Hesitant Fuzzy-Stochastic Data Envelopment Analysis (HF-SDEA) Model for Benchmarking

The Data Envelopment Analysis (DEA) method is a method commonly used in benchmarking. The Dynamic Data Envelopment Analysis (DDEA) method was proposed to improve the DEA method in the benchmarking process. The DDEA method proposed can determine the effectiveness of the Decision Making Unit (DMU). The disadvantage of the DDEA model is that it cannot handle problems that involve benchmarking for stochastic data. To improve the DDEA method, the Stochastic Data Envelopment Analysis (SDEA) method is proposed which can be used for benchmarking involving stochastic data. The SDEA method itself has weaknesses in dealing with noise and uncertainty problems that will appear in the assessment process. The purpose of the research conducted by the researcher was to use the Hesitant Fuzzy method in optimizing the SDEA method so that the Hesitant Fuzzy model Stochastic Data Envelopment Analysis (HF-SDEA) could be carried out benchmarking process in a situation where the assessment contained many elements of uncertainty. The results of this study are benchmarking methods that can do benchmarking for stochastic data on conditions that contain elements of uncertainty. Keywords—Data envelopment analysis; dynamic data envelopment analysis; stochastic data envelopment analysis; hesitant fuzzy. Manuscript received 11 Sep. 2020; revised 15 Dec. 2020; accepted 2 Feb. 2021. Date of publication 31 Mar. 2021. International Journal on Informatics Visualization is licensed under a Creative Commons Attribution-Share Alike 4.0 International License.


I. INTRODUCTION
Data Envelopment Analysis (DEA) is a widely used tool for determining viability or productivity in fields such as decision-making, benchmarking, and organizational research. This approach uses a number of Decision Making Units (DMUs) made up of Inputs and Outputs to determine the degree of performance [1]. Crisp-formatted inputs and outputs are required in the DEA. However, in fact, due to conditions containing elements of ambiguity, crisp input and output can often be difficult to obtain [2]. As a result, a number of researchers have created DEA to address the DEA's uncertainty issues. Kahraman and Tolga [3] conducted research that included fuzzy elements in the calculation of efficiency in the benchmarking process, and the approach they suggest is known as Fuzzy Data Envelopment Analysis (FDEA). Furthermore, the FDEA method is based on a study conducted by Tavana et al. [4] can be grouped into 4 (four) groups, namely: Tolerance Approach, Fuzzy Ranking Approach, The α-level based approach, and The Possibility Approach.
Guo et al. [5] Charnes et al. [6] with a probability approach in a Fuzzy Data Envelopment Analysis. They take into account the possibility and requirement in their approach so that the ideal measurement results in the triangular membership function can be obtained. Yousefi et al. [7] propose the Dynamic Data Envelopment Analysis (DDEA) mode, which is a refinement of the DEA system in a benchmarking framework for determining the Decision Making Unit's performance (DMU). The DDEA model has the drawback of being unable to handle problems involving stochastic data benchmarking. The Stochastic Data Envelopment Analysis (SDEA) model proposed by Olesen and Petersen [8] can be used for benchmarking with stochastic data. Olesen and Petersen's research [8] was unable to address the noise and ambiguity problems that would arise during the evaluation process.
Ahmadvand and Pishvaee [9] suggested the Fuzzy CSWDEA system, which is based on a reputation approach and can be used in decision-making. Peykani et al. [10] proposed the Robust Fuzzy DEA (RFDEA), a Fuzzy-DEA model that bases the benchmarking model on the probability, requirement, and credibility in the efficiency measurement method. On the other hand, in real life, decision making is particularly difficult in the process of benchmarking with DEA, where, in addition to dealing with DMU Inputs and Outputs that include elements of ambiguity, one must also deal with circumstances in which one's judgment is clouded by doubt. This is the context in which the Hesitant Fuzzy method was developed [11]. Fuzzy hesitant models proposed by Ashtiani and Asgomi [12] can be used for measurement processes in a state of noise with several elements of uncertainty. The researcher will use the fuzzy hesitant approach to refine Stochastic Data Envelopment Analysis (SDEA) so that it can manage valuations with elements of uncertainty due to the many parties involved in the benchmarking phase. This study would be interesting because benchmarking can only be performed if the appraiser believes their truth fully, but benchmarking cannot be done in situations where there is doubt or where the appraiser is unsure. Furthermore, in the benchmarking phase, fuzzy hesitant methods are required, particularly in determining assessment consistency [13].
As a consequence, HF-SDEA model is a benchmarking model that can solve this by taking into account the degree of ambiguity and hesitation when making a decision. The study's findings, in the form of an HF-SDEA model, are expected to provide efficiency values based on assessor evaluations with elements of doubt and uncertainty. The remainder of the paper is organized in the following manner. Our approach is presented in Section II. Section III describes the HF-SDEA experimental procedure. In Section IV, the findings are presented along with the experimental procedure. Finally, Section V brings our paper to a close and provides ideas for potential study.

II. MATERIAL AND METHOD
Wang et al. is the first to propose the Stochastic Data Envelopment Analysis Method [14]. The basic principle is to establish quantile functions that can avoid crossing quantiles while also proposing estimates for stochastic frontier measurements [15]. Due to restricted knowledge from many parameters, the probability theory is used in the benchmarking model, which is one of the key factors in developing the Stochastic Data Envelopment Analysis [16]. The focus of Stochastic Data Envelopment Analysis research then shifts to deciding the upper and lower bounds for output and input, but no researchers have addressed the stochastic issue that includes a situation where the assessor gives an uncertain assessment [17]. Since a number of researchers recognize that there are unknown inputs and outputs, and humans are more at ease making decisions in the form of linguistic variables, the Fuzzy Data Envelopment Analysis (FDEA) method was created [18].
Many FDEA models have been developed such as the ideal-seeking FDEA [19], the tolerance and possibility FDEA [20] the FDEAwith double frontiers [21] and the crossefficiency FDEA [22]. However, sometimes qualitative data sourced from linguistic variables are inaccurate and the time available for decision makers is limited so that doubts arise. In this situation, Hesitant Fuzzy developed into Hesitant Fuzzy Data Envelopment Analysis can be used [23]. This research will develop a Hesitant Fuzzy DEA model which in addition can perform the benchmarking process on the stochastic problem, it can also benchmark the conditions that contain Hesitant Fuzzy elements.
The benchmarking process will be carried out to measure the efficiency of the study programs at Malikussaleh University using the HF-SDEA method. There are a number of DMUs with input and output that are qualitative in nature so that they require measurements involving the Hesitant Fuzzy method and Stochastic Data Envelopment Analysis. The stages of research can be seen in Figure 1.  Figure 1 shows that the Hesitant Fuzzy approach is required in the benchmarking phase where there are inputs and outputs that contain elements of uncertainty, where the evaluation cannot be given in the form of crisp values and the assessor has reservations about giving an assessment. The assessors will begin by conducting a benchmarking process and providing an evaluation in the form of a linguistic form for Fuzzy Hesitant. In the linguistic envelope type, all of the assessments will be combined. The crisp value of each input and output for each DMU will then be determined using the linguistic values found in the linguistic envelope. The next step will be to perform a Stochastic Data Development Analysis, which will produce measurement results indicating which DMU is efficient and which DMU is inefficient, particularly if the data contains stochastic data.

A. Linguistic Form
As can be seen in Equation 1, linguistic forms are represented as a collection of linguistic words.
One of the benefits of Hesitant Fuzzy is that assessors can provide evaluation results in linguistic form during the benchmarking process. It can be shown in Equation 1 that there are many words that can be used to make an evaluation.

B. Envelope Form
Equation 2 shows the envelope form, which is a linguistic set of intervals.
( ) , !, ≤ Where and are defined as follows: The envelope type incorporates a negative and positive evaluation in one unit, as shown by Equations 2 and 3. If assessors are reluctant to evaluate an evaluation value, they may include an assessment in the form of a variety of assessments during the benchmarking process. This will assist the appraiser in making an evaluation and increase the accuracy of the assessment during the benchmarking process.

C. Calculating the Envelope Linguistic
Limit or constraint function: G , ≥ 0; r = 1, ..., k; s = 1,.., l Where: α = Efficiency object s k = observed output object s $ = the number of outputs I produced as a result of object s # % = number of inputs i used by object s $ = the output weight i produced by object s % = the input weight i given by object s The aim of the above equation is to find the maximum number of outputs from DMUn that are weighted, by holding the number of inputs weighted on a value less than or equal to one and the ratio of outputs weighted by the input weighted, of all DMUs.

E. Stochastic Data Envelopment Analysis (SDEA)
SDEA model proposed by Olesen and Petersen [8] can be used for benchmarking with stochastic data. The benchmarking method is a stochastic phenomenon in the form of probability that can often only be measured by its frequency distribution and can be approached by an interval function whose shape will mimic, i.e. it will reach a maximum value at some times while others will reach a minimum point. Equations 7 and 8 can be used to measure the efficiency of each DMU using the SDEA equation.
ln K L( , ) + ln + ln (7) Where C is the total cost required, w is the input vector, y is the output vector, and e = u + v is the term error. Where u is a variable that can be managed and represents inefficiency. v is an uncontrollable (random) factor as well as a noise word. The performance ratio can be expressed in the following way.
The determination of variable v, which is a random entity with noise, has its own set of issues. Especially if there are many people involved in the evaluation and each one has a different point of view. This is a distinct fault in the SDEA process.

F. Hesitant Fuzzy -Stochastic Data Envelopment Analysis
(HF-SDEA) Equation 9 illustrates the use of the fuzzy reluctant approach in deciding the index ranking of the Decision Making Unit (DMU). ?
Where is K j$P for Equation 8 based on Equation 9 that can be seen in Equation 10.

A. Decision Making Unit (DMU)
As shown in Table I, the DMU used in this analysis is in the context of Malikussaleh University study programs.  Table 1 displays inputs and outputs in the form of direct values that can be calculated, but there are also inputs in the form of stochastic values that include uncertainties, such as: The university environment for input and output is measured by graduate users' satisfaction levels. As a result, HF-SDEA can be used to quantify input values for the university setting and output values for stakeholder satisfaction. Assume the results of the calculations are as shown in Table II.

B. Testing resuts
The following is the complete type of programming with HF-SDEA.
The efficiency testing results for each DMU based on the HF-SDEA are shown in Table III.  Table 3 shows that the most powerful DMUs were Information Technology, Civil Engineering, Chemical Engineering, Aquaculture, Communication Science, Political Science, Anthropology, Medical, and Management, with a total of 9 (nine) DMUs.
The test results show that the HF-SDEA conducted a good benchmarking process for stochastic data under conditions that included elements of uncertainty and hesitancy. Centered on the HF-SDEA, the results of this study also include an effective and inefficient DMU. Future research should consider feasibility, need, and reputation.

IV. CONCLUSIONS
The results showed that the Hesitant Fuzzy model -Stochastic Data Envelopment Analysis (HF-SDEA) can be used to benchmark stochastic data under uncertain conditions. Future research should be able to assess the productive rating of each DMU in order to determine the university's future growth priorities.