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.

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 [1,2]. In fact research has provided many medical applications that have taken advantage of different fuzzy approaches [3,4], and of models of vague knowledge born from alternative narratives [5]. Feng et al. [6,7] generalized intuitionistic fuzzy soft sets and related multi attribute decision making methods. By inspiration of these applied studies, we have selected a powerful and flexible model for the representation of undertain knowledge. It has been popularized by Yager et al. [8,9], who coined the term Pythagorean fuzzy sets (PFSs). PFSs improve the performance of intuitionistic fuzzy sets, which for each alternative, provide a fuzzy assessment of both membership (μ) and nonmembership (ν) in such a way that μ+ν≤1. In PFSs however, assessments that meet the relaxed condition μ2+ν2≤1 are acceptable. The PFS model has become one of the most influential notions in fuzzy modeling, since it allows us to accept more uncertain appraisals than the intuitionistic fuzzy or fuzzy sets, and consequently, its applications are potentially wider. The next summary of recent results testifies to this claim.

Yucesan et al. [10] presented the ideas of Pythagorean fuzzy analytic hierarchy method and Pythagorean fuzzy method for order decision by comparison to the ideal solution, in order to present an exact decision-making technique for estimating hospital service quality. Guleria et al. [11] proposed a new (R, S)-norm discriminant measure of Pythagorean fuzzy sets and proved some interesting features. Their monotonicity with respect to the parameters R&S were studied too. This information measure is utilized in some problems related to medical diagnosis or pattern recognition. Extensions and hybrid models based on PFSs have been put forward too. Rahman [12] extended the spirit of some popular aggregation operators to the more general framework of interval-valued Pythagorean fuzzy numbers, thus producing the so-called GIVPFWA, GIVPFOWA, and GIVPFHA operators. They discussed their properties, and argued that their generalized operators are more reliable and accurate than the existing aggregation operators. Zulqarnain et al. [13] proposed the averaging and geometric operators of Pythagorean fuzzy hypersoft sets. They also introduced a novel TOPSIS method in this new environment. They applied this methodology to a case-study of selection of multipurpose masks for protection against COVID-19. Yue et al. [14] proposed novel score function of hesitant fuzzy numbers and prove the validity of this function using an example. Yue [15] proposed a novel bilateral matching (BM) decision-making method for knowledge innovation management considering the matching willingness of bilateral enterprises. Ejegwa [16] improved composite relation for Pythagorean fuzzy sets and applied it to medical diagnosis. Many other decision-making methods have been suggested in the literature [17–25].

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.

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.

The rest of this paper is structured as follows. Section 2 discusses preliminary work. Section 3 describes the main contributions of the paper. Then Section 4 performs a comparative analysis. Section 5 concludes the proposed work and lays out some future directions for research.

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).

To make the proposed Pythagorean model more understandable, consider the block diagram shown in Fig. 1. The proposed model uses nineteen symptoms as inputs. For each input, a linguistic variable is defined in the Pythagorean fuzzy number. There may be n experts, but we’re only picking three experts here. Their expertise is represented by PFNs. Symptom PFNs and expert weights are input to PF-TOPSIS, PF-entropy, and PFPWG blocks. In these blocks, the algorithm of each approach is executing and generating its final outputs. We compare the results of each method and highlight the signal output. The results of these three methods should be the same.

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 Fig. 2, the results of PF-TOPSIS are displayed. The eight diseases are shown in different eight colors. The length of each bar shows its value obtained from the PF-TOPSIS method. In Fig. 3, the results of the PF entropy are displayed. In this figure, eight diseases are also represented by different colors, and the length of the bar shows the value of each disease for each data set obtained from PF entropy method. Fig. 4 shows the results of the PFPWG method.

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 D1, D2, D1, D4, D8, D6, D1, D2, D1, and D7 diseases, respectively. We can see that each approach highlights the same disease based on the provided patient’s records, which testifies the authenticity of our model.

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:

Industrial automation and Industry 4.0: In Industry 4.0, we connect the devices through the internet to make a network of different things. Then through the proposed approach, the different manufacturing parts of the machines can be controlled without human intervention. We can use our proposed model in Industry 4.0 to make intelligent decisions by taking into account all parameters and making the manufacturing process more productively and efficiently.

Precision agriculture: We need various decision-making systems to automate traditional farming to increase the yields and reduce the potential risks. With our model, precision farming can be made more efficient and productive. We can make timely decisions, and automate the decision process. Through this approach, an irrigation system can be improved, and water wastage could be reduced. Our proposed approach can be useful for designing a pest control system that helps the farmer to save crops from pests timely. This approach can also be useful to monitor soil pH and other ingredients, which require for the proper growth of the crops. Through this procedure, farmers can decide the right amount of fertilizers for the field.

Computer aided diagnosis: These systems help doctors to analyze the medical images and highlight diseases based on symptoms. The proposed approach can help doctors to early detect the chances of any disease, which could happen in the future due to the patient’s routine or changes in his body and enable the doctor to prevent it from spreading more by proper medication or therapies.

Classroom monitoring: The proposed approach could be beneficial for monitoring students’ activities in a large classroom and concluding which student is not attentive in class or how much students in the class are attentive.