According to the World Health Organization (WHO), cancer is the leading cause of death for children in low and middle-income countries. Around 400,000 kids get diagnosed with this illness each year, and their survival rate depends on the country in which they live. In this article, we present a Pythagorean fuzzy model that may help doctors identify the most likely type of cancer in children at an early stage by taking into account the symptoms of different types of cancer. The Pythagorean fuzzy decision-making techniques that we utilize are Pythagorean Fuzzy TOPSIS, Pythagorean Fuzzy Entropy (PF-Entropy), and Pythagorean Fuzzy Power Weighted Geometric (PFPWG). Our model is fed with nineteen symptoms and it diagnoses the risk of eight types of cancers in children. We develop an algorithm for each method and calculate its complexity. Additionally, we consider an example to make a clear understanding of our model. We also compare the final results of various tests that prove the authenticity of this study.
Childhood cancer is the leading cause of death in children, especially in low and middle-income countries. Their likelihood of survival heavily rests on the country where they live. The chance of curing childhood cancer in high-income countries is above eighty percent, whereas in low and in middle-income countries, it is a mere forty-five percent. This difference in curing rate owes to many factors, such as late diagnosing and cancer diagnosed at its late stages due to unavailability of resources, cost of treatment (the treatment cost is rather high in later stages), or incorrect diagnostics and inappropriate treatment. The survival rate can be increased if low and in middle-income countries improve the access to necessary medicines and technologies. Generally speaking, the most productive way to reduce the effects of childhood cancer is by effective and evidence-based therapy with appropriate nurturing care.
The chances of curing childhood cancer, and the cost of treatment with lesser suffering, can be improved if it is identified early, and appropriate treatment is provided immediately. A correct diagnosis is required to treat childhood cancer efficiently with the right treatment, which may include surgeries, radiotherapy, and chemotherapy. For early diagnosing, we should consider the following three aspects:
Parents should know about childhood cancer so that they can perceive its symptoms and consult a medical expert. The medical expert should be skilled enough and investigate the case promptly in order to cater for the right treatment. The patient has accessibility to the right treatment at right time.
When someone is diagnosed with cancer, the chances of recovery and survival increase if it is detected in an early stage, even with the least amount of financial and physical suffering. The low- and middle-income countries should run education campaigns for parents with the help of competent doctors so that in the presence of symptoms in any child, their parents can respond without delay. This task requires the joint effort of civil society and non-governmental organizations. In 2018, the World Health Organization launched a global initiative on cancer in children. As part of this initiative, they provided governments with professional guidance and support to maintain high-quality childhood cancer programs. They aim to increase childhood cancer survival rates, and by 2030 that rate should be at least sixty percent.
Medical information is very sensitive and contains many uncertainties. Every expert may have their personal opinion on a health record, and it may become rather difficult to make an accurate diagnostic based on these reports. In such uncertain situations, fuzzy logic can play an important role in making decisions among different alternative diagnostics. Many extensions of fuzzy sets theory have been proposed [
Yucesan et al. [
The following targets motivate the research contained in this article:
Early diagnosis of cancer in children can reduce overall mortality and expense of treatment, which ultimately reduces the patient’s suffering. Effective handling of vague and uncertain data in a medical context is required. Get opinions of available medical experts and decide the final treatment. Contribution towards WHO’s goal of increasing survival from childhood cancer by least sixty percent before 2030.
Concerning these issues, our contribution to this study is described below:
We develop a novel decision-making system to determine childhood cancer risk at its early stages, thus increasing the survival rate. We use the Pythagorean fuzzy sets (PFS) for decision-making because it is very close to human thinking. It is characteristic of simultaneously focusing on the degree of truth, the degree of non-membership, and the degree of indeterminacy of each alternative to make it more powerful. We design algorithms to demonstrate the entire performance of the model. In addition, we determine their respective time complexities.
The rest of this paper is structured as follows.
This section summarizes some of the introductory concepts that need to be followed to completely benefit from this study. First we overview technical concepts that will help us formulate our theoretical model. Then we summarize some facts about its prospective application (namely, identification of childhood cancer).
Let
Let
The PF-TOPSIS method uses linguistic terms and Pythagorean fuzzy numbers (PFNs) to represent the relative importance of experts and criteria. These linguistic terms and PFNs are predefined and used to rate any expert or criteria. The following equation is used to calculate the weight in crisp form for any Pythagorean fuzzy evaluation. Assume that
Notice that the sum of all weights should be equal to 1.
Suppose that
Let
Let
The Pythagorean Fuzzy Positive Ideal Solution (PFPIS)
The following formula is used to calculate the distance of each alternative from PFPIS and PFNIS:
The relative closeness value of each choice
The maximum relative closeness value is the best choice among all possible choices.
According to [
The score function
The weighted entropy of each criteria is calculated using the next equation:
The support of two PFNs is calculated using the next formula:
The distance between two PFNs can be calculated using the following normalized Hamming distance:
The formula for the weighted support is as follows:
The Pythagorean Fuzzy Power Weighted Geometric (
Let
Many studies have tried to identify the causes of childhood cancer. Some factors are related to the environment, such as radiation exposure and chemical exposure. Some are lifestyle-related, such as drugs, alcohol, cell phone use, and smoking. Some children inherit DNA changes from a parent that increase their risk of a certain type of cancer. Here we list possible risk factors for childhood cancer with a small description of each factor.
Children and teenagers tend to get different types of childhood cancers. The most common childhood cancers are discussed below:
To make the proposed Pythagorean model more understandable, consider the block diagram shown in
In this subsection, we write all of the instructions that must be followed to obtain final results for any input. Each algorithm takes certain inputs and produces certain outputs. There are seven sub-algorithms of the PF-TOPSIS algorithm, namely, Algorithm A, Algorithm B, Algorithm C, Algorithm D, Algorithm E, Algorithm F, and Algorithm G. Each sub-algorithm shows each step of the TOPSIS process. We also write the net time complexity of every algorithm.
After aggregating all complexities, we get the final time complexity of the PF-TOPSIS algorithm, which is
Algorithm PF-Entropy shows the set of instructions that need to follow to find the final results of each childhood cancer.
Algorithm-PFPWG shows the set of instructions of PFPWG decision-making techniques.
To understand the working of the above algorithm, consider the following example and apply the PF-TOPSIS method to it. The PFNs against each linguistic variable are shown in
Very low | [0,1,0] |
Low | [0.2,0.9,0.39] |
Below medium | [0.4,0.6.0.69] |
Medium | [0.65,0.50,0.57] |
Above medium | [0.8.0.45,0.4] |
High | [0.9,0.2,0.39] |
Very high | [1,0,0] |
We shall first show how a decision can be made with the application of the steps described in
D1 | VL | LO | LO | D1 | VH | HI | HI | ||
D2 | LO | LO | VL | D2 | LO | LO | VL | ||
D3 | MED | MED | BM | D3 | VL | LO | LO | ||
S1 | D4 | MED | MED | BM | S11 | D4 | LO | LO | VL |
D5 | MED | BM | BM | D5 | VL | LO | LO | ||
D6 | LO | LO | VL | D6 | LO | VL | VL | ||
D7 | AM | MED | MED | D7 | LO | LO | VL | ||
D8 | VL | LO | LO | D8 | LO | LO | VL | ||
D1 | VH | VH | HI | D1 | VH | HI | HI | ||
D2 | LO | LO | VL | D2 | LO | LO | VL | ||
D3 | MED | MED | BM | D3 | MED | BM | BM | ||
D4 | MED | MED | BM | D4 | LO | LO | VL | ||
S2 | D5 | MED | MED | BM | S12 | D5 | VL | LO | LO |
D6 | MED | BM | BM | D6 | LO | VL | VL | ||
D7 | MED | BM | BM | D7 | LO | LO | VL | ||
D8 | MED | BM | BM | D8 | LO | LO | VL | ||
D1 | VH | HI | HI | D1 | VH | HI | HI | ||
D2 | LO | LO | VL | D2 | LO | LO | VL | ||
D3 | VL | LO | LO | D3 | VL | LO | LO | ||
S3 | D4 | LO | VL | VL | S13 | D4 | LO | LO | VL |
D5 | LO | LO | VL | D5 | MED | BM | BM | ||
D6 | VL | LO | LO | D6 | LO | VL | VL | ||
D7 | LO | VL | VL | D7 | LO | LO | VL | ||
D8 | MED | BM | BM | D8 | LO | LO | VL | ||
D1 | VH | VH | HI | D1 | VH | HI | HI | ||
D2 | LO | LO | VL | D2 | LO | LO | VL | ||
D3 | VL | LO | LO | D3 | VL | LO | LO | ||
S4 | D4 | LO | VL | VL | S14 | D4 | LO | LO | VL |
D5 | LO | LO | VL | D5 | BM | BM | LO | ||
D6 | VL | LO | LO | D6 | LO | VL | VL | ||
D7 | LO | VL | VL | D7 | LO | LO | VL | ||
D8 | LO | LO | VL | D8 | LO | LO | VL | ||
D1 | LO | LO | VL | D1 | VL | LO | LO | ||
D2 | VL | LO | LO | D2 | BM | MED | MED | ||
D3 | LO | LO | VL | D3 | MED | BM | MED | ||
S5 | D4 | VL | LO | LO | S15 | D4 | BM | BM | MED |
D5 | LO | VL | VL | D5 | LO | LO | VL | ||
D6 | LO | LO | VL | D6 | LO | LO | VL | ||
D7 | MED | BM | BM | D7 | LO | LO | LO | ||
D8 | LO | LO | LO | D8 | VL | LO | LO | ||
D1 | VH | HI | HI | D1 | VL | LO | LO | ||
D2 | VL | LO | LO | D2 | LO | LO | VL | ||
D3 | LO | LO | VL | D3 | BM | BM | MED | ||
S6 | D4 | VL | LO | LO | S16 | D4 | LO | VL | VL |
D5 | LO | VL | VL | D5 | LO | LO | VL | ||
D6 | LO | LO | VL | D6 | LO | LO | VL | ||
D7 | LO | LO | VL | D6 | LO | LO | VL | ||
D8 | LO | LO | LO | D8 | MED | BM | BM | ||
D1 | LO | LO | VL | D1 | VH | VH | VH | ||
D2 | MED | MED | MED | D2 | MED | BM | BM | ||
D3 | LO | LO | VL | D3 | LO | VL | VL | ||
D4 | VL | LO | LO | D4 | LO | LO | VL | ||
S7 | D5 | LO | VL | VL | S17 | D5 | BM | BM | BM |
D6 | LO | LO | VL | D6 | MED | BM | BM | ||
D7 | LO | LO | VL | D7 | BM | BM | MED | ||
D8 | LO | LO | LO | D8 | MED | BM | BM | ||
D1 | LO | LO | VL | D1 | VH | HI | HI | ||
D2 | VL | LO | LO | D2 | VL | LO | LO | ||
D3 | LO | LO | VL | D3 | LO | VL | VL | ||
S8 | D4 | VL | LO | LO | S18 | D4 | LO | LO | VL |
D5 | LO | VL | VL | D5 | LO | LO | VL | ||
D6 | LO | LO | VL | D6 | LO | LO | LO | ||
D7 | LO | LO | VL | D7 | VL | LO | LO | ||
D8 | MED | BM | BM | D8 | LO | LO | VL | ||
D1 | VH | HI | HI | D1 | VL | LO | LO | ||
D2 | MED | MED | BM | D2 | LO | VL | VL | ||
D3 | BM | BM | BM | D3 | LO | LO | VL | ||
S9 | D4 | MED | BM | BM | S19 | D4 | LO | LO | VL |
D5 | MED | BM | MED | D5 | MED | BM | MED | ||
D6 | BM | BM | BM | D6 | VL | LO | LO | ||
D7 | MED | BM | BM | D7 | LO | LO | VL | ||
D8 | MED | BM | MED | D8 | VL | LO | LO | ||
D1 | VH | HI | VH | ||||||
D2 | LO | LO | VL | ||||||
D3 | VL | LO | LO | ||||||
S10 | D4 | LO | LO | VL | |||||
D5 | VL | LO | LO | ||||||
D6 | LO | VL | VL | ||||||
D7 | LO | LO | VL | ||||||
D8 | LO | LO | VL |
S1 | [0.158717,0.93621, 0.3133777] | [0.167842,0.928902, 0.330107] | [0.594807,0.52811, 0.606056] | [0.594807,0.52811, 0.606056] |
S2 | [0.167842,0.928902, 0.330107] | [0.594807,0.52811, 0.606056] | [0.594807,0.52811, 0.606056] | [0.594807,0.52811, 0.606056] |
S3 | [1,0,0] | [0.167842,0.928902, 0.330107] | [0.158717,0.936271, 0.313377] | [0.123255,0.96126, 0.2465508] |
S4 | [1,0,0] | [0.167842,0.928902, 0.330107] | [0.158717,0.936271, 0.313377] | [0.123255,0.96126, 0.2465508] |
S5 | [0.167842,0.928902, 0.330107] | [0.158717,0.936271, 0.313377] | [0.167842,0.928902, 0.330107] | [0.158717,0.936271, 0.313377] |
S6 | [1,0,0] | [0.158717,0.936271, 0.313377] | [0.167842,0.928902, 0.330107] | [0.158717,0.936271, 0.313377] |
S7 | [0.167842,0.928902, 0.330107] | [0.65,0.5,0.572276] | [0.167842,0.928902, 0.330107] | [0.158717,0.936271, 0.313377] |
S8 | [0.167842,0.928902, 0.330107] | [0.158717,0.936271, 0.313377] | [0.167842,0.928902, 0.330107] | [0.158717,0.936271, 0.313377] |
S9 | [1,0,0] | [0.594807,0.52811, 0.6060566] | [0.4,0.6,0.6928203] | [0.519722,0.560349, 0.6449015] |
S10 | [1,0,0] | [0.167842,0.928902, 0.330107] | [0.158717,0.936271, 0.313377] | [0.167842,0.928909, 0.330107] |
S11 | [1,0,0] | [0.167842,0.928902, 0.330107] | [0.158717,0.936271, 0.313377] | [0.167842,0.928909, 0.330107] |
S12 | [1,0,0] | [0.167842, 0.928902, 0.330107] | [0.519722, 0.560349, 0.644901] | [0.167842, 0.928909, 0.330107] |
S13 | [1,0,0] | [0.167842, 0.928902, 0.330107] | [0.158717, 0.936271, 0.313377] | [0.167842, 0.928909, 0.330107] |
S14 | [1,0,0] | [0.167842, 0.928902, 0.330107] | [0.158717, 0.936271, 0.313377] | [0.167842, 0.928909, 0.330107] |
S15 | [0.158717, 0.93621, 0.3133777] | [0.579118, 0.535381, 0.614808247] | [0.589672, 0.530523, 0.6089599] | [0.499309, 0.568063, 0.654213] |
S16 | [0.158717, 0.93621, 0.3133777] | [0.167842, 0.928902, 0.330107] | [0.499309, 0.568063, 0.654213] | [0.263522, 0.814217, 0.517307] |
S17 | [1,0,0] | [0.519722, 0.560349, 0.6449015] | [0.123255, 0.96126, 0.2465508] | [0.167842, 0.928909, 0.330107] |
S18 | [1,0,0] | [0.158717, 0.936271, 0.313377] | [0.123255, 0.96126, 0.2465508] | [0.167842, 0.928909, 0.330107] |
S19 | [0.158717, 0.93621, 0.3133777] | [0.123255, 0.96126, 0.24655] | [0.167842, 0.928902, 0.330107] | [0.167842, 0.928909, 0.330107] |
S1 | [0.519722,0.560349, 0.6449015] | [0.167842,0.928909, 0.330107] | [0.718534,0.48063, 0.502695] | [0.158717,0.936271, 0.313377] |
S2 | [0.519722,0.560349, 0.644901] | [0.519722,0.560349, 0.6449015] | [0.519722,0.560349, 0.6449015] | |
S3 | [0.167842,0.928909, 0.330107] | [0.158717,0.936271, 0.313377] | [0.123255,0.96126, 0.2465508] | [0.519722,0.560349, 0.6449015] |
S4 | [0.167842,0.928909, 0.330107] | [0.158717,0.936271, 0.313377] | [0.123255,0.96126, 0.2465508] | [0.167842,0.928902, 0.330107] |
S5 | [0.123255,0.96126, 0.2465508] | [0.167842,0.928909, 0.330107] | [0.519722,0.560349, 0.6449015] | [0.2,0.9,0.387298] |
S6 | [0.123255,0.96126, 0.2465508] | [0.167842,0.928909, 0.330107] | [0.167842,0.928902, 0.330107] | [0.2,0.9,0.387298] |
S7 | [0.123255,0.96126, 0.2465508] | [0.167842,0.928909, 0.330107] | [0.167842,0.928902, 0.330107] | [0.2,0.9,0.387298] |
S8 | [0.123255,0.96126, 0.2465508] | [0.167842,0.928909, 0.330107] | [0.167842,0.928902, 0.330107] | [0.519722,0.560349, 0.6449015] |
S9 | [0.589672,0.530523, 0.6089599] | [0.4,0.6,0.69282] | [0.519722,0.560349, 0.6449015] | [0.589672,0.530523, 0.608959] |
S10 | [0.158717,0.936271, 0.313377] | [0.123255,0.96126, 0.2465508] | [0.167842,0.928902, 0.330107] | [0.167842,0.928902, 0.330107] |
S11 | [0.158717,0.936271, 0.313377] | [0.123255,0.96126, 0.2465508] | [0.167842,0.928902, 0.330107] | [0.167842,0.928902, 0.330107] |
S12 | [0.158717,0.936271, 0.31337] | [0.123255,0.96126, 0.2465508] | [0.167842,0.928902, 0.330107] | [0.167842,0.928902, 0.330107] |
S13 | [0.519722,0.560349, 0.6449015] | [0.123255,0.96126, 0.2465508] | [0.167842,0.928902, 0.330107] | [0.167842,0.928902, 0.330107] |
S14 | [0.354495,0.677608, 0.644344] | [0.123255,0.96126, 0.2465508] | [0.167842,0.928902, 0.330107] | [0.167842,0.928902, 0.330107] |
S15 | [0.167842,0.928909, 0.330107] | [0.167842,0.928909, 0.330107] | [0.2,0.9,0.387298] | [0.158717,0.936271, 0.313377] |
S16 | [0.167842,0.928909, 0.33010] | [0.167842,0.928909, 0.330107] | [0.2,0.9,0.387298] | [0.519722,0.560349, 0.6449015] |
S17 | [0.4,0.6,0.69282] | [0.519722,0.560349, 0.6449015] | [0.499309,0.568063, 0.654213] | [0.519722,0.560349, 0.6449015] |
S18 | [0.167842,0.928909, 0.330107] | [0.2,0.9,0.387298] | [0.1518717,0.936271, 0.313377] | [0.167842,0.928902, 0.330107] |
S19 | [0.65,0.5,0.572276] | [0.158717,0.936271, 0.313377] | [0.167842,0.928902, 0.330107] | [0.158717,0.936271, 0.313377] |
S1 | H | AM | AM | S11 | M | M | H |
S2 | M | H | M | S12 | M | H | M |
S3 | M | H | M | S13 | M | H | M |
S4 | M | M | H | S14 | M | M | H |
S5 | M | M | H | S15 | M | H | M |
S6 | M | H | M | S16 | H | AM | AM |
S7 | M | H | M | S17 | H | M | AM |
S8 | H | AM | AM | S18 | AM | M | H |
S9 | H | AM | AM | S19 | M | H | AM |
S10 | M | H | M |
S1 | (0.846591,0.332005,0.416001) |
S2 | (0.773055,0.371227,0.51437) |
S3 | (0.773055,0.371227,0.51437) |
S4 | (0.765686,0.379829,0.519091) |
S5 | (0.765686,0.379829,0.519091) |
S6 | (0.773055,0.371227,0.51437) |
S7 | (0.773055,0.371227,0.51437) |
S8 | (0.846591,0.332005,0.416001) |
S9 | (0.846591,0.332005,0.416001) |
S10 | (0.773055,0.371227,0.51437) |
S11 | (0.765686,0.379829,0.519091) |
S12 | (0.773055,0.371227,0.51437) |
S13 | (0.773055,0.371227,0.51437) |
S14 | (0.765686,0.379829,0.519091) |
S15 | (0.773055,0.371227,0.51437) |
S16 | (0.846591,0.332005,0.416001) |
S17 | (0.818343,0.34357,0.460732) |
S18 | (0.808373,0.365114,0.461762) |
S19 | (0.806725,0.359677,0.468858) |
S1 | [0.1343,0.9434, 0.3030] | [0.1420,0.937, 0.3191] | [0.5035,0.5986, 0.6229] | [0.5035,0.5986, 0.6229] |
S2 | [0.7730,0.3712, 0.5143] | [0.1297,0.9390, 0.3184] | [0.4598,0.6150, 0.6405] | [0.4598,0.6150, 0.6405] |
S3 | [0.7730,0.3712, 0.5143] | [0.1297,0.9390, 0.3184] | [0.1229,0.9453, 0.3022] | [0.0952,0.9666, 0.2375] |
S4 | [0.7656,0.3798, 0.5190] | [0.1285,0.9394, 0.3175] | [0.1215,0.9457, 0.3013] | [0.0943,0.9664, 0.2368] |
S5 | [0.1285,0.9394, 0.3175] | [0.1215,0.9457, 0.3013] | [0.1285,0.9394, 0.3175] | [0.1215,0.9457, 0.3013] |
S6 | [0.7730,0.3712, 0.5143] | [0.1226,0.9453, 0.3021] | [0.1297,0.9390, 0.3184] | [0.1226,0.9453, 0.3022] |
S7 | [0.1297,0.9390, 0.3184] | [0.5024,0.5944, 0.6278] | [0.1297,0.9390, 0.3184] | [0.1226,0.9453, 0.3022] |
S8 | [0.142,0.9370, 0.3191] | [0.1343,0.9435, 0.3028] | [0.1420,0.9370, 0.3191] | [0.1343,0.9435, 0.3028] |
S9 | [0.8465,0.332, 0.4160] | [0.5035,0.598, 0.622] | [0.338,0.6561, 0.6743] | [0.4399,0.6241, 0.6455] |
S10 | [0.7730,0.3712, 0.5143] | [0.1297,0.9390, 0.3184] | [0.1226,0.9453, 0.3022] | [0.1297,0.9390, 0.3184] |
S11 | [0.7656,0.3798, 0.5190] | [0.1285,0.9394, 0.31755] | [0.1215,0.9457, 0.3013] | [0.1285,0.9394, 0.3175] |
S12 | [0.7730,0.3712, 0.5143] | [0.1297,0.9390, 0.3184] | [0.4017,0.6391, 0.6557] | [0.1297,0.9390, 0.3184] |
S13 | [0.7730,0.3712, 0.5143] | [0.1297,0.9390, 0.3184] | [0.1226,0.9453, 0.3022] | [0.1297,0.9390, 0.3184] |
S14 | [0.7656,0.3798, 0.5190] | [0.1285,0.9394, 0.3175] | [0.1215,0.9457, 0.3013] | [0.1285,0.9394, 0.3175] |
S15 | [0.1226,0.9452, 0.3023] | [0.4476,0.6204, 0.6439] | [0.4558,0.6168, 0.6416] | [0.3859,0.6450, 0.6595] |
S16 | [0.1343,0.9434, 0.3030] | [0.1420,0.9370, 0.3191] | [0.4227,0.6303, 0.6511] | [0.223,0.8367, 0.5001] |
S17 | [0.8183,0.343, 0.4607] | [0.4253,0.6284, 0.6512] | [0.1008,0.9659, 0.2384] | [0.1373,0.9375, 0.319] |
S18 | [0.8083,0.3651, 0.4617] | [0.1283,0.9450, 0.3008] | [0.0996,0.9665, 0.2364] | [0.1356,0.9386, 0.3169] |
S19 | [0.1280,0.9447, 0.3018] | [0.0994,0.9663, 0.2371] | [0.1354,0.938, 0.31790] | [0.1354,0.9384, 0.3178] |
S1 | [0.4399,0.6241, 0.6455] | [0.1420,0.9370, 0.3190] | [0.6083,0.5619, 0.5605] | [0.1343,0.9435, 0.3028] |
S2 | [0.4598,0.6150, 0.6405] | [0.4017,0.6391, 0.6557] | [0.4017,0.6391, 0.6557] | [0.4017,0.6391, 0.6557] |
S3 | [0.1297,0.9390, 0.3184] | [0.1226,0.9453, 0.3022] | [0.0952,0.9666, 0.2375] | [0.4017,0.6391, 0.6557] |
S4 | [0.1285,0.9394, 0.3175] | [0.1215,0.9457, 0.3013] | [0.0943,0.9669, 0.2368] | [0.1285,0.9394, 0.3175] |
S5 | [0.0943,0.9669, 0.236] | [0.1285,0.9394, 0.3175] | [0.3979,0.6426, 0.6547] | [0.1531,0.9151, 0.3730] |
S6 | [0.0952,0.9666, 0.2375] | [0.1297,0.9390, 0.3184] | [0.1297,0.9390, 0.3184] | [0.1546,0.9144, 0.3740] |
S7 | [0.0952,0.9666, 0.2375] | [0.1297,0.9390, 0.3184] | [0.1297,0.9390, 0.3184] | [0.1546,0.9144, 0.3740] |
S8 | [0.1043,0.9656, 0.2381] | [0.1420,0.9370, 0.3190] | [0.1420,0.9370, 0.3191] | [0.4399,0.6241, 0.6455] |
S9 | [0.4992,0.6005, 0.6246] | [0.3386,0.6561, 0.6743] | [0.4399,0.6241, 0.6455] | [0.4992,0.6005, 0.6240] |
S10 | [0.1226,0.9453, 0.3022] | [0.0952,0.9666, 0.2375] | [0.1297,0.9390, 0.3184] | [0.1297,0.9390, 0.3184] |
S11 | [0.1215,0.9457, 0.3013] | [0.0943,0.9667, 0.2368] | [0.1285,0.9394, 0.3175] | [0.1285,0.9394, 0.3175] |
S12 | [0.1226,0.9453, 0.3022] | [0.0952,0.9667, 0.2375] | [0.1297,0.9390, 0.318] | [0.1297,0.9390, 0.3184] |
S13 | [0.4017,0.6391, 0.6557] | [0.0952,0.9666, 0.2375] | [0.1297,0.9390, 0.3184] | [0.1297,0.9390, 0.3184] |
S14 | [0.2714,0.7329, 0.6238] | [0.09437,0.9669, 0.236] | [0.1285,0.9394, 0.3175] | [0.1285,0.9394, 0.3175] |
S15 | [0.1297,0.9390, 0.3184] | [0.1297,0.9390, 0.3184] | [0.1546,0.9144, 0.3740] | [0.1226,0.9453, 0.3022] |
S16 | [0.1420,0.937, 0.3190] | [0.1420,0.9370, 0.3190] | [0.1693,0.9115, 0.3746] | [0.4399,0.624, 0.6455] |
S17 | [0.327,0.6594, 0.6762] | [0.4253,0.6284, 0.6512] | [0.4086,0.6345, 0.6560] | [0.4253,0.6284, 0.6512] |
S18 | [0.1356,0.9386, 0.3169] | [0.1616,0.9139, 0.3721] | [0.1227,0.9450, 0.3031] | [0.1356,0.9386, 0.3169] |
S19 | [0.5243,0.5890, 0.6148] | [0.1280,0.9447, 0.301726] | [0.1354,0.938, 0.3179] | [0.128,0.9447, 0.3017] |
S1 | (0.6083044176, 0.5619336555, 0.56053216) | (0.1343683837, 0.9435067904, 0.3028862392) |
S2 | (0.773055, 0.371227, 0.514369985) | (0.1297510973, 0.9390198435, 0.318443694) |
S3 | (0.773055, 0.371227, 0.514369985) | (0.09528289403, 0.9666909764, 0.2375494187) |
S4 | (0.765686, 0.379829, 0.5190904354) | (0.09437462793, 0.9669448349, 0.2368778499) |
S5 | (0.3979438593, 0.6426208583, 0.6547359142) | (0.09437462793, 0.9669448349, 0.2368778499) |
S6 | (0.773055, 0.371227, 0.514369985) | (0.09528289403, 0.9666909764, 0.2375494187) |
S7 | (0.50248575, 0.5944384864, 0.6278144287) | (0.09437462793, 0.9669448349, 0.2368778499) |
S8 | (0.4399919677, 0.6241858178, 0.6455998243) | (0.1043465737, 0.9656064274, 0.2381512544) |
S9 | (0.846591, 0.332005, 0.4160004311) | (0.3386364, 0.6561596489, 0.6743737123) |
S10 | (0.773055, 0.371227, 0.514369985) | (0.09528289403, 0.9666909764, 0.2375494187) |
S11 | (0.765686, 0.379829, 0.5190904354) | (0.09437462793, 0.9669448349, 0.2368778499) |
S12 | (0.773055, 0.371227, 0.514369985) | (0.09528289403, 0.9666909764, 0.2375494187) |
S13 | (0.773055, 0.371227, 0.514369985) | (0.09528289403, 0.9666909764, 0.2375494187) |
S14 | (0.765686, 0.379829, 0.5190904354) | (0.09437462793, 0.9669448349, 0.2368778499) |
S15 | (0.455848888, 0.6168281917, 0.6416578319) | (0.1226969704, 0.9453087376, 0.3022198604) |
S16 | (0.4399919677, 0.6241858178, 0.6455998243) | (0.1343683837, 0.9434529307, 0.3030539637) |
S17 | (0.818343, 0.34357, 0.4607324489) | (0.1008648665, 0.9659137643, 0.2384048629) |
S18 | (0.808373, 0.365114, 0.4617627745) | (0.09963601412, 0.9665140672, 0.2364809137) |
S19 | (0.52437125, 0.5890888373, 0.6148244741) | (0.09943288988, 0.9663591629, 0.2371983741) |
D1 | 0.275112 | 0.557659 | 0.669642 | 1 |
D2 | 0.562568 | 0.194540 | 0.256951 | 6 |
D3 | 0.546636 | 0.233754 | 0.299534 | 4 |
D4 | 0.562210 | 0.184700 | 0.247285 | 7 |
D5 | 0.527542 | 0.255102 | 0.325949 | 2 |
D6 | 0.579636 | 0.149774 | 0.205336 | 8 |
D7 | 0.547882 | 0.217842 | 0.284491 | 5 |
D8 | 0.525929 | 0.240877 | 0.314130 | 3 |
Now we evaluate the same inputs with our second methodology (cf.,
E1 | 0.7159104962 | E6 | 0.4178064273 | E11 | 0.4076892564 | E16 | 0.6504636906 |
E2 | 0.7960083765 | E7 | 0.5377938017 | E12 | 0.4726217253 | E17 | 0.7162852648 |
E3 | 0.4544188736 | E8 | 0.5308859753 | E13 | 0.4726217431 | E18 | 0.4172182446 |
E4 | 0.3924935986 | E9 | 0.8469676038 | E14 | 0.4600604293 | E19 | 0.5216621254 |
E5 | 0.5410031461 | E10 | 0.4076892564 | E15 | 0.6774897253 |
W1 | 0.03317680263 | W6 | 0.06799014839 | W11 | .0691716591 | W16 | .04081980058 |
W2 | 0.02382269643 | W7 | 0.05397769656 | W12 | .06158866884 | W17 | .03313297819 |
W3 | 0.06371444735 | W8 | 0.05478441131 | W13 | 0.06158866884 | W18 | 0.06805883796 |
W4 | 0.07094624934 | W9 | 0.01787153932 | W14 | 0.06305561114 | W19 | 0.05586159757 |
W5 | 0.05360290059 | W10 | 0.0691716591 | W15 | 0.03766362677 |
S1 | 0.7185344956 | 0.4806300777 | 0.1587167962 | 0.9362709927 |
S2 | 1 | 0 | 0.1678421306 | 0.9289016977 |
S3 | 1 | 0 | 0.123255 | 0.96126 |
S4 | 1 | 0 | 0.123255 | 0.96126 |
S5 | 0.519722 | 0.560349 | 0.123255 | 0.96126 |
S6 | 1 | 0 | 0.123255 | 0.96126 |
S7 | 0.65 | 0.5 | 0.123255 | 0.96126 |
S8 | 0.519722 | 0.560349 | 0.123255 | 0.96126 |
S9 | 1 | 0 | 0.4 | 0.6 |
S10 | 1 | 0 | 0.123255 | 0.96126 |
S11 | 1 | 0 | 0.123255 | 0.96126 |
S12 | 1 | 0 | 0.123255 | 0.96126 |
S13 | 1 | 0 | 0.123255 | 0.96126 |
S14 | 1 | 0 | 0.123255 | 0.96126 |
S15 | 0.5896724049 | 0.5305226244 | 0.158717 | 0.936271 |
S16 | 0.519722 | 0.560349 | 0.1587167962 | 0.9362709927 |
S17 | 1 | 0 | 0.1232545023 | 0.9612601555 |
S18 | 1 | 0 | 0.1232545023 | 0.9612601555 |
S19 | 0.65 | 0.5 | 0.1232545023 | 0.9612601555 |
D1 | 0.2673218011 | 0.7632174342 |
D2 | 0.7701354677 | 0.188677577 |
D3 | 0.7588156448 | 0.2131948401 |
D4 | 0.7738775789 | 0.1547205578 |
D5 | 0.7495318939 | 0.2463499314 |
D6 | 0.7855832056 | 0.1222792276 |
D7 | 0.7662546068 | 0.1935045705 |
D8 | 0.7451043189 | 0.2317462283 |
D1 | 0.7406000743 | 1 |
D2 | 0.1967824469 | 6 |
D3 | 0.2193338893 | 4 |
D4 | 0.1666173468 | 7 |
D5 | 0.2473686387 | 2 |
D6 | 0.1346891589 | 8 |
D7 | 0.2016178381 | 5 |
D8 | 0.237238162 | 3 |
We consider the same example again, but now we follow the PFPWG algorithm (cf.,
S1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |
S2 | 0.07429382481 | 1 | 1 | 1 0.9406123802 | 0.6045895256 | 0.8354170216 | 0.5962338726 | |
S3 | 0.07429382481 | 1 | 0.5368457814 | 0.5081379061 | 0.6045895256 | 0.991644347 | 0.4029425476 | 0.5962338726 |
S4 | 0.07429382481 | 1 | 0.5368457814 | 0.5081379061 | 0.6045895256 | 0.991644347 | 0.4029425476 | 0.9916378446 |
S5 | 0.9916375164 | 0.9916375164 | 0.545208265 | 0.5368462527 | 0.567525526 | 1 | 0.8354170216 | 0.9592938503 |
S6 | 0.07429382481 | 0.9916375164 | 0.545208265 | 0.5368462527 | 0.567525526 | 1 | 0.4400130496 | 0.9592938503 |
S7 | 0.9916375164 | 0.4964063084 | 0.545208265 | 0.5368462527 | 0.567525526 | 1 | 0.4400130496 | 0.9592938503 |
S8 | 0.9916375164 | 0.9916375164 | 0.545208265 | 0.5368462527 | 0.567525526 | 1 | 0.4400130496 | 0.5962338726 |
S9 | 0.07429382481 | 0.5452087848 | 0.8625519566 | 0.9406123802 | 0.9449307707 | 0.6826495033 | 0.8354170216 | 0.5411646433 |
S10 | 0.07429382481 | 1 | 0.5368457814 | 0.5452019058 | 0.5962338726 | 0.9629360004 | 0.4400130496 | 0.9916378446 |
S11 | 0.07429382481 | 1 | 0.5368457814 | 0.5452019058 | 0.5962338726 | 0.9629360004 | 0.4400130496 | 0.9916378446 |
S12 | 0.07429382481 | 1 | 0.9406124118 | 0.5452019058 | 0.5962338726 | 0.9629360004 | 0.4400130496 | 0.9916378446 |
S13 | 0.07429382481 | 1 | 0.5368457814 | 0.5452019058 | 1 | 0.9629360004 | 0.4400130496 | 0.9916378446 |
S14 | 0.07429382481 | 1 | 0.5368457814 | 0.5452019058 | 0.8551970739 | 0.9629360004 | 0.4400130496 | 0.9916378446 |
S15 | 1 | 0.5582837365 | 0.9956816017 | 0.9258593552 | 0.6045895256 | 1 | 0.4723570439 | 1 |
S16 | 1 | 1 | 0.9258584902 | 0.6657996631 | 0.6045895256 | 1 | 0.4723570439 | 0.5962338726 |
S17 | 0.07429382481 | 0.6045958532 | 0.5081372495 | 0.5452019058 | 0.9219400223 | 0.6045895256 | 0.8206639966 | 0.5962338726 |
S18 | 0.07429382481 | 0.9916375164 | 0.5081372495 | 0.5452019058 | 0.6045895256 | 0.9676495033 | 0.4305879033 | 0.9916378446 |
S19 | 1 | 0.9629289845 | 0.545208265 | 0.5452019058 | 0.8918099777 | 0.991644347 | 0.4400130496 | 1 |
D1 | 0.1194823486 |
D2 | 0.2670615473 |
D3 | 0.196587252 |
D4 | 0.1866896644 |
D5 | 0.2115133567 |
D6 | 0.2750992294 |
D7 | 0.1617538985 |
D8 | 0.2559919525 |
S1 | 0.01396530372 | S11 | 0.01621589327 |
S2 | 0.01624052684 | S12 | 0.01624052684 |
S3 | 0.01624052684 | S13 | 0.01624052684 |
S4 | 0.01621589327 | S14 | 0.01621589327 |
S5 | 0.01621589327 | S15 | 0.01624052684 |
S6 | 0.01624052684 | S16 | 0.01396530372 |
S7 | 0.01624052684 | S17 | 0.01516553483 |
S8 | 0.01396530372 | S18 | 0.0148661582 |
S9 | 0.01396530372 | S19 | 0.01515533886 |
S10 | 0.01624052684 |
S1 | 0.04757226825 | 0.04719906613 | 0.04737572284 | 0.04740046969 | 0.04733855366 | 0.04717920819 | 0.04746287952 | 0.04722336609 |
S2 | 0.05462660171 | 0.05501188798 | 0.05521778653 | 0.05524662971 | 0.05512210018 | 0.05464126637 | 0.05517386925 | 0.05468505886 |
S3 | 0.05462660171 | 0.05501188798 | 0.05480908316 | 0.05481236679 | 0.05482581458 | 0.05498140024 | 0.05479153667 | 0.05468505886 |
S4 | 0.05454364459 | 0.0549271147 | 0.05472523149 | 0.05472854815 | 0.05474184762 | 0.05489668488 | 0.05470788958 | 0.05494806034 |
S5 | 0.05535403485 | 0.05491978515 | 0.05473258846 | 0.05475381779 | 0.05470926579 | 0.05490400536 | 0.0550890632 | 0.0549196969 |
S6 | 0.05462660171 | 0.05500453615 | 0.05481646251 | 0.05483771326 | 0.05479313368 | 0.05498874297 | 0.05482430915 | 0.05500440529 |
S7 | 0.05543945597 | 0.05456915647 | 0.05481646251 | 0.05483771326 | 0.05479313368 | 0.05498874297 | 0.05482430915 | 0.05500440529 |
S8 | 0.04756678906 | 0.04719362992 | 0.04707896944 | 0.04709810219 | 0.04705658376 | 0.04717920819 | 0.04709681361 | 0.04696075437 |
S9 | 0.04696573542 | 0.04690341962 | 0.04728603745 | 0.0473616988 | 0.04730264896 | 0.04697299468 | 0.04735529091 | 0.04692493704 |
S10 | 0.05462660171 | 0.05501188798 | 0.05480908316 | 0.05484509043 | 0.05481844704 | 0.05495617208 | 0.05482430915 | 0.05503285497 |
S11 | .05454364459 | 0.0549271147 | 0.05472523149 | 0.0547611726 | 0.05473450242 | 0.0548715332 | 0.05474056271 | 0.05494806034 |
S12 | 0.05462660171 | 0.05501188798 | 0.05516538086 | 0.05484509043 | 0.05481844704 | 0.05495617208 | 0.05482430915 | 0.05503285497 |
S13 | 0.05462660171 | 0.05501188798 | 0.05480908316 | 0.05484509043 | 0.05517446476 | 0.05495617208 | 0.05482430915 | 0.05503285497 |
S14 | 0.05454364459 | 0.0549271147 | 0.05472523149 | 0.0547611726 | 0.0549621491 | 0.0548715332 | 0.05474056271 | 0.05494806034 |
S15 | 0.05544686593 | 0.05462355566 | 0.05521397583 | 0.05518117125 | 0.05482581458 | 0.05498874297 | 0.05485290312 | 0.0550402103 |
S16 | 0.04757226825 | 0.04719906613 | 0.04732734522 | 0.04718228875 | 0.04708074919 | 0.04717920819 | 0.04711795702 | 0.04696075437 |
S17 | 0.0510066918 | 0.05101308249 | 0.05112979126 | 0.0511850345 | 0.05140784028 | 0.05099161504 | 0.05146478003 | 0.05103293237 |
S18 | 0.04999867869 | 0.05028219687 | 0.05011289268 | 0.05016648925 | 0.05014483695 | 0.05024337859 | 0.05014763937 | 0.05030808439 |
S19 | 0.05168666771 | 0.05125280746 | 0.05112364047 | 0.05115034017 | 0.05134966675 | 0.0512532187 | 0.05113670654 | 0.05130758994 |
[0.5194678133, 0.7224007634] | |
[0.2114815744, 0.9036659282] | |
[0.227599577, 0.8913426691] | |
[0.2089763783, 0.9037153385] | |
[0.2324989425, 0.8878193748] | |
[0.1809496915, 0.9227676788] | |
[0.2225946569, 0.8940073968] | |
[0.242201083, 0.8764918485] |
0.373991973 | |
0.1140561733 | |
0.1286549069 | |
0.1134848568 | |
0.132916258 | |
0.09062130094 | |
0.1251495779 | |
0.1452117021 |
We confirm that each approach highlights the same disease based on the patient record provided.
In this section, we compare the results of the Pythagorean Fuzzy TOPSIS (PF-TOPSIS), the Pythagorean Fuzzy Entropy (PF-Entropy), and the PFPWG method. To do this, we took ten different data sets and applied PF-TOPSIS, PF-Entropy, and the PFPWG techniques. The results obtained from each technique are represented by drawing bar graphs. In
Now we compare these three bar charts for each data set. We can see that the data set 1,2,3,4,5,6,7,8,9, highlighted
According to the World Health Organization (WHO), around 400,000 children are diagnosed with cancer each year and the rate of cure in low and middle-income countries is only 45 percent, which is highly unsatisfactory. To improve this percentage, WHO has launched a global initiative and provided appropriate professional guidance and resources. Their goal is to increase the survival rate up to sixty percent by the end of 2030. To help achieve this goal, we have proposed a novel model that allows doctors to diagnose the type of childhood cancer early, so that appropriate treatment can be given at the right time. This ultimately reduces the physical and financial suffering of the patient and their parents. Our model takes nineteen symptoms as inputs and determines the type of cancer. We have used Pythagorean fuzzy decision-making techniques for diagnostic purposes. We designed three algorithms, namely, Pythagorean fuzzy TOPSIS method, Pythagorean fuzzy entropy, and PFPWG. We have determined their respective time complexities. To test them, we have taken ten data sets and compared the results of the different approaches. Also, we have set forth a numerical example to make each of their steps understandable.
There are many other applications where decision-making takes place and our approaches can provide assistance. Our system is applicable when data is fuzzy and decisions must be made. So, some future directions of our work are discussed below: