Arash Method

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The Arash Method is a research framework used in the field of social sciences, particularly in qualitative research, that emphasizes participatory approaches and community engagement. It focuses on co-creating knowledge with stakeholders to ensure that research outcomes are relevant and beneficial to the communities involved.

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A new method in data envelopment analysis to find efficient decision making units and rank both technical efficient and inefficient DMUs together

by Dariush Khezrimotlagh and

2012, Applied Mathematical …

The inefficient DMUs are usually arranged after the technical efficient ones by DEA methods, however, it is possible that a technical efficient DMU neither be efficient nor be more efficient than some inefficient ones. This study... more

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How to deal with numbers of decision making units and variables in data envelopment analysis

by Dariush Khezrimotlagh

Sufficient numbers of Decision Making Units (DMUs) in comparison with the number of input and output variables has been a concern of using Data Envelopment Analysis (DEA) in the last three decades. There are several studies in the... more

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Nonlinear Arash Model in DEA

by Dariush Khezrimotlagh and

2013, Maxwell

This study illustrates the nonlinear Arash Model (AM) in data envelopment analysis (DEA) to distinguish between technical efficient DMUs and arrange both technical efficient and inefficient DMUs at the same time. It demonstrates how the... more

From the above illustrations the following proposition is proved. Indeed, if the weights are available, the extended nonlinear « -AM minimizes the score of linear « -AM. Moreover, when they are not available it is quite the same as the previous nonlinear e- AM and the proof follows from Proposition 2. with some technical efficient ones which do not dominated them. Res. J. Appl. Sci. Eng. Technol., 5(17): 4268-4273, 2013

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Comparing Arash Model with SBM in DEA

by Dariush Khezrimotlagh and

2012, Applied Mathematical …

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Benchmarking Inefficient Decision Making Units in DEA

by Dariush Khezrimotlagh and

2012

Data Envelopment Analysis (DEA) is a non-parametric approach to operations research for assessing the relative efficiencies of a set of peer units, called Decision Making Units (DMUs), with multiple inputs and multiple outputs. DEA... more

Figure 1: Example of two inputs and one constant output. sz = 13. In comparison, SBM yields s;* = 3 ands;* = 4, orsy*+s7* = amount of 3/4 + 4/16(= 1) is greater than the amount of 0/4 + 13/16 calculate the maximum summation of slacks. 7 (Figure 4). In other words, the (= 0.8125), and SBM cannot

Figure 5 demonstrates the differences between DEA models for benchmarking A6 and A15. In addition, the optimal slacks of applying DEA models are shown in Table 4. Furthermore, since SBM and CCR may not suitably benchmark technically inefficient DMUs with the unity scale for inputs and outputs, they may not be acceptable for the variety of weights and scale of DMUs, as well.

Figure 5: Benchmarking by DEA models (CRS or VRS).

Figure 3: Benchmarking by ADD (CRS or VRS) Figure 2: Benchmarking by Input-Oriented CCR or BCC.

Figure 6: Example of two inputs and one constant output.

Figure 7 demonstrates the results of applying CCR model (radial model) for benchmarking technically inefficient DMUs such as A18, A13, and A12 to a virtual technically efficient DMU next to Al. As can be seen, A12 uses the same input in comparison with A5 and they also have the same quantity for output 1. This obviously suggests that A 12 requires only increasing the quantity of output 2 to be the most efficient DMU in comparison with other DMUs, whereas CCR model benchmarks it to a worse technically efficient DMU. From the figures, there are seven technically efficient DMUs A1, A2, A3, A5, A7, A8 and A10, which all models are able to characterize. However, the benchmarking of technically inefficient DMUs is different.

Figure 9: Benchmarking by SBM or ERM (CRS or VRS). Figure 10: Benchmarking by DEA models (CRS or VRS).

Figure 12: Benchmarking by 1-KM (CRS or VRS). Therefore, it is quite obvious that ADD is more significant than other DEA models to benchmark technically inefficient DMUs. In order to remove the shortcomings of ADD to give efficiency scores and distinguish between technically efficient DMUs, Khezrimotlagh et al. [3] proposed a significant technique, called Kourosh and Arash Method (KAM) which is completely flexible for any purpose in DEA. It not only gives an efficiency score for each DMU, but also is able to distinguish between technically efficient DMUs which CCR, BCC, ADD, SBM, ERM and other current DEA models are not able to do it. From their proposed methods (AM, and KM), all the DMUsin Tables 2 and 5 are benchmarked to A7 and A5, respectively (Figures 11 and 12). The rank of all DMUs is suggested as the same rank by the definition of efficiency, that is, output/input. The Kourosh Model (KM) is also as follows:

Figure 11: Benchmarking by 1-AM (CRS or VRS).

In contrast to CCR (also applies to BCC), the non-radial model, called Additive DEA model (ADD) introduced by Charnes et al. [7], considers the possibility of simultaneous input decreases and output increases There is no weak technical efficiency in the targets of ADD, whereas the previous models may reflect weal technical efficiency on the Farrell frontier. The weak technical efficiency means that there are some non-zer optimal slacks for a DMU whereas the models show that the DMU is technically efficient. However, ADI does not have the property of units’ invariance. The model also does not give a technical efficiency score o: between 0 and 1. In addition, if there is no weak technical efficiency in the targets of BCC, a DMU is VRS ADD-technically efficient if and only if it is BCC-technically efficient [8]. This is also the case for those tha are CCR-technically efficient in relation to CRS ADD-technically efficient. under, at most, the present input consumption. Besides, CCR and BCC models are invariant to the units of measurement and they describe a technical efficiency score of between 0 and 1. The unit invariance property means that, the technical efficiency scores of DMUs are independent of the units in which the inputs and outputs are measured provided these units are the same in every DMU. It can also be defined by replacing (a@;xi;, Bx¥ix) with (x;;,y;,) for inputs and outputs of DMUs, where the technical efficiency score of DMUs is not changed, fora; > 0.8, >0,i=1.2.....n,j=12.....mandk =12.....v.

Table 2: Example of two inputs and one constant output.

Obviously, the above example demonstrates that the abilities and benefits of SBM or ERM, overall, may not comprehend all inefficiencies, whereas ADD can identify them clearly. They may also suggest the worst reference sets for technically inefficient DMUs in comparison with ADD. However, ADD does not give an efficiency score for each DMU, which is the most important part of assessing the performance evaluation of DMUs.

Table 5: Example of one constant input and two outputs.

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How to detect outliers in data envelopment analysis by Kourosh and Arash method

by Dariush Khezrimotlagh

One of the concerns about using non-parametric estimators such as Data Envelopment Analysis (DEA), is the presence of outliers. There are a good number of studies that mention this assessment in the literature of DEA, however, there is no... more

The targets of KAM, that is, the highest (x*, y*), the best technical (x*, y*) and the lowest (x*, y*) efficient targets with e-Degree of Freedom (€-DF), the highest KA?!, the best technical KA*! and the lowest KA*! efficiency scores as well as sensitivity score 52! with e-DF, respectively are given by:

Let us suppose 100 DMUs depicted in Figure 1 with one input and one output when at least one of the types (1)-(IV) happen. This example was selected to be similar to the work of Simar (2007) and Simar and Zelenyuk (2011) which claimed the weakness of DEA to detect outliers. From the figure, it is clear that there are some outliers and extreme values which effect on the frontier.

Now let us see which DMUs are A-R in the sample. Figure 3 depicts DMUs A-R among observed DMUs. As can be seen, KAM nicely detects outliers while it simultaneously ranks and benchmarks them without the computational complexities of current hybrid technologies. We also selected the half values of minimum input and minimum output of DMUs to define ey and ¢;', respectively, in order to have the same commensurate effects on the Farrell frontier for evaluating each DMU (Khezrimotlagh et al. 2013a), and again found DMUs A-R as outliers according to the parts (1), (11) and (iii).

Figure 2: KAM outcomes for 100 DMUs. Khezrimotlagh, D. (2015). How to detect outliers in data envelopment analysis by Kourosh and Arash method. arXiv preprint arXiv: 1503.02548

Figure 4: Detecting best efficient DMUs by KAM. Khezrimotlagh, D. (2015). How to detect outliers in data envelopment analysis by Kourosh and Arash method. arXiv preprint arXiv: 1503.02548. The polygon chart of technical efficiency score with 0-DF is gradually decreased. As it depicted with two dotted vertical ellipses, DMUs ‘a’-‘d’, ‘g’ and ‘h’ have simultaneously a good value of technical efficiency score with 0-DF, a good value of best technical and lowest efficiency score with 0.1-DF, a lesser difference between technical efficiency score with 0-DF and 0.1-DF, and not a great value of sensitivity score with 0.1- DF among other DMUs. Therefore, 0.1-KAM suggests these DMUs as the best performers among remainder DMUs. 0.1-KAM also suggests DMUs ‘e’, ‘f’, ‘i! and ‘r’ as outliers, because there is a rapid difference between their technical efficiency scores with 0-DF and 0.1-DF.

The 0.1-KAM lowest targets, shown by small multiplication signs, illustrate the effects on the Farrell frontier by introducing 10 percentage errors in DMUs’ input and output values. Indeed, the effects on the Farrell frontier were commensurately considered for each evaluated DMU by selecting ey and ef as 0.1 x x,, and 0.1 x y;,, respectively, and can be considered as variations of the sample, which was one of the concerns of Grosskopf (1996). Khezrimotlagh, D. (2015). How to detect outliers in data envelopment analysis by Kourosh and Arash method. arXiv preprint arXiv: 1503.02548

We also applied KAM when ¢; and e;* were 0.5 X min{x;1: xj, # 0,i = 1,2, ...,100} and 0.5 xX min{y;1: viz # 0,i = 1,2,...,100}, respectively, and wy = 1/x,, and wf = 1/y,,. In this case, the effect on the Farrell frontier was the same for evaluated DMUs. There were no significant differences to arrange and benchmark DMUs. DMU ‘a’ still had the first rank, but it was suggested as an outlier due to have a very great sensitivity score by 0.5-DF. Moreover, the rank of technical efficient DMUs were respectively as ‘a’, ‘c’, ‘d’, ‘b’, ‘f and ‘e’. In order to have a clear figure, ¢; and ej were defined the same as 0.3 of unit and KAM was applied. Figure 7 depicts the 0.3-KAM benchmarks. From the figure, 0.3- KAM technical targets are between DMUs ‘a’ to ‘c’ on the Farrell frontier, which is the

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A Review on Arash Method in Data Envelopment Analysis

by Dariush Khezrimotlagh and

Arash Method (AM) is a new technique in Data Envelopment Analysis (DEA), which estimates the performance of Decision Making Units (DMUs) with flexible linear programming based on Additive DEA model (ADD). It is simultaneously able to... more

~ In order to depict the shortcoming in Pareto-K oopmans definition of efficiency obviously, let us suppose that there are five DMUs with one input and one output according to Table 1 and Figure 1 in Variable Returns to Scale (VRS) [17]. From Figure 1 and the forth column of Table 1, the technically efficient DMUs A, B and C are fully (100%) efficient by Pareto-Koopmans definition of efficiency, however, none of DMUs A and C are more efficient than other inefficient DMUs D and E as can be seen in Figure 2 and the last column of Table 1. In other words, the technically efficient DMUs A and C are more inefficient than inefficient DMUs D and E. This simple clear example obviously proves that the Pareto-Koopmans definition of efficiency is not valid to call a technically efficient DMU as fully (100%) efficient.

Figure 5: Differences between L.AM and NL.AM and SBM. Khezrimotlagh et al. [13] considered 18 DMUs with five inputs and two outputs to depict the differences between linear AM (L.AM) and non-linear AM (NL.AM) and SBM when «¢ is 0, 0.0001, 0.001, 0.01 and 0.1. Figure 5 depicts the results of these models clearly where ¢ = ¢/w;, w; = 1/x,;, for j = 1,2,...,m and ¢ = 0.1, wi = 1/y,, for k = 1,2,...,p,. For instance, inefficient DMU A16 is more efficient than technically efficient DMUs A06, All, A13, Al0, Al4, A08, A15, A03, AO7 and A04. Non-linear AM also discriminates the differences between A01, A02 and A05 in comparison with Linear AM in constant returns to scale.

They also considered the example of public libraries with four inputs (floor area, number of books, staffs and population of wards) and two outputs (number of registered residents and borrow books) in the 12 Wards of the Tokyo Metropolitan A rea in 1986 [6]. The forth input was considered as non-controllable data. Figure 6 depicts the results of Non-controllable model (NCN) [6], 0.00002-AM and Extended of 0.00002-AM [14]. There are also a number of good examples in [15, 16, 31] which identify the robust technique of AM in comparison with current DEA techniques to assess the performance evaluation of DMUs inclusive integer-valued and non-controllable data whether the necessary information of data is available or not. In other words, the technique of AM is able to cover many subjects regarding DEA.

If the potential increase of outputs, thatis, s’s, are eliminated in the linear AM objective and the score is considered as A* = 7", w; x;;/ Lit, w; xj, the cost-efficiency outcome is the same as the result of e-linear AM when « is large enough [10]. Similarly AM can be extended to the Kourosh Method (KM) to measure the revenue-efficiency. The combination of KM and AM, called the Kourosh and Arash Method (KAM) is also able to measure the profit efficiency.

As can be seen, 0.1-AM clearly discriminates between DMUs A and B and depicts that A is less efficient than inefficient DMUs D and E with 0.1-DF by the scale of the non-zero minimum input. This means that the technically efficient DMU A in comparison with other technically efficient DMUs B and C, should increase its input to find a better place on the Farrell frontier. Likewise, 0.1-KM discriminates B and C, and suggests that C should decrease its output to get a more efficient place on the Farrell frontier. The minimum values of input and output are 2 for DMUs in Table 2, therefore, ¢; = 2e. Table 3 illustrates the lowest efficiency scores [16] of AM, KM and KAM when € is 0, 0.1 and 055. It is calculated by highest efficient targets x/; = x,; — sj“, and yj, = Viz + Sy". Here, 0-AM is the same as 0-KM and 0-KAM. Sj

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Profit Efficiency with Kourosh and Arash Model

by Dariush Khezrimotlagh

This article illustrates how Kourosh and Arash Model (KAM) is able to measure profit-efficiency of Decision Making Units (DMUs) in Data Envelopment Analysis (DEA). The advantages of KAM in comparison with the current profit efficiency DEA... more

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Airport Efficiency with Arash Method in Data Envelopment Analysis

by Dariush Khezrimotlagh and

2012

Airport efficiency with data envelopment analysis (DEA) is one of the common research studies in the last decade. DEA is a nonparametric method in operations research which has been provided some significant tools to estimate the... more

The constant returns to scale means a proportionate increase in inputs results in the same proportionate increase in output. The CCR has two stages, first minimizing @ and second maximizing the slacks s; (potential reduction of input) and s (potential increase of output), for j =1,2,...,m,and k = 1,2,...,p. The following notations are also used in the models: A DMU is called technical efficient if and only if the performances of other DMUs do not show that some of its inputs or outputs can be improved without worsening some of its other inputs or outputs. This definition is called Pareto-K oopmans definition in DEA and unfortunately it is wrongly interpreted as efficiency i.e., doing the jobs right, where there is no any units price or costs information for data. Khezrimotlagh et al. [7] recently illustrated that the definition of Pareto-Koopmans in DEA is only able to characterize the technical efficient DMUs and a technical efficient DMU may neither be efficient nor be more efficient than an inefficient one. They proposed a significant method to distinguish between efficient and technical efficient DMUs where the weights, units price or costs information of DMUs are not available and defined that an efficient DMU is a technical efficient DMU which has the best performer in comparison with other technical efficient DMUs, i.e., it has the best combination of inputs and outputs among other DMUs or the ratio of its produced output to its used input (output/input) among other DMUs does not muc depend h change where a very little error happens in its data. Their proposed definition does not to the weights or units price of data and it is the same as the definition of efficiency in economics where costs information are available. In order to illustrate the models, let us assume that there are n DMUs (DMU;,,i = 1,2,...,n) with m nonnegative inputs (x;;,j = 1,2,...,m) and p nonnegative outputs (y;,,k = 1,2,...,p) for each DMU which at least one of its inputs and one of its outputs are not zero. The CCR and SBM in constant returms to scale (CRS) are as following where DMU, (i = 1,2,...,n) is evaluated.

Table 2 depicts the scores of CCR, SBM, 0-AM, 0.0001-AM, 0.001-AM and 0.01-AM in constant returns to scale (CRS). There are 11 technical efficient DMUs (A01 to A11) which characterized by applying CCR, SBM and 0-AM, and other DMUs are inefficient. There are no any differences between the scores of SBM and 0-AM for the inefficient DMUs in this example and 0-AM similar to other conventional DEA models is not able to identify the differences between technical efficient DMUs. However, 0.0001-AM clearly shows that only one ten thousandth errors in each input are enough to characterize the instability of the efficiency of technical efficient DMUs as the fifth column of Table 2 illustrates it. In fact, A02 and A09 are two technical efficient DMUs which have the best combination of their data in comparison with other technical efficient DMUs.

The above discussions can be similarly examined in variable returns to scale by adding the convexity constraint, i.e, 07,4; = 1, to the constraints of those models. For example, Table 3 depicts the scores of BCC, SBM, 0-AM, 0.0001-AM, 0.001-AM and 0.01-AM in variable returns to scale (VRS). From the table, only four DMUs A14, A15, A16 and A17 are inefficient and other DMUs are technical efficient. Likewise, A11 has the worst combination of its input and output data in comparison with all other DMUs due to this fact that just 0.0001 errors in its input data fail its efficiency score to 0.0665.

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Arash Method

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