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1 European Journal of Operational Research 202 (2010) Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: Production, Manufacturing and Logistics Deriving the DEA frontier for two-stage processes Yao Chen a, Wade D. Cook b, Joe Zhu c, * a College of Management, University of Massachusetts at Lowell, Lowell, MA 01845, USA b Schulich School of Business, York University, Toronto, Ontario, Canada M3J 1P3 c Department of Management, Worcester Polytechnic Institute, Worcester, MA 01609, USA article info abstract Article history: Received 19 July 2008 Accepted 11 May 2009 Available online 18 May 2009 Keywords: Data envelopment analysis (DEA) Efficiency Two-stage Intermediate measure Frontier Data envelopment analysis (DEA) is a method for measuring the efficiency of peer decision making units (DMUs). Recently DEA has been extended to examine the efficiency of two-stage processes, where all the outputs from the first stage are intermediate measures that make up the inputs to the second stage. The resulting two-stage DEA model provides not only an overall efficiency score for the entire process, but as well yields an efficiency score for each of the individual stages. Due to the existence of intermediate measures, the usual procedure of adjusting the inputs or outputs by the efficiency scores, as in the standard DEA approach, does not necessarily yield a frontier projection. The current paper develops an approach for determining the frontier points for inefficient DMUs within the framework of two-stage DEA. Ó 2009 Elsevier B.V. All rights reserved. 1. Introduction Data envelopment analysis (DEA) is an approach for identifying best practices of peer decision making units (DMUs), in the presence of multiple inputs and outputs (Charnes et al., 1978). DEA provides not only efficiency scores for inefficient DMUs, but also provides for efficient projections for those units onto an efficient frontier. In recent years, a number of DEA studies have focused on two-stage processes. For example, Seiford and Zhu (1999) develop a DEA approach for evaluating US commercial banks in a two-stage process characterized by profitability and marketability. In their study, profitability is measured using labor and assets as inputs, and the outputs are profits and revenue. In the second stage for marketability, the profits and revenue are then used as inputs, while market value, returns and earnings per share are used as outputs. Zhu (2000) applies the same two-stage process to the Fortune Global 500 companies. Sexton and Lewis (2003) study the Major League Baseball performance in a two-stage process. In their paper, Seiford and Zhu apply the standard DEA approach to each stage, but do not address potential conflicts between the two stages arising from the intermediate measures. For example, the second stage may have to reduce its inputs (intermediate measures) in order to achieve an efficient status. Such an action would, however, imply a reduction in the first stage outputs, thereby reducing the efficiency of that stage. * Corresponding author. addresses: Yao_Chen@uml.edu (Y. Chen), wcook@schulich.yorku.ca (W.D. Cook), jzhu@wpi.edu (J. Zhu). A number of DEA studies have been developed in an attempt to address this type of conflict. For example, the network DEA approach of Färe and Whittaker (1995), Färe and Grosskopf (1996), and Färe and Grosskopf (2000) is used by Lewis and Sexton (2004) to compute the efficiency scores of sub-units. The approach of Sexton and Lewis (2003) is to apply a methodology similar to that of Seiford and Zhu (1999) (although using an output-oriented DEA model), and then suggest as an overall efficiency score, using the ratio of outputs from the second stage to the projected outputs from the first stage. Related work is given in Pendharkar et al. (2000) and Pendharkar and Rodger (2009) using Bayesian network classifiers, in the context of DEA, for mining breast cancer patterns. Kao and Hwang (2008) develop a different approach where the entire two-stage process can be decomposed into the product of the efficiencies of the two sub-processes. As a result, both the overall efficiency and each stage s efficiency are obtained. There are also other DEA studies that deal with multiple stages and/or DMUs with internal structures. See Castelli et al. (in press) for an excellent review of various DEA approaches in dealing with the internal structures of DMUs. In the current study we focus on a particular, and typical case of a two-stage process where all outputs from the first stage are intermediate measures (nothing leaves the system at that point), becoming the inputs to the second stage. As pointed out by a number of authors, including Kao and Hwang (2008), Lewis and Sexton (2004), and Castelli et al. (in press), adjusting the inputs and outputs by the efficiency scores in a two-stage process is generally not sufficient to yield a frontier projection. Chen et al. (2009) present a model similar to that of Kao and Hwang (2008), but in an additive format. However, as with the multiplicative setup, the usual input and output adjustments do /$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi: /j.ejor

2 Y. Chen et al. / European Journal of Operational Research 202 (2010) not yield the efficient frontier here either. For many of the cases addressed in the DEA literature dealing with structured DMUs, this will be the situation. In the sections to follow we show that the overall efficiency scores resulting from Kao and Hwang (2008) are not direct indicators of potential input reductions or output increases not realized by the inefficient DMUs, e.g. how much more output each DMU can produce given its present inputs, or how much each DMU could reduce its input-use while still producing the same output. In other words, the resulting DEA scores do not provide complete information on how to project inefficient DMUs onto the DEA frontier for a specific two-stage process. Although we know the efficiency scores, we still do not know where the DEA frontier is. The current paper develops an approach for determining the DEA frontier or DEA projections for inefficient DMUs under the framework of Kao and Hwang (2008). Section 2 presents the generic two-stage process and the Kao and Hwang s (2008) model. In Section 3 we revisit their application involving Taiwanese non-life insurance companies and illustrate the fact that none of the DMUs are efficient, meaning that the usual DEA projections fail to identify the frontier. We then develop an approach for determining the DEA frontier for the two-stage processes, and illustrate this using the aforementioned data set. Conclusions are given in Section Input- and output-oriented DEA models for two-stage processes Consider a generic two-stage process as shown in Fig. 1, for each of a set of n DMUs. We assume each DMU j ðj 1; 2;...; nþ has m inputs x ij ði 1; 2;...; mþ to the first stage, and D outputs z dj ðd 1; 2;...; DÞ from that stage. These D outputs then become the inputs to the second stage, hence behaving as intermediate measures. The outputs from the second stage are y rj ðr 1; 2;...; sþ Input orientation For DMU j we denote the efficiency ratios for the first stage as h 1 j and the second as h 2 j. Based upon the input-oriented DEA model of Charnes et al. (1978), we define P D w P s dz dj P m and h v 2 j u ry rj P i x D ij ~w ; ð1þ dz dj h 1 j where v i ; w d ; ~w d, and u r are unknown non-negative weights. It is assumed that w d are set equal to ~w d as in Kao and Hwang (2008). Asa result, the two-stage overall efficiency ratio is defined as h 1 j h P 2 j s which is equal to h j ury ro P m v. To calculate the overall efficiency i x io of h j, Kao and Hwang (2008) present the following model P s Max h 1 j h 2 j u ry ro Pm v i x io ð2þ h 1 j < 1 and h 2 j 6 1 and w d ~w d : Applying the usual transformation, model (2) can be converted into the following linear program Stage 1 DMU j, j = 1,2,..., n x ij, i = 1,2,,,, m z dj, d = 1,2,..., D Fig. 1. Two-stage process. Stage 2 y rj, r = 1,2,..., s Max Xs X m u r y ro w d z dj Xm v i x io 1; w d z dj 6 0; v i x ij 6 0; j 1;2;...;n; j 1;2;...;n; w d P 0; d 1;2;...;D; v i P; i 1;2;...;m; u r P;r 1;2;...;s: ð3þ Note that constraints P s u ry rj P m v i x ij 6 0 are redundant in Kao and Hwang s (2008) model, because P s u ry rj P D w dz dj 6 0 and P D w dz dj P m v i x ij 6 0 imply P s u ry rj P m v i x ij 6 0. The dual to model (3) can be expressed as h k j x ij 6 hx io ; i 1; 2;...; m; y rj P y ro ; r 1; 2;...; m; ðk j Þz dj P 0; d 1; 2;...; D; k j ; P 0; h 6 1: 2.2. Output orientation The output-oriented version of the above model is given by P m v i x ij0 Ps u ry rj0 ð5þ h 1 j < 1 and h 2 j 6 1 for al; w d ~w d for all d; which is equivalent to the linear programming formulation Xm v i x io w d z dj Xm u r y ro 1; w d z dj 6 0; j 1; 2;...; n; v i x ij < 0; j 1; 2;...; n; w d ; d 1; 2;...; D; v i ; i 1; 2;...; m; u r ; r 1; 2;...; s P 0: ð6þ The dual to model (6) can be expressed as Max / k j x ij 6 x io ; i 1; 2;...; m; y rj P /y ro ; r 1; 2;...; m; ðk j Þz dj P 0; d 1; 2;...; D; k j ; P 0; / P 1: ð4þ ð7þ

3 140 Y. Chen et al. / European Journal of Operational Research 202 (2010) Note that both the input- and output-oriented two-stage DEA models (4) and (7) are identical to the standard DEA envelopment model if k j. To illustrate their model, Kao and Hwang (2008) present a numerical example with three DMUs A, B, and C which use 2, 4, and 5 units of input X to produce 1.5, 4, and 4 units of intermediate product Z at stage 1, which in turn become inputs to stage 2 to produce 1.5, 5, and 6 units of output Y, respectively. It is shown that DMU A has an overall score of 0.5, indicating that this DMU would be able to produce twice as much if it used its inputs efficiently. In this example the overall DEA efficiency scores show how much more output each DMU can produce given its present inputs, or how much each DMU could reduce its input-use while still producing the same output. i.e., the overall efficiency scores obtained from model (4) (or (7)) are intended to identify the DEA frontier points. The following discussion shows that this is not true and that we, therefore, need alternative models to determine the frontier points for two-stage processes. 3. The DEA frontier We here revisit the two-stage application involving 24 Taiwanese non-life insurance companies studied in Kao and Hwang (2008). The two stages represent premium acquisition and profit generation respectively. The inputs to the first stage are operational expenses and insurance expenses, and the outputs from the second stage are underwriting profit and investment profit. There are two intermediate measures between the two stages, namely direct written premiums and reinsurance premiums (see Kao and Hwang (2008) for detailed discussion on these measures). The data appear in Table 1. The second and third columns of Table 2 report the overall efficiency scores obtained from models (4) and (7), respectively. It can be seen that h j 1=/ j for all the DMUs, as expected under the condition of constant returns to scale (CRS). As in the standard DEA approach, we first calculate the DEA projections by multiplying the current inputs (outputs) with the related optimal values h ð/ Þ while keeping the intermediate measures constant. We then apply these DEA projections to models (4) and (7) to examine whether they are efficient. Note that the current paper does not consider potential DEA slacks in the projections. We consider two cases. In Case I, all the DMUs are first replaced by the projected DMUs and then models (4) and (7) are applied. Namely, in model (4), for al, x 1j and x 2j are replaced with h j x 1j and h j x 2j and all the intermediate measures and outputs are kept unchanged, and in model (7), for al, y 1j and y 2j are replaced with / j y 1j and / j y 2j, and all the intermediate measures and inputs are kept unchanged. The fourth column of Table 2 reports the input-oriented scores obtained from model (4) and the fifth column of Table 2 displays the output-oriented scores obtained from model (7). It can be seen that none of the projected DMUs are efficient. It can also be seen that the newly obtained h j is no longer equal to 1=/ j. In Case II, we only replace the DMU under evaluation by its projection in models (4) and (7). Namely, we apply models (4) and (7) to the projected DMU o, and the data for other DMU j ðj oþ remain unchanged. The seventh column of Table 2 reports the input-oriented scores obtained from model (4) and the eighth column of Table 2 displays the output-oriented scores obtained from model (7). It can be seen that except for DMUs 3 and 22 (under the input orientation), and DMUs 9, 12, 15 and 19 (under the output orientation), none of the projected DMUs are efficient. Also, models (4) and (7) do not identify the same efficient DMUs. It can also be seen that the output-oriented score is not equal to the reciprocal of the input-oriented score for all the DMUs, as expected under the condition of CRS. Under the standard DEA approach, Case I and Case II should yield identical results. However, Case I and Case II yield very different results under both models (4) and (7). Thus, models (4) and (7) only yield an overall efficiency score, but fail to provide the complete information on how to project inefficient DMUs on to the DEA frontier. Table 1 Data set. DMU Operation expenses (X1) Insurance expenses (X2) Direct written premiums (Z1) Reinsurance premiums (Z2) Underwriting profit (Y1) Investment profit (Y2) 1 Taiwan Fire 1,178, ,512 7,451, , , ,687 2 Chung Kuo 1,381,822 1,352,755 10,020,274 1,812,894 1,228, ,754 3 Tai Ping 1,177, ,790 4,776, , , ,428 4 China Mariners 601, ,259 3,174, , , ,331 5 Fubon 6,699,063 3,531,614 37,392,862 1,753,794 7,851,229 3,925,272 6 Zurich 2,627, ,363 9,747, ,326 1,713, ,058 7 Taian 1,942,833 1,443,100 10,685, ,412 2,239, ,039 8 g Tai 3,789,001 1,873,530 17,267,266 1,134,600 3,899, ,868 9 Central 1,567, ,432 11,473, ,337 1,043, , The First 1,303,249 1,298,470 8,210, ,528 1,697, , Kuo Hua 1,962, ,414 7,222, ,178 1,486,014 18, Union 2,592, ,952 9,434,406 1,118,489 1,574, , Shingkong 2,609,941 1,368,802 13,921, ,343 3,609, , South China 1,396, ,888 7,396, ,509 1,401, , Cathay Century 2,184, ,063 10,422, ,893 3,355, , Allianz President 1,211, ,071 5,606, , , , Newa 1,453,797 1,085,019 7,695, ,489 3,144, , AIU 757, ,997 3,631, , , , North America 159, ,338 1,141, , ,121 46, Federal 145,442 53, , , ,624 26, Royal & 84,171 26, ,888 40,542 51, Sunalliance 22 Aisa 15,993 10,502 52,063 14,574 82, AXA 54,693 28, ,910 49, , Mitsui Sumitomo 163, , , , ,370 16,976 Source: Kao and Hwang (2008).

4 Y. Chen et al. / European Journal of Operational Research 202 (2010) Table 2 DEA scores. DMU Input-oriented Output-oriented Efficiency scores for projected DMUs Case I a Case II b Input-oriented Output-oriented 1/Output-oriented score Input-oriented Output-oriented 1/Input-oriented score a b Case I: All DMUs are replaced with their projections. Case II: Only the DMU under evaluation is replaced with its projection, and the data for other DMUs remain unchanged. To address this deficiency, we propose a model that is equivalent to model (4) (or model (7)) and generates a set of new inputs, outputs and intermediate measures that constitute an efficient point (projection) under model (4) (or (7)). To this end, consider the input-oriented model (4). For each DMU o, we introduce ~z do ðd 1;...; DÞ, representing a set of new intermediate measures to be determined. We then break the constraints P n ðk j Þz dj P 0 into two new sets of constraints z dj 6 ~z do ; d 1;...; D: The first new set of constraints treats the ~z do as outputs, and the second set treats the ~z do as inputs. We now propose the following DEA type model: min ~ h k j x ij 6 ~ hx io ; i 1;...; m; y rj P y ro ; r 1;...; s; z dj 6 ~z do ; d 1;...; D; ~z do P 0; d 1;...; D; k j P 0; j 1;...; n; P 0; j 1;...; n; ~h 6 1: We will show that model (8) and model (4) yield the same efficiency score, and model (8) provides an efficient projection. ð8þ The dual to model (8) can be expressed as: Max Xs u r y ro w 1 d z dj Xm w 2 d z dj 6 0; j 1;2;...:;n; v i x ij 6 0; j 1;2;...;n; Table 3 Optimal intermediate measures for DEA frontier. DMU Input-oriented (model 8) Output-oriented (model 10) z1 z2 z1 z2 1 5,129, , ,335, , ,287, , ,063,742 1,324, ,776, ,244 6,922, , ,332, , ,379, , ,127,364 4,177,166 39,280,374 5,446, ,807, , ,769,924 1,117, ,738, , ,516,290 2,364, ,553,015 1,009,007 20,180,210 3,666, ,166, , ,703,191 1,575, ,417, , ,480,468 1,440, , , ,745,796 1,608, ,166, , ,434,406 1,118, ,649, , ,748,722 3,100, ,749, , ,526,595 1,575, ,663, , ,226,915 1,033, ,899, , ,932, , ,504, , ,735,516 2,014, ,356, , ,242, , , ,242 1,154, , , , , , , , , ,063 14,574 88, , , , , , , , ,184, ,366.5 ð9þ

5 142 Y. Chen et al. / European Journal of Operational Research 202 (2010) X m v i x io 1; w 2 d w1 d 6 0; d 1;2;...;D; w 1 d ; w2 d P 0; d 1;2;...;D; v i P; i 1;2;...;m; u r P; r 1;2;...;s Note that in model (8), the constraints ~z do P 0 are redundant, hence we can omit ~z do P 0, and let the ~z do be unrestricted variables. Then, the constraint w 2 d w1 d 6 0; d 1; 2;...; D in model (9) becomes w 2 d w1 d meaning that we put the same weights w d on the z dj in both settings (as outputs from stage 1 and as inputs to stage 2), as in model (2). Thus, problem (9) above is identical to problem (3). This further indicates that model (8) yields the same overall efficiency score as model (4). Based upon model (8), the projection point for DMU o is given by ð ~ h x io ; ~z dj ; y roþ which is efficient under models (8) and (4), namely the optimal objective function value for model (8) is equal to unity for this projection. In a similar manner, we can show that in the output-oriented case, model (7) is equivalent to the following model max ~ / k j x ij 6 x io ; i 1;...; m; y rj P ~ /y ro ; r 1;...; s; z dj 6 ~z do ; d 1;...; D; ~z do P 0; d 1;...; D; k j P 0; j 1;...; n; P 0; j 1;...; n; / P 1: ð10þ Model (10) yields a set of new inputs, outputs and intermediate measures that render the DMU efficient under model (7). From model (8) or (10), it can be seen that in addition to the overall efficiency scores, we have to obtain a set of optimal intermediate measures (z). Models (4) and (7) do not immediately yield the set of optimal z values that are on the DEA frontier, rather we have to use model (8) or (10) to determine the frontier points for two-stage processes. Table 3 reports the optimal intermediate measures for both orientations, obtained from models (8) and (10). If we apply the new (projected) DMUs with the intermediate measures reported in Table 3 and h j x 1j and h j x 2j (or / j y 1j and / j y 2j) to model (4) (or (7)) under either Case I or Case II, the overall efficiency scores are all equal to one. Note that under the input-oriented models, for DMUs 3 and 22, model (8) indicates the newly obtained intermediate measures are equal to the original values. This is confirmed by the fact that model (4) identifies DMUs 3 and 22 as efficient under the Case II. The same situation is found for DMU12 under the output-oriented models. While DMUs 9, 15 and 19 are efficient under model (7), model (10) identifies a set of different values on intermediate measures for DMUs 9, 15 and 19 (see the last two columns of Table 3). This indicates that multiple optimal solutions exist for intermediate measures in models (8) and (10). In fact, if we calculate model (10) using the original intermediate values and / j y 1j and / j y 2j for DMUs 9, 15, and 19, model (10) yields an efficiency score of Conclusions This paper develops models for determining the DEA frontier points (projections) for inefficient DMUs under the framework of the DEA model for two-stage processes. The current study is based upon the assumption of constant returns to scale (CRS). Chen et al. (2009) develop a two-stage DEA model under the condition of variable returns to scale (VRS) wherein the overall efficiency is expressed as a (weighted) sum of the efficiencies of the individual stages. Although overall VRS efficiency scores as well as scores for individual stages can be obtained by using Chen et al. (2009), adjusting the inputs or outputs by the efficiency scores is not sufficient to yield VRS frontier projections. Because Chen et al. (2009) focus on the additive efficiency decomposition, the newly developed approach in the current study cannot be (directly) applied. Further study is then needed to develop models for determining the DEA frontier points for VRS inefficient DMUs. The presented model has important applications in areas such as supply chain management. With the need to better understand supply chains and to be able to evaluate the performance of supply chain members, tools such as that provided herein can provide new insights. The model has the advantage of providing a clear indication of where the strengths and weaknesses exist in such multi-stage settings. In the banking environment, there is also a need for tools that allow one to uncover inefficiencies in different components of the operation. Many of the bank s operations take the form of two-stage processes. Customer deposits can be viewed as outputs in regard to the service being supplied by branch staff, but at the same time deposits are also inputs to the consumer and business loan part of the bank s operation. To address the many practical problems encountered in settings such as these, we suggest that more work be done on multi-stage processes to enhance the usefulness of tools of the type presented herein. Acknowledgement The authors thank three anonymous reviewers for their comments and suggestions. References Castelli, L., Pesenti, R., Ukovich, W., in press. A classification of DEA models when the internal structure of the Decision Making Units is considered, Annals of Operations Research. Charnes, A., Cooper, W.W., Rhodes, E., Measuring the efficiency of decision making units. European Journal of Operational Research 2, Chen, Y., Cook, W.D., Li, N., Zhu, J., Additive efficiency decomposition in twostage DEA. 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