Profit Efficiency Evaluation: A composed Approach of DEA and multi- objective programming
الموضوعات :
Soheila Seyedboveir
1
,
Fatemeh Mehregan
2
,
Mahnaz Maghbouli
3
1 - Department of Statistics and Mathematics, Arvand International Branch, Islamic Azad University, Abadan, Iran
2 - Department of Statistics and Mathematics, Arvand International Branch, Islamic Azad University, Abadan, Iran
3 - Department of Mathematics, Aras Branch, Islamic Azad University, Jolfa, Iran
الکلمات المفتاحية: Data Envelopment Analysis, Profit Efficiency, Multi-objective Programming,
ملخص المقالة :
Data envelopment analysis (DEA) is a nonparametric method for evaluating the relative efficiency of decision making units (DMUs) described by multiple inputs and multiple outputs. The issue of measuring the cost, revenue and profit efficiency in manufacturing and economic systems is one of the most important issues for managers. In this research, using Data envelopment analysis and multi-objective programming an attempt is made to provide a model for evaluating profit efficiency of banking industry. We apply data envelopment analysis (DEA) and multi-objective programming (MOP) models to measure profit efficiency as cost and revenue scores are as close as possible to their best scores and as far away as possible to their worst scores. The results showed that composing these two models, can directly affect the result and also findings of research distinguished the differences between the efficient DMUs from the point of view of DEA. In this study, Profit efficiency score has been obtained from a fairer perspective than the previous models. A numerical example of Iranian banking industry is used to illustrate the proposed model.
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[3] CAMANHO, A.S. & DYSON, R.G. (2005). Cost efficiency measurement with price uncertainty: a DEA application to bank branch assessments. European journal of operational Research, 161,432-446.
[4] FARELL, M.J. (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society (Series A) 129, 253-351.
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[6] FARE, R., GROSSKOPF.S. & LOVELL, C.A.K (1994). Production Frontiers. Cambridge University Press. Southern Illinois university, Carbondale.
[7] FARE, R., GROSSKOPF.S. & WEBER, W. (2004). The effect of Risk-based capital requirements on profit efficiency in banking. Applied Economics, 36, 1731-1743.
[8] FUKUYAMA, H., & MATOUSEK, R. (2017). Modeling bank performance: A network DEA approach. European Journal of Operational Research, 259, pp 721-732.
[9] FUKUYAMA H. & WEBER, W. L. (2008). Profit inefficiency of Japanese securities firms. Journal of Applied Economics, 11, 281-303.
[10] JAHANSHAHLOO, G.R. INAATEEOT- HEEEOAD, M., MOSTAFAEE, A. (2008). A simplified version of the DEA cost efficiency model. European Journal of Operational Research. 184, 814-815.
[11] JAHANSHAHLOO, G.R. ETRHADREO, S.M., VAKILI.J. (2011). an interpretation of the cost model in Data Envelopment Analysis. Journal of Applied Sciences, 11(2), 389-392.
[12] MOGHADDAS, Z & VAEZ GHASEMI, M. (2022). Revenue Efficiency Evaluation in a Two-stage Network with Nonlinear Prices in Data Envelopment Analysis. Journal of Decisions and operations Research, 6, 1–9.
[13] PARK, K.S.M., & ODN J.W. (2011). Pro-efficiency: Data speak more than technical efficiency. European Journal of Operational Research, 215, 301-308.
[14] PORTELA, M.C.A.S. &, THANASSOULIS, E. (2007). Developing a decomposable measure of profit efficiency using DEA. Journal of the Operational Research Society, 58, 481-490.
[15] SEYEDBOVEIR, S. KORDROSTAMI, S. DANESHIAN. (2018). Revenue-Profit Measurement in Data Envelopment Analysis with Dynamic Network Structures: A Relational Model. Int. J. Industrial Mathematics, 10.
[16] TOLOO, M. & ERTAY, T. (2014). The most cost efficient automotive vendor with price uncertainty: A new DEA approach. Measurement, 52,135-144.
Islamic Azad University Rasht Branch ISSN: 2588-5723 E-ISSN:2008-5427
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Optimization Iranian Journal of Optimization Volume 16, Issue, 2 2024,95-102 Research Paper |
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Online version is available on: https://sanad.iau.ir/journal/ijo
Profit Efficiency Evaluation: A composed Approach of DEA
And multi- objective programming
]
Soheila Seyedboveir*, Fatemeh Mehregan, Mahnaz Maghbouli
Department of Statistics and Mathematics, Arvand International Branch, Islamic Azad University, Abadan, Iran.
Department of Statistics and Mathematics, Arvand International Branch, Islamic Azad University, Abadan, Iran,
Department of Statistics and Mathematics, Aras Branch, Islamic Azad University, Jolfa, Iran.
Revise Date: 19 April 2025 Abstract
*Correspondence E‐mail: |
Keywords: Data Envelopment Analysis (DEA) Profit Efficiency Multi-objective Programming |
*Correspondence E‐mail: s.seyedboveir@ gamil.com |
INTRODUCTION
Data Envelopment Analysis (DEA) has been originated for measuring the relative efficiencies of a set of homogeneous decision making units (DMUs) that applies multiple inputs to generate multiple outputs. Currently, there has been a growing interest among decision makers in application of non-parametric techniques like DEA which extends markedly beyond the task of evaluating cost and revenue efficiency. Cost efficiency was first pioneered by Farrell (1957) and then extended by Fare et.al (1985). The following approach for modeling cost efficiency goes back to Camanho and Dyson (2005). The authors developed the traditional cost efficiency model into two various situations including precise known prices and incomplete price situations. Their model estimated upper and lower bound for cost efficiency evaluation in presence of price uncertainty. Jahanshahloo et.al (2008) continued their debate and refined the model with reducing the number of constraints and variables. In this respect, Jahanshahloo et.al (2011) offered an interpretation of cost models and introduced an alternative model foe assessment of cost efficiency assuming that the input prices of each DMU are accessible. In this framework, Amirteimoori et.al (2006) improved the cost efficiency interval of a DMU by adjusting its observed inputs and outputs. Camanho and Dyson (2005) contributed to this topic by obtaining cost efficiency from optimistic and pessimistic viewpoints including uncertain price. Considering uncertain price, Toloo and Ertay (2014) applied an alternative cost efficiency model based on DEA approach posits finding the most efficient unit. The concept of revenue efficiency was first debated in Fare et.al (1985). In their 1994 study [6], the authors improved the overall output price with the goal of maximizing revenue. As another instance, Fukuyama and Matousek (2017) expanded the environmental revenue function based on directional distance function in two-stage network structures. Another recent studies in revenue efficiency, Mogaddas and Vaez Ghasemi (2022) applied DEA approach to compute a specific set of weights to evaluate cost efficiency in a two-stage network system. A review of the DEA literature demonstrates that Fare et.al (2004) studies are widely recognized as a seminal reference in profit efficiency research. Fare et.al (2004) investigated two sources of inefficiency in assessing profit efficiency, including technical inefficiency and allocative inefficiency. Portela and Thanassoulis (2007) highlighted the drawbacks of existing approaches in the literature and suggested another measure of profit efficiency which is grounded in the geometric mean of input/output adjustments to achieve maximum profitability. Several researchers have proposed methods to address profit efficiency. A new indicator of profit inefficiency was suggested by Fukuyama and Weber (2008) emphasizing the choices made by decision-makers regarding the allocation of funds to inputs and the revenue derived from outputs, rather than the physical measurement of input and output quantities. Park and Cho (2011) introduced a linear programming model, for the evaluation of profit efficiency. The main focus of their paper was on approximation of profit efficiency in the absence of price information. Aparicio et.al (2013) illustrated the utility of DEA for measuring and decomposing revenue inefficiency. Their study considered all sources of technical waste with a specific emphasis on the Spanish quality waste sector. In a current study of all industries, the utilization of profit efficiency is becoming increasingly crucial especially in bank branch activities. In analyzing the literature, it is evident that several studies have advocated profit efficiency from the optimistic perspective. With respect to non-parametric Data Envelopment Analysis (DEA) models, the attention of this study has been given to both optimistic and pessimistic standpoint. This study examines the reasonable and equitable amount for profit regarding to costs incurred and generated revenues. The remainder of this study is organized as follows. Section 2 provides a brief overview of cost, revenue and profit efficiency. Then Section 3 formulates an alternative model for assessing profit efficiency as a Multi-Objective Programming (MOP) task. To clarify the details of the proposed method, a real case in banking sector is given in Section 4. Eventually, Section 5 concludes the paper.
PRELIMINARIES
According to Farrell [4] efficiency consist of two elements: technical efficiency (TE) and allocative efficiency (AE). TE refers to production where the best available technologies are applied and AE refers to allocation of inputs and products to different producers. Together, these efficiencies are named the economic efficiency, (EE defined. The (EE) is expressed in). The (EE) is expressed is different manners, depending on how the best available production technology is terms of cost minimization, revenue maximization or profit maximization. If cost minimization is assumed, The (EE) is expressed as (CE). In this case, CE constitutes a combination of inputs that generates the minimum possible cost. In a similar manner, the (EE) is expressed as (RE), RE constitutes a combination of outputs that generates the maximum possible revenue and if maximization of profit is of concern, the (EE) is expressed as profit efficiency (PE), that is, the amount of output that maximizes profit (Seyedboveir et.al, 2018).
Cost Efficiency
Suppose that there is a set of n decision-making units 
. Let 
be the numbers of inputs and outputs respectively. The term 
is applied in the input resource 
to to produce the output
, that is, the output product
from. Also let the unit price of all input be known, and 
shows the price of input from. Given these assumptions, the cost efficiency model can be written as follows:
Model (1) is a constant return to scale (CRS), the observed cost obtained through 
is presented as 
. The cost efficiency of (
is measured through :
In which 
is the optimal solution of model (1).
Revenue Efficiency
Let
be the price of the under evaluated unit (
output r, then DEA model of revenue maximization is:
Model (2) is a constant return to scale (CRS). The revenue obtained through the is equal to
. The revenue efficiency of (
is measured through:
Profit Efficiency
According to the assumptions of the previous two parts, the profit maximization problem is solved as follows:
Where 
is the price of input and 
is the price of output 
of the profit obtained by the is 
and profit efficiency (
of is measure as follows:
PROPOSED METHODS
Surveying the previous studies on profit efficiency concept, the optimistic point of view is highlighted in their evaluation. However, the pessimistic perspective was disregarded. Although, the existing studies has its merits but addressing the fairest and the most appropriate amount of profit continues to elicit questions. To obtain reliable results and improve applicability, a modifications appears warranted. Hence, for achieving the best profit to the decision maker, this paper aims to determine the most appropriate amount of cost and revenue, simultaneously. Toward this end, the combination of data envelopment analysis (DEA) and multi-objective programming (MOP) is becoming increasingly crucial as it utilizes mutual attributes of each model. Models (1) and (2) discussed in the previous sections, solely provide minimum cost 
and maximum revenue 
as the best values of cost and revenue. As mentioned before, both models focus on optimistic points of view. In order to consider both optimistic and pessimistic perspectives in a linear model, it is logical to obtain the worst values, i.e., maximum cost 
and minimum revenue
. For this modification, it seems sufficient to maximize model (1) and minimize model (2) respectively. Considering the concept of multi-objective programming (MOP), dual perspectives are replaced in the proposed model. To achieve the fair and reasonable values of cost and revenue the multi-objective model is applied. The proposed model ensures the values as close as possible to the most optimistic values 
and 
and sufficiently far from values 
and
. Again assume that there are n decision-making units 
, the data for 
on vectors of inputs and outputs are represented and, respectively. Applying the modified objective function, the objective function of model (3) can be rewritten as a bi-objective function as follows:
According to the amount of cost and revenue and employing the objective function (4), the fairest amount of profit can be achieved. Among different approach to solve a MOP, the Min-Max weighted format is adopted for our goal. In the following, we develop the proposed model with aggregating the two objective functions, which transforms the bi-objective function in a single weighted objective function. Aggregating the objective function (4) along with the constraints of Model (3), the following single-objective program is proposed.
It should be noted that, the weights 
defined in Model (5) are positive parameters. Also, solving the weighted Min-max formulation (5) means searching for a solution to obtain the most appropriate cost and revenue. In this sense, the above model (5) aligns more effectively with the concept of “the best” and “the worst” in the assessments. Getting advantages of employing the dual perspectives, the weights can be defined as follows:
The proposed Model (5) transforms to Model (7) using the substitution of the above weights 
and
. The modified Model (7) has the following format:
Upon close examination, all the constraints within the modified Model (7), support the idea of achieving the fairest amount of profit. The first and second constraints in Model (7) guarantee a fair cost and revenue that is as close as possible to the most optimistic and and sufficiently far from and. Therewith, Model (7) addresses some issues. First, two optimistic and pessimistic perspectives are integrated in a combined model. Second, defining appropriate weights
and in the objective function of Model (7) provide the most appropriate and fair amount of profit for each unit. The last but not least, as the objective function denotes, parameter 
, minimizes the deviation between the best value and the worst value. The rest of constraints are same as in Model (3). Considering model (7) searches for a solution where the deviation are equal and minimized. The main advantages of Model (7) over Model (3) is that it provides a suitable criterion for determining the difference between efficient DMUs. It can be easily demonstrated that Model (7) is always feasible.
Definition 1: The unit in the evaluation with model (7) is considered efficient if the optimal value of the model is equal to 1.
NUMERICAL EXAMPLES
The applicability of the proposed Model (7) is demonstrated using a real data set including ten units. The data set are related to one of the Iranian bank. Input and output indicators have been considered according to past researches and experts' opinions. There are three inputs characterized by deposits
, operating expenses 
and facilities
. The three outputs are reported by revenue from commissions
, annual net profit 
and transactions
. The input and output data are given in Table 1.
Table 1: Data Set of Inputs and Outputs
DMU |
|
|
|
|
|
| |
1 | 0.948 | 1 | 0.337 | 0.879 | 0.437 | 0.537 | |
2 | 1.330 | 0.993 | 0.180 | 0.538 | 0.282 | 0.280 | |
3 | 0.621 | 0.675 | 0.198 | 0.911 | 0.098 | 0.658 | |
4 | 1.783 | 0.897 | 0.491 | 0.570 | 0.391 | 0.461 | |
5 | 1.892 | 1.290 | 0.372 | 1.086 | 0.472 | 0.372 | |
6 | 0.990 | 0.856 | 0.253 | 0.722 | 0.263 | 0.153 | |
7 | 0.151 | 0.987 | 0.241 | 0.509 | 0.131 | 0.441 | |
8 | 0.108 | 0.203 | 0.097 | 0.619 | 0.097 | 0.267 | |
9 | 1.364 | 0.432 | 0.380 | 1.023 | 0.380 | 0.470 | |
10 | 1.992 | 0.956 | 0.178 | 0.769 | 0.176 | 0.288 |
DMU |
|
|
|
| Profit Model(3) | New Profit (Model7) |
|
1 | 0.921
| 8.602 | 1.501 | 0 | -.039 | 0.698 | 0.07 |
2 | 0.523
| 4.595 | 0.802 | 0 | -0.835 | 0.497 | 0.12 |
3 | 0.934
| 5.054 | 13.386 | 0 | 1 | 1 | 0.06 |
4 | 1.231
| 10.941 | 11.932 | 0 | -.0695 | 0.995 | 0.08 |
5 | 1.069
| 9.496 | 9.628 | 0 | -.0411 | 0.937 | 0.06 |
6 | 0.720
| 6.458 | 3.960 | 0 | -0.201 | 1 | 0.09 |
`7 | 0.604
| 3.462 | 11.414 | 0 | 1 | 1 | 0.07 |
8 | 0.432 | 2.476 | 6.911 | 0 | 1 | 1 | 0.00 |
9 | 1.053
| 5.269 | 12.165 | 0 | 1 | 1 | 0.05
|
10 | 0.625 | 4.544 | 7.454 | 0 | -1.230 | 0.567 | 0.07 |
