The Application of Fuzzy Analytic Hierarchy Process in High School Math Teachers Ranking

Assessment in the field of Education like other areas as a ancient history and it has various applications such as selecting "The eminence teachers" conceived. Selection of teachers depends on performance for selection criteria to generate a final ranking. The Performance criteria of teacher's are evaluated by experts and it is always ambiguous. This ambiguity of human decision-making can be deal with the fuzzy decision support model. In this paper by fuzzy analytical hierarchy process (fuzzy-AHP) a model is developed to search the criteria’s for the evaluation of best middle school math teachers, which can tolerate vagueness and uncertainty of human judgment. At the end, a case study is presented to make this model more understandable.


Background
According to Moayeri et al. (2015), the most important organizational system in every country which has a great influence on the countries' future is the Educational system.Moayeri et al. (2015) considers students and teachers as the two main pillars in the academic institutions, without which the development of the education system is impossible.Accordingly, the process of choosing and distributing the teachers are very important since they play the most important role in the field of education that therefore make the teachers' employment status an important issue to pay attention to.Although the evaluation and selection of the eminent math teachers is a common process being practiced in academic institutions, it is yet a complex practice (Moayeri et al., 2015).As Moayeri et al. (2015) reported, lack of qualified math teachers and the need to identify the qualified ones from a pool of math teachers for giving them benefits such as financial one has made the Iranian secondary education to face problem.Identifying the qualified good teachers can http://www.ispacs.com/journals/metr/2016/metr-00084/International Scientific Publications and Consulting Services be done by using a decision support system (DSS) made by the help of some appropriate techniques or by the help of human experts (Moayeri et al., 2015).According to Moayeri et al. (2015), Secondary education enhances the quality of the individuals' lives, and the performance and the quality of both teachers and students by providing knowledge.Moayeri et al. (2015) considers the evaluation of math teachers important.According to him, Multi criteria decision making (MCDM) appears as the result of the math teachers' several conflicting criteria and difficulty in deciding about their ranking.Moayeri et al. (2015) used the criteria such as creativity, attitude and personality for the evaluation of the middle school math teachers ranking through presenting a Fuzzy-AHP model.

Aims of the study
The main aims of the study conducted by Moayeri et al. (2015) was first to find the performance measurement indicators to evaluate the Iranian best math teachers, teaching at middle schools, and second to create a suitable Fuzzy-AHP model to evaluate the middle school math teachers.

Introduction to Analytic Hierarchy Process
According to Moayeri et al.(2015), in many stages, the more preferable way to reach the best decision is to use an analytical way.As he asserted, the reason why an analytical way is required for making a successful decision is that most of the time qualitative variables exist beside the measurable variables, or people are expected to give preferences to the best choices among the many choices available.In case of determining the different evaluations and factor weights, the decision makers may often face difficulties in some situations (Moayeri et al., 2015).In such situations, as Moayeri et al. (2015) recommended, the Analytic hierarchy process (AHP) can be applied.According to him, in Analytical Hierarchy Process, the decision makers begin by making wise comparisons that leads to the determination of factor weights.According to Saaty (1980) [11], the best alternative to choose is the one with the highest total weighted score.

Triangular Fuzzy Number and Fuzzy set
In dealing with the human thought's vagueness, Zadeh (1975) [16,17] introduced the theory of Fuzzy set that was based on rationality of uncertainty.As Moayeri et al. (2015) elaborated on the fuzzy theory, the ability of showing vague data is the main role of the fuzzy set theory.She also defined fuzzy set as "a class of objects with a membership function ranging between zero and one [which] was specifically designed to mathematically represent uncertainty and vagueness" (p.2-3).Fuzzy set theory based on her definition carries out the groups of data with unclearly-defined boundaries (i.e.fuzzy).According to Moayeri et al. (2015), any theory or methodology that uses "crisp" definitions like arithmetic, programming, or classical set theory may be "fuzzified" if the concept of a crisp set be generalized to a fuzzy set with vague boundaries.According to Bohui (2007) [3] the strength in solving real-world problems is the benefit of the analysis methods and extending crisp theory over the fuzzy techniques, which predictably requires some extent of imprecision in the parameters and variables measured and processed for the case of application.As Moayeri et al. (2015) stated, "A triangular fuzzy number (TFN) is the special class of fuzzy number whose membership is defined by three real numbers, expressed as (l, m, u)" (p.3). Figure 1 illustrates the structure of a Triangular Fuzzy Number (TFN).Lee et al. (2009) [12] exemplified the triangular membership function as follow.

Organization of the Study
This study has been conducted to create a Fuzzy-AHP decision making model in order to evaluate the best middle school math teachers.To do so, the following sections include review of the literature, research methodology containing the stages of creating the model, findings of the empirical study and eventually the conclusion.

Review of existing work
As Moayeri et al. (2015) asserted, the observations have shown that compared to other methodologies, the Fuzzy-AHP method was used widely in decision making.Pan (2008) [10] cleared that the reason of using this method was to choose the best bridge construction method from the pool of alternative methods, while avoiding the existing inconsistency.In most of the studies as Moayeri et al. ( 2015) declared, fuzzy-AHP has been extremely used in finding the solution for many difficult decision making problems.According to Yasemin (2006) [14], the reason of creating the Fuzzy-AHP and its extensions was to choose the key abilities in technology management.One of the usages of the fuzzy AHP approach, as Kahraman at el (2004) [8] declared was in evaluating the computer integrated manufacturing alternatives.According to to Bohui (2007) [3] and Hua (2008) [7], Fuzzy Integrated Analytic Hierarchy Process Approach was also used for the case of vendor selection and Multi-criteria Supplier Evaluation.According to Buckley (1985) [4] and Chang (1996) [6], in many studies it has been observed that Fuzzy AHP provides evidences that compared to the Classical AHP method, shows more efficiency in treating human judgments.; R.I. is random index and the value of R.I. can be got with Table 2.  alternatives she used in her study were the eminent math teachers in Kerman city.Table 3 represents the description of the selected criteria and sub-criteria in the study by Moayeri et al. (2015).

Construction of the Detailed Hierarchy of the Problem
Moayeri et al. ( 2015) introduced a hierarchy that was built from the top level (evaluating the math teachers' ranking) through the intermediate levels (sub-criteria and main criteria that subsequent levels are depend on them) to the bottom level (list of math teachers).The hierarchy is described in detail in Figure2.Incorporating the whole criteria, sub-criteria and alternatives to the research problem, the hierarchy was created.

Results and Discussions
The detail of the steps of Fuzzy-AHP model described in section 2.1.2to 2.1.4are explained elaborately using the data collected from eminence math teachers in Kerman.

Illustration of the Fuzzy-AHP Model
Once the hierarchy was established and a series of questions were asked to direct pair wise comparisons, each expert performed a pair wise comparison.Hence pair wise comparison matrix and main criteria weights from the first expert's judgment can be expressed in Table 4.The results indicate that the priority of Personality is the maximum followed by math teachers.following the same procedure the weights of the sub-criteria are calculated and Further the sub-criteria overall weights are multiplied by the corresponding main criteria weighs to obtain final weight of the sub-criteria as results are described below in tables 5,   The results of the overall sub-criteria weights indicate that the priorities are highest in Responsibility followed by personality criteria .teachersfeedbacks of three alternative math teachers are collected with http://www.ispacs.com/journals/metr/2016/metr-00084/International Scientific Publications and Consulting Services respect to each of the sub-criteria using fuzzy linguistic preference scale and the corresponding weights are generated as described in table 9.

Findings and Discussions
From the main criteria and sub-criteria weights in the tables is can be inferred that there exists variation between the priorities of the main and sub criteria mentioned in model.It is further observed that the priority of the main criteria "Personality'' is highest followed by ''math teachers''.In case of sub criteria the priority is highest for ''Cognitive'' under ''Creativity'', ''Responsibility'' among ''Personality'' and ''matching jobs and career'' among ''Attitude''.when it comes to the alternative math teachers it is found that the Responsibility of teacher 1, Cognitive of teacher2 and non-cognitive of teacher3 are the best.Finally from the final score of the alternative math teachers it has been observed that the overall score of teacher1.

Conclusion
Decision making is so essential and common in our everyday life, since we need to take different economic, social and other types of decisions in every moment of our life (Moayeri et al, 2015).In this situation, as Moayeri et al.  2015) used a pair wise comparison between alternative to alternative for each criteria and sub criteria and eventually the obtained weights were used for decision making about teachers ranking as "T1, T2 and T3 means the teacher T1 is the best" (p.15), this study can be broadly worked upon in future for the greater numbers of teachers and criteria and the comparison with other fuzzy MCDM methods such as fuzzy TOPSIS method can be made (Moayeri et al.2015)

2 . 1 . 3 .
importance of each criteria in Pair wise comparison and , are the minimum value, most plausible value & maximum value of the triangular fuzzy number.Generation of Criteria and Sub-Criteria weight According to Buckley et al.(1985) [5], the Normalization of the Geometric Mean (NGM) method is used to calculate weights from the fuzzy pair wise comparison matrices that is computed by: (2.6) (3) The consistency of comparison matrix Moayeri et al. (2015) declared that while comparison matrix lead to mathematical critical thinking, critical thinking and the consistency of comparison matrix must be tested.Accordingly, after the estimation of the importance of indexes by experts, the consistency and therefore reliability of the estimation results obtained by the different experts must be checked.Hence, in order to check the consistency and consequently the reliability of the different experts' critical thinking, after the use of AHP, the comparison matrix should be computed (Moayeri et al, 2015).As Moayeri et al. (2015) suggested, consistency ratio C.R. can be used to check the consistency of comparison matrix.CR = CI/ RI (2.7) C.I. is consistency index and (2.8) Determining all the important criteria and their relationship with the decision variables is one of the essential steps of the proposed model by Moayeri et al. (2015), due to the probable influence of the selected criteria and sub criteria on the final choice.Moayeri et al. (2015) in her project selected the criteria and sub-criteria based on the related literature and by the help of the expert's opinion.The http://www.ispacs.com/journals/metr/2016/metr-00084/International Scientific Publications and Consulting Services

Figure 2 :
Figure 2: Detailed hierarchy of the problem (2015) suggested, when criteria are conflicting in nature, the multi criteria decision making (MCDM) can be useful in helping us making decisions about ranking.Moayeri et al. (2015) defined AHP as "an effective problem solving multi criteria decision making method"(p.15).According to Moayeri et al. (2015), different factors may involve in the heart of decision problem that must be evaluated by the help of linguistic variables.Accordingly, the numerical values of linguistic variables are directly used for evaluation in classical AHP.If the decision making process happens in a fuzzy environment, we use fuzzy numbers to evaluate some of the decision makers' deviations (Moayeri et al, 2015).This situation as Moayeri et al. (2015) declared, can be controlled very well by the Fuzzy AHP (FAHP).In order to make decision about the math teachers ranking in secondary, FAHP method was used by Moayeri et al. (2015).For demonstration purpose, Moayeri et al. (2015) considered only 3 criteria and 3 alternatives.Moayeri et al. (

2 Methodology 2.1. Development of Fuzzy-AHP model in multi criteria decision making 2.1.1. Conceptual Hierarchy of Fuzzy -AHP model
Analytical Hierarchy Process, which is built from the top level (the general purpose of the problem) to intermediate levels (i.e.criteria and sub-criteria that subsequent levels depend on them) and finally to the bottom level (the alternatives' list), as Moayeri et al. (2015) declared, begins by designing the whole decision making problem hierarchy.Accordingly, in the lower level of hierarchy, each criterion is compared based on the criteria in the upper level of the hierarchy.For the criteria in the same level, as Moayeri et al. (2015) stated, pair wise comparison is employed.The hierarchy of a decision making problem is described in Figure2.

.2. Fuzzy Pair Wise Comparison Method According
toMoayeri et al. (2015), the pair wise comparison evaluation is done after the creation of the hierarchy.Accordingly the comparison takes place between the whole criteria on the same level of the hierarchy and each of the criterions on the previous level.Using Fuzzy linguistic terms in the scale of 1-9 displayed by the Triangular Fuzzy Numbers in Table1, a pair wise comparison was done in the study conducted byMoayeri et al. (2015).

Table 4 :
Calculated final weigh of main criteria

Table 5 :
Calculated overall and final weigh of creativity sub -criteria

Table 6 :
Calculated overall and final weigh of personality sub -criteria International Scientific Publications and Consulting Services

Table 7 :
Calculated numerical and final weigh of attitude sub -criteria

Table 9 :
Weights of alternatives Fuzzy Score of alternative eminence math teachers, namely teacher1, teacher 2 and teacher3 of Kerman along with the final score are expressed in table10.

Table 10 :
Final weights of alternatives International Scientific Publications and Consulting Services