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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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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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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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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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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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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Comparing Arash Model with SBM in DEA

by Dariush Khezrimotlagh and

2012, Applied Mathematical …

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Arash method and Uncontrollable Data in DEA

by Dariush Khezrimotlagh and

2012, Applied Mathematical …

The uncontrollable data such as environmental variables are a kind of data which cannot be controlled by managers or users, but they may affect in assessing the performance evaluation of decision making units (DMU). In this paper, the... more

Table 2: The outcomes of the models without uncontrollable data and where Input4 is uncontrollable.
From the fifth column of Table 2, 0.00002-AM identifies that AO2, A03 and
A06 are efficient with 0.00002-degree of freedom in input and arranges all DMUs
together. In addition, when Input4 is considered as uncontrollable, four technical
efficient DMUs AO1, A02, A03 and A06 can be efficient with 0.00002-degree of
freedom in input and they have the most significant combination of their data
where 0.00002 errors happen in those inputs by improved AM as the last column
of Table 2 depicts it. However, it can arrange I DMUs A02, AO3 and A06.

Sy ee, a: a re, ee a) a a es ee a ae

Table 1: The example of 12 DMUs with one uncontrollable input.
Table 2 depicts the scores of CCR and ¢-AM for those DMUs where the
data are controllable and when only Input4 is uncontrollable.

Table 3: The targets and scores of proposed model (Extended 0.00001-AM) and Ap 90002:

Table 4: The slacks and the amount of one hundred thousandth errors in input values by 0.00001-AM.

As it can be seen, the optimum values of s;’s in Table 4 cannot be more
than €x,. However, the values of s,* for DMUs A01, A02, AO3 and A11 are 0.
This outcome means that the combinations of their controllable data can be
improved even if those DMUs increase their uncontrollable Input4. For example,
the proposed extension 0.00001-AM suggests AO1 (technical efficient DMU) to
decrease its Input] to 11380.43 and increase its output2 to 758750.31, even if it
increases 0.001% of its input2 and input4. Although, AO1 is able to reject
increasing the input2 and input4 values, it is recommended to improve its input
and output2 in order to have a good combination of its data. In other words, AM
suggests AQ1 to concentrate to decrease its input] and increase output2. This
capability of AM clearly represents its worth in comparison with other current
DEA models.

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Cost-Efficiency by Arash Method in DEA

by Dariush Khezrimotlagh and

2012, Applied Mathematical …

Arash Method (AM) was recently proposed to distinguish between efficient decision making units (DMUs) and technical efficient ones where there are no any prices, weights or other assumptions between inputs and outputs of DMUs. This paper... more

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Arash Method

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