Profit Efficiency Evaluation: A composed Approach of DEA and multi- objective programming

Subject Areas : Operation Research

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

Received: 2025-02-09 Accepted : 2025-06-08 Published : 2025-01-13

Keywords: Data Envelopment Analysis, Profit Efficiency, Multi-objective Programming,

Abstract :

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.

References:

[1] AMIRTEIMOORI, A., KORDROSTAMI, S. & RABETIEZER, A. (2006). An improvement to the cost efficiency interval: A DEA based approach. Applied mathematics and computation, 81,775-781.

[2] APARICIO, J. BORRAS, F. &, PASTOR, J.T., VIDAL, F. (2013). Accounting for slacks to Measure and decompose revenue efficiency in the Spanish Designation of Origin wines with DEA European Journal of Operational Research, 231,443-451.

[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.


[5] FARE, R., GROSSKOPF.S. & LOVELL, C.A.K. (1985). The measurement of efficiency of Production. Boston: Kluwer Publication.

[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.

Full-Text:



Islamic Azad University Rasht Branch ISSN: 2588-5723        E-ISSN:2008-5427

Optimization Iranian Journal of Optimization Volume 16, Issue, 2 2024,95-102  Research Paper

 

 

                                        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:

Accept Date: 08 June 2025         

Keywords: Data Envelopment Analysis (DEA)  Profit Efficiency  Multi-objective Programming

   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.

 

 

 

 

 

 

 

 

 

*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

 

By implementing the cost and revenue efficiency models, namely Models (1) and (2), the minimum cost (and maximum revenue ( values ​​are shown in the second and third columns of Table 2.  In order to have the worst amount for cost and revenue, Models (1) and(2) are re-solved with maximization and minimization objective functions, respectively. The results ​​of the maximum cost and minimum revenue are indicated by and . The results of Models (1) and (2) are reported under the heading of ( and ( in Table 2. The fourth and fifth column of Table 2 are reporting the amount of   and, respectively. Applying Model (3) on the data set of Table 1 are represented in the sixth column of Table 2. The last two columns of Table 2 Show the amount of profit efficiency and the value of parameter. As the seventh column of Table 2 under the heading of “New Profit(Model(7)” shows, there are five units ,namely units# 3, 6, 7, 8 and 9 have the efficiency equal to one. In other words, in evaluation with Model (7), these units are efficient. On the other hand, Model (3) estimates four efficient units, units # 3, 7, 8 and 9 have been evaluated as efficient. As concerns the results obtained by the min-max Model (7), one can see that the profit efficiency score are greater than the counterpart Model (3). The fact that unit six, show the profit efficiency as unity, whilst Model (3) presents the amount of -0.201. What’s more, existence of the deviation parameter 𝛿 in Model (7) advocates that the proposed Model (7) outperforms its counterpart Model (3). That is to say, the proposed Model (7) performs effectively among efficient units than Model (3) does. The effect of integrating the smallest distance between and  and between and, are reported as parameter 𝛿 in the last column of Table 2. The value of parameter 𝛿 in Model (7) are reported as positive number except for the unit #8. It is revealed that Unit#8 has better performance among efficient units. In a nutshell, Model (7) outperforms Model (3) in this real example.

 

 

CONCLUSION

 

In spite of numerous advancement about profit efficiency in DEA, this study contributes the continuing debate in the realm of profit efficiency. To be specific, the existing studies consider the optimistic perspective in profit evaluation. Equipped with multi-objective models, a combined model was introduced for evaluation the best amount of profit efficiency. The main advantages of the proposed model was integrating dual perspectives in one model. Some issues are addressed in the proposed model. First, the proposed model aimed to minimize the distance between the minimum cost and the maximum cost, at the same time, the distance between the maximum income and the minimum income is also minimized. Second, according to the amount of cost and income, the proposed model can identify units with the best performance among profit efficient units. Additionally, the proposed model may elucidate the factors of profit inefficiency and provide valuable insights into challenges posed by inefficiency.

 

 

 

 

 

 

Table 2: Results of Numerical Example

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

 

 

 

 

REFERENCES

Amirteimoori, A, Kordrostami, S., & Rabetiezer, A. (2006). An improvement to the cost efficiency interval: A DEA based approach. Applied Mathematics and Computation 81,775-781..

Aparicio, B., Borras, F., Pastor, J.T., &Vidal, F. (2013). Accounting for slacks to measure and decompose revenue efficiency in the Spanish Designation of Origin wines with DEA, European Journal of Operation Research, 231,443-451.

Camanho, A.S., & Dyson, R.G. (2005). Cost efficiency measurement with price uncertainty: a DEA application to bank branch assessments, European Journal of Operation Research ,161,432-446

Farell, M.J. (1957). The measurement of productive efficiency. Journal of the Royal Statistical Society(Series A) 129, 253-351. 

Fare, R., Grosskopf, S., & Lovell, A.C.K. (1994). Production Frontiers. Cambridge University Press. Southern Illinois University, Carbondale.

Fare, R., Grosskopf, S., & Weber, W. (2004). The effect of Risk-based capital requirements on profit efficiency in bankining. Applied Economics, 36, 1731-1743

Fukuyama, H., & Matousek, R. (2017). Modeling bank performance: A network DEA approach. European Journal of Operation Research, 259, 721-732

Fukuyama, H., &Weber, W.L.(2008). Profit Efficiency of Japanese securities firms, Journal of Applied Economics, 11, 281-303.

Jahanshahloo, G.R., Inateeot- Heeoad, M., & Mostafaee, A. (2008).  A simplified version of the DEA cost efficiency model, European Journal of Operation Research, 184, 814-815.

Jahanshahloo, G.R., Etrhadero, S.M., & Vakili,J.(2011).  An interpretation of the cost model in Data Envelopment Analysis. Journal of Applied Science,11(2),389-392. 

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.

Park, K.S.M., & ODN J.W. (2011).  Pro-efficiency: Data speak more than technical efficiency. European Journal of Operational Research, 215, 301-308.

Portela, M.C.A.S. &, Thanassoulis E. (2007). Developing a decomposable measure of Profit Efficiency using DEA. Journal of Operation Research Society, 58,481-490.

 Seyedboveir, S., Kordrostami,S.,& Daneshian,B., (2013). Revenue-Profit Measurement Data Envelopment Analysis with Dynamic Network Structures: A Relational Model, International Journal of Industerial Mathematics, 1(10), 103-114.  

Toloo, M., & Ertay, T. (2014). The most cost efficient automotive vendor with price uncertainty: A new DEA approach. Measurement, 52,135-144. 

 

 

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

Iranian Journal of Optimization, 16(2), 95-102, June 2024 ,2023  2022  2022   22

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