# A new inverse data envelopment analysis model for mergers with negative data

A new inverse data envelopment analysis model for mergers with negative data Abstract One of the important issues in a merger between two or more decision making units is identification of the levels of inherited inputs and outputs from merging units. This paper introduces new inverse data envelopment analysis models for target setting of a merger in the presence of negative data. It enables the merged entity to identify the levels of required inputs and outputs from merging units to realize an efficiency target. The applicability of the proposed method in this paper is illustrated through an example from banking; however, it can be used in other merger contexts. 1. Introduction Mergers and acquisitions have shown the facilities of combining decision making units (DMUs) as an effective strategic corporate restructuring for increasing the production capabilities of the involved units (Hagedoorn & Duysters, 2002). One of the research areas that has been successfully used for mergers performance evaluation in different contexts is data envelopment analysis (DEA) (Emrouznejad et al., 2008; Liu et al., 2013; Gattoufi et al., 2014). DEA applications in mergers are found in different areas such as banking (Wheelock & Wilson, 2000; Luo, 2003; Sherman & Rupert, 2006; Hahn, 2007; Wu et al., 2011; Wu & Birge, 2012; Wu et al., 2015; Moradi-Motlagh & Babacan, 2015; Du & Sim, 2016), non-bank institutions like credit units (Fried et al., 1999; Worthington, 2004; Jin et al., 2015; Halkos et al., 2016), healthcare (Kristensen et al., 2010; Leleu et al., 2012), forestry (Bogetoft & Bo, 2003) agriculture (Bogetoft & Wang, 2005), and airlines (Kong et al., 2012). In all of these cases, the standard DEA methodology is used for estimation of merger gains under the existing levels of inputs and outputs (Bogetoft & Wang, 2005; Lozano & Villa, 2010). The classic DEA models do not reveal the levels of inputs and outputs that would allow the merged entity to realize a specific efficiency target. A standard DEA model then cannot deal with such a problem and as a result the Inverse DEA (InvDEA) is a more appropriate method (Wei et al., 2000; Gattoufi et al., 2014). Unlike the objective of a conventional DEA model that obtains the efficiency scores of DMUs (Emrouznejad et al., 2008; Farzipoor Saen, 2011; Yeung & Azevedo, 2011), InvDEA aims to find the levels of inputs and outputs that are required to realize a given efficiency target. The idea of InvDEA was first introduced to estimate inputs and/or outputs for a resource allocation problem (Wei et al., 2000). The DEA literature was highly enriched by studies that implemented InvDEA in different contexts. Li & Cui (2008) suggested a comprehensive InvDEA approach for resource allocation. Lertworasirikul et al. (2011) discussed the possibility of simultaneous increase of some outputs and decrease of the other outputs by means of InvDEA. Gattoufi et al. (2014) proposed an InvDEA methodology to identify the maximum levels of inputs anticipated to be saved bythe synergy of the merging DMUs. They have also suggested an InvDEA model for determining the maximum additional outputs that can be produced after the merger. Given an efficiency target for the merged entity, the proposed input-oriented InvDEA identifies the optimal levels of inputs inherited from merging DMUs. Recent years have seen various applications of InvDEA in different areas of DEA. Jahanshahloo et al. (2015) proposed an inter-temporal application of InvDEA. Ghiyasi (2015) introduced InvDEA models for variable return to scale phenomena. Moreover, Lim (2016) suggested an InvDEA methodology for a new product target setting by assuming frontier change. To the best of our knowledge, there is no application of InvDEA in mergers except the recent work of Gattoufi et al. (2014). The authors developed the concept of InvDEA in M{&}A analysis in order to find the optimal levels of inputs and outputs that are needed from the merging entities. By using the InvDEA, the authors enabled the merged entity to realize a given efficiency target. The merged entity can realize a given efficiency target if it removes redundant inputs or produces additional outputs depending on the selected orientation InvDEA model. Nevertheless, the proposed method in Gattoufi et al. (2014) cannot be used for target setting of the merged entity when there is negative data. In this paper we develop a new InvDEA method by generalizing the concept of InvDEA. In our proposed InvDEA method, there is at least one negative variable among inputs and outputs. Under this model, the merged entity can set its inputs and outputs to realize a given efficiency target. More specifically, the proposed method identifies the maximum improvement on the negative outputs of merging DMUs after a merger. We choose banking as a business application to show the usefulness of the proposed method in an area yet studied. Another main advantage of this study is that there is no alternative DEA or similar method for target setting of a merger when there are negative variables such demonstrated in our case on banking. The remainder of this paper is as follows. Section 2 reviews the literature of DEA models suggested for negative data. This is followed by a brief explanation of the InvDEA models for mergers in Section 3. Section 4 develops new InvDEA models for target setting after merger with negative data. In Section 5, we use a real case of bank mergers to show the practical contribution of the proposed method. Finally, Section 6 gives the conclusions. 2. DEA with negative data In the presence of negative data, classical DEA models cannot be used for performance evaluation. There are many research studies in the literature that have developed new DEA models for dealing with negative variables. Data transformation is one of the introduced approaches in such literature that transforms all negative data to positive values; see e.g. Lovell (1995); and Seiford&Zhu (2002). Scheel (2001) proposed a method that treats the absolute values of negative outputs as inputs and the absolute values of negative inputs as outputs. A modified slack-based measure for handling both negative outputs and negative inputs is introduced in Sharp et al (2007). Portela et al. (2004) have also suggested range directional measure DEA models for dealing with positive and negative values variables. More recently, Emrouznejad et al. (2010a,b) proposed a semi-oriented radial measure (SORM) DEA model which can yield a measure of efficiency by handling variables that take positive values for some and negative values for other DMUs In this paper, we use the SORM model as the base DEA model and suggest new InvDEA models for M&A target setting. We used this combination due to the similarity between the SORM and the classical DEA models in providing an efficiency value. Nevertheless, the base DEA model can be any other forms of DEA models that can handle negative data. Let us assume an output variable $${{\bf Y}}_{k}$$ is positive for some DMUs and negative for others. Emrouznejad et al. (2010a,b) defined the following two variables $${{\bf Y}}_{k}^{1}$$ and $${{\bf Y}}_{k}^{2}$$  Ykj1={Ykjif Ykj⩾00Otherwise Ykj2={−Ykjif Ykj<00Otherwise  where, $${{\bf Y}}_{kj}^{1} \geqslant 0$$, $${{\bf Y}}_{kj}^{2} \geqslant 0$$ and $${{\bf Y}}_{kj} ={{\bf Y}}_{kj}^{1} -{{\bf Y}}_{kj}^{2}$$ for all $$j=1,... ,n$$, and treated $${{\bf Y}}_{k}^{1}$$ as an output and $${{\bf Y}}_{k}^{2}$$ as an input. The same decomposition is used for an input variable that is positive for some DMUs and negative for others. That is   Xkj1={Xkjif Xkj⩾00Otherwise Xkj2={−Xkjif Xkj<00Otherwise  where, $$X_{kj}^{1} \geqslant 0$$, $$X_{kj}^{2} \geqslant 0$$ and $$X_{kj} =X_{kj}^{1} -X_{kj}^{2}$$ for all $$j=1,... ,n$$, and treated $$X_{k}^{1}$$ as an input and $$X_{k}^{2}$$ as an output. Assume $$I_{1}$$ denotes the set of indices for positive inputs and $$I_{2}$$for the set of inputs that is negative for at least one DMU. Moreover, let $$R_{1}$$ be the set of indices for positive outputs and $$R_{2}$$ for the set of outputs that is negative for at least one unit. The output-oriented SORM DEA model can be written as follows (Emrouznejad et al., 2010a)    maxhs.t. ∑j=1nxijλj⩽xik,i∈I1,∑j=1nxij1λj⩽xik1,i∈I2,∑j=1nxij2λj⩾xik2,i∈I2 ∑j=1nyrjλj⩾hyrk,r∈R1,∑j=1nyrj1λj⩾hyrk1,r∈R2,∑j=1nyrj2λj⩽hyrk2,r∈R2 ∑j=1nλj=1λj⩾0,j=1,...,n (1) In this paper we use an output-oriented SORM DEA model however, the input-oriented SORM DEA model can be similarly written. 3. Merging DMUs using InvDEA For simplicity in modelling, let us consider the case of a merger when there are two merging units. As we demonstrate, this assumption in the modelling can be easily extended to more than two merging DMUs. Suppose we have $$n$$ DMUs each consuming $$m$$ inputs for producing $$s$$ outputs. More specifically, let $$x_{ij}$$ and $$y_{rj}$$ denote, respectively, the $$i$$th input and the $$r$$th output for the $$j$$th DMU ($$i=1,... ,m,\, r=1,... ,s$$,$$j=1,... ,n)$$. Assume that DMUs $$k$$and $$l$$decide to go through a merger for producing a new entity, say M. In an input-oriented model, as assumed in Gattoufi et al. (2014), the merged entity M keeps the entire outputs of DMUs $$k$$ and $$l$$ and seeks the minimum levels of inherited inputs to realize a given efficiency target $$\bar{{\theta}}$$. Let us define the decision variables $$\alpha_{ik}$$ and $$\alpha_{il}$$ for the levels of the $$i$$th input that is going to be kept by the merged entity M from the merging DMUs $$k$$ and $$l$$, respectively, implying the $$i$$th input of the merged entity M to be $$\alpha_{ik} +\alpha_{il}$$ ($$i=1,... ,m)$$. The corresponding input-oriented InvDEA model suggested in Gattoufi et al. (2014) can be rewritten as follows:   min∑i=1m(αik+αil)s.t.∑j∈Fxijλj+(αik+αil)λM−(αik+αil)×θ¯⩽0i=1,...,m∑j∈Fyrjλj+(yrk+yrl)λM⩾(yrk+yrl)r=1,...,s∑j∈Fλj+λM=10⩽αik⩽xik,0⩽αil⩽xili=1,...,mλj⩾0,j∈F,λM⩾0 (2) where $$F$$ is the set of available peers in the post-merger evaluation process. In a merger there are two possible cases (1) only the acquiring DMU survives in the market and continues to operate by its former name, or (2) both merging DMUs consolidate in one new entity with a new name. Therefore, $$F$$ may contain one of the merging DMUs $$k$$ or $$l$$ or neither, depending on the form of the merger. In the above InvDEA model the merged entity is denoted by M regardless of the form of the merger and the name of the merged entity. In fact, the important issue here is what should be the levels of the inherited inputs of the post-merger entity? As discussed in Gattoufi et al. (2014), model (2) is a non-linear programming problem that can be linearized by excluding $$M$$ from the set of its evaluating peers, or equivalently taking $$\lambda_{M} =0$$. Therefore, the relaxed linear programming form of model (1) can be written as follows:   min∑i=1m(αik+αil)s.t.∑j∈Fxijλj−(αik+αil)×θ¯⩽0i=1,...,m∑j∈Fyrjλj⩾(yrk+yrl)r=1,...,s∑j∈Fλj=10⩽αik⩽xik,0⩽αil⩽xili=1,...,mλj⩾0,j∈F (3) In the InvDEA model (1) and the corresponding relaxed model (3) it is assumed that all inputs have the same measurement units. Nevertheless, inputs with different units can be unified by using the formula$$\sum\nolimits_{i=1}^m {p_{i} (\alpha_{ik} +\alpha_{il} )}$$as the objective function, where $$p_{i}$$ is the cost per unit of the $$i$$th input ($$i=1,... ,m)$$. Similar to the input orientation, Gattoufi et al. (2014) suggested the following output-oriented InvDEA model.   max∑r=1sβrs.t.∑j∈Fxijλj+(xik+xil)λM⩽(xik+xil)i=1,...,m∑j∈Fyrjλj+(yrk+yrl+βr)λM−(yrk+yrl+βr)h¯⩾0r=1,...,s∑j∈Fλj+λM=1βr⩾0,r=1,...,sλj⩾0,j∈F,λM⩾0 (4) where, $$\beta_{r}$$ is the decision variable for the $$r$$th output ($$r=1,... ,s)$$ that the merged entity should produce additional than the outputs of the merging DMUs in order to realize a given efficiency target $$\bar{{h}}$$. Model (4) can be linearized using the same relaxation explained for the input orientation model. The relaxed output-orientation InvDEA model is as follows:   max∑r=1sβrs.t.∑j∈Fxijλj+(xik+xil)λM⩽(xik+xil)i=1,...,m∑j∈Fyrjλj−(yrk+yrl+βr)h¯⩾0r=1,...,s∑j∈Fλj=1βr⩾0,r=1,...,s,λj⩾0,j∈F (5) Both the relaxed InvDEA models (3) and (5) might be infeasible. The infeasibility occurs when the merged entity $$M$$ falls outside the pre-merger production possibility set (PPS). We show this issue for the proposed new InvDEA models in the following section. 4. Modelling mergers with negative data The proposed InvDEA models in Gattoufi et al. (2014) fail to realize target setting for a merger when there is negative data and demonstrate in fact the limitation of the base DEA model used in their modelling. In this section, we extend InvDEA models that can be used for evaluation of a merger in the presence of negative data. More precisely, this paper develops an output-oriented InvDEA model for target setting of a merged entity when there is negative variable. In the following, the base DEA model for assessing DMUs with negative data is the VRS SORM model suggested in Emrouznejad et al. (2010a). We propose the following output-oriented InvDEA model.   max∑r∈R1βr+∑r∈R2(βr1−βr2)s.t.∑j∈Fxijλj+(xik+xil)λM⩽xik+xili∈I1∑j∈Fxij1λj+(xik1+xil1)λM⩽xik1+xil1i∈I2∑j∈Fxij2λj+(xik2+xil2)λM⩾xik2+xil2i∈I2∑j∈Fyrjλj+(yrk+yrl+βr)λM−h¯(yrk+yrl+βr)⩾0r∈R1∑j∈Fyrj1λj+(yrk1+yrl1+βr1)λM−h¯(yrk1+yrl1+βr1)⩾0r∈R2∑j∈Fyrj2λj+(yrk2+yrl2+βr2)λM−h¯(yrk2+yrl2+βr2)⩽0r∈R2∑j∈Fλj+λM=1λj⩾0,j∈F,λM⩾0,βr⩾0,r∈R1,βr1⩾0,βr2⩾0,r∈R2 (6) where the objective is to find the maximum additional outputs that the merged entity should produce in order to realize a given efficiency target $$\bar{{h}}$$. This objective is separated into two terms, one to find the maximum additional for positive outputs ($$\beta_{r} \,for\,\,r\in R_{1} \,\And \,\,\,\beta_{r}^{1} \geqslant 0\,\,for\,\,r\in R_{2} )$$ and the second term to find the minimum amount for negative outputs ($$\beta _{r}^{2} \,for\,\,r\in R_{2} )$$. The non-linear model (6) can be converted to the following linear programming problem by excluding the merged entity M from the group of peers.   max∑r∈R1βr+∑r∈R2(βr1−βr2)s.t.∑j∈Fxijλj⩽xik+xili∈I1∑j∈Fxij1λj⩽xik1+xil1i∈I2∑j∈Fxij2λj⩾xik2+xil2i∈I2∑j∈Fyrjλj−h¯(yrk+yrl+βr)⩾0r∈R1∑j∈Fyrj1λj−h¯(yrk1+yrl1+βr1)⩾0r∈R2∑j∈Fyrj2λj−h¯(yrk2+yrl2+βr2)⩽0r∈R2∑j∈Fλj=1λj⩾0,j∈F,βr⩾0,r∈R1,βr1⩾0,βr2⩾0,r∈R2 (7) The target setting for the merged entity M will be identified after solving the above InvDEA model (7). The objective function of the proposed InvDEA models (6) and (7) is to improve the values of all outputs that are positive for some DMUs and negative for others. It worth nothing that the proposed InvDEA model (7) is the result of extending the InvDEA model (5) taking the output-oriented SORM DEA model (1) as the base model. The first three sets of the constraints in the output-oriented InvDEA model (7), are introduced to allow the merged entity to be able to keep the entire inputs, both positive and negative variables. In addition, the second three sets of constraints guarantee that the merged entity would produce additional outputs as much as possible. This can be interpreted as the power of the synergy. The following theorems show the property of the proposed InvDEA model for negative data. Theorem 1 The InvDEA model (7) is feasible if and only if the merged entity M falls inside the pre-merger PPS. Proof. The relaxed output-oriented InvDEA model (7) is obtained from the non-linear programming problem (6) by excluding the merged entity M from the group of peers. Therefore, the feasibility of model (7) implies that the merged entity M can be presented, even on the frontier, in terms of efficient DMUs in the pre-merger PPS. This completes the proof. □ The feasibility of the InvDEA model (7) happens when the merged entity M has the entire inputs and outputs of the merging DMUs and it is a feasible unit in the pre-merger PPS. On the other hand, falling outside the pre-merger PPS means that the merged entity M cannot be presented in terms of other DMUs. This is usually the result of merging some big DMUs and such a merger can change the post-merger frontier and does not need producing additional outputs. Theorem 2 The InvDEA model (7) is infeasible if and only if the merged entity M falls outside the pre-merger PPS. Proof. This is a direct conclusion of Theorem 1. □ Similar InvDEA model can be formulated for input-orientation in the presence of negative data. It is quite possible that the merged entity would be looking for resource minimization. In this case, we propose the following input-oriented InvDEA model.    min∑i∈I1(αik+αil)+∑i∈I2[(αik1+αil1)+(αik2+αil2)]s.t. ∑j∈Fxijλj−(αik+αil)θ¯⩽0i∈I1 ∑j∈Fxij1λj−(αik1+αil1)θ¯⩽0i∈I2 ∑j∈Fxij2λj−(αik2+αil2)θ¯⩾0i∈I2 ∑j∈Fyrjλj⩾yrk+yrlr∈R1 ∑j∈Fyrj1λj⩾yrk1+yrl1r∈R2 ∑j∈Fyrj2λj⩽yrk2+yrl2r∈R2 ∑j∈Fλj=10⩽αik⩽xik,0⩽αil⩽xili∈I10⩽αik1⩽xik1,0⩽αil1⩽xil1,0⩽αik2⩽xik2,0⩽αil2⩽xil2i∈I2λj⩾0,j∈F (8) The above InvDEA model (8) extends the input-oriented InvDEA model (3). The objective function in model (8) aims to save the inputs of merging DMUs as much as possible. It is easy to show that the proposed InvDEA model (8) saves both positive and negative input variables as much as possible. Similar to the output-orientation, we have the following theorem. Theorem 3 The InvDEA model (8) is feasible if and only if the merged entity M falls inside the pre-merger PPS. Proof. This is similar to the proof in Theorem 1. □ 5. A numerical illustration In this section, we consider the dataset of 46 banks in Gulf Cooperation Council (GCC) countries which consists of six counties, Bahrain, Kuwait, Oman, Qatar, Saudi Arabia and United Arab Emirates. The dataset, three inputs and four outputs all in million US$, is collected from Bankscope and given in Appendix Table A1. The illustration of this section can be employed for any other merger context. Two main approaches can be used to determine what constitutes a bank inputs and outputs: the intermediation approach and the production approach. In the intermediation approach, the selection is based on the bank’s assets and liabilities. In Bogetoft et al. (1997), bank inputs are purchased funds, core deposits and labour. Outputs are consumer loans, business loans and securities. This is the same method applied in Rezvanian & Mehdian (2002). The inputs are borrowed funds (time deposits and other borrowed funds) and other inputs (labour and capital) and the outputs are total loans, securities and other earning assets. Cavallo & Rossi (2002) also viewed labour, capital and deposits as bank inputs. In contrast, the production approach considers the bank as a producer just like producers in the product market. Inputs, therefore, are physical entities such as labour and capital. In relation to deposits, proponents of this approach argue that all deposits should be treated as outputs since they are associated with liquidity, safekeeping and are involved in generating value added. Without loss of generality, the selection of inputs and outputs in this study is based on the intermediation approach. Similar to Al-Muharrami (2007, 2008), the inputs are personnel expenses, fixed assets and total deposits. The total deposits are made up of demand deposit, saving deposit and fixed deposit. The output variables are gross loans, other operating income, other earning assets and off balance sheet items. As to the outputs, gross loans included all types, such as real estate loans, commercial and industrial loans, and consumer loans. Net deposits reflected the value of all deposits derived from the sum of demand and savings deposits. The other operating income is the sum of net gains (losses) on trading and derivatives, net gains (losses) on assets at future value through income statement, net fees and commissions, and remaining operating income. Therefore, the other operating income could be negative as shown in Appendix Table A1 for some banks. For instance, the other operating income for B$$_{\mathrm{06}}$$ is $$-$$15.71 million US$ which is the sum of 39.30, $$-$$3.60, 147.30 and $$-$$198.71. There are three positive inputs in the dataset given in Appendix Table A1, i.e. $$I_{1} =\{1,\,2,\,3\}$$ and $$I_{2} =\varphi$$ or no negative input. There are also three positive outputs and one output which is negative for some DMUs, $$R_{1} =\{1,\,2,\,3\}$$ and $$R_{2} =\{4\}$$. Nevertheless, the proposed InvDEA method can be used for the case of more than one negative variable. Three merger cases are illustrated in the following: First, consider the merger of two merging banks B$$_{\mathrm{40}}$$ and B$$_{\mathrm{43}}$$ producing the merged bank M. Suppose that the merged bank M aims to be fully efficient, or $$\bar{{h}}=1$$. An optimal solution of the proposed InvDEA model (7), corresponding to this merger, is as follows:   λ21∗=0.63,λ43∗=0.36,andλj∗=0.00∀j≠21,43β1∗=92.70,β2∗=779.40,β3∗=0.00,β41∗=40.29,β42∗=0.00 The targets for the outputs of the merged bank are shown in Table 1. Table 1 Three merger scenarios: additional outputs Merging  Gross  Other earning  Off-balance  Other operating  DMUs  loans  assets  sheet items  incomes  B$$_\mathrm{40}$$, B$$_{\mathrm{43}}$$  92.70  779.40  0.00  40.29  B$$_{\mathrm{01}}$$, B$$_{\mathrm{40}}$$  The InvDEA model (7) is infeasible  B$$_{\mathrm{39}}$$, B$$_{\mathrm{44}}$$, and B$$_{\mathrm{46}}$$  900.55  747.51  922.25  64.05  Merging  Gross  Other earning  Off-balance  Other operating  DMUs  loans  assets  sheet items  incomes  B$$_\mathrm{40}$$, B$$_{\mathrm{43}}$$  92.70  779.40  0.00  40.29  B$$_{\mathrm{01}}$$, B$$_{\mathrm{40}}$$  The InvDEA model (7) is infeasible  B$$_{\mathrm{39}}$$, B$$_{\mathrm{44}}$$, and B$$_{\mathrm{46}}$$  900.55  747.51  922.25  64.05  The optimal solution of the InvDEA model gives the additional outputs that can be produced by the synergy of the merging banks, which in turn allows the merged bank to be fully efficient. This is especially noticeable for the last output which is negative for both merging banks. As shown in Table 1, the other operating incomes for the merged bank need to be improved by $$\beta_{4}^{1 \ast }-\beta_{4}^{2\ast} = 40.29$$ million US$. This additional value can be reached by selecting the right investment and keeping the right assets. As the second case, consider the merger of two merging banks B$$_{\mathrm{01\thinspace }}$$and B$$_{\mathrm{40}}$$. The output-oriented InvDEA model (7) is infeasible and according to Theorem 2 the merger of banks B$$_{\mathrm{01\thinspace }}$$and B$$_{\mathrm{40}}$$ implies a merged entity outside the pre-merger PPS. The reason for the infeasibility is because of the size of bank B$$_{\mathrm{01}}$$. It is a big bank in the GCC region and merger or acquisition of this bank with B$$_{\mathrm{40}}$$ falls outside the pre-merger PPS. If such a merger happens it affects the performance of the banks in the market. Infeasibility indicates that the occurrence of the related merger affects the market and hence the principals and/or authorities might not approve such a merger. Finally, consider the merger between three banks B$$_{\mathrm{39}}$$, B$$_{\mathrm{44}}$$ and B$$_{\mathrm{46}}$$. This case simply shows that the proposed InvDEA model (7) can be easily used for a merger of more than two merging DMUs. The target for additional output is shown in the last row of Table 1. The implication of the proposed method, in this paper, is to get the maximum benefit from mergers among the merging DMUs. The result of this study can help policy makers to choose the best alternative among potential merging DMUs. Therefore, they can select the right DMUs to merge with in order to reach the maximum benefit either by saving inputs or maximizing outputs. To the best of our knowledge, there is currently no approach that can be used to compare the results of our proposed method presented in this paper. The only available work based on InvDEA suggested in Gattoufi et al. (2014) cannot be used for a merger in the presence of negative data. The main assumption in the modelling of the proposed method is that a merger would create synergy. This means that two or more firms can combine their resources to produce some additional outputs. This assumption is the main limitation to the applicability of the suggested InvDEA method in this paper. 6. Concluding remarks The problem of target setting for a generated entity from a merger has been discussed in this paper. It is shown that, when there is negative data, the standard inverse data envelopment analysis (InvDEA) method cannot be used. Therefore, we have developed in this study a new InvDEA model for this purpose. A numerical illustration in the banking sector was used to show the applicability of the proposed method for target setting of the merged entity. 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Google Scholar CrossRef Search ADS Appendix Table A1 46 GCC banks with negative output, year 2011 (all the amounts in million US$    Inputs  Outputs     Bank  Personnel  Fixed  Deposits and  Gross loans  Other earning  Off-balance  Other operating     expenses  assets  S.T. funding     assets  sheet items  incomes  B$$_{\mathrm{01}}$$  173.32  251.35  49399.73  36612.23  25774.34  21299.37  520.14  B$$_{\mathrm{02}}$$  188.96  156.35  25557.92  21991.87  7546.21  14667.79  350.88  B$$_{\mathrm{03}}$$  248.43  336.21  25678.80  18239.31  9266.85  7132.43  368.24  B$$_{\mathrm{04}}$$  142.30  373.10  21446.10  14910.30  10780.10  2945.60  229.90  B$$_{\mathrm{05}}$$  112.39  293.68  10368.52  9490.80  7056.90  5206.43  254.84  B$$_{\mathrm{06}}$$  222.68  177.84  18476.92  16372.53  6091.03  6607.52  –15.71  B$$_{\mathrm{07}}$$  215.87  159.54  15631.99  13677.14  3092.72  3310.01  120.08  B$$_{\mathrm{08}}$$  248.00  122.00  21218.00  12754.00  10873.00  9848.00  279.00  B$$_{\mathrm{09}}$$  97.29  176.05  12032.79  7967.21  5064.25  3435.14  198.15  B$$_{\mathrm{10}}$$  262.19  374.32  16017.64  12102.57  8678.31  25704.75  521.17  B$$_{\mathrm{11}}$$  119.65  99.71  17656.36  15726.73  5715.43  10354.72  164.47  B$$_{\mathrm{12}}$$  147.72  194.54  11551.37  10908.19  3280.88  3981.53  168.01  B$$_{\mathrm{13}}$$  88.99  199.44  11229.63  8794.51  5568.21  1487.81  130.16  B$$_{\mathrm{14}}$$  113.79  132.05  11856.16  9652.53  4151.04  4657.33  177.79  B$$_{\mathrm{15}}$$  80.58  101.81  10616.15  8163.52  6667.56  1953.82  109.81  B$$_{\mathrm{16}}$$  61.40  37.10  9649.10  8152.40  8990.50  2149.90  67.60  B$$_{\mathrm{17}}$$  73.17  28.45  7606.54  4206.40  2230.52  1129.04  101.78  B$$_{\mathrm{18}}$$  79.84  154.74  6273.49  4229.35  1604.82  1946.63  69.73  B$$_{\mathrm{19}}$$  102.56  91.17  4617.28  3446.08  1974.91  792.93  125.60  B$$_{\mathrm{20}}$$  25.55  59.37  3882.97  2551.51  3086.37  387.20  39.84  B$$_{\mathrm{21}}$$  40.44  18.27  611.00  511.00  1905.63  38.74  62.88  B$$_{\mathrm{22}}$$  78.19  77.39  4762.77  3614.36  2867.29  1432.18  88.03  B$$_{\mathrm{23}}$$  125.77  210.18  4550.00  4485.00  881.17  1473.90  161.50  B$$_{\mathrm{24}}$$  44.53  48.41  5756.21  4256.04  2481.32  3694.59  77.91  B$$_{\mathrm{25}}$$  47.40  16.75  3784.39  3024.95  1118.81  399.86  34.93  B$$_{\mathrm{26}}$$  59.30  58.26  3789.86  3725.36  664.50  1498.83  57.48  B$$_{\mathrm{27}}$$  50.80  42.82  5319.95  2571.01  3583.24  277.66  51.86  B$$_{\mathrm{28}}$$  27.77  50.25  4218.49  3201.54  1256.24  2088.79  32.23  B$$_{\mathrm{29}}$$  35.75  61.89  4030.31  3449.45  1531.22  1130.81  36.11  B$$_{\mathrm{30}}$$  48.11  22.89  3473.08  3467.88  811.18  994.80  33.81  B$$_{\mathrm{31}}$$  48.99  213.59  4337.56  3669.08  1171.27  1095.03  103.99  B$$_{\mathrm{32}}$$  56.01  223.53  3010.27  2672.51  1182.00  202.53  44.79  B$$_{\mathrm{33}}$$  40.52  23.47  2553.00  2479.00  326.24  3449.09  60.86  B$$_{\mathrm{34}}$$  31.47  79.84  2140.70  1864.24  1060.34  831.99  26.27  B$$_{\mathrm{35}}$$  30.43  35.89  2750.07  2378.15  903.25  1010.14  11.96  B$$_{\mathrm{36}}$$  36.97  113.30  1739.00  1303.00  1764.63  118.62  72.07  B$$_{\mathrm{37}}$$  42.07  57.24  2629.24  2533.78  483.51  1347.86  49.26  B$$_{\mathrm{38}}$$  33.55  32.51  2017.17  1774.77  552.15  1788.56  46.55  B$$_{\mathrm{39}}$$  17.60  8.00  704.50  77.10  713.10  23.10  –0.90  B$$_{\mathrm{40}}$$  41.20  7.10  394.00  236.00  332.10  3.40  –101.90  B$$_{\mathrm{41}}$$  9.84  40.96  194.41  150.00  3.19  27.13  19.41  B$$_{\mathrm{42}}$$  3.56  0.36  385.96  17.82  525.84  1.00  –4.99  B$$_{\mathrm{43}}$$  8.42  10.98  8.63  7.00  163.85  63.09  –12.51  B$$_{\mathrm{44}}$$  52.00  9.00  421.00  41.21  56.28  2.00  –1.39  B$$_{\mathrm{45}}$$  5.00  3.90  11.00  9.00  70.90  1.00  –62.10  B$$_{\mathrm{46}}$$  1.10  3.80  6.90  6.00  34.40  2.00  –2.30     Inputs  Outputs     Bank  Personnel  Fixed  Deposits and  Gross loans  Other earning  Off-balance  Other operating     expenses  assets  S.T. funding     assets  sheet items  incomes  B$$_{\mathrm{01}}$$  173.32  251.35  49399.73  36612.23  25774.34  21299.37  520.14  B$$_{\mathrm{02}}$$  188.96  156.35  25557.92  21991.87  7546.21  14667.79  350.88  B$$_{\mathrm{03}}$$  248.43  336.21  25678.80  18239.31  9266.85  7132.43  368.24  B$$_{\mathrm{04}}$$  142.30  373.10  21446.10  14910.30  10780.10  2945.60  229.90  B$$_{\mathrm{05}}$$  112.39  293.68  10368.52  9490.80  7056.90  5206.43  254.84  B$$_{\mathrm{06}}$$  222.68  177.84  18476.92  16372.53  6091.03  6607.52  –15.71  B$$_{\mathrm{07}}$$  215.87  159.54  15631.99  13677.14  3092.72  3310.01  120.08  B$$_{\mathrm{08}}$$  248.00  122.00  21218.00  12754.00  10873.00  9848.00  279.00  B$$_{\mathrm{09}}$$  97.29  176.05  12032.79  7967.21  5064.25  3435.14  198.15  B$$_{\mathrm{10}}$$  262.19  374.32  16017.64  12102.57  8678.31  25704.75  521.17  B$$_{\mathrm{11}}$$  119.65  99.71  17656.36  15726.73  5715.43  10354.72  164.47  B$$_{\mathrm{12}}$$  147.72  194.54  11551.37  10908.19  3280.88  3981.53  168.01  B$$_{\mathrm{13}}$$  88.99  199.44  11229.63  8794.51  5568.21  1487.81  130.16  B$$_{\mathrm{14}}$$  113.79  132.05  11856.16  9652.53  4151.04  4657.33  177.79  B$$_{\mathrm{15}}$$  80.58  101.81  10616.15  8163.52  6667.56  1953.82  109.81  B$$_{\mathrm{16}}$$  61.40  37.10  9649.10  8152.40  8990.50  2149.90  67.60  B$$_{\mathrm{17}}$$  73.17  28.45  7606.54  4206.40  2230.52  1129.04  101.78  B$$_{\mathrm{18}}$$  79.84  154.74  6273.49  4229.35  1604.82  1946.63  69.73  B$$_{\mathrm{19}}$$  102.56  91.17  4617.28  3446.08  1974.91  792.93  125.60  B$$_{\mathrm{20}}$$  25.55  59.37  3882.97  2551.51  3086.37  387.20  39.84  B$$_{\mathrm{21}}$$  40.44  18.27  611.00  511.00  1905.63  38.74  62.88  B$$_{\mathrm{22}}$$  78.19  77.39  4762.77  3614.36  2867.29  1432.18  88.03  B$$_{\mathrm{23}}$$  125.77  210.18  4550.00  4485.00  881.17  1473.90  161.50  B$$_{\mathrm{24}}$$  44.53  48.41  5756.21  4256.04  2481.32  3694.59  77.91  B$$_{\mathrm{25}}$$  47.40  16.75  3784.39  3024.95  1118.81  399.86  34.93  B$$_{\mathrm{26}}$$  59.30  58.26  3789.86  3725.36  664.50  1498.83  57.48  B$$_{\mathrm{27}}$$  50.80  42.82  5319.95  2571.01  3583.24  277.66  51.86  B$$_{\mathrm{28}}$$  27.77  50.25  4218.49  3201.54  1256.24  2088.79  32.23  B$$_{\mathrm{29}}$$  35.75  61.89  4030.31  3449.45  1531.22  1130.81  36.11  B$$_{\mathrm{30}}$$  48.11  22.89  3473.08  3467.88  811.18  994.80  33.81  B$$_{\mathrm{31}}$$  48.99  213.59  4337.56  3669.08  1171.27  1095.03  103.99  B$$_{\mathrm{32}}$$  56.01  223.53  3010.27  2672.51  1182.00  202.53  44.79  B$$_{\mathrm{33}}$$  40.52  23.47  2553.00  2479.00  326.24  3449.09  60.86  B$$_{\mathrm{34}}$$  31.47  79.84  2140.70  1864.24  1060.34  831.99  26.27  B$$_{\mathrm{35}}$$  30.43  35.89  2750.07  2378.15  903.25  1010.14  11.96  B$$_{\mathrm{36}}$$  36.97  113.30  1739.00  1303.00  1764.63  118.62  72.07  B$$_{\mathrm{37}}$$  42.07  57.24  2629.24  2533.78  483.51  1347.86  49.26  B$$_{\mathrm{38}}$$  33.55  32.51  2017.17  1774.77  552.15  1788.56  46.55  B$$_{\mathrm{39}}$$  17.60  8.00  704.50  77.10  713.10  23.10  –0.90  B$$_{\mathrm{40}}$$  41.20  7.10  394.00  236.00  332.10  3.40  –101.90  B$$_{\mathrm{41}}$$  9.84  40.96  194.41  150.00  3.19  27.13  19.41  B$$_{\mathrm{42}}$$  3.56  0.36  385.96  17.82  525.84  1.00  –4.99  B$$_{\mathrm{43}}$$  8.42  10.98  8.63  7.00  163.85  63.09  –12.51  B$$_{\mathrm{44}}$$  52.00  9.00  421.00  41.21  56.28  2.00  –1.39  B$$_{\mathrm{45}}$$  5.00  3.90  11.00  9.00  70.90  1.00  –62.10  B$$_{\mathrm{46}}$$  1.10  3.80  6.90  6.00  34.40  2.00  –2.30  © The authors 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. permissions@oup.com http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png IMA Journal of Management Mathematics Oxford University Press

# A new inverse data envelopment analysis model for mergers with negative data

, Volume 29 (2) – Apr 1, 2018
13 pages

/lp/ou_press/a-new-inverse-data-envelopment-analysis-model-for-mergers-with-mvoR0jX60p
Publisher
Oxford University Press
ISSN
1471-678X
eISSN
1471-6798
D.O.I.
10.1093/imaman/dpw016
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### Abstract

Abstract One of the important issues in a merger between two or more decision making units is identification of the levels of inherited inputs and outputs from merging units. This paper introduces new inverse data envelopment analysis models for target setting of a merger in the presence of negative data. It enables the merged entity to identify the levels of required inputs and outputs from merging units to realize an efficiency target. The applicability of the proposed method in this paper is illustrated through an example from banking; however, it can be used in other merger contexts. 1. Introduction Mergers and acquisitions have shown the facilities of combining decision making units (DMUs) as an effective strategic corporate restructuring for increasing the production capabilities of the involved units (Hagedoorn & Duysters, 2002). One of the research areas that has been successfully used for mergers performance evaluation in different contexts is data envelopment analysis (DEA) (Emrouznejad et al., 2008; Liu et al., 2013; Gattoufi et al., 2014). DEA applications in mergers are found in different areas such as banking (Wheelock & Wilson, 2000; Luo, 2003; Sherman & Rupert, 2006; Hahn, 2007; Wu et al., 2011; Wu & Birge, 2012; Wu et al., 2015; Moradi-Motlagh & Babacan, 2015; Du & Sim, 2016), non-bank institutions like credit units (Fried et al., 1999; Worthington, 2004; Jin et al., 2015; Halkos et al., 2016), healthcare (Kristensen et al., 2010; Leleu et al., 2012), forestry (Bogetoft & Bo, 2003) agriculture (Bogetoft & Wang, 2005), and airlines (Kong et al., 2012). In all of these cases, the standard DEA methodology is used for estimation of merger gains under the existing levels of inputs and outputs (Bogetoft & Wang, 2005; Lozano & Villa, 2010). The classic DEA models do not reveal the levels of inputs and outputs that would allow the merged entity to realize a specific efficiency target. A standard DEA model then cannot deal with such a problem and as a result the Inverse DEA (InvDEA) is a more appropriate method (Wei et al., 2000; Gattoufi et al., 2014). Unlike the objective of a conventional DEA model that obtains the efficiency scores of DMUs (Emrouznejad et al., 2008; Farzipoor Saen, 2011; Yeung & Azevedo, 2011), InvDEA aims to find the levels of inputs and outputs that are required to realize a given efficiency target. The idea of InvDEA was first introduced to estimate inputs and/or outputs for a resource allocation problem (Wei et al., 2000). The DEA literature was highly enriched by studies that implemented InvDEA in different contexts. Li & Cui (2008) suggested a comprehensive InvDEA approach for resource allocation. Lertworasirikul et al. (2011) discussed the possibility of simultaneous increase of some outputs and decrease of the other outputs by means of InvDEA. Gattoufi et al. (2014) proposed an InvDEA methodology to identify the maximum levels of inputs anticipated to be saved bythe synergy of the merging DMUs. They have also suggested an InvDEA model for determining the maximum additional outputs that can be produced after the merger. Given an efficiency target for the merged entity, the proposed input-oriented InvDEA identifies the optimal levels of inputs inherited from merging DMUs. Recent years have seen various applications of InvDEA in different areas of DEA. Jahanshahloo et al. (2015) proposed an inter-temporal application of InvDEA. Ghiyasi (2015) introduced InvDEA models for variable return to scale phenomena. Moreover, Lim (2016) suggested an InvDEA methodology for a new product target setting by assuming frontier change. To the best of our knowledge, there is no application of InvDEA in mergers except the recent work of Gattoufi et al. (2014). The authors developed the concept of InvDEA in M{&}A analysis in order to find the optimal levels of inputs and outputs that are needed from the merging entities. By using the InvDEA, the authors enabled the merged entity to realize a given efficiency target. The merged entity can realize a given efficiency target if it removes redundant inputs or produces additional outputs depending on the selected orientation InvDEA model. Nevertheless, the proposed method in Gattoufi et al. (2014) cannot be used for target setting of the merged entity when there is negative data. In this paper we develop a new InvDEA method by generalizing the concept of InvDEA. In our proposed InvDEA method, there is at least one negative variable among inputs and outputs. Under this model, the merged entity can set its inputs and outputs to realize a given efficiency target. More specifically, the proposed method identifies the maximum improvement on the negative outputs of merging DMUs after a merger. We choose banking as a business application to show the usefulness of the proposed method in an area yet studied. Another main advantage of this study is that there is no alternative DEA or similar method for target setting of a merger when there are negative variables such demonstrated in our case on banking. The remainder of this paper is as follows. Section 2 reviews the literature of DEA models suggested for negative data. This is followed by a brief explanation of the InvDEA models for mergers in Section 3. Section 4 develops new InvDEA models for target setting after merger with negative data. In Section 5, we use a real case of bank mergers to show the practical contribution of the proposed method. Finally, Section 6 gives the conclusions. 2. DEA with negative data In the presence of negative data, classical DEA models cannot be used for performance evaluation. There are many research studies in the literature that have developed new DEA models for dealing with negative variables. Data transformation is one of the introduced approaches in such literature that transforms all negative data to positive values; see e.g. Lovell (1995); and Seiford&Zhu (2002). Scheel (2001) proposed a method that treats the absolute values of negative outputs as inputs and the absolute values of negative inputs as outputs. A modified slack-based measure for handling both negative outputs and negative inputs is introduced in Sharp et al (2007). Portela et al. (2004) have also suggested range directional measure DEA models for dealing with positive and negative values variables. More recently, Emrouznejad et al. (2010a,b) proposed a semi-oriented radial measure (SORM) DEA model which can yield a measure of efficiency by handling variables that take positive values for some and negative values for other DMUs In this paper, we use the SORM model as the base DEA model and suggest new InvDEA models for M&A target setting. We used this combination due to the similarity between the SORM and the classical DEA models in providing an efficiency value. Nevertheless, the base DEA model can be any other forms of DEA models that can handle negative data. Let us assume an output variable $${{\bf Y}}_{k}$$ is positive for some DMUs and negative for others. Emrouznejad et al. (2010a,b) defined the following two variables $${{\bf Y}}_{k}^{1}$$ and $${{\bf Y}}_{k}^{2}$$  Ykj1={Ykjif Ykj⩾00Otherwise Ykj2={−Ykjif Ykj<00Otherwise  where, $${{\bf Y}}_{kj}^{1} \geqslant 0$$, $${{\bf Y}}_{kj}^{2} \geqslant 0$$ and $${{\bf Y}}_{kj} ={{\bf Y}}_{kj}^{1} -{{\bf Y}}_{kj}^{2}$$ for all $$j=1,... ,n$$, and treated $${{\bf Y}}_{k}^{1}$$ as an output and $${{\bf Y}}_{k}^{2}$$ as an input. The same decomposition is used for an input variable that is positive for some DMUs and negative for others. That is   Xkj1={Xkjif Xkj⩾00Otherwise Xkj2={−Xkjif Xkj<00Otherwise  where, $$X_{kj}^{1} \geqslant 0$$, $$X_{kj}^{2} \geqslant 0$$ and $$X_{kj} =X_{kj}^{1} -X_{kj}^{2}$$ for all $$j=1,... ,n$$, and treated $$X_{k}^{1}$$ as an input and $$X_{k}^{2}$$ as an output. Assume $$I_{1}$$ denotes the set of indices for positive inputs and $$I_{2}$$for the set of inputs that is negative for at least one DMU. Moreover, let $$R_{1}$$ be the set of indices for positive outputs and $$R_{2}$$ for the set of outputs that is negative for at least one unit. The output-oriented SORM DEA model can be written as follows (Emrouznejad et al., 2010a)    maxhs.t. ∑j=1nxijλj⩽xik,i∈I1,∑j=1nxij1λj⩽xik1,i∈I2,∑j=1nxij2λj⩾xik2,i∈I2 ∑j=1nyrjλj⩾hyrk,r∈R1,∑j=1nyrj1λj⩾hyrk1,r∈R2,∑j=1nyrj2λj⩽hyrk2,r∈R2 ∑j=1nλj=1λj⩾0,j=1,...,n (1) In this paper we use an output-oriented SORM DEA model however, the input-oriented SORM DEA model can be similarly written. 3. Merging DMUs using InvDEA For simplicity in modelling, let us consider the case of a merger when there are two merging units. As we demonstrate, this assumption in the modelling can be easily extended to more than two merging DMUs. Suppose we have $$n$$ DMUs each consuming $$m$$ inputs for producing $$s$$ outputs. More specifically, let $$x_{ij}$$ and $$y_{rj}$$ denote, respectively, the $$i$$th input and the $$r$$th output for the $$j$$th DMU ($$i=1,... ,m,\, r=1,... ,s$$,$$j=1,... ,n)$$. Assume that DMUs $$k$$and $$l$$decide to go through a merger for producing a new entity, say M. In an input-oriented model, as assumed in Gattoufi et al. (2014), the merged entity M keeps the entire outputs of DMUs $$k$$ and $$l$$ and seeks the minimum levels of inherited inputs to realize a given efficiency target $$\bar{{\theta}}$$. Let us define the decision variables $$\alpha_{ik}$$ and $$\alpha_{il}$$ for the levels of the $$i$$th input that is going to be kept by the merged entity M from the merging DMUs $$k$$ and $$l$$, respectively, implying the $$i$$th input of the merged entity M to be $$\alpha_{ik} +\alpha_{il}$$ ($$i=1,... ,m)$$. The corresponding input-oriented InvDEA model suggested in Gattoufi et al. (2014) can be rewritten as follows:   min∑i=1m(αik+αil)s.t.∑j∈Fxijλj+(αik+αil)λM−(αik+αil)×θ¯⩽0i=1,...,m∑j∈Fyrjλj+(yrk+yrl)λM⩾(yrk+yrl)r=1,...,s∑j∈Fλj+λM=10⩽αik⩽xik,0⩽αil⩽xili=1,...,mλj⩾0,j∈F,λM⩾0 (2) where $$F$$ is the set of available peers in the post-merger evaluation process. In a merger there are two possible cases (1) only the acquiring DMU survives in the market and continues to operate by its former name, or (2) both merging DMUs consolidate in one new entity with a new name. Therefore, $$F$$ may contain one of the merging DMUs $$k$$ or $$l$$ or neither, depending on the form of the merger. In the above InvDEA model the merged entity is denoted by M regardless of the form of the merger and the name of the merged entity. In fact, the important issue here is what should be the levels of the inherited inputs of the post-merger entity? As discussed in Gattoufi et al. (2014), model (2) is a non-linear programming problem that can be linearized by excluding $$M$$ from the set of its evaluating peers, or equivalently taking $$\lambda_{M} =0$$. Therefore, the relaxed linear programming form of model (1) can be written as follows:   min∑i=1m(αik+αil)s.t.∑j∈Fxijλj−(αik+αil)×θ¯⩽0i=1,...,m∑j∈Fyrjλj⩾(yrk+yrl)r=1,...,s∑j∈Fλj=10⩽αik⩽xik,0⩽αil⩽xili=1,...,mλj⩾0,j∈F (3) In the InvDEA model (1) and the corresponding relaxed model (3) it is assumed that all inputs have the same measurement units. Nevertheless, inputs with different units can be unified by using the formula$$\sum\nolimits_{i=1}^m {p_{i} (\alpha_{ik} +\alpha_{il} )}$$as the objective function, where $$p_{i}$$ is the cost per unit of the $$i$$th input ($$i=1,... ,m)$$. Similar to the input orientation, Gattoufi et al. (2014) suggested the following output-oriented InvDEA model.   max∑r=1sβrs.t.∑j∈Fxijλj+(xik+xil)λM⩽(xik+xil)i=1,...,m∑j∈Fyrjλj+(yrk+yrl+βr)λM−(yrk+yrl+βr)h¯⩾0r=1,...,s∑j∈Fλj+λM=1βr⩾0,r=1,...,sλj⩾0,j∈F,λM⩾0 (4) where, $$\beta_{r}$$ is the decision variable for the $$r$$th output ($$r=1,... ,s)$$ that the merged entity should produce additional than the outputs of the merging DMUs in order to realize a given efficiency target $$\bar{{h}}$$. Model (4) can be linearized using the same relaxation explained for the input orientation model. The relaxed output-orientation InvDEA model is as follows:   max∑r=1sβrs.t.∑j∈Fxijλj+(xik+xil)λM⩽(xik+xil)i=1,...,m∑j∈Fyrjλj−(yrk+yrl+βr)h¯⩾0r=1,...,s∑j∈Fλj=1βr⩾0,r=1,...,s,λj⩾0,j∈F (5) Both the relaxed InvDEA models (3) and (5) might be infeasible. The infeasibility occurs when the merged entity $$M$$ falls outside the pre-merger production possibility set (PPS). We show this issue for the proposed new InvDEA models in the following section. 4. Modelling mergers with negative data The proposed InvDEA models in Gattoufi et al. (2014) fail to realize target setting for a merger when there is negative data and demonstrate in fact the limitation of the base DEA model used in their modelling. In this section, we extend InvDEA models that can be used for evaluation of a merger in the presence of negative data. More precisely, this paper develops an output-oriented InvDEA model for target setting of a merged entity when there is negative variable. In the following, the base DEA model for assessing DMUs with negative data is the VRS SORM model suggested in Emrouznejad et al. (2010a). We propose the following output-oriented InvDEA model.   max∑r∈R1βr+∑r∈R2(βr1−βr2)s.t.∑j∈Fxijλj+(xik+xil)λM⩽xik+xili∈I1∑j∈Fxij1λj+(xik1+xil1)λM⩽xik1+xil1i∈I2∑j∈Fxij2λj+(xik2+xil2)λM⩾xik2+xil2i∈I2∑j∈Fyrjλj+(yrk+yrl+βr)λM−h¯(yrk+yrl+βr)⩾0r∈R1∑j∈Fyrj1λj+(yrk1+yrl1+βr1)λM−h¯(yrk1+yrl1+βr1)⩾0r∈R2∑j∈Fyrj2λj+(yrk2+yrl2+βr2)λM−h¯(yrk2+yrl2+βr2)⩽0r∈R2∑j∈Fλj+λM=1λj⩾0,j∈F,λM⩾0,βr⩾0,r∈R1,βr1⩾0,βr2⩾0,r∈R2 (6) where the objective is to find the maximum additional outputs that the merged entity should produce in order to realize a given efficiency target $$\bar{{h}}$$. This objective is separated into two terms, one to find the maximum additional for positive outputs ($$\beta_{r} \,for\,\,r\in R_{1} \,\And \,\,\,\beta_{r}^{1} \geqslant 0\,\,for\,\,r\in R_{2} )$$ and the second term to find the minimum amount for negative outputs ($$\beta _{r}^{2} \,for\,\,r\in R_{2} )$$. The non-linear model (6) can be converted to the following linear programming problem by excluding the merged entity M from the group of peers.   max∑r∈R1βr+∑r∈R2(βr1−βr2)s.t.∑j∈Fxijλj⩽xik+xili∈I1∑j∈Fxij1λj⩽xik1+xil1i∈I2∑j∈Fxij2λj⩾xik2+xil2i∈I2∑j∈Fyrjλj−h¯(yrk+yrl+βr)⩾0r∈R1∑j∈Fyrj1λj−h¯(yrk1+yrl1+βr1)⩾0r∈R2∑j∈Fyrj2λj−h¯(yrk2+yrl2+βr2)⩽0r∈R2∑j∈Fλj=1λj⩾0,j∈F,βr⩾0,r∈R1,βr1⩾0,βr2⩾0,r∈R2 (7) The target setting for the merged entity M will be identified after solving the above InvDEA model (7). The objective function of the proposed InvDEA models (6) and (7) is to improve the values of all outputs that are positive for some DMUs and negative for others. It worth nothing that the proposed InvDEA model (7) is the result of extending the InvDEA model (5) taking the output-oriented SORM DEA model (1) as the base model. The first three sets of the constraints in the output-oriented InvDEA model (7), are introduced to allow the merged entity to be able to keep the entire inputs, both positive and negative variables. In addition, the second three sets of constraints guarantee that the merged entity would produce additional outputs as much as possible. This can be interpreted as the power of the synergy. The following theorems show the property of the proposed InvDEA model for negative data. Theorem 1 The InvDEA model (7) is feasible if and only if the merged entity M falls inside the pre-merger PPS. Proof. The relaxed output-oriented InvDEA model (7) is obtained from the non-linear programming problem (6) by excluding the merged entity M from the group of peers. Therefore, the feasibility of model (7) implies that the merged entity M can be presented, even on the frontier, in terms of efficient DMUs in the pre-merger PPS. This completes the proof. □ The feasibility of the InvDEA model (7) happens when the merged entity M has the entire inputs and outputs of the merging DMUs and it is a feasible unit in the pre-merger PPS. On the other hand, falling outside the pre-merger PPS means that the merged entity M cannot be presented in terms of other DMUs. This is usually the result of merging some big DMUs and such a merger can change the post-merger frontier and does not need producing additional outputs. Theorem 2 The InvDEA model (7) is infeasible if and only if the merged entity M falls outside the pre-merger PPS. Proof. This is a direct conclusion of Theorem 1. □ Similar InvDEA model can be formulated for input-orientation in the presence of negative data. It is quite possible that the merged entity would be looking for resource minimization. In this case, we propose the following input-oriented InvDEA model.    min∑i∈I1(αik+αil)+∑i∈I2[(αik1+αil1)+(αik2+αil2)]s.t. ∑j∈Fxijλj−(αik+αil)θ¯⩽0i∈I1 ∑j∈Fxij1λj−(αik1+αil1)θ¯⩽0i∈I2 ∑j∈Fxij2λj−(αik2+αil2)θ¯⩾0i∈I2 ∑j∈Fyrjλj⩾yrk+yrlr∈R1 ∑j∈Fyrj1λj⩾yrk1+yrl1r∈R2 ∑j∈Fyrj2λj⩽yrk2+yrl2r∈R2 ∑j∈Fλj=10⩽αik⩽xik,0⩽αil⩽xili∈I10⩽αik1⩽xik1,0⩽αil1⩽xil1,0⩽αik2⩽xik2,0⩽αil2⩽xil2i∈I2λj⩾0,j∈F (8) The above InvDEA model (8) extends the input-oriented InvDEA model (3). The objective function in model (8) aims to save the inputs of merging DMUs as much as possible. It is easy to show that the proposed InvDEA model (8) saves both positive and negative input variables as much as possible. Similar to the output-orientation, we have the following theorem. Theorem 3 The InvDEA model (8) is feasible if and only if the merged entity M falls inside the pre-merger PPS. Proof. This is similar to the proof in Theorem 1. □ 5. A numerical illustration In this section, we consider the dataset of 46 banks in Gulf Cooperation Council (GCC) countries which consists of six counties, Bahrain, Kuwait, Oman, Qatar, Saudi Arabia and United Arab Emirates. The dataset, three inputs and four outputs all in million US$, is collected from Bankscope and given in Appendix Table A1. The illustration of this section can be employed for any other merger context. Two main approaches can be used to determine what constitutes a bank inputs and outputs: the intermediation approach and the production approach. In the intermediation approach, the selection is based on the bank’s assets and liabilities. In Bogetoft et al. (1997), bank inputs are purchased funds, core deposits and labour. Outputs are consumer loans, business loans and securities. This is the same method applied in Rezvanian & Mehdian (2002). The inputs are borrowed funds (time deposits and other borrowed funds) and other inputs (labour and capital) and the outputs are total loans, securities and other earning assets. Cavallo & Rossi (2002) also viewed labour, capital and deposits as bank inputs. In contrast, the production approach considers the bank as a producer just like producers in the product market. Inputs, therefore, are physical entities such as labour and capital. In relation to deposits, proponents of this approach argue that all deposits should be treated as outputs since they are associated with liquidity, safekeeping and are involved in generating value added. Without loss of generality, the selection of inputs and outputs in this study is based on the intermediation approach. Similar to Al-Muharrami (2007, 2008), the inputs are personnel expenses, fixed assets and total deposits. The total deposits are made up of demand deposit, saving deposit and fixed deposit. The output variables are gross loans, other operating income, other earning assets and off balance sheet items. As to the outputs, gross loans included all types, such as real estate loans, commercial and industrial loans, and consumer loans. Net deposits reflected the value of all deposits derived from the sum of demand and savings deposits. The other operating income is the sum of net gains (losses) on trading and derivatives, net gains (losses) on assets at future value through income statement, net fees and commissions, and remaining operating income. Therefore, the other operating income could be negative as shown in Appendix Table A1 for some banks. For instance, the other operating income for B$$_{\mathrm{06}}$$ is $$-$$15.71 million US$ which is the sum of 39.30, $$-$$3.60, 147.30 and $$-$$198.71. There are three positive inputs in the dataset given in Appendix Table A1, i.e. $$I_{1} =\{1,\,2,\,3\}$$ and $$I_{2} =\varphi$$ or no negative input. There are also three positive outputs and one output which is negative for some DMUs, $$R_{1} =\{1,\,2,\,3\}$$ and $$R_{2} =\{4\}$$. Nevertheless, the proposed InvDEA method can be used for the case of more than one negative variable. Three merger cases are illustrated in the following: First, consider the merger of two merging banks B$$_{\mathrm{40}}$$ and B$$_{\mathrm{43}}$$ producing the merged bank M. Suppose that the merged bank M aims to be fully efficient, or $$\bar{{h}}=1$$. An optimal solution of the proposed InvDEA model (7), corresponding to this merger, is as follows:   λ21∗=0.63,λ43∗=0.36,andλj∗=0.00∀j≠21,43β1∗=92.70,β2∗=779.40,β3∗=0.00,β41∗=40.29,β42∗=0.00 The targets for the outputs of the merged bank are shown in Table 1. Table 1 Three merger scenarios: additional outputs Merging  Gross  Other earning  Off-balance  Other operating  DMUs  loans  assets  sheet items  incomes  B$$_\mathrm{40}$$, B$$_{\mathrm{43}}$$  92.70  779.40  0.00  40.29  B$$_{\mathrm{01}}$$, B$$_{\mathrm{40}}$$  The InvDEA model (7) is infeasible  B$$_{\mathrm{39}}$$, B$$_{\mathrm{44}}$$, and B$$_{\mathrm{46}}$$  900.55  747.51  922.25  64.05  Merging  Gross  Other earning  Off-balance  Other operating  DMUs  loans  assets  sheet items  incomes  B$$_\mathrm{40}$$, B$$_{\mathrm{43}}$$  92.70  779.40  0.00  40.29  B$$_{\mathrm{01}}$$, B$$_{\mathrm{40}}$$  The InvDEA model (7) is infeasible  B$$_{\mathrm{39}}$$, B$$_{\mathrm{44}}$$, and B$$_{\mathrm{46}}$$  900.55  747.51  922.25  64.05  The optimal solution of the InvDEA model gives the additional outputs that can be produced by the synergy of the merging banks, which in turn allows the merged bank to be fully efficient. This is especially noticeable for the last output which is negative for both merging banks. As shown in Table 1, the other operating incomes for the merged bank need to be improved by $$\beta_{4}^{1 \ast }-\beta_{4}^{2\ast} = 40.29$$ million US$. This additional value can be reached by selecting the right investment and keeping the right assets. As the second case, consider the merger of two merging banks B$$_{\mathrm{01\thinspace }}$$and B$$_{\mathrm{40}}$$. The output-oriented InvDEA model (7) is infeasible and according to Theorem 2 the merger of banks B$$_{\mathrm{01\thinspace }}$$and B$$_{\mathrm{40}}$$ implies a merged entity outside the pre-merger PPS. The reason for the infeasibility is because of the size of bank B$$_{\mathrm{01}}$$. It is a big bank in the GCC region and merger or acquisition of this bank with B$$_{\mathrm{40}}$$ falls outside the pre-merger PPS. If such a merger happens it affects the performance of the banks in the market. Infeasibility indicates that the occurrence of the related merger affects the market and hence the principals and/or authorities might not approve such a merger. Finally, consider the merger between three banks B$$_{\mathrm{39}}$$, B$$_{\mathrm{44}}$$ and B$$_{\mathrm{46}}$$. This case simply shows that the proposed InvDEA model (7) can be easily used for a merger of more than two merging DMUs. The target for additional output is shown in the last row of Table 1. The implication of the proposed method, in this paper, is to get the maximum benefit from mergers among the merging DMUs. The result of this study can help policy makers to choose the best alternative among potential merging DMUs. Therefore, they can select the right DMUs to merge with in order to reach the maximum benefit either by saving inputs or maximizing outputs. To the best of our knowledge, there is currently no approach that can be used to compare the results of our proposed method presented in this paper. The only available work based on InvDEA suggested in Gattoufi et al. (2014) cannot be used for a merger in the presence of negative data. The main assumption in the modelling of the proposed method is that a merger would create synergy. This means that two or more firms can combine their resources to produce some additional outputs. This assumption is the main limitation to the applicability of the suggested InvDEA method in this paper. 6. Concluding remarks The problem of target setting for a generated entity from a merger has been discussed in this paper. It is shown that, when there is negative data, the standard inverse data envelopment analysis (InvDEA) method cannot be used. Therefore, we have developed in this study a new InvDEA model for this purpose. A numerical illustration in the banking sector was used to show the applicability of the proposed method for target setting of the merged entity. 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Google Scholar CrossRef Search ADS Appendix Table A1 46 GCC banks with negative output, year 2011 (all the amounts in million US$    Inputs  Outputs     Bank  Personnel  Fixed  Deposits and  Gross loans  Other earning  Off-balance  Other operating     expenses  assets  S.T. funding     assets  sheet items  incomes  B$$_{\mathrm{01}}$$  173.32  251.35  49399.73  36612.23  25774.34  21299.37  520.14  B$$_{\mathrm{02}}$$  188.96  156.35  25557.92  21991.87  7546.21  14667.79  350.88  B$$_{\mathrm{03}}$$  248.43  336.21  25678.80  18239.31  9266.85  7132.43  368.24  B$$_{\mathrm{04}}$$  142.30  373.10  21446.10  14910.30  10780.10  2945.60  229.90  B$$_{\mathrm{05}}$$  112.39  293.68  10368.52  9490.80  7056.90  5206.43  254.84  B$$_{\mathrm{06}}$$  222.68  177.84  18476.92  16372.53  6091.03  6607.52  –15.71  B$$_{\mathrm{07}}$$  215.87  159.54  15631.99  13677.14  3092.72  3310.01  120.08  B$$_{\mathrm{08}}$$  248.00  122.00  21218.00  12754.00  10873.00  9848.00  279.00  B$$_{\mathrm{09}}$$  97.29  176.05  12032.79  7967.21  5064.25  3435.14  198.15  B$$_{\mathrm{10}}$$  262.19  374.32  16017.64  12102.57  8678.31  25704.75  521.17  B$$_{\mathrm{11}}$$  119.65  99.71  17656.36  15726.73  5715.43  10354.72  164.47  B$$_{\mathrm{12}}$$  147.72  194.54  11551.37  10908.19  3280.88  3981.53  168.01  B$$_{\mathrm{13}}$$  88.99  199.44  11229.63  8794.51  5568.21  1487.81  130.16  B$$_{\mathrm{14}}$$  113.79  132.05  11856.16  9652.53  4151.04  4657.33  177.79  B$$_{\mathrm{15}}$$  80.58  101.81  10616.15  8163.52  6667.56  1953.82  109.81  B$$_{\mathrm{16}}$$  61.40  37.10  9649.10  8152.40  8990.50  2149.90  67.60  B$$_{\mathrm{17}}$$  73.17  28.45  7606.54  4206.40  2230.52  1129.04  101.78  B$$_{\mathrm{18}}$$  79.84  154.74  6273.49  4229.35  1604.82  1946.63  69.73  B$$_{\mathrm{19}}$$  102.56  91.17  4617.28  3446.08  1974.91  792.93  125.60  B$$_{\mathrm{20}}$$  25.55  59.37  3882.97  2551.51  3086.37  387.20  39.84  B$$_{\mathrm{21}}$$  40.44  18.27  611.00  511.00  1905.63  38.74  62.88  B$$_{\mathrm{22}}$$  78.19  77.39  4762.77  3614.36  2867.29  1432.18  88.03  B$$_{\mathrm{23}}$$  125.77  210.18  4550.00  4485.00  881.17  1473.90  161.50  B$$_{\mathrm{24}}$$  44.53  48.41  5756.21  4256.04  2481.32  3694.59  77.91  B$$_{\mathrm{25}}$$  47.40  16.75  3784.39  3024.95  1118.81  399.86  34.93  B$$_{\mathrm{26}}$$  59.30  58.26  3789.86  3725.36  664.50  1498.83  57.48  B$$_{\mathrm{27}}$$  50.80  42.82  5319.95  2571.01  3583.24  277.66  51.86  B$$_{\mathrm{28}}$$  27.77  50.25  4218.49  3201.54  1256.24  2088.79  32.23  B$$_{\mathrm{29}}$$  35.75  61.89  4030.31  3449.45  1531.22  1130.81  36.11  B$$_{\mathrm{30}}$$  48.11  22.89  3473.08  3467.88  811.18  994.80  33.81  B$$_{\mathrm{31}}$$  48.99  213.59  4337.56  3669.08  1171.27  1095.03  103.99  B$$_{\mathrm{32}}$$  56.01  223.53  3010.27  2672.51  1182.00  202.53  44.79  B$$_{\mathrm{33}}$$  40.52  23.47  2553.00  2479.00  326.24  3449.09  60.86  B$$_{\mathrm{34}}$$  31.47  79.84  2140.70  1864.24  1060.34  831.99  26.27  B$$_{\mathrm{35}}$$  30.43  35.89  2750.07  2378.15  903.25  1010.14  11.96  B$$_{\mathrm{36}}$$  36.97  113.30  1739.00  1303.00  1764.63  118.62  72.07  B$$_{\mathrm{37}}$$  42.07  57.24  2629.24  2533.78  483.51  1347.86  49.26  B$$_{\mathrm{38}}$$  33.55  32.51  2017.17  1774.77  552.15  1788.56  46.55  B$$_{\mathrm{39}}$$  17.60  8.00  704.50  77.10  713.10  23.10  –0.90  B$$_{\mathrm{40}}$$  41.20  7.10  394.00  236.00  332.10  3.40  –101.90  B$$_{\mathrm{41}}$$  9.84  40.96  194.41  150.00  3.19  27.13  19.41  B$$_{\mathrm{42}}$$  3.56  0.36  385.96  17.82  525.84  1.00  –4.99  B$$_{\mathrm{43}}$$  8.42  10.98  8.63  7.00  163.85  63.09  –12.51  B$$_{\mathrm{44}}$$  52.00  9.00  421.00  41.21  56.28  2.00  –1.39  B$$_{\mathrm{45}}$$  5.00  3.90  11.00  9.00  70.90  1.00  –62.10  B$$_{\mathrm{46}}$$  1.10  3.80  6.90  6.00  34.40  2.00  –2.30     Inputs  Outputs     Bank  Personnel  Fixed  Deposits and  Gross loans  Other earning  Off-balance  Other operating     expenses  assets  S.T. funding     assets  sheet items  incomes  B$$_{\mathrm{01}}$$  173.32  251.35  49399.73  36612.23  25774.34  21299.37  520.14  B$$_{\mathrm{02}}$$  188.96  156.35  25557.92  21991.87  7546.21  14667.79  350.88  B$$_{\mathrm{03}}$$  248.43  336.21  25678.80  18239.31  9266.85  7132.43  368.24  B$$_{\mathrm{04}}$$  142.30  373.10  21446.10  14910.30  10780.10  2945.60  229.90  B$$_{\mathrm{05}}$$  112.39  293.68  10368.52  9490.80  7056.90  5206.43  254.84  B$$_{\mathrm{06}}$$  222.68  177.84  18476.92  16372.53  6091.03  6607.52  –15.71  B$$_{\mathrm{07}}$$  215.87  159.54  15631.99  13677.14  3092.72  3310.01  120.08  B$$_{\mathrm{08}}$$  248.00  122.00  21218.00  12754.00  10873.00  9848.00  279.00  B$$_{\mathrm{09}}$$  97.29  176.05  12032.79  7967.21  5064.25  3435.14  198.15  B$$_{\mathrm{10}}$$  262.19  374.32  16017.64  12102.57  8678.31  25704.75  521.17  B$$_{\mathrm{11}}$$  119.65  99.71  17656.36  15726.73  5715.43  10354.72  164.47  B$$_{\mathrm{12}}$$  147.72  194.54  11551.37  10908.19  3280.88  3981.53  168.01  B$$_{\mathrm{13}}$$  88.99  199.44  11229.63  8794.51  5568.21  1487.81  130.16  B$$_{\mathrm{14}}$$  113.79  132.05  11856.16  9652.53  4151.04  4657.33  177.79  B$$_{\mathrm{15}}$$  80.58  101.81  10616.15  8163.52  6667.56  1953.82  109.81  B$$_{\mathrm{16}}$$  61.40  37.10  9649.10  8152.40  8990.50  2149.90  67.60  B$$_{\mathrm{17}}$$  73.17  28.45  7606.54  4206.40  2230.52  1129.04  101.78  B$$_{\mathrm{18}}$$  79.84  154.74  6273.49  4229.35  1604.82  1946.63  69.73  B$$_{\mathrm{19}}$$  102.56  91.17  4617.28  3446.08  1974.91  792.93  125.60  B$$_{\mathrm{20}}$$  25.55  59.37  3882.97  2551.51  3086.37  387.20  39.84  B$$_{\mathrm{21}}$$  40.44  18.27  611.00  511.00  1905.63  38.74  62.88  B$$_{\mathrm{22}}$$  78.19  77.39  4762.77  3614.36  2867.29  1432.18  88.03  B$$_{\mathrm{23}}$$  125.77  210.18  4550.00  4485.00  881.17  1473.90  161.50  B$$_{\mathrm{24}}$$  44.53  48.41  5756.21  4256.04  2481.32  3694.59  77.91  B$$_{\mathrm{25}}$$  47.40  16.75  3784.39  3024.95  1118.81  399.86  34.93  B$$_{\mathrm{26}}$$  59.30  58.26  3789.86  3725.36  664.50  1498.83  57.48  B$$_{\mathrm{27}}$$  50.80  42.82  5319.95  2571.01  3583.24  277.66  51.86  B$$_{\mathrm{28}}$$  27.77  50.25  4218.49  3201.54  1256.24  2088.79  32.23  B$$_{\mathrm{29}}$$  35.75  61.89  4030.31  3449.45  1531.22  1130.81  36.11  B$$_{\mathrm{30}}$$  48.11  22.89  3473.08  3467.88  811.18  994.80  33.81  B$$_{\mathrm{31}}$$  48.99  213.59  4337.56  3669.08  1171.27  1095.03  103.99  B$$_{\mathrm{32}}$$  56.01  223.53  3010.27  2672.51  1182.00  202.53  44.79  B$$_{\mathrm{33}}$$  40.52  23.47  2553.00  2479.00  326.24  3449.09  60.86  B$$_{\mathrm{34}}$$  31.47  79.84  2140.70  1864.24  1060.34  831.99  26.27  B$$_{\mathrm{35}}$$  30.43  35.89  2750.07  2378.15  903.25  1010.14  11.96  B$$_{\mathrm{36}}$$  36.97  113.30  1739.00  1303.00  1764.63  118.62  72.07  B$$_{\mathrm{37}}$$  42.07  57.24  2629.24  2533.78  483.51  1347.86  49.26  B$$_{\mathrm{38}}$$  33.55  32.51  2017.17  1774.77  552.15  1788.56  46.55  B$$_{\mathrm{39}}$$  17.60  8.00  704.50  77.10  713.10  23.10  –0.90  B$$_{\mathrm{40}}$$  41.20  7.10  394.00  236.00  332.10  3.40  –101.90  B$$_{\mathrm{41}}$$  9.84  40.96  194.41  150.00  3.19  27.13  19.41  B$$_{\mathrm{42}}$$  3.56  0.36  385.96  17.82  525.84  1.00  –4.99  B$$_{\mathrm{43}}$$  8.42  10.98  8.63  7.00  163.85  63.09  –12.51  B$$_{\mathrm{44}}$$  52.00  9.00  421.00  41.21  56.28  2.00  –1.39  B$$_{\mathrm{45}}$$  5.00  3.90  11.00  9.00  70.90  1.00  –62.10  B$$_{\mathrm{46}}$$  1.10  3.80  6.90  6.00  34.40  2.00  –2.30  © The authors 2016. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) For permissions, please e-mail: journals. permissions@oup.com

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IMA Journal of Management MathematicsOxford University Press

Published: Apr 1, 2018

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