easier to improve the results of PFSMs with dynamic parameters. Symptoms are often more dynamic in the research of medical diagnosis. Therefore, it is necessary to consider the MS and NMS values of all other diseases to determine the probable origin of each symptom. The standardization procedure helps the researchers do a more thorough analysis by reducing the degree of fuzziness in these data.
Disease analysis algorithm
Let there be a patient who is experiencing multiple health problems. Let \(G=\{d_{1},~d_{2},~\dots ,~d_{m}\} \subseteqq \mathfrak {R}\) represents the universe of possible diagnoses for these health problems. Consider the n symptoms, denoted as \(S=\{s_{1},~s_{2},~\ldots ,~s_{n}\}\) represents the precise medical condition that the patient is experiencing and serves as the parameter set. Let k physicians be represented as \(H=\{p_{1},~p_{2},~\dots ,~p_{k}\}\), who serve as the DMs in this GDM process. Each physician has their own unique set of selection criteria (CPS). The objective is to identify the disease with the highest degree of similarity to the symptoms in the CPSs that best represent the opinions of the physicians. We have examined this objective for two types of PFSMs: non-standard and standard.
CASE 1: non-standard PFSMs
Step 1:
Record the physician’s opinions (H) regarding diseases (G) related to the symptoms (S) of the patient in PFSMs.
Step 2:
Combining the CPS of the respective physicians with those of the remaining physicians yields the combined CMs for each physician.
Step 3:
Determine the max-min product of the non-standard PFSMs from step 1 and the corresponding CMs from step 2.
Step 4:
Combine all the matrices obtained in step 3.
Step 5:
Compute the MS weight of every disease by adding the MS grades of the entries in the row corresponding to that disease in the matrix of step 4. The disease with the greatest MS weight is the one with the highest probability. If a unique disease is obtained, the procedure concludes; otherwise, it continues with step 6.
Step 6:
Compute the NMS weight of every disease by adding the MS grades of the entries in the row for that disease in step 4’s matrix. The disease with the lowest NMS weight is the best choice among those with equal MS weights. If a unique disease is obtained, the procedure concludes; otherwise, the process continues with step 7.
Step 7:
Repeat the same procedure with indeterminacy weights in place of NMS weights to determine the disease with the highest probability. If the equivocal situation persists after applying indeterminacy weights, then any disease with the lowest indeterminacy weight is the optimal disease.
CASE 2: standard PFSMs
Step 1:
Construct the PFSMs based on the physician’s opinions (H) regarding the diseases (G) associated with the patient’s symptoms (S).
Step 2:
Derive the combined CMs for each physician from their CPS by adding the CPS of the respective physicians with those of the remaining physicians.
Step 3:
Use Eq. (12), to standardize the PFSMs regarding the opinions of physicians obtained in step 1.
Find the max-min product of the standard PFSMs obtained in step 3 and the corresponding CMs in step 2.
Step 5:
Compute the combined value of all the matrices in Step 3.
Step 6:
The best disease to choose is the one with the highest MS weight. The MS weight of a disease is equal to the sum of the MS grades of all the entries in the row for that disease in the matrix from Step 5. The process stops here when a unique result is obtained. Otherwise, it proceeds with step 6.
Step 7:
Find the sum of the NMS grades of each disease from the combined matrix obtained in step 5. The most preferable disease with the same MS weight is the disease with the least NMS weight. Stop the algorithm if a unique disease has been identified; otherwise, proceed to the next step.
Step 8:
Repeat the procedure for indeterminacy weights to break the tie. If the equivocal situation still exists even after indeterminacy weight, then any disease with the least indeterminacy weight is the most preferable choice.
The proposed distance function-based algorithms are explained in Fig. 2.
Fig. 2
Network flow of the proposed algorithm
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The proposed decision-making methodology offers a novel approach to addressing complex GDM problems, providing distinct advantages in terms of efficiency and effectiveness. However, like any methodology, it comes with its own set of challenges and limitations that must be carefully considered for informed DMs. Table 3 outlines the advantages and disadvantages of the proposed methodology, highlighting areas for improvement and its applicability.
Table 3
Advantages and disadvantages of proposed methodology
Advantages
Disadvantages
The CM generated data for DMs and implemented the GDM process by discrediting CPs that other DMs had initially disregarded
The construction of such a CM is a tiresome process
This standardization concept necessitates the use of PFSMs to enhance the precision of results in dynamic situations; this methodology is suitable for GDM
This algorithm is exclusively applicable to systems characterized by a higher degree of dynamism and is not viable for implementation in other system types
It demonstrates proficiency in integrating the judgments of numerous DMs within a dynamic setting, thereby managing unpredictability
This method disregards other alternatives and fails to account for minor human errors, despite its close relation to the result
Medical case study: disease analysis
A patient is experiencing multiple health conditions, including temperature, headache, body pain, cough, and fatigue. Four hospital physician’s clinical data are collected, and these data are used to diagnose the disease based on the patient’s symptoms. Based on their medical knowledge and clinical observations so far, the likely diseases that the patient might be suffering from are malaria, viral fever, typhoid, chikungunya, and dengue. Let \(G=\{d_{1},~d_{2},~d_{3},~d_{4},~d_{5}\}\) be a set of diagnoses that represent malaria, viral fever, typhoid, chikungunya, and dengue, respectively. Let \(S=\{s_{1},~s_{2},~s_{3},~s_{4},~s_{5}\}\) be a set of symptoms that represent temperature, headache, body pain, cough, and fatigue, respectively. Let \(H=\{p_{1},~p_{2},~p_{3},~p_{4}\}\) be a group of four physicians, each carrying out the diagnosis of diseases by considering their own choice of parameters (i.e., symptoms), which are the CPS here. Each Physician has their own opinions on ‘these diseases being related to these symptoms’ according to their medical knowledge and observations so far, which can be modelled by PFSSs, \(({\rho _{1}},~S)\), \(({\rho _{2}},~S),\)\(({\rho _{3}},~S),\) and \(({\rho _{4}},~S)\) for physicians \(p_{1},~p_{2},~p_{3},\) and \(p_{4}\) respectively, where \({\rho _{1}},~{\rho _{2}},~{\rho _{3}},\) and \({\rho _{4}}\) are mappings defined respectively by
$$\begin{aligned}{} & {} {\rho _{1}}: S \rightarrow \textrm{PFS} (G) \\{} & {} {\rho _{2}}: S \rightarrow \textrm{PFS} (G) \\{} & {} {\rho _{3}}: S \rightarrow \textrm{PFS} (G) \\{} & {} {\rho _{4}}: S \rightarrow \textrm{PFS} (G) \end{aligned}$$
The \(\textrm{PFS}(G)\) represents the collection of all PFSs over G, which comprises all feasible pairings of MS and NMS grades for each disease \(d_{i} \in G\). The image of an element (symptom) \(s_j\) in S under \(\sigma :S \rightarrow \textrm{PFS}(G)~\text {where}~~\sigma =({\rho _{1}},~{\rho _{2}},~{\rho _{3}},~{\rho _{4}})\) is a \(\textrm{PFS}(G)\) in which each element (disease) \(d_i\) in G has a MS and NMS grade. These MS and NMS grades give the degree to which \(s_{j}\) is the symptom of the disease \(d_{i}\). Thus, the PFSMs \(\left[ {{\tilde{\varrho }}}_{1(ij)}\right] ,~\left[ {\tilde{\varrho }}_{2(ij)}\right] ,~\left[ {{\tilde{\varrho }}}_{3(ij)}\right] \), and \(\left[ {{\tilde{\varrho }}}_{4(ij)}\right] \) can be constructed for each physician, \(p_{1},~p_{2},~p_{3}\), and \(p_{4}\) by PFSSs \(({\rho _{1}},~S),~({\rho _{2}},~S),~({\rho _{3}},~S),~({\rho _{4}},~S)\) respectively. The PFSSs are utilized because physicians may hold the opinion that the sum of MS and NMS grades of the symptom \(s_{j}\) of the disease \(d_{i}\) is greater than 1, which the PFS is capable of accommodating. The information of four hospital physicians’ clinical data is collected from a private hospital in Mayiladuthurai, Tamil Nadu, India.
CASE 1: non-standard PFSMs
Step 1:
The PFSMs \(\tilde{\varrho _1},~\tilde{\varrho _1},~\tilde{\varrho _1}\), and \(\tilde{\varrho _1}\) be the opinions of four physicians \(p_{1},~p_{2},~p_{3}\), and \(p_{4}\) respectively, where
The \((i,j)\textrm{th}\) element in the matrix \({{{\tilde{\varrho }}}_k}\) is the opinion of physician \(p_{k}\) about a disease \(d_{i}\) for the symptom \(s_{j}\), where \(i,~j=1,~2,~3,~4,~5\) and \(k=1,~2,~3,~4\).
Step 2:
The combined CMs are constructed for each Physicians based on the CPS.
In \(\textrm{CM}_{P_1},~\textrm{CM}_{P_2},~\textrm{CM}_{P_3}\), and \(\textrm{CM}_{P_4}\) rows denote the elements of \(S_{p_1},~S_{p_2},~S_{p_3}\), and \(S_{p_4}\) respectively and columns denote the elements of \(S_{(p_2\wedge p_3\wedge p_4)},\)\(S_{(p_1\wedge p_3\wedge p_4)},\)\(S_{(p_1\wedge p_2\wedge p_4)}\), and \(S_{(p_1\wedge p_2\wedge p_3)}\) respectively, where \(S_{p_i}\) is the CPS of Physician \(p_{i}\) and \(S_{(p_i\bigwedge p_j\bigwedge \ p_k)}\) is the intersection of the CPSs of physicians \(p_{i},~p_{j},~p_{k}~(i,~j,~k=1,~2,~3,~4)\). The \((i,~j)\textrm{th}\) element of \(\textrm{CM}_{p_k} \)\(={\left\{ \begin{array}{ll} (1,~0),~\text {if}~i\textrm{th}~\text {and}~j\textrm{th}~\text {both parameters are the choice} \\ \text {parameters of}~ p_{k}, \\ (0,~1),~\text {if at least one of}~i\textrm{th}~\text {or}~j\textrm{th}~\text {parameters be}\\ \text {not under choice of}~p_{k}. \end{array}\right. }\)
Step 3:
The opinion matrices \({{\tilde{\varrho }}_{1}},~{{{\tilde{\varrho }}}_{2}},~{{{\tilde{\varrho }}}_{3}}\), and \({{{\tilde{\varrho }}}_{4}}\) are multiplied with the combined CMs: \(\textrm{CM}_{p_1}\), \(\textrm{CM}_{p_2}\), \(\textrm{CM}_{p_3}\), and \(\textrm{CM}_{p_4}\), respectively.
The disease with the highest MS weight is \(d_{3}\), specifically typhoid. Since a unique result is obtained, the process concludes at this step, and the unique result represents the optimal disease selection.
CASE 2: standard PFSMs
Step 1:
The PFSMs \({{\tilde{\varrho }}}_{1},~{\tilde{\varrho }}_{2},~{{\tilde{\varrho }}}_{3}\), and \({{\tilde{\varrho }}}_{4}\) be the opinion matrices of four physicians \(p_{1},~p_{2},~p_{3},\) and \(p_{4}\) respectively, where
The \((i,j)\textrm{th}\) element in the matrix \({{{\tilde{\varrho }}}_k}\) is the opinion of Physician \(p_{k}\) about a disease \(d_{i}\) for the symptom \(s_{j}\), where \(i,~j=1,~2,~3,~4,~5\) and \(k=1,~2,~3,~4\).
Step 2:
Based on the CPSs, the combined CMs are constructed for each of the physicians.
In \(\textrm{CM}_{P_1},~\textrm{CM}_{P_2},~\textrm{CM}_{P_3}\), and \(\textrm{CM}_{P_4}\), rows denote the elements of \(S_{p_1},~S_{p_2},~S_{p_3}\), and \(S_{p_4}\) respectively and columns denote the elements of \(S_{(p_2\wedge p_3\wedge p_4)},\)\(S_{(p_1\wedge p_3\wedge p_4)}\), \(S_{(p_1\wedge p_2\wedge p_4)}\), and \(S_{(p_1\wedge p_2\wedge p_3)}\) respectively, where \(S_{p_i}\) is the CPS of physician \(p_{i}\) and \(S_{(p_i\bigwedge p_j\bigwedge p_k)}\) is the intersection of the CPSs of physicians \(p_{i},~p_{j},~p_{k}~(i,~j,~k=1,~2,~3,~4)\). The \((i,~j)^th\) element of
\({CM}_{p_k}={\left\{ \begin{array}{ll} (1,~0),~\text {if}~i\textrm{th}~\text {and}~j\textrm{th}~\text {both parameters are}\\ \text {the choice parameters of}~p_{k}, \\ (0,~1),~\text {if at least one of}~i\textrm{th} \text { or}~j\textrm{th}~\text { parameters}\\ \text { be not under choice of}~p_{k}. \end{array}\right. }\).
Step 3:
The standard PFSMs of \({\tilde{\varrho }}_{1},~{{\tilde{\varrho }}}_{2},~{{\tilde{\varrho }}}_{3},\) and \({{\tilde{\varrho }}}_{4}\) are computed respectively \({\tilde{\varrho }}_{{1}(\varsigma )},~{{\tilde{\varrho }}}_{{2}(\varsigma )}\), \({{\tilde{\varrho }}}_{{3}(\varsigma )}\), and \({\tilde{\varrho }}_{{4}(\varsigma )}\) respectively.
The range of the elements of the resulting matrices are small and even some of MS and NMS grades become zero, facilitating the max-min product with CMs.
Step 4:
The standard opinion matrices \({\tilde{\varrho }}_{{1}(\varsigma )},~{\tilde{\varrho }}_{{2}(\varsigma )},~{{\tilde{\varrho }}}_{{3}(\varsigma )}\), and \({{\tilde{\varrho }}}_{{4}(\varsigma )}\) are multiplied with the combined CMs \(\textrm{CM}_{p_1},~\textrm{CM}_{p_2},~\textrm{CM}_{p_3}\), and \(\textrm{CM}_{p_4}\) respectively.
Each entry in the resulting matrix gives the maximum MS and minimum NMS among the four physicians, while considering the combined CPS (i.e., corresponding physicians’ CPS being combined with others’).
Step 6:
The MS weights of each disease are: \(W(d_{1})=0.7+0.7+0.52+0.26+0.12=2.3\)\(W(d_{2})=0.41+0.41+0.4+0.41+0.24=1.87\)\(W(d_{3})=0.69+0.69+0.3+0.69+0.04=2.41\)\(W(d_{4})=0.48+0.48+0.11+0.46+0.48=2.01\)\(W(d_{5})=0.44+0.44+0.43+0.44+0.32=2.07\) The disease with the highest MS weight is \(d_{3}\) i.e., typhoid. Because a unique result is obtained, the process is terminated at this point, and the unique result is the optimal choice of diseases.
Sensitivity analysis
The sensitivity of the proposed method has been examined in conjunction with the proposed and existing distance functions. The results produced by the method have been computed and are shown in Table 4.
Table 4
Sensitivity analysis of current and projected Mabala distances to parameter variation
The proposed analytical approach aids in quantifying the durability and consistency of distance measurements, providing a valuable understanding of their responsiveness across different input scenarios, as shown in Fig. 3.
Fig. 3
Comparative analysis of existing and projected distance functions according to parameter variations
×
The insightful comments on the sensitivity analysis through Table 4 and Fig. 3 are drawn as follows:
1.
The proposed distance measure differentiates significantly more than Euclidean and Hamming distances, but not more than Juthika’s distance, between two sets whose MS and NMS grades of each element are approximately 0.5.
2.
The proposed distance measure distinguishes better than all other measures except Hamming distances for two sets, one with both grades (MS and NMS) at the lowest extreme and the other at the highest extreme.
3.
The presented distance measure demonstrates less distinction among other measures for two sets, one with MS grade at the highest extreme and NMS grade at the lowest extreme for each element, and the other set vice versa.
4.
For two sets,
both grades are the least extreme in both sets for some elements.
others have one of the grades at the most extreme in one group but not both, and both at the lowest extreme in the other.
It varies far less than the other metrics.
5.
For two sets such that,
In one set, both grades (MS and NMS) are at different ends of the spectrum for certain elements, and vice versa in the other.
Either one of the grades, but not both, is at the highest extreme in one set, or both grades are at the lowest extreme in the remaining sets.
The MS grades are near zero and the NMS grades are near one on both sets.
The proposed Mabala distance shows comparatively less distinction among the other measures.
The proposed distance measure employs two categories of ratios: first, the ratio of the difference between two sets to the sum of the two sets’ values, and then the ratios of each MS grade, NMS grade, and indeterminacy grade. A variation between MS grades and NMS grades exists in the difference between the two sets, which is not accounted for in the other inferences. As a result, inferences 4 and 5 are extremely sensitive to this factor.
Distance function comparison
This section presents two types of comparison analysis to check the consistency of the proposed Mabala distance measure. First, a comparison is calculated for the opinions of each physician of the disease \(d_{i},~i=1,~2,~3,~4,~5\) that exhibits symptoms \(s_{j},~(j=1,~2,~3,~4,~5)\). This comparison is conducted using four different distance measures on the PFSS data set, including the suggested Mabala’s distance. Second, the proposed approach is compared with non-normalized and normalized PFSM-based algorithms in terms of the range of variation in weights of the diseases.
The disease with the shortest distance from hospital information is most accurately mapped with the physician’s opinion for overall symptoms. In general, the degree of properness (in terms of precision) of a disease decreases as the value of distance measurements increases. The GDM process selects the disease that appears to have the most accurate representation of overall symptoms, based on the majority opinion of physicians, as presented in Table 5.
Table 5
Hospital data pertaining to the symptoms of potential diseases
Temperature
Headache
Body pain
Cough
Fatigue
Malaria
(0.6, 0.3)
(0.5, 0.4)
(0.3, 0.7)
(0.2, 0.8)
(0.8, 0.2)
Viral fever
(0.7, 0.2)
(0.5, 0.4)
(0.2, 0.8)
(0.4, 0.7)
(0.7, 0.3)
Typhoid
(0.8, 0.1)
(0.7, 0.3)
(0.7, 0.2)
(0.6, 0.5)
(0.6, 0.3)
Chikungunya
(0.7, 0.3)
(0.6, 0.5)
(0.2, 0.7)
(0.2, 0.6)
(0.1, 0.8)
Dengue
(0.3, 0.6)
(0.4, 0.5)
(0.8, 0.3)
(0.7, 0.2)
(0.1, 0.8)
Table 6 displays the results of two types of Hamming distance measurements. The inferences from the results regarding the correctness of the physicians’ opinions are then discussed.
Table 6
Hamming distance measurements of the diseases based on the opinions of four physicians
Malaria
Viral fever
Typhoid
Chikungunya
Dengue
\(\textrm{HD}_1\)
\(\textrm{HD}_2\)
\(\textrm{HD}_1\)
\(\textrm{HD}_2\)
\(\textrm{HD}_1\)
\(\textrm{HD}_2\)
\(\textrm{HD}_1\)
\(\textrm{HD}_2\)
\(\textrm{HD}_1\)
\(\textrm{HD}_2\)
Physician \(p_{1}\)
0.433
0.429
0.247
0.235
0.182
0.191
0.253
0.252
0.313
0.303
Physician \(p_{2}\)
0.257
0.256
0.315
0.303
0.253
0.233
0.282
0.296
0.222
0.232
Physician \(p_{3}\)
0.402
0.390
0.304
0.313
0.192
0.173
0.253
0.257
0.299
0.316
Physician \(p_{4}\)
0.222
0.220
0.250
0.244
0.253
0.217
0.272
0.281
0.301
0.290
It is observed from the Table 6, physician \(p_{1}\) and \(p_{3}\) had the most proper opinion on typhoid and physician \(p_{2}\) on dengue according to both \(\textrm{HD}_{1}\) and \(\textrm{HD}_{2}\). physician \(p_{4}\)’s most proper opinion is on malaria according to \(\textrm{HD}_{1}\) and typhoid according to \(\textrm{HD}_{2}\). The diseases in the decreasing order of their degree of properness in opinion PFSSs of each of the physicians according to \(\textrm{HD}_{1}\) are:
According to \(\textrm{HD}_{1}\), the most proper opinion of a majority of two physicians is Typhoid. According to \(\textrm{HD}_{2}\), the most proper opinion of a majority of three physicians is also Typhoid. Hence, according to both kinds of Hamming distances, the disease having the most proper opinion regarding its related symptoms by the GDM process is Typhoid. Also, for Physician \(p_{1}\) and \(p_{2}\), the decreasing order of the diseases in terms of the degree of properness according to \(\textrm{HD}_{2}\) is the same as that according to \(\textrm{HD}_{1}\). For Physician \(p_{3}\) and \(p_4\), the order according to \(\textrm{HD}_{2}\) is almost the same up to a considerable level as that according to \(\textrm{HD}_{1}\) with some minor changes. But the order varies among various physicians. By comparing both kinds of Hamming distance measurements, it is observed that they do not differ much and give approximately the same distance for the opinions of each of the physicians on each disease regarding its symptoms.
The results of two types of Euclidean distance measurements are tabulated in Table 7, and the inferences drawn from them regarding the correctness of the physicians’ opinions are then discussed.
Table 7
Euclidean distance measurements of the diseases based on the opinions of four physicians
Malaria
Viral fever
Typhoid
Chikungunya
Dengue
\(\textrm{ED}_{1}\)
\(\textrm{ED}_2\)
\(\textrm{ED}_1\)
\(\textrm{ED}_{2}\)
\(\textrm{ED}_{1}\)
\(\textrm{ED}_2\)
\(\textrm{ED}_1\)
\(\textrm{ED}_{2}\)
\(\textrm{ED}_{1}\)
\(\textrm{ED}_2\)
Physician \(p_{1}\)
0.401
0.409
0.235
0.217
0.185
0.205
0.248
0.265
0.293
0.286
Physician \(p_{2}\)
0.308
0.316
0.295
0.281
0.246
0.224
0.274
0.286
0.252
0.285
Physician \(p_{3}\)
0.398
0.397
0.323
0.345
0.216
0.200
0.259
0.283
0.301
0.297
Physician \(p_{4}\)
0.298
0.297
0.274
0.277
0.275
0.237
0.291
0.311
0.312
0.386
It is observed that Physicians \(p_{1},~p_{2}\), and \(p_{3}\) had the most proper opinion on Typhoid according to both kinds of Euclidean distances. Physician \(p_4\) had the most proper opinion on Viral fever according to \(\textrm{ED}_{1}\) and on Typhoid according to \(\textrm{ED}_{2}\). The diseases, in decreasing order of their degree of properness in the opinion of the PFSSs of each of the physicians according to \(\textrm{ED}_{1}\) are:
According to \(\textrm{ED}_{1}\), the most proper opinion of a majority of three physicians is typhoid. According to \(\textrm{ED}_{2}\), the most proper opinion of a majority of all four physicians is also Typhoid. Hence, according to both kinds of Euclidean distance measurements, the disease having the most proper opinion regarding its related symptoms by the GDM process is Typhoid. Furthermore, for Physicians \(p_{1}\) and \(p_{3}\), the decreasing order of diseases by degree of appropriateness according to \(\textrm{ED}_{2}\) is identical to that according to \(\textrm{ED}_{1}\). The ranking of physicians \(p_{2}\) and \(p_{4}\) based on \(\textrm{ED}_{2}\) closely resembles their ranking based on \(\textrm{ED}_{1}\), with only minor variations. But the order varies among various physicians. By comparing both kinds of Euclidean distance measurements, it is observed that they do not differ much and give approximately the same distance for the physicians’ opinions on each disease regarding its symptoms. Our findings indicate that our distance measure surpasses existing measures in terms of accuracy and robustness.
The results of Juthika et al. [44] distance are tabulated in Table 8, and the inferences from its results regarding the correctness of opinions of the physicians are then discussed.
Table 8
Juthika’s distance measurements of the diseases based on the opinions of four physicians
Malaria
Viral fever
Typhoid
Chikungunya
Dengue
Physician \(p_{1}\)
0.59
0.34
0.23
0.33
0.44
Physician \(p_{2}\)
0.36
0.46
0.32
0.45
0.35
Physician \(p_{3}\)
0.53
0.45
0.21
0.34
0.44
Physician \(p_{4}\)
0.31
0.36
0.32
0.39
0.39
Physician \(p_{1}\), Physician \(p_{2}\), and Physician \(p_{3}\) commonly had the most proper opinion on Typhoid, but Physician \(p_{4}\)‘s most proper opinion is on Malaria. The diseases, in decreasing order of their degree of properness in opinion of the PFSSs of each of the physicians, according to Juthika’s distance, are:
(Note: Chikungunya and Dengue are in the same position in the order.)
The most proper opinion of the majority of the three physicians is Typhoid. Hence, the disease having the most proper opinion regarding its related symptoms by the GDM process is Typhoid. But the decreasing order of degrees of properness varies among various physicians.
The results of the proposed Mabala’s distance are tabulated in Table 9, and the inferences drawn from its results regarding the properness of the opinions of the physicians are discussed.
Table 9
Mabala’s distance measurements of the diseases based on the opinions of four physicians
Malaria
Viral fever
Typhoid
Chikungunya
Dengue
Physician \(p_1\)
0.5
0.31
0.26
0.31
0.38
Physician \(p_2\)
0.31
0.38
0.30
0.42
0.28
Physician \(p_3\)
0.43
0.37
0.22
0.31
0.45
Physician \(p_4\)
0.26
0.29
0.28
0.34
0.34
Both Physician \(p_{1}\) and Physician \(p_{3}\) had the most proper opinion commonly on Typhoid. Physician \(p_{2}\) had the most proper opinion on Dengue, and Physician \(p_{4}\) on Malaria. The diseases, in decreasing order, of each of the physicians according to the proposed Mabala’s distance are:
Physician \(p_{1}\) - Typhoid > Viral fever > Chikungunya Dengue > Malaria (Note: Viral fever and Chikungunya are in the same position in the order)
(Note: Chikungunya and Dengue are in the same position in the order.)
The most proper opinion of a majority of two physicians is Typhoid. Hence, the disease having the most proper opinion regarding its related symptoms by the GDM process is typhoid. But the decreasing order of degrees of properness varies among various physicians. The opinions of the four physicians in terms of these distances are shown in Figs. 4, 5, 6 and 7.
Fig. 4
Prediction of Physician \({{{\tilde{\varrho }}}_{1}}\) regarding a patient’s symptoms based on six distance measures
Fig. 5
Prediction of Physician \({{{\tilde{\varrho }}}_{2}}\) regarding a patient’s symptoms based on six distance measures
Fig. 6
Prediction of Physician \({{{\tilde{\varrho }}}_{3}}\) regarding a patient’s symptoms based on six distance measures
Fig. 7
Prediction of Physician \({{{\tilde{\varrho }}}_{4}}\) regarding a patient’s symptoms based on six distance measures
Table 10
Comparison of the proposed Method with existing methodologies
These results of various distance measures between the views of four physicians lead us to conclude that
The two types of Euclidean distances, the two types of Hamming distances, Juthika’s distance, and the proposed Mabala’s distance all identify Typhoid as the disease with the most accurate opinion regarding its associated symptoms according to the GDM procedure.
According to all distance measurements up to a valid level, the decreasing order of maladies in terms of degree of appropriateness for each physician is nearly identical. In actuality, it is identical based on all distance measurements for Physician \(p_{1}\).
It is to be noted that
(i)
The proposed Mabala’s distance, like the other distance measurements, gives the same disease (Typhoid) as the disease having the most proper opinion regarding its related symptoms by the GDM process.
(ii)
Concerning the opinions of individual physicians, the diseases having the most proper opinion according to Mabala’s distance exactly coincide with that of \(\textrm{HD}_{1}\).
(iii)
The decreasing order of the diseases in terms of the degree of properness for each physician almost coincides with the other distance measurements.
Comparison of proposed method with existing methods
This section provides a comparison between the outcomes of the proposed standard PFSMs and those of the non-normalized and normalized PFSMs. Typhoid emerges as the most probable disease, according to each algorithm. The non-normalized algorithm, on the other hand, has produced contradictory, identical outcomes. Conversely, the order of the results produced by the standardization-based algorithm and the normalized algorithm is identical. Furthermore, in comparison to the alternative approaches, the standardization-based algorithm’s proposed distance measure has generated the lowest MS weight. The findings are presented in Table 10.
The PFSMs, which lack normalization and standardization, determined typhoid to be the most advantageous disease option. This method, nevertheless, has its limitations. Specific diseases possess similar MS weights, which hinders the effectiveness of this technique in differentiating them. Conversely, the normalized PFSMs algorithm [32] produces more precise differentiation among diseases. While the outcome of the suggested approach also includes typhoid, it improves the differentiation between diseases by producing strict inequality. Using standard PFSMs circumvents the reduced range of variation in weights, which is another limitation of the proposed method.
The degree of variability among diseases is comparatively extensive when compared to existing approaches. This attribute substantially enhances the capacity to distinguish between diseases and assures precise outcomes.
Conclusion
This study presents a newly devised distance measure for PFSs, which incorporates the ratio of the discrepancy in values between two sets to the combined sum of values in both sets. Our measure is designed to consider the specific values of MS degree, NMS degree, and indeterminacy, allowing it to effectively capture sets that exhibit distinct MS and NMS behaviors. The variance of PFSs is computed, and a formula for variance-based standardization of PFSMs. The process of standardizing PFSM is simple, so it does not add any extra burden to the proposed algorithm. A novel GDM algorithm has been developed to demonstrate the practicality of the proposed Mabala distance measure for both standard and non-standard PFSMs.
We conducted a thorough case study of disease diagnosis involving four physicians and five diseases. The case study of disease diagnosis provides empirical evidence supporting the advantages of standardization, highlighting the consistent outcomes that were unaffected by the implementation of standard procedures. The sensitivity analysis is conducted to examine numerous scenarios with changing parameters, and the results demonstrate that certain decisions are particularly sensitive to the difference between MS and NMS grades. The proposed algorithm is relatively straightforward and capable of producing accurate results. The proposed distance measure can be used when the MS and NMS grades exhibit distinct behaviors that necessitate differentiation.
The proposed Mabala’s distance function demonstrates superior performance compared to traditional metrics, including \(\textrm{ED}_1\), \(\textrm{ED}_2\), \(\textrm{HD}_1\), \(\textrm{HD}_2\), and Juthika’s distance. The proposed distance measure effectively distinguished differences between data points, resulting in more accurate and meaningful findings for medical disease diagnosis. This makes it a robust and versatile distance metric that can be used in various applications, especially in the field of GDM problems. The ability to accurately measure dissimilarity between data points is crucial for decision-making and analysis. Furthermore, the flexibility of this distance metric allows for customization and adaptation to different problem domains, making it a valuable tool for addressing complex GDM challenges.
In the future, this research could be used to develop distance measures for various fuzzy environments in order to determine the distance between two fuzzy sets. The formulation of statistical measures for PFSs will be analogous to that of crisp set data. These derived metrics, which can reveal distinguishing characteristics within PFS-type data, will be used to investigate uncertain data sets. These advancements will be used in a variety of real-world applications, including transportation, travel issues, image synthesis, and mathematical analysis.
Declarations
Conflict of interest
The corresponding author, on behalf of the other co-authors, declares no conflict of interest regarding any financial or personal relationships with other people or organizations.
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