Research Article, J Comput Eng Inf Technol Vol: 12 Issue: 6

Integrated Fuzzy Analytic Hierarchy Process with Fuzzy VIKOR: A Case Study of Early English Childhood

Ahmed Mohammed^* and Laith Abualigah

Department of Manufacturing and Civil Engineering, Norwegian University of Science and Technology, Trondheim, Norway

*Corresponding Author:

Ahmed Mohammed
Department of Manufacturing and Civil Engineering,
Norwegian University of Science and Technology,
Trondheim,
Norway;
E-mail: ahmedalahmed3800@gmail.com

Received date: 16 September, 2022, Manuscript No. JCEIT-22-74909; Editor assigned date: 19 September, 2022, PreQC No. JCEIT-22-74909 (PQ); Reviewed date: 04 October, 2022, QC No. JCEIT-22-74909; Revised date: 27 January, 2023, Manuscript No. JCEIT-22-74909 (R); Published date: 03 February, 2023, DOI: 10.4172/2324-9307.1000258.

Citation: Mohammed A, Abualigah L (2023) Integrated Fuzzy Analytic Hierarchy Process with Fuzzy VIKOR: A Case Study of Early English Childhood. J Comput Eng Inf Technol 12:2.

Abstract

Keywords: Fuzzy sets; Fuzzy Multi-Criteria Decision Making (FMCDM); Mobile application; English learning application

Introduction

The various degrees of uncertainty contained in real-world problems have a significant influence on decision making today. The use of decision-making strategies to tackle complicated issues with many quantitative and qualitative criteria has been improved. Overall, the majority of the criteria are in conflict with one another and as a result, numerous techniques known as Multi Criteria Decision Making (MCDM) have been developed to assist overcome these issues. The best alternative or combination of the right alternatives is selected using these techniques, which are based on mathematical reasoning and assess multiple options depending on several criteria.

This field of operational research known as MCDM, the optimal outcomes in complicated problems with numerous indicators, conflicting alternatives and criteria are determined [1]. MCDM is a systematic framework that represents technical knowledge and requires expert judgment to combine numerous criteria and address unclear information. Many disciplines that involve the processing of significant levels of information and knowledge have widely used MCDM approaches [2]. MCDM is an expert system approach that includes a variety of distinct choice alternatives and criteria. It comprises structuring, planning and solving decision-making issues utilizing a range of criteria. In comparison to traditional methodologies, MCDM is significantly getting importance due to its ability to improve decision quality through a more transparent, reasonable and efficient procedure [3].

Due to incomplete knowledge about the situation, uncertainty occurs in many aspects of life and it is critical to adapt the fuzzy results to the reality around us. The ability to specify the level of uncertainty is a key aspect of fuzzy logic. As a result, utilizing fuzzy logic to simulate a real world situation is simple. Uncertainty related problems are highly prevalent in the actual world, yet they are exceedingly difficult to solve because of how complex they are to model and handle. However, there are also circumstances when the uncertainty is not probabilistic in nature, but rather inaccurate or ambiguous. Various solutions have been suggested to handle those challenges. To manage unclear and imprecise information, other models, such as fuzzy logic and fuzzy sets theory, have been effectively employed.

Fuzzy Multi-Criteria Decision Making (FMCDM) is a key issue in expert systems and operations research since it has several decision alternatives and criteria. On the basis of the stated criteria, FMCDM seeks to determine the most suitable alternative(s) from a set of alternatives. Economics, engineering, social problems and management may all be solved using FMCDM methodologies which are proposed as a viable solution to complicated issues with high uncertainty and complexity [4].

According to the findings of the literature study, the Fuzzy Analytic Hierarchy Process (FAHP) and Fuzzy Vlsekriterijumska Optimizacija I Kompromisno Resenje (FVIKOR) are the best decision making strategies for dealing with uncertainty and subjectivity issues in a fuzzy environment.

Materials and Methods

Research problem

Making decisions is a challenge that individuals frequently face in daily life and that can take many different forms and finding the optimal compromise solution that is impacted by the decision-making preferences for both quantitative and qualitative objectives is the main purpose of the MCDM technique.

Decision Makers (DMs) in classic MCDM problems typically use specific numbers to assess alternatives. However, in many real world decision making scenarios, the problems may be too complicated or ill-defined to be answered by quantitative expressions. In these circumstances, DMs could favor using linguistic variables to describe the assessment data. Although linguistic variables are less accurate than numerical ones, they are more compatible with a person's cognitive process.

Moreover, Selection is often based on unreliable data or personal judgment because of the ambiguity of a human being’s thought which leads to wrong and biased decisions. The uncertainty and subjectivity problems can result in weighting errors and difficulties in the process of criterion weight acquiring [5]. Uncertainty and insufficient knowledge about the problem dimensions are major challenges in the decision-making process [6]. This possible confusion or uncertainty may lead to a naïve approach of simply adding up pluses and minuses. Whilst estimating preferences, such uncertainty may also introduce cognitive dissonance i.e. the holding of two contradictory beliefs simultaneously.

Due to the general effect of subjective perceptions, the decision makers may rely on their cognition and experience to evaluate the complicated criteria of alternatives in decision making problems.

Therefore, the traditional MCDM techniques are considered insufficient for dealing with linguistic uncertainty. As a result, it is recommended to use MCDM approaches with fuzzy sets to deal with ambiguity in decision-making. Furthermore, these fuzzy approaches allow for more precise outcomes and provided a solution to numerous ambiguities and imprecise information in people's judgments. Decision makers (or experts) employ linguistic variables to evaluate the criteria and alternatives, which are then transformed to appropriate fuzzy numbers for FMCDM procedures. FMCDM techniques can suitably explain the decision maker evaluation of existing alternatives for selecting the best alternative when the criteria have subjective perceptions. Therefore, the evaluation process preferably solved under a fuzzy environment in order to consider the linguistic variables.

Research questions

The following research questions have been framed to set the direction for this research:

• What are the suitable FMCDM techniques that can solve the uncertainty and subjective problems in decision making?

• Is the proposed FAHP-FVIKOR technique valid systematically?

Research objectives

The objectives of this research are listed as follows:

• To apply the integrated FAHP-FVIKOR for triangular type-1 fuzzy set to the early childhood English education case study, through ranking the available young learners' english learning mobile apps in a suitable way.

• To validate and evaluate the ranking results using statistical methods and benchmarking.

When evaluating two alternatives in MCDM environment, it is extremely difficult for decision makers to reflect their feelings with accurate positive real numbers [7]. Real world information can be unclear or imprecise, implying that it is untrustworthy or it might be delivered in pieces, with ambiguity in the facts or contradicting information, all of which can cause to uncertainty. Therefore, the main reasons that researchers employ the fuzzy set in the MCDM are uncertainty and a lack of knowledge.

The Fuzzy Sets (FS) theory is an effective technique for dealing with the imprecise and vague information provided by experts or decision-makers in an MCDM process. It is a field of modern mathematics that models the ambiguity inherent in human cognitive processes, providing a more natural representation for real-world problems. The advantage of fuzzy sets in addressing ambiguity in issues involving decision making. This theory has been shown to be an effective tool for solving challenges with decision-making processes involving vague or incomplete information, as well as for modeling subjective and inaccurate information in many environments [8].

Moreover, Fuzzy sets that have been effectively used in fuzzy forecasting, fuzzy control, similarity measures, artificial intelligence, control mechanisms, analysis, data processing and expert systems. Fuzzy numbers are a subset of fuzzy sets that are used to represent values of real world parameters when precise values are not measured owing to a lack of knowledge or inadequate information. Part of this uncertainty may be managed by utilizing type-1 fuzzy sets and type-2 fuzzy sets, in which the uncertainty is represented by values that are normally in range (0, 1). They are also highly effective in dealing with uncertainty when the membership functions of a fuzzy set can be identified precisely by a specific numeric value.

Triangular Fuzzy Number (TFN)

In order to interpret linguistic variables as probability distributions and give membership degrees to them, fuzzy set theory employs fuzzy numbers. Even so, fuzzy numbers can take the form of trapezoidal, triangular or Gaussian forms. But the most preferred fuzzy number is Triangular Fuzzy Numbers (TFN) [9]. Triangular fuzzy numbers represent membership by the function, which may more precisely describe the knowledge of the decision maker in complicated decision-making problems. They are frequently employed in fuzzy environments. Thus, they are successfully utilized to deal with FMCDM. Instead of exact values, triangular fuzzy numbers have been effectively used to express decision makers' assessments. As a result, they were used to improve judgment and offer flexibility to decision making. When dealing with problems that are expressed in quantitative terms, the linguistic variable is quite effective. Fuzzy numbers can be used to express linguistic values. So, to indicate such Fuzzy numbers, the triangular fuzzy number in particular is frequently employed. Therefore, because it is simple for decision makers to use and compute TFNs technique, TFNs are used for the evaluation of criteria weights and alternatives rating. Furthermore, TFN modeling has shown to be an effective technique for developing decision making problems especially when the given information is arbitrary and vague.

Related works

This section displays the related works of fuzzy AHP and fuzzy VIKOR integration in multi-criteria decision making. Many studies have shown the integration among fuzzy AHP, ANP, TOPSIS, BWM, VIKOR and others. And what is relevant to this research is fuzzy AHP-VIKOR integration.

The study by provides a hybrid multi criteria optimization approach that may be used for the evaluation and choosing of logistics service providers based on a variety of economic, environmental and social considerations. There are two stages to the research process. First, Fuzzy AHP is applied to evaluate each criteria and determine the relative significant fuzzy weight. Fuzzy VIKOR, the second step, is initiated to rate the alternatives. Presented integrated fuzzy AHPVIKOR technique was used to measure the degree of safety level in hot and humid environments. Based on three primary criteria and ten supporting criteria, a framework for safety evaluation was developed. In the process of risk assessment, the fuzzy VIKOR approach was used to prioritize the risk associated with various work stations while the fuzzy AHP method was employed to determine the weight of the criteria. In spherical fuzzy AHP and spherical fuzzy VIKOR techniques are combined. In the first stage, SF-AHP is used to calculate the weights and SF-VIKOR is utilized in the second stage to choose which advertising should be shown. The decision model took into account four factors, the importance of the location, the advertisements' relevance to the person, the potential of the campaign and the quality of the material. The model is applied to hypothetical scenarios and the decision makers discuss the chosen advertising. The strategy produces good results in terms of portraying the decision makers' point of view, according to the decision makers' judgments. The researcher applied on a case study related with water consumption to measure the effect of advertising on water consumption. To remove the level of uncertainty in the data, the suggested technique is based on a group decision of consumer opinion according to trapezoidal fuzzy numbers. The fuzzy AHP method is then adopted to determine the important weight of the effective criteria. Then, the Fuzzy VIKOR was used to evaluate the priority of the different alternatives. The study by an intelligent, integrated approach is offered to assist instructors pick the ideal simulation software package. Choosing the ideal simulation software package for educational purposes is a complicated Multi Criterion Decision Making (MCDM) problem with numerous possibly conflicting criteria. Fuzzy AHP is applied to derive the fuzzy criterion weights in the proposed fuzzy AHP-VIKOR technique and Fuzzy VIKOR is used to rank simulation software package choices in relation to these criteria. In this case study, different educational simulation software programs in Turkey are ranked and evaluated.

English childhood (case study)

The case study is adopted from the criteria are identified from the 2016 KSPK standard, which includes main and sub-criteria as shown below.

• In the 2016 KSPK standard, speaking and listening skills are treated as one component (stimulus given, rimes, poems and rhymes, stories, favorite things and activities, oral texts, familiar activities and experiences, stories heard, daily situations) criteria.

• Reading (Alphabet letters, simple phrases, simple sentences, texts) criteria.

 

• Writing (copy legible phrases, copy legible sentences, ideas and information communication and legible writing) criteria.

MCDM techniques and methods

Several theories of multi criteria decision making have been investigated. The most popular MCDM techniques that make use of different principles are the following. Weighted Product Method (WPM), Weighted Sum Model (WSM), Multiplicative Exponential Weighting (MEW), Simple Additive Weighting (SAW), Hierarchical Adaptive Weighting (HAW), Analytic Network Process (ANP), Technique for Order Performance by Similarity to Ideal Solution (TOPSIS), Vlsekriterijumska Optimizacija I Kompromisno Resenje (VIKOR) and Analytic Hierarchy Process (AHP). The following are the benefits, drawbacks and suggestions for prominent MCDM techniques.

HAW and WSM are simple to learn and apply. However, attribute weights are assigned arbitrarily and both techniques are difficult to implement when there are multiple criteria. Another disadvantage of these systems is that the final score is calculated using standard numerical scaling.

The benefits of WPM and MEW include the flexibility to remove any item to be measured and the use of relative values rather than exact ones. These two strategies, on the other hand, do not provide any solution with identical decision matrix weight. SAW takes into account all criteria, performs basic calculations and makes decisions naturally. All criterion values, however, must be positive and maximal. Furthermore, SAW does not always reveal the true scenario.

TOPSIS is linked to discrete alternative problems. This strategy is one of the most effective ways for dealing with real world problems. TOPSIS's main advantage is its capacity to swiftly identify the best alternative. TOPSIS' significant shortcomings, on the other hand, are the lack of a mechanism to weigh elicitation and assess the consistency of judgments.

The ANP technique, in contrast, offers a thorough explanation of the significance degree that a criterion might have in respect to its connection with other criterion. The benefit of the ANP technique is that it facilitates the assignment of weights by decomposing the problem into more manageable components so that a panel of experts may have a productive discussion where just two criteria are compared in making judgments. ANP, on the other hand, has two drawbacks. First, even for experts, it might be hard to provide the proper network structure among the criteria and different designs provide varied outcomes. Second, the creation of a super matrix needs the costly and unnatural pairwise comparison of every criteria with every other criteria.

VIKOR method has been shown to be an efficient and useful tool for dealing with MCDM problems. In the situations of competing and incommensurable (different units) criteria, VIKOR attempts to discover the best compromise solution by weighing all alternatives against all set criteria.

AHP is the most widely used multi criteria decision making technique for handling complex decision problems. It can assist decision-makers in making the best selection for their aims and preferences. AHP breaks down a decision problem into its essential elements and creates criteria hierarchies.

Fuzzy set theory is used in both VIKOR and AHP methods to deal with the uncertainty and ambiguity that occur in expert opinion and are defined by linguistic variables. Fuzzy AHP is an extension of the traditional AHP approach that takes into account the decision maker's fuzziness [10].

In FAHP, precise numbers are substituted by fuzzy numbers that reflect the linguistic terms in fuzzy AHP in order to address the imprecision in traditional AHP. By giving membership degrees to specific numbers to indicate how much these numbers belong to an expression, this tolerates the imprecise judgments.

FVIKOR, on the other hand, is functionally linked to discrete alternative issues. This method is one of the most useful for dealing with real world issues. FVIKOR's advantage is its capacity to quickly locate the optimal alternative in a fuzzy environment. Thus, FVIKOR is appropriate for ranking many alternatives among others techniques. The lack of facilities for weight elicitation and judgment consistency testing is, however, FVIKOR's fundamental flaw. As a result, FVIKOR requires an efficient method for determining the relative importance of various criteria in relation to the objectives and FAHP provides such a method. However, FAHP is used to set weights for objectives based on opinion former preferences.

Using a single FMCDM technique to rank the solution alternatives may raise questions about whether other findings may have been obtained using a different methodology. So, the most recent trend in the usage of MCDM methods is to combine two or more techniques to compensate for shortcomings in a single methodology [11].

Therefore, the fuzzy AHP-VIKOR is a middle-ground method that can convert subjective and unclear linguistics into objective and clear results.

As a result, FAHP and FVIKOR are combined to gain the benefits of both approaches. In the proposed fuzzy AHP-VIKOR. First, FAHP is applied to obtain the weights of the evaluation criteria, since FAHP can determine valid importance weights through pairwise comparisons. Then, FVIKOR is applied to rank the competing alternatives, using the fuzzy criteria weights acquired with FAHP, since this technique is commonly used to seek a compromise solution (s).

Results and Discussion

The methodology of the proposed research consists of four phases: The identification phase, integration phase, validation phase and evaluation phase. Each phase performs a set of specific objectives. The methodology of the proposed research is illustrated in Figure 1.

Identification phase (case study)

This phase presents early childhood english education as a case study, which has an uncertainty and subjectivity decision making problem. In this case study, young learners' english learning mobile applications will be presented as alternatives and different evaluation criteria will be utilized.

However, the goal of this phase is to select a decision matrix based on the intersection of multi-evaluation criteria for LSRW skills and young learners' english applications. Existing significant terminology, such as alternatives, criteria and the decision or evaluation matrix, must be specified in each MCDM problem. This phase consists of the following.

Determine the evaluation criteria based on LSRW skills: The criteria used in this study are identified from the 2016 KSPK standard, which includes main and sub-criteria Table 1. Shows the main criteria and sub-criteria with their definitions.

Identify mobile applications (alternatives) for english learning: Six early childhood English learning applications have been chosen as alternatives for this study. These applications, including Lingokids, Fun English, Fun With-Flupe, First Words, Montessori and Spelling Bee, are appropriate for students aged 5 and higher.

Construct the decision matrix: An intersection is established between mobile applications (alternatives) and identified characteristics, which are LSRW. Table 3 illustrates the data presentation required at this stage to complete the suggested decision matrix.

Table 1: The decision matrix.

Evaluate the decision matrix: A group of specialists (English teachers) evaluates the six English learning applications in accordance with the established LSRW skill criteria.

Expert selection: This step describes experts' selection, checklist form for evaluation and evaluation procedures undertaken for English learning mobile apps based on LSRW skill criteria. The panel of experts consists of three English-learning teachers who have been teaching English for more than ten years in various schools in Baghdad. So they have excellent experience in their field. Checklist forms are delivered amongst them.

The experts are involved in three stages. First, they review and examine the decision matrix and the checklist form's structure and content. Second, as participants in this study, they assist us in acquiring data for analysis by answering the checklist forms. Finally, assist us in acquiring the weights for evaluation criteria by filling out the FAHP standard.

The checklist is applied to evaluate the alternatives (applications) with respect to the criteria (LSRW) and consists of 21 questions that are broken up into three sections to evaluate LSRW skills. The forms are distributed and collected independently of any other person and the respondents should complete the form questions in order to avoid bias from the study Table 2 illustrates the checklist form.

Table 2: The checklist form.

To protect confidentiality, the forms are delivered and collected manually rather than online. The evaluation data obtained in the checklist form is analyzed using the FVIKOR technique, which is necessary for the ranking of English-learning mobile applications.

Integration phase (FAHP-VIKOR)

This phase presents a mathematical model design of the integration between FAHP and FVIKOR techniques. The FAHP technique is suitable for calculating the weight for the evaluation criteria and FVIKOR is the best technique for ranking the alternatives. Then, the integration FAHP-FVIKOR will be applied on the basis of the triangular fuzzy set.

Fuzzy AHP: Fuzzy set theory to deal with uncertainty MCDM problems. Equation (1), a Triangular Fuzzy Number (TFN) is defined as (a,b,c) which represents lower bound, middle value and upper bound. Figure 2 shows the triangular membership.

Equation (2) shows the illustration of each level of membership function.

Linguistic words converted into fuzzy numbers which are qualitative terms or of a natural language that express the subjective views of experts [12]. In this research, triangular fuzzy numbers are used as shown in Table 3 for rating the criteria.

Table 3: Linguistic scale in triangular fuzzy number.

The limitations of the original Analytical Hierarchy Process (AHP) are solved by the Fuzzy Analytical Hierarchy Process (FAHP), which also solves several MCDM issues. The procedures of FAHP are presented below.

Step 1: We assume a decision group contains K experts. Using the geometrical aggregation, Equation (3) used to create an integrated fuzzy pairwise comparison matrix.

Step 2: Equation (4) is used to calculate each criterion's fuzzy geometric mean.

Step 3: Equation (5) is used to calculate each criterion's fuzzy weight.

Step 4: As shown in equation (6), defuzzify the fuzzy weight using the average weight criteria.

Step 5: Equation (7) calculates the normalized weight criteria .

By performing the final steps of fuzzy AHP. Now it is important to calculate the consistency ratio to ensure whether the results of FAHP method is satisfactory or not. In order to check the consistency ratio in applying FAHP model. The biggest eigenvector (λ_max) is calculated to determine the Consistency Index (CI), the Random Index (RI) and the Consistency Ratio (CR) is computed in the equation (8).

After determining the weights of criteria based on the fuzzy AHP method, we now apply the fuzzy VIKOR method for ranking the competing alternatives.

Fuzzy VIKOR: The fuzzy VIKOR, an evolution of traditional VIKOR under uncertain conditions, employs linguistic terms to evaluate the competing alternatives. The following lists the FVIKOR protocols. The fuzzy VIKOR procedures are described in detail by the following steps.

Step 1: Define the criteria's fuzzy weights.

In this study, the fuzzy weights of criteria are calculated from the FAHP model.

Step 2: Build the fuzzy decision matrix according to the linguistic variable values in Table 4, equations (9) and (10).

Table 4: Linguistics variable values of alternatives.

The value of v is assigned as 0.5.

Step 7: Defuzzify triangular fuzzy number of the worse group score values ~Qj into the clear values. Rank the competing alternatives, ascending the order of the values S,R,Q.

Typically, the ranking order of S,R,Q is used to provide a group of compromised solutions.

Validation phase

This phase achieves the third objective of this research, which is to validate the ranking results using statistical methods. Validation is an important measure of numerous empirical studies to evidence the validity and accuracy of their results. The objective validation method is used to validate the findings of the English learning app rating. The statistical approaches (mean and standard deviation) will be used at this step to guarantee that the English learning app rank is organized correctly.

As shown in equation, the mean is calculated as the sum of all observed outcomes from the sample divided by the total number.

Standard Deviation (SD) is a measure that is used to quantify the amount of variation or dispersion of a set of data values, as presented in equation (21);

The mean and standard deviation will be used to ensure whether the proposed decision matrix ranking result is valid and undergoing systematic ranking or not. To validate the result, the six apps' scores are divided into three groups according to the ranked result based on integration of FAHP-VIKOR methods, such as that in the study. Each group includes two apps. Each group's results are expressed as a mean. By calculating the mean and standard deviation in comparison to the other groups, the validation procedure must demonstrate that the first group has the greatest score value. The second group’s mean and standard deviation values have to be lower than or equal to those of the first group. The third group’s mean and standard deviation values need to be lower than those of the first and second groups or equal to those of the second group. According to the systematic ranking results, the first group should be statistically proven as the highest among all groups.

Evaluation phase

This phase represents the final objective of this research. The evaluation phase is presented to demonstrate the performance of this study. The proposed method has been compared with the study of which is considered the most relevant study in the research area. To illustrate the comparison points and issues in the checklist benchmarking, the terms descriptions of the checklist comparison points are presented as shown below.

• LSRW skills: This point demonstrates the four skills of evaluation criteria in the English language and it is important to be included in the benchmarking checklist.

• Expert opinion: Selecting the right group of experts is a critical point for ensuring an accurate outcome.

• MCDM techniques: This point of comparison demonstrates whether this technique is included in the benchmarking checklist or not. This technique is regarded as a highly useful tool for assessing difficult real world situations since it allows for the comparison of many options based on predetermined criteria and improves decision quality through a more transparent, reasonable and efficient procedure [13].

• Ranking method: This point indicates whether the study addressed methods for ranking the alternatives.

• Weighting method: This point shows the technique used for gaining the weighting of criteria.

• Fuzzy set: It is viewed as an effective strategy for handling the ambiguous and imprecise information supplied by experts or decision makers in an MCDM process. Particularly when the provided information is arbitrary and ambiguous, triangular fuzzy numbers have demonstrated their effectiveness at addressing these problems. Therefore, this point of comparison is considered vital, especially when an expert’s opinions are subjective.

• Validation: It is an important stage to prove that the results of this model are valid and consist of the following sub methods.

• Mean: This point ensures whether the ranking result is valid and undergoing systematic ranking or not.

• Standard deviation: This statistical approach is regarded to be the one that is most frequently utilized to evaluate a method's efficiency. Consequently, the standard deviation must always be included when using the mean technique [14]. As a result, it is critical to include a checklist with this method.

• Evaluation: This point shows whether the benchmarking has been provided.

FAHP results

By calculating the consistency ratio based on the Equation mentioned in the research methodology, the (CR) is less than (0.1). Which means that the pairwise comparison matrix is reliable and the FAHP model's output is adequate and satisfactory [15].

In this research, three experts (english teachers) were asked for their preferences to evaluate the multiple criteria of english learning application by using the FAHP technique. As mentioned before in research methodology, the expert’s preferences, which are linguistic terms such as "Extremely strong", "Intermediate", etc. are transformed into triangular fuzzy numbers. For example, when such an expert chooses the term "Strong" which means will be represented as in triangular fuzzy numbers and so on for the remaining terms. These fuzzy preference weights of all three experts are grouped together and divided by the number of these experts in order to get the average weights (Grouping weights). Then, these weights will be employed in the FVIKOR model to rank alternatives Table 5. Illustrates the FAHP results that were obtained from all experts [16].

Table 5: FAHP results.

The Table above shows that the main criteria listening and speaking with its sub criteria has the major weight (0.67) by comparing it with other main criteria. Followed by criteria reading with its sub criteria with (0.22) weight. Last but not least, writing criteria with also criteria with weight (0.11) (Figure 3).

In order to give more importance to the main criteria, we multiply every main criteria with its sub criteria’s weight. As a result, we get the new weight for each criteria. Table 6 shows the new weights.

Table 6: The new weights of criteria.

FVIKOR results

The FVIKOR method's basic principle is to rank various alternatives based on the compromise solution method [17]. The checklist form in research methodology, Table 7 is transformed into as shown below.

Table 7: Triangular decision matrix.

In the above decision matrix, experts are considered as a committee to evaluate the alternatives. As mentioned in research methodology in in FVIKOR techniques, triangular linguistics terms are used to express the rating of each alternative and the weight of each criteria [18]. For example, when such an expert selects a "High" term, this will be transformed into a triangular fuzzy number which corresponds to (0.3, 0.5, 0.75) and so on for the rest of the alternatives.

After the previous step, it is important to calculate S and R (equation 15-19 respectively in Chapter 3) which leads to compute the Q value, which represents the final rank. Tables 8 to 12, respectively, show the values of S, R, Q and the final rank.

Table 8: S results.

Table 9: R results.

Table 10: Q procedures.

Table 11: Q procedures cont.

Table 12: Deffuzification and final rank.

According to the final rank in the Table above, the best English application for young english students is Montessori, which takes the first place among the other applications, followed by spelling bee, fun english, lingokids, first words and fun with-flupe, respectively.

Validation

Validation is an important stage to prove that the results of this model are accurate and valid. To make sure that the english learning applications are ranked in a methodical manner, the statistical approaches known as the mean and standard deviation are used as mentioned in the research methodology, equations.

The suggested decision matrix results are systematically ranked and validated using the mean. According to the ranking results using the FVIKOR approach, which is similar to the research the ranking of the six applications is separated into three groups. The average for each group is used to represent the results.

By computing the mean and comparing it with that of the other groups, the validation procedure must demonstrate that the first group has the greatest score value. The second group's mean must be lower than the first group's. The third group's mean, however, must be lower than that of the first and second groups [19]. The findings of the systematic ranking indicate that the first group should be statistically demonstrated to be the highest of all groups.

The ranking results of english learning applications are validated by splitting the ranking result into three equal groups of two applications. To provide a systematic ranking of english learning applications, the mean is computed for each group. The validation results for the external aggregation group decision making after the normalization and weighting processes for the data of the first, second and third groups of english learning applications show that the first group's mean value (0.68) is greater than the second and third groups' mean values (0.55 and 0.54, respectively). The second group's mean value (0.55) is greater than the third group's (0.54). As a result, the rankings of internal and external group decision-making are systematic and objectively valid. Table 13 illustrates the validation results Figure 4.

Table 13: Validation results.

Evaluation

After determining and defining the checklist comparison issues in section 3.6 of research methodology, the proposed method was compared to the most relevant study in this research field on these particular issues. According to the findings of the literature review, the study of is the most relevant study. Following the description of the checklist comparison issues, the checklist comparison between the proposed and benchmark studies is shown in Table 14.

Table 14: Checklist benchmarking.

The checklist benchmarking points have been discussed between the proposed work and the related study. According to this table, two issues were found: The fuzzy set issue, which is considered the most important comparison point because of its ability to manipulate the subjectivity in an expert's preferences and the standard deviation issue, which represents an extra validation method to make the results more valid and concrete.

As a result, the findings show the superiority of the proposed work over the benchmark method with a (22.3%) score. On the other hand, the proposed work addressed all the issues with a percentage of 100%. As shown below in Table 15.

Table 15: Comparison between the benchmark and proposed research.

Conclusion

In this study, an integrated FMCDM method was presented for evaluating and ranking six English learning applications based on the KSPK standards: Listening, speaking, reading and writing criteria as well as their sub-criteria. This hybrid method is made up of FAHP and FVIKOR. A triangular fuzzy number is adopted in both methods to make the evaluation procedure more precise and flexible for decision makers. First, FAHP is employed to evaluate each criteria and determine the relative significant fuzzy weight. FVIKOR is used in the second step to rank the alternatives. According to the results of this study, the best English application S for young english students is Montessori, which takes first place among the other applications, followed by spelling bee, fun english, lingokids, first words and fun with-flupe, respectively. Then, the findings were objectively verified by statistical methods and benchmarking. However, the main contribution of this research is evaluating and ranking English learning applications for learners who are more than five years old. As a result, this research helps both the learners and the instructors to select the right and most reliable application.

Limitations

Because of the increasing complexity of decision making problems, a small group of experts may not be able to provide a comprehensive evaluation. So finding more english experts with excellent skills and enough experience in teaching English to young children was the limitation of this study.

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