A comparison of classical and fuzzy data envelopment analyses in measuring and evaluating school activities

Transcription

1 TJFS: Turkish Journal of Fuzzy Systems (eissn: ) An Official Journal of Turkish Fuzzy Systems Association Vol.5, No.1, pp , A comparison of classical and fuzzy data envelopment analyses in measuring and evaluating school activities Emre Demir* University of Hitit, Osmancık Ömer Derindere Vocational School Department of Technical Programs Çorum, Turkey emredemir@hitit.edu.tr *Corresponding author Abstract Data Envelopment Analysis (DEA) is an exceedingly useful method in measuring performance but limit of its efficacy is fairly affected by mistakes and uncertainties in the data due to the fact that it is a very responsive method. Some potential mistakes in the data may largely change the activity measurements of Decision Making Units (DMU). Classical DEA assumes that input and output values should be certain values. But it is not always possible to work with certain values due to various reasons. Therefore, classical DEA models are not sufficient. Fuzzy Data Envelopment Analysis (FDEA) by Zadeh's Fuzzy Theory is beneficial in terms of getting more consistent activity scores. Basic aim of the current study is to compare classical DEA and FDEA by means of an application for educational researches. For that purpose, relative activities of the 25 high schools (S) in the education year were calculated by means of DEA and FDEA, and the results were compared. Fixed yield Charnes-Cooper- Rhodes (CCR) Model (1978) for input measurement and variable yield Banker- Charnes-Cooper (BCC) Model (1984) for the scale were enjoyed in the study. As a result of the study, Fuzzy Theory is strongly recommended to be practiced for DEA problems with uncertain data in the schools in order to get more secure results in activity measurements. Keywords: Fuzzy data envelopment analysis, interval data, efficiency, productivity, high schools. 1. Introduction Secondary schools are significant parts of the Turkish Education System and have important roles to play in development and improvement of the country. First and foremost, these schools should work more efficiently for the sake of providing better education. Therefore, it is a must that educational institutions should always be renovated and current conditions in these schools should be determined. Having no limitless resources puts forth that they should be used more efficiently and productively. 37

2 DEA was initially designed by Farrell's (1957) Frontier Production Function and improved by Charnes, Cooper and Rhodes (CCR) in order to measure and compare the technical productivity of the state institutions. Now it is one of the methods to use to compare efficacy of the DMUs. In the first DEA application (Charnes et al., 1978), the purpose was to measure comparable productivity of the schools. Basic assumption of the DEA in measurement of the total technical efficacy of the DMU is the assumption of constant returns to scale (CRS). According to this assumption, the DEA model is also called CCR Model. The assumption was later changed by Banker, Charnes and Cooper and variable return to scale (VRS) was improved. In this model, measurement of pure technical activities was achieved by cleansing scale differences of the DMU. Banker, Charnes and Cooper (BCC) was developed in 1984 and the leading difference between BCC and CCR is that there is not an obligation for DMU's scale active in CCR model. As a result, BCC models measure only local technical efficacy for each DMU. While in order for DMU to be active in CCR model, it should be both technique active and scale active, it is enough to be technique active in BCC model. In this regard, CCR model measures total efficacy under constant returns to scale but BCC model measures technical efficacy under variable returns to scale (Bowlin, 1998). Lots of approaches have been recommended for fuzzy effectiveness measurement. However, interval data FDEA technique depending on α- intercept method was used in this study due to present uncertain data. During the data collection process, all the data which would be used for all of the inputs, were initially gotten from provincial directorate of national education and later from the schools. Then the collected data were conclusively compared. However, it was noticed that there were differences in the same data. DEA is a data-sensitive method and so, from the perspective of any existing mistake in the data, relative efficacies of the schools were primarily calculated by means of classical DEA. And then interval data FDEA technique was enjoyed and the results were compared. EMS (Version 1.3) package program was run for two different conditions and efficacy values were determined. For the purpose of efficacy measurement, numbers of the students, teachers and classes were described as inputs, and Transition to Higher Education Examination (YGS), Undergraduate Placement Exam (LYS) success (placement) rates, YGS point averages, all points of the LYS Maths-Science (MS), Turkish-Maths (TM), and Turkish-Social (TS) Sciences were described as outputs. The terms of productivity, efficacy, classical DEA and FDEA and the theoretical bases were located in the second and third parts, which are actually theoretical part of the study. In the application section in the fourth and fifth parts, twenty-five high schools in Çorum, from which the data belonging to education year were gotten, were selected as DMUs, and the related data were analyzed by means of classic DEA and FDEA techniques. Since variable return to scale is mostly used to compare profitoriented businesses, the evaluations as a result model were made according to CCR model under the assumption of constant return to scale, which is commonly used in the comparison of public performance. But BCC model efficacy measurement results were 38

3 also provided for the FDEA and classical DEA comparison. The conclusive sixth section, the efficacy rates of the schools were evaluated and both methods were compared in terms of educational researches. 2. Data envelopment analysis (DEA) Even though the terms, productivity and efficacy, are interchangeably used, they are different concepts. Drucker's definition is commonly enjoyed to determine the difference between these terms (Isaac-Henry et al., 1993). According to Drucker, efficacy is related to the correct completion of the tasks (Drucker, 1994). A productively completed work does not mean that the work was done in an efficient way and efficacy cannot be realized without productivity. In this regard, it is simply claimed that there is essentially efficacy on the focus of productivity (Kök, 1991). While productivity is interested in how well the production bases are employed, efficacy regards to what extent the purposes are actualized Productivity When a general definition is needed, productivity is the relationship between the output of the production system or service system and the input which is used to create that output. Therefore, productivity is an efficient usage of resources in the production of various goods and services (Prokopenko, 2001). Productivity is a comprehensive measurement of to what extent an institution taking a close position to the following standards below. Aims: Degree of the actualization of the purposes. Effectiveness: To what extent the resources are used efficiently to get useful outputs. Efficacy: Those which are actualized in comparison with those which are potentially to be actualized. Comparability: Actualization of productivity performance within time. The term, productivity, which was firstly used in an article by Quesnay in 1776, is mostly defined as Productivity = Output/Input (Prokopenko, 2001). When the condition of single input-single output is regarded, productivity of any DMU is termed as the output's ratio on the input. In other words, the pitch of the ray which starts from the point (0,0) and goes through the point representing the DMU, provides productivity rate for that DMU. Increase in the pitch of the ray shows the rise in the productivity. In the Figure-1, in which single input-single output condition is represented, it is indicated that even though less input is used and more output is produced when compared to the others, F DMU has got the highest productivity rate among various DMUs. F DMU, which can produce the most output by using the least input, is of the highest productivity. The scale size of the DMU having the highest productivity rate is defined as the most Productive Scale Size (MPSS) by Banker. 39

4 Figure 1. Productivity (Tarım, 2001) The approach of simple ration becomes insufficient in measuring productivity of multiple input and multiple output processes. The term of total factor productivity is profited in order to avoid aforementioned inconveniences of the simple productivity measurement, which is a ratio analysis. In total factor productivity, inputs of the production process are added and limited into single-input factor (virtual input) and sum of the outputs to the single-output factor (virtual output). Then, they are evaluated by regarding the ratios of the total input and output factors. The weakest link of this approach is that it does not provide any hint dealing with how to sum the input and output factors having different properties. To put it differently, the reason is actually the unknown parameters which are to be applied for the factors (Tarım, 2001). Total Factor Productivity = Total Output/Total Input Most of the DMUs produces more than one output by using multiple inputs. Therefore, relying on the inconvenient sides of the approach, DEAs have been put into practice since DEA is of a suitable structure for performance measurement in production environments with multiple inputs and outputs (Kettani et al., 1992) Efficacy The efficacy, which is in close relationship with the terms, productivity and effectiveness, describes that how good output can be produced by using available inputs. Efficacy determines degree of optimal use of the resources in producing the outputs (Akal, 2000). Technical efficacy, which is defined as part of production function, is a production type having no waste. (Akdoğan, 2001). Efficacy indicates the degree of producing the needed output by means of the available input and the present degree of usage. Efficacy measurement clarifies the relationship between the input and output and the degree of the use of the resources in comparison to total quality. That indicator should figure out where the non-productiveness is derived from (Prokopenko, 2001). Technical efficacy may be defined like that a high production of output from a certain degree of input or production of the same output by using the least input. (Töngür, 2001). The scale efficacy is also described as the production in a suitable scale. (Abbott and Doucoulıagos, 2003). 40

5 The success in the production having the optimum scale is called as scale efficacy and technical efficacy together with scale efficacy forms DEA efficacy (total efficacy). Accordingly, it is represented as follows (Cingi and Tarım, 2000): DEA Efficacy (Total Efficacy) = Technical Efficacy * Scale Efficacy Figure 2. Technical Efficacy and Productivity (Tarım, 2001) A and B observations given in the Figure 2 is located in the production frontier and defined as technical efficient. The observation P actualized the same output level with A by needing more inputs. On the other hand, even though P as DMU used the same amount of input with B, it produced less output. Hence, it is possible to say that P was set as in technical inefficiency. The productivity of these three observations are calculated in the ratio of input/output and, as a result, B is more productive in comparison with P and A, and P is the least productive DMU. Despite the fact that the observation A is regarded as technical efficient, its efficacy is less than B's (Tarım, 2001). The observation P may increase its technical efficacy and productivity by slipping forward to the observation B since sequentially it is closing to production frontier and its input/output ratio is rising. The observation A is slipping to the observation B and may increase its productivity by conserving its technical efficacy and having an advantage from the scale. Relatively the most productive observation C is of the most productive scale size (MPSS) as defined by Banker. When DMUs, C and D, are compared, it is observed that D wastes the resource due to the fact that D is not above production frontier. Nevertheless, D is of the same input scale with C having the most productive scale size. As a result of that, D is obviously in the optimum scale but cannot use the resources in an efficient way. As consequence of comparison of A and D, it is clear that A is technically efficient but D is not. On the other hand, D's productivity is above A's. In other words, when a technically efficient observation is compared to the one with technically inefficient, it may seem as nonproductive. By means of this simple sample, it has been seen that the notions, technically efficient and productivity, do not involve each other. When the observation F is analyzed, it may be commented that its productivity will increase only if it preserves its technical efficacy, and its scale is broaden. This condition is called increasing returns to scale (IRS). The observation E will increase its 41

6 productivity when it keeps its technical efficacy and decrease its scale. And this condition is named as decreasing returns to scale (DRS). In the production frontier, the state of having all increasing, decreasing and constant returns to scale together is called as variable return to scale (VRS) (Tarım, 2001) Data envelopment analysis Ratio Analysis, Regression Analysis and DEA are the leading techniques that are mostly used in the productivity and efficacy analyses. DEA is a non-parametric activity and of a mathematical linear programming basis. Non-parametric efficacy criterion are mainly grouped in terms of input and output oriented ones. The input oriented ones look for to what extent they should decrease the inputs of the inactive DMUs for any output levels. Similarly, output oriented ones regard to what extent they will decrease the outputs of the inactive DMUs for any input combinations. (Yolalan, 1993). The input and output DEA models are not basically alike, and input oriented DEA models are to be used for determining the most suitable input combinations to produce a certain output combination in a most actively way. But output oriented DEA models are enjoyed to seek for how many output combinations they will get by certain number of input combination (Charnes et al., 1981). DEA analysis is mostly used for non-parametric methods. This method compares production units which are assumed as homogeneous. Just after efficacy frontier is regarded as the best observation, the other observations are evaluated according to that efficient/active observation. Therefore, efficacy frontier is not an assumed state but a real observation actualized. Because the efficacy frontier is described in that way, random error is not employed in this method (Banker and Thrall, 1992). DEA's measurement of relative efficiency is of two stages as follows (Yolalan, 1993): 1) It determines the best observations (or the DMU forming the efficacy frontier) producing the most output combinations by using the least input combination in an observation mass. 2) It takes the aforementioned frontiers as reference and rationally measures the distance of the inactive DMUs to that frontier (or their efficiency levels). DEA is a mathematical technique which is used to measure the performance of hospital, schools and mass of banks. Such kind of similar units are known as Decision Making Units (DMU). DMU may include both the institutions like universities, schools, branch banks, hospitals, police stations, tax offices, jails and individual applications like activities of a doctor (Ramanathan, 2003). DEA may define the reason behind the efficacy condition while it may also describe the amount of the inefficiency of the DMUs. That will contribute the process of strategy 42

7 finding just after the measurement of the efficacy. Thanks to that particularity of the method, the administrator can decide on to what extent the amounts of the input and output will be increased in the inefficient unit(s) by just checking the results of the study. By means of that approach, all the units are enveloped by the efficient frontier and there is not any unit out of that frontier. Piecewise linear active frontier created by the DEA made it possible that it is known as data envelopment since it consists of all points dealing with data analysis (Cooper et al. 2000). Banker, Charnes and Cooper (1984) asserted alternative sets of the assumption of the variable return to scale in their studies while Charnes, Cooper and Rhodes (1978) supported a model based on the assumption of input oriented and constant return to scale. As a consequence, DEA has been used both under the constant and variable returns to the scale. Likewise, that method measures the efficacy levels according to the approaches referring both to get maximum data output by means of the input and to attain data output by the least input (Coelli et al. 1998). The constant return to the scale, Charnes-Cooper-Rhodes (CCR) Model (1978), which has been commonly employed in the comparison of public performance and for the evaluations in the current study, is mentioned below Input oriented CCR model and formulation for linear programing Fractional programing model created by Charnes et.al and co-linear (CCR) are given below. As following these models, a dual model including some important managing information has been formed (Tarım, 2001). Y rk : The number of the output by the DMU, k X ik : The amount of the input used by the DMU, k u rk : The weight of the output, r v ik : The weight of the output, i objective function: subject to: max h s i1 s r1 k m i1 u v rk ik rk rj r1 m u v ik Y X ij 1 Y X rk ik j = 1,, N, r = 1,,s, i = 1,, m urk 0, vik 0 (1) 43

8 The fractional programing model given in Model (1) can be turned into linear programing model given in Model (2) which soluble by means of Simplex algorithm. That model in primal form is known as multiplier model in DEA literature. objective function: max k s r1 u rk Y rk subject to: s u m i1 Y v ik X ik m rk rj r1 i1 1 v ik X ij 0 j = 1,, N, r = 1,,s, i = 1,, m urk ε, vik ε (2) When model (2) is made as dual, the available result becomes model (3). That dual model is called as envelopment model in DEA literature (Tarım, 2001). objective function: subject to: N Yrj jk j1 rk min k Y r = 1,, s k X - X 0 i = 1,, m ik N j1 ij jk 0 j = 1,, N (3) jk The aforementioned problem is processed n times in order to define all the efficacy scores of the DMU. The weighted input and outputs are chosen in order to make the best all efficacy scores of each DMU. In general, if efficacy score of a DMU is 1, it becomes active and if it is below 1, it is inactive (Talluri, 2000). On the other hand, when each model is turned into dual and solved, it also becomes clear that according to which units they are inactive and what they need to be active in input and output levels (Lang et al. 1995, Ulucan, 2002). 44

9 3. Fuzzy data envelopment analysis (FDEA) Cooper et al. is one of the leading scientists who studied on DEA with fuzzy data. The model that they created is called IDEA. Model IDEA can change the non-linear programing problems into linear programing problem by means of scale conversions and variable changes (Cooper et al. 1999). Sengupta (1992, 1993) suggested two membership functions, Linear Membership Function and Non-linear Membership Function, for fuzzy mathematical programing model. DEA is in linear state; Notation ~ indicates that target function and n constraint are fuzzy. By making the constraints as fuzzy, there may occur tolerations. It also assumed that there is an order level ( ) for efficacy score and maximum levels for violation of the tolerations (Kahraman and Tolga, 1998). There are four different approaches such as Tolerance Approach, Fuzzy Recovery Approach, α- level Oriented Approach, Fuzzy Gradation Approach dealing with solving FDEA problems (Güngör and Demirgil, 2005). Oruç (2008) analyzed ten FDEA models including Despotis-Smirlis, Cook-Kress-Seiford, Cooper-Park-Yu, Kao-Liu, Saati- Memariani- Jahanshahloo, Saati-Memariani, Lertworasirikul, Lertworasirikul- Fang- Joines-Nuttle, Guo-Tanaka, Leon-Liern-Ruiz- Sirvent in his study. Kao and Liu (2000, 2003) suggested to change fuzzy data into offset data by using α- level masses, and so could create a solution that they could take advantage of a family of classical certain DEA models. Saati et al. (2002) took that suggestion into consideration and made fuzzy CCR model into an offset programing model by defining it as programing problem and using α-level. Wang et al. (2005) improved an interval data DEA couple by enjoying DEA technique in the offset data and noted a fuzzy efficacy measurement. Therefore, interval data programing model can be solved like a definitive linear programing model and for each DMU, an efficacy score can be made by means of each α-level (Deniz, 2009). Kao and Liu (2000) improved a method for DMU with fuzzy observations, which would made FDEA model into certain DEA model series. That method could also employ α-levels and Zadeh's (1965) extension principle to measure efficacies. They changed DEA model into a linear model family by means of α-levels. (Güneş 2006, Deniz 2009, Demir and Demir 2013) 3.1. FDEA Linear programing formulation In a condition that all inputs and outputs are not totally available due to uncertainties, L L these values are only known as x 0 and y 0 and [ x, x ] and [ y, ] and refer ij that they are between these top-down limits. FDEA with fuzzy interval data, model in rj L ij U ij L rj U y rj 45

10 which limited data is used for efficacy measurement, is defined as following: (Kao and Liu, 2000, Wang et.al. 2005, Güneş, 2006). (5) max U j0 s r1 m i1 u v r y U rj 0 L i xij0 s.t s U u r y rj U 1 j r 1 m L vi xij i1, j = 1,, n, ur ε > 0, vi ε > 0, r, i (6) s.t max U j L j0 s s r1 m i1 u v r y L rj0 U i xij0 r1 1 m u r v x i i1 y U rj L ij (7), j = 1,, n, ur ε > 0, vi ε > 0, r, i (8) When all DMUs are in the best production frontier, while the most probable lower limit shows relative efficacy, the most probable upper limit indicates relative efficacy. These potentially form the best relative efficacy interval data [ ] (Despotis and Simirlis 2002, Entani et al. 2002, Wang et al. 2005). 4. Application of the data envelopment analysis DEA, which was theoretically mentioned above, was applied in twenty-five high schools located in Çorum and the data belonging to education year was used in the research. Three inputs and five outputs were defined in these schools. The constant return to scale, CCR model (1978) and variable return to scale, BCC model (1984) provided by the package program, EMS, were used in the study. The input oriented models were chosen since direct intervention in the input variables of the schools was not possible. 46

11 4.1. Choosing the DMUs In terms of the reliability of the research, the number of the DMUs, which were taken into consideration, should be at least m+p+1 or twofold of the sums of the inputs and outputs when the number of the inputs was named as m, and the number of the outputs was called p (Boussofiane et.al. 1991). According to Vassiloglou and Giokas, DMU should be threefold of the sums of the inputs and outputs. (Vassiloglou and Giokas, 1990). The twenty-five schools in Çorum was determined as DMUs, which are of homogeneous structure, similar bodies, and whose efficacies would relatively be counted. Some schools whose data was not available and students could not get enough LYS scores from their fields of studies like Maths-Science, Turkish-Maths, Turkish- Social Sciences to enter universities in Choosing the Inputs and Outputs Physical infrastructure, classroom sizes, the number of the teachers are significant parts of the current study in terms of comparing the schools which are having similar conditions. Initially, six inputs and seven outputs, which were already used in the literature, were described in the efficacy measurement of the DMU schools, which would be the bases for the research. However, the inputs and outputs were decreased by considering necessary correlations and dissociation abilities. The variables of the inputs and outputs were given in Table-1. Table 1. The variables of the inputs and outputs used in the literature input variables output variables - Number of total student -Number of Graduate student - Number of total teacher -The average final grade for graduate students - Number of Classroom -YGS MS score averages - Number of Classroom -YGS TM score averages branches -YGS TS score averages - Number of Laboratories -Number of the students entered in a university - Number of Teaching -The average of the students entered in a materials university The classroom branches, the number of the teachers, students, and classrooms were defined at first hand, which are the factors to describe the quality of the education at the schools. But because of higher correlation between the numbers of the classrooms and branches, only the number of the branches was taken into consideration. Even though a schools has got high quality teachers and teaching materials, if a teacher is of a hundred students with only one material, there is not obviously a quality education at that school. (Demir and Durakoğlu, 2013) The number of the teaching materials and laboratories were not received for consideration due to the fact that multiple choice exams are used for the student selection after high education in Turkey and the content-scope of the questions are only 47

12 problem solving and memorization. The number of the teaching materials and laboratories would clearly be non-realistic from that perspective. The score averages of YGS and LYS, and the average of those who entered in a university program after the exams were determined. The most important factor for success from the perspective of the families is to get higher marks from the exams to enter a prestigious university program. The number of high school graduates was not regarded because there is not any benefits in getting a high school diploma in today's Turkey Calculation of the Correlation between the Input and Output Variables Choosing the input and output variables are challenging because they are the bases for the comparison of DMUs. If an important variable is disregarded in the model, the efficacy of the DMU will decrease. But if more inputs and outputs are used in the application, dissociation ability of the DEA will also decrease and necessitate an increase in the number of the DMU. If the input and output couples are of higher positive correlations, the number of the inputs and outputs could be decreased in the analysis by disregarding one of the couples. SPSS 15.0 was used for the calculation of the correlations in the study. Table 2. Correlation table between the score based YGS couples Ygs1 Ygs2 Ygs3 Ygs4 Ygs5 Ygs6 Ygs1 1 Ygs2 0,998* 1 Ygs3 0,951* 0,939* 1 Ygs4 0,945* 0,932* 0,999* 1 Ygs5 0,977* 0,967* 0,994* 0,991* 1 Ygs6 0,996* 0,990* 0,975* 0,969* 0,992* 1 *0.01 level of significance Table 3. Correlation table between the score based LYS couples LYS MS LYS TM LYS TS LYS MS 1 LYS TM 0,703* 1 LYS TS 0,310 0,168 1 *0.01 level of significance Table 4. Pearson Correlation Table between the Classrooms and Branches Number of Number of the Classrooms classroom branches Number of Classrooms 1 Number of the classroom branches 0,864* 1 *0.01 level of significance 48

13 DMU Number of teacher Number of student Number of Classroom branches YGS-LYS success rates YGS point averages LYS-score MS LYS-score TM LYS-score TS 4.4. The Inputs and Outputs defined after the Correlation Process According to ÖSYM results, there are six YGS score types (ÖSYM, 2012). It is seen that, as a result of correlation analysis, there is a very high positive relation in the YGS couples mentioned in the Table 2. Because of that reason, all averages of YGS score types were taken. Due to the higher relation between the classrooms and branches given in the Table 4, only the number of the branches were taken into the analysis. As a result of the correlation analysis, the data belonging to input and output variables of the DMU was provided in the Table 5. Table 5. Input and output variables and the data INPUTS OUTPUTS S ,18 397, , , ,372 S , , , ,346 S ,17 304, , , ,108 S ,48 318, , , ,058 S ,77 309, , , ,963 S ,62 285, , , ,905 S ,47 257, , , ,787 S ,17 271, , , ,212 S ,67 274, , , ,325 S ,06 282, , , ,603 S ,38 206, , , ,492 S ,93 224, , , ,182 S ,45 213, , , ,845 S ,15 209, , , ,100 S ,66 213, , , ,618 S ,67 210, , , ,395 S ,43 204, , , ,788 S ,75 197, , , ,994 S ,79 208, , , ,079 S , , , ,751 S ,48 196, , , ,464 S ,05 201, , , ,435 S ,02 180, , , ,271 S , , , ,623 S ,06 229, , , ,933 49

14 The model has been solved for all DMUs and the efficacy values were cleared by means of EMS under the assumption of constant return to scale. As a result of the dissociation, the efficacy ratios found by means of classical DEA and input oriented CCR and BCC models were mentioned in the Tables 6 and 7. Table 6. CCR efficacy table with classical DEA for high schools Table 7. BCC efficacy table for high schools 50

15 When the Table 6 is closely examined, it becomes clear that twenty-one high schools are not efficient, and only four high schools are efficient and work productively. When it comes to the Table 7, according to BCC model results, it is obvious that sixteen high schools are not efficient and nine schools are efficient. 5. Application of the fuzzy data envelopment analysis In the second part of the study FDEA model with interval data was applied. According that model, the analysis was realized by changing the data into interval data. Standard errors of the twenty five variables were used to define the data as interval data. Upper and lower frontier data were calculated by adding and removing standard errors to each variable, and so each data was turned into interval data. While upper frontier efficacy score was calculated, upper frontier values of the output data and lower frontier values of the input data were used. When it came to the lower frontier efficacy scores, lower frontier values of the output data and upper frontier values of the input data were enjoyed Changing the data into interval data (Upper frontier data) = (Available data) + (Standard Error) (Lower frontier data) = (Available data) - (Standard Error) Number of the Teacher Number of the Student Number of the classroom branches Table 8. Statistical information dealing with the input data Number of the School Minimum Maximum Mean Standard Error ,4 5, ,32 57, ,6 2,03715 Table 9. Statistical information dealing with the output data School Standard Minimum Maximum Mean Number Error YGS-LYS success rates 25 11,15 73,17 35,3364 3,54397 YGS point averages ,08 397,20 250, ,13165 LYS (MS) ,39 393,74 194, ,98894 LYS (TM) ,54 288,68 219,5453 7,35307 LYS (TS) ,49 245,37 203,2660 3,

16 While upper frontier efficacy values were being calculated, lower frontier input values, which were formed by adding standard errors in Table 9 to output data in Table 5 and removing the input data in Table 5 from the standard errors in Table 8., were used. The analysis result with the new data was given in Table 10 with CCR model and in Table 11 with BCC model. Table 10. Upper frontier efficacy with input oriented CCR model Table 11. Upper frontier efficacy with output oriented BCC model While lower frontier efficacy values were being calculated, lower frontier input values, which were formed by removing standard errors in Table 9 from output data in Table 5 and adding the input data in Table 5 to the standard errors in Table 8., were used. The 52

17 analysis result with the new data was given in Table 12 with CCR model and in Table 13 with BCC model. Table 12. Lower frontier efficacy with input oriented CCR model Table 13. Lower frontier efficacy with input oriented BCC model 53

18 efficient schools Number of the efficient school Table 14. Comparison of DEA and FDEA efficacy results DEA FDEA CCR BCC CCR BCC lower frontier efficient upper frontier efficient upper and lower frontier efficient lower frontier efficient upper frontier efficient S3 S8 S11 S17 S1 S2 S3 S8 S9 S10 S11 S17 S18 S1 S3 S8 S11 S17 S3 S11 S17 S3 S11 S17 S1 S2 S3 S8 S9 S10 S11 S17 S18 S1 S2 S3 S8 S9 S10 S11 S17 S18 upper and lower frontier efficient S1 S2 S3 S8 S9 S10 S11 S17 S Conclusions and discussion Having no limitless resources necessitates that they should be used in a more productive and efficient ways. DEA is one of the mostly used methods to compare the efficacies of the DMUs. In this study, efficacy measurements of the high schools were calculated by using DEA and FDEA. EMS was run two times both for classical and fuzzy applications. Initially, input oriented CCR model results were given under constant return to scale and then input oriented BCC model results were presented under variable return to scale. In order to measure efficacy levels of the twenty five schools, three inputs such as the number of the teachers, students and branches, five outputs like YGS-LYS success rates, YGS averages, LYS Maths-science, Turkish-Maths, Turkish-Social Sciences were determined. As a result of the analysis, four high schools were efficient according to CCR model while twenty-one schools were inefficient. The scale was active in the efficient schools, S3, S8, S11 and S17. S1 was the closest school to the efficacy frontier with %97,01 while S3 was the farthest one to the relative efficacy with % 12,59. According to BCC model, the schools, S1, S2, S3, S8, S9, S10, S11, S17, and S18 were efficient and the rest, 16 schools, were inefficient. The reason behind the difference between CCR and BCC model results is that DMU should be both technically active and scale active (total active) in order to be efficient but in the BCC model it is enough to be technically active. Additionally, because variable return to scale is mostly used to compare profit-oriented businesses, the evaluations as a result model were made under 54

19 the assumption of constant return to scale in the current study. Therefore, primarily CCR model was regarded and only efficacy results were provided for the BCC model. In the table 14, there are input oriented constant and variable return, classical DEA and FDEA lower and upper frontier efficacy values with interval data. According to these results, four high schools work efficiently when the input oriented CCR modeled classical DEA result is regarded. As for FDEA result, it is seen that three high schools are of upper frontier efficient and five schools are of lower frontier efficient. In that it has been found that three schools which are both upper and lower frontier work efficiently. Considering the result of the classical DEA measurement, the school, S8, which is efficient, is inefficient according to FDEA. In the BCC mode nine schools, which stand efficient in both analyses, is seen efficient. In conclusion, there are not so strict difference and it shows that the evaluations are more realistic when the potential errors in the data are regarded. Another point that should be considered, all the schools, which are efficient in the fuzzy model, are also efficient in the classical model but those which are efficient in the classical model, are not efficient in the fuzzy model. As a result, results of the analyses prove that the evaluations by means of FDEA are more selective when compared to those done in the classical model. During the efficacy analyses in the education, the results can be more correctly evaluated by FDEA when the errors originated from the data structure of the DMUs' were considered. When some of the input and output data are uncertain, the efficacy scores become indefinite. DEA is a sensitive technique towards various errors in data collection processes. When such kind of conditions are taken into consideration, FDEA is very important in terms of regarding such errors. Accordingly, Fuzzy Theory is highly recommended to use to DEA problems with fuzzy data in the efficacy measurements in educational researches. When the analysis results are evaluated in terms of the education, it attracts utmost attention that there are great differences in the numbers of the students getting education in these schools. The high number of the students considerably affect the efficacy scores. The schools, which are not efficient, should have less students to turn into efficient levels. These schools should be reconstructed and required regulations should be done in parallel to social References Abbott, M., Doucouliagos, C., The efficiency of Australian universities: a data envelopment analysis. Economics of Education Review, 22 (1), 89-97, Akal, Z., İşletmelerde performans ölçüm ve denetimi: çok yönlü performans göstergeleri, Milli Prodüktivite Merkezi Yayınları, Ankara, Akdoğan, M., Veri zarflama analizi tekniği ile sigorta şirketlerinin etkinlik ölçümü Türkiye örneği, M.S. thesis, Hacettepe University, Ankara, Turkey,

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