Modified Fuzzy Data Envelopment Analysis Models

This paper examines the use of data envelopment analysis (DEA) in the conduct of efficiency measurement involving fuzzy (interval) input-output values. Data envelopment analysis is a linear programming method for comparing the relative productivity (or efficiency) of multiple service units. Standard DEA models assume crisp data for both the input and output values. In practice however, input and output values may be uncertain, vague, imprecise or incomplete. New pairs of fuzzy DEA (FDEA) models are presented which differ from existing fuzzy DEA models handling uncertain data. In this approach, upper bound interval data are used exclusively to obtain the upper frontier values while lower bound interval data are used exclusively to obtain the lower frontier values. The outcome, when compared with the outcome of existing approach, based on the same set of data, shows a swap in the upper and lower frontier values with exactly the same number of efficient decision making units (DMUs). This new approach therefore clears the ambiguity occasioned by the mixture of upper and lower bound values in the determination of the upper and lower frontier efficiency scores respectively. The modified FDEA models make application and interpretation of results easy. The most efficient units, for each of the models, have efficiency score of 1 with equivalent ranking score of 1. These efficient units also serve as reference sets to the inefficient units. The inefficient units have efficiency scores less than 1 for all the models. The most inefficient unit is S13 for all the models and it has the least efficiency score in each case and a ranking score of 25.


Introduction
This paper extends the technique of fuzzy (interval) data envelopment analysis (FDEA). In particular, the paper seeks to present modified fuzzy Charnes, Cooper and Rhodes (FCCR) and fuzzy Banker, Charnes and Cooper (FBCC) models for use with interval fuzzy numbers. This paper also compares the conventional DEA, the FDEA presented by Zeidan et al. (2016) and Demir (2014). For ease of comparison, the data set for 25 high schools in the 2012-2013 education year in Demir (2014) is used.
Data envelopment analysis (DEA) was first presented by Charnes, Cooper and Rhodes (1978) leveraging on the 1957 seminal paper of Farrell whose main purpose was the estimation of technical efficiency and efficiency frontiers. DEA has become one of the most widely used techniques for measuring the efficiency of decision making units (DMU). A basic assumption of DEA for the measurement of the total technical efficiency of a DMU is that of constant returns to scale (CRS). This was later modified by Banker, Charnes and Cooper (1984) to become variable returns to scale (VRS) (Demir, 2014). According to Zeidan et al., (2016), Data envelopment analysis is a nonparametric technique for evaluating and measuring the relative efficiency of decision making units characterized by multiple inputs and multiple outputs.
The basic DEA works with crisp values for both the input and output values. Being a very responsive method, its efficiency is easily affected by errors bothering on imprecise data, incomplete data, judgment data, forecasting data or ambiguous data. In general, imprecise data can be presented in form of fuzzy numbers. It is therefore worthwhile to study how to evaluate the efficiency of a set of data in fuzzy form. In such a situation, FDEA becomes a useful method to overcome the shortcomings of basic DEA. Wang et al. (2005) studied how to conduct efficiency assessment in interval and/or fuzzy input-output environments in a simple, rational and effective way using data envelopment analysis. They constructed a new pair of interval DEA models on the basis of interval arithmetic, which differs from the existing DEA models handling interval data. Demir (2014) compared classical DEA and FDEA based on α-intercept method by means of an application for educational researches. He compared the relative activities of 25 high schools in the 2012-2013 education year by means of DEA and FDEA and strongly recommends that fuzzy theory be practiced for DEA problems with uncertain data in order to get more secure results in activity measurements. Zeidan, et al. (2016) presented a technique to improve a statistical method based on arithmetic operations to solve fuzzy data envelopment analysis models. They transformed the original data into interval data in the form of lower and upper frontier data and used them to obtain the interval DEA efficiency scores. Their method requires that data should be distributed as a normal distribution. Thus, the technique assumes that the variables are normally distributed. This position is however at variance with the fact that DEA, being a non-parametric technique, does not assume any specific functional form relating inputs to outputs (Zhu, 2002). Mahmudah and Lola (2016) applied the fuzzy DEA approach to measure the Indonesian universities performances under imprecise inputs and outputs. Their empirical results show that 36% of universities perform efficiently under the constant returns to scale model. For the variable returns to scale model, 52% of the universities were efficient. They discovered that the well-known universities obtained relatively low scores indicating the need for them to improve their performances in publishing scientific work in addition to providing useful information to the public through the official websites. They concluded that the results of the VRS model are better than the CRS model for both the DEA and FDEA methods. Tlig and Hamed (2017) accessed the efficiency of commercial Tunisian Banks using two approaches of fuzzy data envelopment analysis, namely, the possibility approach and the approach based on relations between fuzzy numbers (BRONF). They evaluated the efficiency of the banks in terms of several crisp and imprecise data. Their results indicate that in a competitive environment, no-financial inputs and outputs should be considered in order to have credible and realistic efficiency scores. Gökşen et al. (2015) used Data Envelopment Analysis to determine the performance levels of departments in Dokuz Eylul University (Turkey). Their study discussed the technical scores and scale scores of departments and revealed the main cause of inefficiency. The input and output goals of departments were fixed for a better efficiency.
Fatimah and Mahmudah (2017)  The paper further provided evidence that some high literacy rate but low technical efficiency scores were found after comparing literacy rates and technical efficiency scores of the districts, indicating that high literacy rate does not necessarily mean that districts are technically efficient.
The rest of the paper is organized as follows: Basic models of DEA and (FDEA) fuzzy Data Envelopment Analysis models, and the suggested modification to the fuzzy (interval) DEA, are discussed in the second, third and fourth sections covering the theoretical aspect of the study. Section five deals with the application of the modified fuzzy DEA model and its comparison with the model by Wang et al., (2005). Section six presents the summary of results and concludes the work.

Basic Models of Data Envelopment Analysis
Many authors have studied the technique of data envelopment analysis. Originally, DEA was designed to measure the relative efficiency of non-for-profit organizations. Due to its ability to model multiple input and multiple output relationships without a priori underlying functional form assumption, data envelopment analysis has also been applied to other areas which are profit oriented (Zhu, 2003). Development of new methods and models have evolved due to wide application. This paper will however, present only Charnes, Cooper and Rhodes (CCR) and Banker, Charnes and Cooper (BCC) DEA models for the purpose of understanding the fundamentals of DEA.

Charnes, Cooper, and Rhodes DEA model
The CCR DEA model by Charnes et al. (1978) is given below in fractional form. yro: The number of the output by the DMU, o xio: The amount of the input used by the DMU, o ur: The weight of the output, r vi: The weight of the input, i Objective function: Transformation of fractional CCR DEA model (1) into linear form: Objective function: Efficiency Frontier of the CCR DEA Model

The Banker, Charnes and Cooper DEA Model
The BCC model was introduced by Banker, Charnes and Cooper in 1984. It is an extension of the CCR model. The major difference between the two models lies in the establishment of returns to scale. While constant returns to scale is assumed in CCR which means that increase in inputs results to commensurate increase in outputs, variable returns to scale is assumed in BCC implying that increase in inputs does not result to commensurate increase in outputs. Accordingly, the BCC model is more robust than the CCR model (Zeidan et al., 2016). The CCR and BCC radial models are depicted pictorially in Fig. 1 and Fig. 2, respectively.
The BCC model in fractional form differs from the CCR model (1) by an additional variable as presented below: Objective function: Where the new variable separates scale efficiency from technical efficiency in CCR model.
The approach to change fuzzy data into offset data using α-level mass to create a solution that could take advantage of a family of classical DEA models was made by Liu (2000, 2003). Leveraging on the approach, Saati et al., (2002) made fuzzy CCR model as an offset programming model through defining it as programming problem using α-level. An improvement on interval data DEA was made by Wang et al., (2005) by employing DEA technique in the offset data and established a fuzzy efficiency measurement. Cooper et al., (1999) created interval data envelopment analysis model (IDEA). The IDEA model can change the non-linear programing problems into linear programing problem through scale conversions and variable changes (Demir, 2014). Using Wang et al's technique, interval data programing model can be solved like a definitive linear programming model for each DMU and an efficiency score can be made by means of each α-level (Deniz, 2009).
The technique of DMU with fuzzy data which could convert FDEA model into certain DEA model series was improved by Kao and Liu (2000).

Existing FDEA Linear Programing Formulation
Given that all inputs and outputs are incomplete as a result of uncertainties, let these values be known as > 0 and > 0 and [ , ] and [ , ] and they are between these top-down limits. To deal with such uncertain situation, Kao and Liu, 2000, Wang et al., 2005, Güneş, 2006, and Demir, 2014 defined FDEA model with fuzzy interval data in which limited data is used for efficiency measurement to generate upper and lower bounds for each DMU, as follows: Upper Bound , ≥ 0; = 1,2, ⋯ , ; = 1,2, ⋯ , Observe that, in the fractional programing model (5), a mixture of the upper output values and lower input values were used to obtain the upper bound of the best possible relative efficiency of DMUo, ℎ 0 . Similarly, for model (6), a mixture of the lower output values and upper input values were used to obtain the lower bound of the best possible relative efficiency of DMUo, ℎ 0 . However, the ratio of upper and lower bound values cannot logically give rise to ℎ 0 , neither can the ratio of lower and upper bound values logically give rise to ℎ 0 . Hence the need for a modification.

Application and Comparison of Classical DEA Models, existing Fuzzy Models and the Modified Models
Models (9) and (10) will be solved by first transforming the crisp data into interval data using the approach of Demir (2014). Standard errors for each variable will be added to obtain the upper frontier data, while standard errors for each variable will be subtracted to obtain the lower frontier data. For the upper frontier efficiency scores, the upper frontier values of both the output and input data will be used. To obtain the lower frontier efficiency scores, the lower frontier values of both the output and input data will be used.
To evaluate and compare results from classical DEA models, existing interval DEA models and the modified interval DEA models; real data set of 25 high schools in the 2012 -2013 education year is taken from Demir (2014). The data description is as follows: inputs (numbers of students, teachers and classes), outputs (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 (Zeiden et al., 2016). See Appendix 1. The DEA models are solved using DEA-SOLVER-LV8.
The most inefficient unit, S13, has least efficiency score of 0.1272 and a ranking score of 25. It has efficient units S3, S11 and S17 as reference set (Lambda). In order words, it should emulate what these efficient units are doing in order to become efficient. Notice that each efficient unit serves as its own reference (Lambda).   Demir 2014 In Table 3, the efficient DMUs due to the modified FDEA lower bound CCR model are the same as the efficient DMUs due to the FDEA upper bound CCR model in Table 4 from Demir (2014). S9 1 1 S9 1 10 S10 1 8 S10 1 11 S11 1 1 S11 1 12 S12 0.1641 22 S3 0.198 S11 0.245 S17 0.557 13 S13 0.09 25 S3 0.085 S11 0.641 S17 0.275 14 S14 0.1608 23 S1 0.038 S11 0.323 S17 0.639 15 S15 0.2073 20 S1 0.064 S3 0.026 S11 0.18 S17 0.731 16 S16 0.1558 24 S3 0.053 S11 0.348 S17 0.599 17 S17 1 1 S17 1 18 S18   Demir 2014 In Table 5, the efficient DMUs due to the modified FDEA lower bound BCC model are the same as the efficient DMUs due to the FDEA upper bound BCC model in Table 6 from Demir (2014).   Table 7, the efficient DMUs due to the modified FDEA upper bound CCR model are the same as the efficient DMUs due to the FDEA lower bound CCR model in Table 8 from Demir (2014).  In Table 9, the efficient DMUs due to the modified FDEA upper bound BCC model are the same as the efficient DMUs due to the FDEA lower bound BCC model in Table 10, from Demir (2014).

Results and Conclusion
Results of the efficient decision making units (DMUs) due to classical DEA models, fuzzy DEA models proposed by Wang et al., (2005) and adopted by Demir (2014) and the modified fuzzy DEA models are presented in summary form in tables 11 and 1 Lower and upper efficient FDEA (Modified) L U L U S3 S1 S3 S3 S1 S8 S3 S11 S11 S3 S11 S8 S17 S17 S8 S17 S11 S11 S17 S17 Lower and upper efficient FDEA (Modified) L U L U S1 S1 S1 S1 S1 S9 S9 S9 S9 S10 S10 S10 S10 S10 S11 S11 S11 S11 S11 S17 S17 S17 S17 S17 S18 S18 S18 S18 S18 Table 11 presents the efficient DMUs from the three DEA models. A major finding in the case of CCR, when the results of Demir and that of the modified model are compared is that, the efficient DMUs when the upper bound model (Model 5) is applied, corresponds to the efficient DMUs when the lower bound modified model (Model 10) is applied. Similarly, when the lower bound model (Model 6) is applied, the result corresponds to that of the upper bound modified model (Model 9).
The implication of this finding is that, the ambiguity created by the mixture of upper bound and lower bound values to generate efficiency scores in each of Models 5 and 6 can be avoided. Instead, the modified Models 9 and 10, where upper bound values are used exclusively to generate upper efficiency scores and lower bound values are used exclusively to generate lower efficiency scores can be adopted to avoid the ambiguity.
In the case of BCC, Table 12, the efficient DMUs are the same for all the models compared. This is not unexpected since BCC is more robust and adopts variable returns to scale (VRS) frontier as against the more restrictive CCR which adopts the constant returns to scale (CRS) frontier.