Merger analysis using inverse DEA: the case of variable returns to scale
Subject Areas : Data Envelopment Analysis
1 - Faculty of Industrial Engineering and Management Science, Shahrood University of Technology
Keywords: Inverse DEA, Merger Analysis, Semi-additive technology, financial institute,
Abstract :
In a dynamic economy, mergers and consolidations of economic and finance sectors likebanks, etc. becoming more common. One of the applications of the inverseDEA is the merger analysis of a series of production units. Data envelopment analysis (DEA) measures the efficiency score of a decision making unit (DMU) considering its input and output level. On the other side, the inverse DEA approach aims to find required input levels for DMU to produce a presumed output level preserving the efficiency score. In a rather recent paper, (Gattoufi, Amin, & Emrouznejad, 2014) introduced an interesting application of inverse DEA models for merger analysis. The current paper extends this work by developing a generalized inverse DEA model assuming variable returns to scale. In contrast with (Gattoufi et al., 2014), it is shown that proposed models are always feasible and bounded. The idea is illustrated using a methodical argument and two numerical examples. An empirical study of financial institutes shows the strength and the applicability of the proposed methods.
Merger analysis using inverse DEA: the case of variable returns to scale
Abstract
In a dynamic economy, mergers and consolidations of economic and finance sectors like
banks, etc. becoming more common. One of the applications of the inverse
DEA is the merger analysis of a series of production units. Data envelopment analysis (DEA) measures the efficiency score of a decision making unit (DMU) considering its input and output level. On the other side, the inverse DEA approach aims to find required input levels for DMU to produce a presumed output level preserving the efficiency score. In a rather recent paper, (Gattoufi, Amin, & Emrouznejad, 2014) introduced an interesting application of inverse DEA models for merger analysis. The current paper extends this work by developing a generalized inverse DEA model assuming variable returns to scale. In contrast with (Gattoufi et al., 2014), it is shown that proposed models are always feasible and bounded. The idea is illustrated using a methodical argument and two numerical examples. An empirical study of financial institutes shows the strength and the applicability of the proposed methods.
Keywords: Inverse DEA; Merger Analysis; Semi-additive technology; financial institute
1. Introduction
Data envelopment analysis (DEA) is a mathematical programming based approach for efficiency analysis of a group of decision making units (DMUs) proposed by (Charnes, Cooper, & Rhodes, 1978). An extension for considering variable returns to scale (VRS) was proposed by (Banker, Charnes, & Cooper, 1984) that has been used in many applications. The strength of the former is letting the production space have different types of returns to scale, thus it has rather been more interested of users unless there exists specific prior information about returns to scale of the production. The traditional DEA model considered estimating the relative efficiency of DMUs based on observed input and output data. In a different path, the inverse DEA models keep relative efficiencies unchanged and then try to find (a) required input for a provided output (input oriented) or (b) producible output for a given level of input. It is assumed that the efficiency score is fixed and the aim is to find required input (output) levels for a given perturbed outputs (inputs) level. (Wei, Zhang, & Zhang, 2000) were inspired by inverse optimization and the work of (Zhang & Cui, 1999) to start this new path in DEA literature that yields inverse DEA models. Different extensions of this method have been done in the literature so far. (Jahanshahloo, Vencheh, Foroughi, & Matin, 2004) investigated the input and output estimation when some of the output is undesirable. (Hadi-Vencheh, Foroughi, & Modelling, 2006) uses inverse DEA models for generalized resource allocation problems. (Jahanshahloo, Soleimani-Damaneh, & Ghobadi, 2015) considered the inter-temporal dependence DEA model and proposed an inverse framework for this case. (Lertworasirikul, Charnsethikul, & Fang, 2011) considered the variable returns to scale (VRS) properties like (Banker et al., 1984) for the production technology in the inverse DEA problem. However, there exist some drawbacks in their model that were pointed out and revised by (Mojtaba Ghiyasi, 2015). (Mojtaba Ghiyasi, 2017) dealt with the cost and revenue efficient in the inverse literature. (Eyni, Tohidi, & Mehrabeian, 2017) dealt with cone constraint inverse DEA modeling in the presence of undesirable output. (Mojtaba Ghiyasi & Zhu, 2020) dealt with the negative data in the inverse DEA modeling.
In a dynamic economy, mergers and consolidations of economic and finance sectors like
banks, etc. becoming more common. One of the attractive applications of the DEA models is in the banking sector. We refer the readers to an interesting review of DEA models applied in the banking sectors by (Paradi & Zhu, 2013). A few research also used the DEA models for the merger analysis in the banking sectors. See for instance (Wheelock & Wilson, 2000) which studied the characteristics of U.S. individual banks to be acquired. (Sherman & Rupert, 2006) considered the merger issue for bank branches and analyzed the potential of avoiding the operational costs using merger analysis of branches. In another application, bi-level programming models were used by (Wu, Luo, Wang, & Birge, 2016) based on a leader-follower game model for the merger effects of banks. The bootstrap DEA approach was utilized by (Moradi-Motlagh & Babacan, 2015) for analysing the merger’s impact on the Australian banks during the period of the financial crisis.
In an interesting application of the inverse DEA approach, (Gattoufi et al., 2014) utilized the inverse DEA models for merger analysis of the world bank. This recent contribution also is extended by different studies. (Amin & Al-Muharrami, 2016) proposed an inverse DEA model for merger analysis capable of dealing with negative data. (Amin & Boamah, 2020) dealt with the cost efficiency concept in the merger and inverse DEA analysis. (Amin, Al-Muharrami, & Toloo, 2019) combined goal programming and inverse DEA model for target setting and merge. (Amin & Boamah) proposed an inverse DEA model for estimating potential merger gain, considering cost efficiency measurement of DMUs. By this, they distinguished the technical and cost efficiency measure in the merger analysis using the inverse DEA based models.
The returns to scale is an important characteristic of the production technology and may affect the level of production for a different level of the input. This issue becomes more important when we deal with the merger problem. Consider the merger of two banks. If we assume the constant returns to scale for the production technology, then we face a merged bank that still operates in constant returns to scale region. However, one of the main issues of the merger analysis in the scale effect of the merger that should be considered in the analysis. This can be done using a production technology with variable returns to scale properties.
In the current paper, we show that we need more care when we deal with the merger analysis in case of the variable returns to scale. The current paper shows the models of (Gattoufi et al., 2014) may not be feasible in some situations. The main source of infeasibility is in fact due to the variable returns to scale properties of their model, not because of the inverse structure of the models. Although they categorized mergers as consequence feasible and infeasible mergers by minor and major consolidation in another recent paper (Amin, Emrouznejad, & Gattoufi, 2017), but the infeasibility of the merger in the variable returns to scale is still an important issue. In the current paper, problematic issues are described and the source of the obstacle is scrutinized and then some new models are proposed to extend the work of (Gattoufi et al., 2014) and tackle problematic issues. Proposed models are motivated and illustrated using mathematical arguments and two numerical examples. Moreover, we applied the proposed models for the merger analysis of financial institute in Iran. Section 2 reviews relative DEA and inverse DEA. Section 3 describes the existing problems in the inverse merger DEA model and then some new models are developed as an extension of existing models in the literature for overcoming the aforementioned existing problems.
2. DEA and inverse DEA
Suppose there are n DMUs that are using m inputs to produce p outputs. Let 
be i-th input of j-th unit and 
be r-th output of j-th unit,
. The following DEA model measures the efficiency of DMUo, that is, the DMU under evaluation
In a different path, the following inverse DEA model finds the required input level 
for producing a perturbed given level of output, preserving the relative efficiency of this DMU.
An interesting application of the inverse DEA model was proposed by (Gattoufi et al., 2014) for merger analysis. Considering DMUk and DMUl to be merged to a new DMU, let us call it DMUM, the following model finds the minimum required input of merged DMUs for producing the aggregated output of the new DMUM.
, where 
is the index set of remaining units in the market that can be either 
or 
. In the former DMUk remains in the market but in the latter setting, both DMUk and DMUl vanish after merging.
3. Motivation and extension of the inverse DEA for merger analysis
In this section, we first describe and highlight the shortcoming of the merger model (Gattoufi et al., 2014) by mathematical argument and a numerical example and then propose a new model that tackles existing shortcoming.
3.1. Theoretical Motivation
As stated by (Gattoufi et al., 2014) model (2.3) is feasible in their theorem 1. They yielded this result by saying that (2.3) is bounded and its dual is feasible as follows:
However, having linear programming bounded and its dual is feasible we cannot conclude that the primal is feasible. This ambiguity yields a wrong result in the theorem 1 of (Gattoufi et al., 2014). This problem is illustrated in the following example.
Numerical example1. Consider four DMUs with one input and two outputs listed in table 1.
Table 1: Input-output data of numerical example 1
| A | B | C | D |
Input | 7 | 6 | 9 | 5 |
First Output | 4 | 8 | 11 | 5 |
Second Output | 10 | 9 | 5 | 13 |
Consider DMU A and suppose DMU B aims to merge with DMU A, thus the aggregated output levels of the first and second output are 12 and 19 respectively. (Gattoufi et al., 2014) considered two cases. In The first case, both DMUk and DMUl disappear, that is, both DMU A and DMU B disappear, then the exiting DMUs are 
. In the second case, only DMUk remains, that is, DMU A remains and then the set of existing DMUs is 
. Considering the first case we yield the following linear programming
And considering the second case, we get the following linear programming
Regardless of the value of the predefined value of 
, we see both above models are infeasible and therefore model (2.3) that is model (5) of (Gattoufi et al., 2014) needs a revision. This is due to the fact that the merging unit falls out of the production set and therefore, it is not possible to find a new input level for producing the output level of the merging unit with any efficiency level.
3.2. A generalized inverse DEA model for merger analysis
The source of infeasibility in the model (2.3) is because even the classical DEA model considering VRS assumption may fail on merger analysis. In other words, classical DEA models, specifically the well-known BCC model with VRS assumption may be infeasible when assessing merged units. See (Mojtaba Ghiyasi, 2016) for more details. However, (Mojtaba Ghiyasi, 2016) proposed a new production set called semi-additive with VRS properties that does not face any problem in terms of infeasibility. The following model gauges the relative efficiency of DMUo using semi-additive technology
, where 
is the index set of all aggregated, but not self-aggregated units. See (Mojtaba Ghiyasi, 2016) for more detail about the characteristics of semi-additive technology.
The above model is capable of measuring the efficiency of any aggregated (merged) unit for the VRS case, without any concern about the infeasibility problem. The following inverse DEA model for merger analysis is developed using semi-additive production technology and we make sure it has no problem in terms of infeasibility.
, where 
is the index set of reaming units that can be either 
(ignoring both DMUk and DMUl)or 
(keeping DMUk ).
Theorem1 . The linear programming model of (3.2) is always feasible and bounded.
Proof. For merger analysis of DMUk and DMUl and regardless of the selection of we know that the aggregated unit of DMUk and DMUl and its index exist in the index set of . Thus, considering 
is a feasible solution of model (3.2) such that 
, 
and 
is the efficiency of aggregated DMUM using model (3.1). This guarantees the feasibility of the model (3.2). The objective value of the aforementioned feasible solution, namely, is , that is, input summation of DMUk and DMUl .
Numerical example 2. Considering the same data set as numerical example 3, we take DMU A into consideration and merge DMU B with DMU A. the following model finds the required input level for this merger provided by a predefined efficiency score of .
The above model ignores both DMU A and DMU B. However, one may consider the case that DMU A stays in the market and for this case, we can use the following model.
In the proposed model of (3.2), we used the full semi-additive technology for the inverse merger model. However, we can use partial semi-additive technology. Considering DMUo for the merger analysis, we can only think through those aggregated units that include DMUo. For this case, we just need to update the index set of 
to include observed DMU and aggregated DMUs that consist of DMUo, in the proposed model of (3.2). We prevent rewriting associated models. For the sake of clarification, for the numerical example in this case we have the following models.
3.3. Output estimation in the merger analysis
Taking the output oriented inverse DEA model for merger analysis for the case of VRS, we get the following feasible model for output estimation.
, where 
is the desired output efficiency level for the merged unit by DMUk and DMUl . The above model finds the maximum output level that can be produced using the sum of input level of DMUk and DMUl, that is, 
. The following model shows the feasibility of model (3.3) and it also shows that this model is bounded too.
Theorem 2. The linear programming model of (3.3) is always feasible and bounded.
Proof. It is similar to the proof of theorem 1 with some minor modifications.
4. An application
In this section, we perform a performance assessment and then a merger analysis for 19 branches of a financial institute in Iran. Branches’ area (m2 )and total cost (1000 Iranian Rial) are considered as inputs. On the other side, a number of transactions and deposited value (1000 Iranian Rial) are considered as outputs. Table 2 reports the statistical description of the data.
Table 2: Summary statistics of data
Variable name | Branches’ area | Total cost | Number of transactions | Deposited value |
Mean | 211.7368421 | 3538.052632 | 55353.73684 | 193960.5263 |
Max | 296 | 7303 | 93898 | 285459 |
Min | 103 | 1209 | 16003 | 105069 |
Standard error | 67.98052516 | 1757.384674 | 22055.06285 | 59334.74128 |
In the first part of the analysis, we gauge the efficiency measure of the branches using the model ((3.1). The results are reported in the second column of Table 3 which shows only three efficient branches. However, the mean efficiency score is about 70 percent, considering all branches. This shows a 30 percent possibility of improvement in the system.
Table 3: Efficiency score of branches
Branches | Efficiency scores |
B1 | 0.906890176 |
B2 | 0.456379301 |
B3 | 0.481927626 |
B4 | 0.69320566 |
B5 | 1 |
B6 | 0.825405847 |
B7 | 0.497660997 |
B8 | 0.807301362 |
B9 | 1 |
B10 | 0.615041543 |
B11 | 0.476851852 |
B12 | 0.911504425 |
B13 | 0.546131971 |
B14 | 0.437497773 |
B15 | 0.830645161 |
B16 | 0.881354412 |
B17 | 0.38576779 |
B18 | 1 |
B19 | 0.556756757 |
In the next run, we perform a merger analysis of branches. In this analysis, we consider the merger of those branches that are potentially possible in reality. We used the ideas of the institute’s top managers at the province in this step. There are possibilities of having ten potential mergers based on the opinion of the managers. Then we performed the merger analysis for these branches using the proposed model (3.2) and the results are reported in table 4. The required inputs for producing the aggregated level of the merging branches with a given efficiency level are reported in the second and third columns of Table 4. Merging branches of B6 and B17 requires the highest area while merging branches of B11 and B17 suffers the highest cost. However, these poetical mergers switch their places when we look at the second highest cost, and area then we see B17 and B11. Therefore, these two mergers are the most resource-demanding in merger planning. It is important to note that model (2.3) falls into infeasibility for analysis of some potential mergers like B1-B10.
Table 4: The merger analysis
Merging branches | Required 1th input | Required 2th input | Expected efficiency |
B1 & B10 | 180.7641 | 2523.7959 | 0.91 |
B5 & B12 | 193.6139 | 3220.2301 | 0.9036 |
B6 & B17 | 538.0289 | 6451.9006 | 0.5 |
B4 & B11 | 180.5104 | 3156.2816 | 0.9700 |
B7 & B15 | 328.0361 | 4361.7827 | 0.58 |
B10 & B19 | 355.5545 | 4325.99 | 0.6 |
B11 & B17 | 530.0199 | 6588.7803 | 0.35 |
B13 & B2 | 222.154 | 4415.0414 | 0.8 |
Table 5 reports the input share of each branch involved in the merger process. We see for instance in the merger plan of B1-B10, B4-B11, and B13-B2 that more efficient branches cover the less efficient peer branch. B1 has more efficiency level compared with B10, thus only B1 brings more share of the first input in the merger. We have the same finding for B4-B11 and B13-B2. however, this may not always be the case. In some merger plans, the less efficient branch should put more effort into the efficient ways of using resources. See for instance B7-B15, where B15 with the lower efficiency should bring more resources into the merger. These sort of resources that could have been wasted by B15 should be used more efficiently by merging with a more efficient branch.
Table 5: Input share of merging branches
Merging branches | The first input share of merging branches | The first input share of merging branches | The second input share of merging branches | The second input share of merging branches |
B1 & B10 | 180.7641 | 0.0000 | 244.0000 | 2279.7959 |
B5 & B12 | 121.0000 | 72.6139 | 113.0000 | 3107.2301 |
B6 & B17 | 142.0000 | 396.0289 | 267.0000 | 6184.9006 |
B4 & B11 | 180.5104 | 0.0000 | 291.0000 | 2865.2816 |
B7 & B15 | 267.0000 | 61.0361 | 124.0000 | 4237.7827 |
B10 & B19 | 244.0000 | 111.5545 | 185.0000 | 4140.9900 |
B11 & B17 | 216.0000 | 314.0199 | 267.0000 | 6321.7803 |
B13 & B2 | 222.1540 | 0.0000 | 288.0000 | 4127.0414 |
5. Conclusion
This paper extended the merger inverse DEA models in the case of VRS for the production technology. Some problematic issues in the merger analysis using the inverse DEA models are pointed out. This highlights the importance of using the methodology in real-world problems. Then a generalized inverse DEA model for merger analysis is proposed that considers the VRS and it is capable of dealing with all merger analyses without any concern about infeasibility. Proposed models are illustrated using numerical examples and their applicability is shown in a real word problem.
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