Two-stage network DEA: when intermediate measures can be treated as outputs from the second stage

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1 Journal of the Operational Research Society (2015) 66, Operational Research Society Ltd. All rights reserved /15 Two-stage network DEA: when intermediate measures can be treated as outputs from the second stage Sonia Aviles-Sacoto 1, Wade D Cook 2 *, Raha Imanirad and Joe Zhu 4 1 Doctoral Program in Engineering Sciences, ITESM-Monterrey, Mexico; 2 Schulich School of Buisness, York University, Toronto, Canada MJ1P; Doctoral Program, Harvard Business School, Soldiers Field, Boston, MA 0216; and 4 International Center for Auditing and Evaluation, Nanjing Audit University, Nanjing, P.R. China; School of Business, Worcester Polytechnic Institute, Worcester, USA This paper investigates efficiency measurement in a two-stage data envelopment analysis (DEA) setting. Since 1978, DEA literature has witnessed the expansion of the original concept to encompass a wide range of theoretical and applied research areas. One such area is network DEA, and in particular two-stage DEA. In the conventional closed serial system, the only role played by the outputs from Stage 1 is to behave as inputs to Stage 2. The current paper examines a variation of that system. In particular, we consider settings where the set of final outputs comprises not only those that result from Stage 2, but can include, in addition, certain outputs from the previous (first) stage. The difficulty that this situation creates is that such outputs are attempting to play both an input and output role in the same stage. We develop a DEA-based methodology that is designed to handle what we term timestaged outputs. We then examine an application of this concept where the DMUs are schools of business. Journal of the Operational Research Society (2015) 66(11), doi: /jors Published online 15 April 2015 Keywords: DEA; networks; two-stage efficiency; time-staged outputs 1. Introduction Data envelopment analysis (DEA), developed by (Charnes et al, 1978), is a methodology for evaluating the relative efficiencies of a set of comparable decision-making units. Since its inception, DEA literature has witnessed the expansion of the original concept to encompass a wide range of theoretical and applied research areas. One such area is network DEA (Färe and Grosskopf, 1996), and in particular two-stage DEA. This latter, and its extensions to multi-stage situations, has been particularly influential in such settings as supply chain management. See, for example, Kao (2009b), Tone and Tsutsui (2009), Liang et al (2006), and Cook and Zhu (2014). A survey of network models is provided by Cook et al (2010). One of the common two-stage structures investigated in the DEA literature is the serial process wherein the outputs from the first stage become the inputs to the second stage. Some variants of this permit outputs from Stage 1 to leave the system and inputs to the second stage to enter the system at that point; other two-stage systems are closed in that nothing enters or leaves the system between the stages. It is this latter system that *Correspondence: Wade D. Cook, Schulich School of Business, York University, Room S7M, 4700 Keele Street, Toronto, Ontario, Canada MJ1P. wcook@schulich.yorku.ca we consider herein. The usual objective, if one regards the two-stage process as the DMU, is to view that process as one where inputs enter the DMU at Stage 1 and outputs exit from Stage 2. Various methods have been suggested for evaluating the (sub) efficiencies of each of the two stages, and for then combining these to get the overall DMU efficiency. In the conventional closed serial system, the only role played by the outputs from Stage 1 is that of inputs to Stage 2. The current paper examines a variation of that system. In particular, we consider settings where the set of final outputs comprises not only those that emanate from Stage 2, but can include, in addition, certain outputs from the previous (first) stage. The difficulty that this situation creates is that such outputs are attempting to play both an input and output role in the same stage. Section 2 details an application of this concept, where the DMUs are schools of business. In Section we develop a DEA-based methodology that is designed to handle such situations. Section 4 applies the new methodology to 7 business schools, primarily located in the United States. Conclusions are presented in Section Undergraduate business programmes viewed as twostage processes Numerous studies of efficiency have been undertaken regarding educational institutions. These include Charnes et al (1981), Bessent et al (1982), Beasley (1990), Johnes

2 Sonia Aviles-Sacoto et al Two-stage network DEA 1869 Figure 1 A two-stage process. and Johnes (199), Athanassopoulos and Shale (1997), Johnes and Yu (2008), Kao and Hung (2008), Tyagi et al (2009), and (Chang et al (2012). Generally, these studies are based on settings where technical efficiency focuses on inputs such as teaching staff and research funding, and outputs measured in terms of courses taught and students supervised. In the current settingweexamineasomewhat different situation involving student recruitment. Business schools around the world compete annually for the best and brightest students. Currently worldwide, there has been a rapid growth in the offerings of universities and colleges, and every year when students are applying to these higher education institutions, they focus on the results that they could gain after their studies. Therefore, institutions within the higher education market need to adopt or increase their interests in those features, characteristics, and competitive differentiators, which are attractive to applicants. In a recent study by Avilés-Sacoto (2012) and captured in a working paper (Avilés-Sacoto et al, 201), a set of 41 undergraduate business programmes was surveyed to gain an understanding of best practices in terms of admissions criteria and various outcome factors. Forty of the business schools are in the United States, while the 41st was the business school, ENCSH at ITESM, in Monterrey, Mexico. Many previous studies in this area have been undertaken, and there is much debate about what factors are most relevant for efficiency evaluation purposes. We have attempted to choose factors that define the quality of the students (hence of the institution as well), and factors which reflect the achievements of the students as a result of being at that institution. With this in mind, the following are the evaluation factors used. The notation x, z, y used herein to designate these factors will become evident below. Top 25% at class (x 1j ). This factor is defined as the percentage of students applying to school j who were in the top 25% of their high school classes. Percentage of applicants rejected, hence not sent offers of admission (x 2j ). Academic rating (x j ). This rating is a measure of how hard students work at the school and how much they get back for their efforts. Percentage of applicants enroled (z 1j ). Of those students accepted, this is the percentage of those who are enroled. Percentage receiving institutional scholarships (z 2j ).This is the percentage of students earning/holding scholarships. Percentage receiving internships (z j ). This is the percentage of enroled students who obtained overseas internships while in the programme. Percentage of students who get jobs (y 1j ). This is the percentage of students who obtained employment in their chosen fields within 6 months of graduation. Theviewwetakehereinisthatefficiency can be modelled as a two-stage process. This is displayed schematically in Figure 1: Stage 1 can be looked upon as the process of attracting applicants to programmes, and realizing the resulting outcomes once applicants are enroled. The inputs to this stage areintendedtoreflect the combined quality of the institution and the students who apply there; the factors x 2j and x j are measures of institutional reputation and quality, while Top 25% (x 1j ) captures an important dimension of the quality of students applying. Outputs here are taken as those factors that reflect the combined accomplishments of the institution and the enroled students. Specifically, z 1j measures the success of the jth institution in converting applicants to enroled students, while students receiving scholarships (z 2j ) and internships (z j ) are important indicators of both institutional and student achievements while the student is in the programme. Stage 2 captures the students (and institutions ) accomplishments following graduation. The inputs to this stage are the outputs from Stage 1 while the percentage of students getting jobs following graduation (y 1j ) is the major output from the students perspective. A problem with viewing efficiency measurement in the current setting as a two-stage process is that the final output y 1j does not represent the only outcome that the institution wishes to emphasize. From the perspective of marketing business programmes, an additional important statistic is the percentage of students who obtain international internships. A business school that can claim success in placing its students in overseas settings has an advantage in comparison to institutions that are less successful on this dimension. Students, as well, view

3 1870 Journal of the Operational Research Society Vol. 66, No.11 INPUTS (Stage 1) % of applicants rejected OUTPUTS (Stage 1) INPUTS (Stage 2) % of accepted applicants enrolled OUTPUTS (Stage 2) Academic Rating Stage 1 % receiving institutional scholarships Stage 2 % of students who get jobs % students in top 25% of their classes % receiving internships % receiving internships Figure 2 Modified two-stage process. internships as a key success factor in that it affords them opportunities to gain valuable experience in an international setting, and as well opens doors in the job market. This latter aspect points, however, to a modelling complication in that internships serve both as an input to Stage 2, because it impacts students abilities to gain employment, and at the same time it holds a status equivalent to job success as a final outcome, thus can be viewed as an output from the second stage. In this way, one can argue that the two-stage process can be viewed as shown in Figure 2. There are a number of DEA-based approaches for modelling two-stage processes, and these approaches can be categorized into several groups: (1) all outputs from the first stage become inputs to the second stage (see a review by Cook et al, 2010); (2) two-stage processes that take the form of parallel relations (see Kao, 2009b); () two-stage processes that are part of a dynamic system where each stage has its own inputs and outputs (see, eg, Chen and van Dalen, 2010); (4) two stages that have shared resources (see, eg, Chen et al, 2010); (5) each of the two-stage processes has unstructured internal systems (see, eg, Castelli et al, 2010). Overlaps exist among these groups. The reader is referred to the recent publication of Cook and Zhu (2014) for a comprehensive resource of network DEA developments. Given the above comments, we argue that none of the existing approaches address the issue when some outputs from the first stage are both inputs to the second stage and outputs from the second stage. In other words, existing approaches cannot be used directly to model the dual roles of some outputs from the first stage. This being the case, we propose that the unique contribution of the current paper is the development of a methodology that allows for final outputs in a multistage network to consist not only of those from the last stage, but as well selected outputs from previous intermediate stages. In the following section we investigate the modelling of efficiency for both types of two-stage processes as depicted in Figures 1 and 2, respectively. We first examine the conventional serial process (Figure 1) where all outputs from the first stage serve the sole purpose of being inputs to the second stage. Hence, we ignore the second stage output role of internships in this first analysis. We then examine efficiency modelling in the context of the particular circumstances described in the previous section (Figure 2), where internships must be acknowledged as serving both as an output from the first stage and as a factor to be combined with final stage outputs.. Modelling efficiency in two-stage processes The conventional two-stage serial process As indicated above the modelling of efficiency in two-stage processes has been the subject of numerous papers including Chen et al (2009), Chen et al (2010), Chen et al (2009), Kao (2009a, b), Kao and Hwang (2008), Liang et al (2008), and Liang et al (2006). A recent paper by Cook et al (2010) provides a comprehensive survey of developments in this area. As background for the development of the model for the process pictured in Figure 1, we briefly review the methodology for one of the most common structures for two-stage serial processes. We begin by looking at the conventional two-stage model as depicted in Figure 1. Referring again to the variables x, z, y as defined in the previous section, we let x ij,, 2, denote the inputs to Stage 1, y 1j the single output (jobs) from Stage 2, and z dj,, 2, the intermediate variables that serve simultaneously as outputs from the first stage and inputs to the second stage. Their corresponding multipliers are denoted by v i, u r, η d, respectively. In the development below we focus on the applicationspecific structure and the variables therein. We point out that this special case is immediately extendable to a general setting where multiple intermediate variables can be included with second stage outputs to generate the set of final outputs. Furthermore, these ideas can be extended to include multiple (more than two) stages. To evaluate the overall efficiency of schools of business, we argue that it is appropriate to adopt an output-oriented (rather than input-oriented) model since it is outcomes such as jobs and internships that organizations wish to enhance, as opposed to reducing inputs that, in the current case, take the form of measures of quality. Furthermore, we propose to use the variable returns to scale (VRS) model of Banker et al (1984), as opposed to the CRS model, since the data appear in the form of ratios and percentages.

4 Sonia Aviles-Sacoto et al Two-stage network DEA 1871 In this setting one can represent the output-oriented VRS efficiency for Stage 1 as the solution to the radial projection model: P min ν i x io + u 1 ^e 1 o = η d z do subject to ν i x ij + u 1 i = 1 P 1; 8j η d z dj ν i ; η d 0; u 1 unrestricted in sign ð1þ The Stage 2 model is given by η d z do + u 2 ^e 2 o = min subject to ½u 1 y 1o Š η d z dj + u 2 u 1 y 1j 1; 8j u 1 ; η d 0; u 2 unrestricted in sign ð2þ A number of approaches have been suggested in the literature for deriving an overall score for the two stages combined, and from these, deriving scores for the individual stages. Two of the methodologies are those based on game theoretic principles, namely the cooperative and non-cooperative game models (Cook et al, 2010). The non-cooperative or non-centralized approach views the two stages as players in a game and adopts a leader follower methodology. This methodology, often referred to as a Stackelburg game, involves choosing one of the two stages as the leader, and then deriving multipliers for the inputs and outputs that yield the best possible score for that stage. The efficiency score for the other stage or player (the follower) is then derived by finding the best possible weights for its inputs and outputs, but with the restriction that the score of the leader is not compromised. The cooperative game (or centralized) methodology derives the best aggregate efficiency score for the two stages combined. Then, one sets out to derive scores for the two stages separately, which are such that when put together yields the overall score. One of the original models for two-stage process was that put forward by Kao and Hwang (2008), and is a form of the cooperative or centralized approach. In their model, the overall efficiency is derived by the product e o = e o1 e 2 o. The Kao and Hwang (2008) approach is designed for those settings where a CRS methodology is appropriate. To accommodate VRS settings, as is needed in the application addressed herein, Chen et al (2009) proposed an additive or arithmetic mean approach for combining the two stages, as opposed to the geometric-type methodology suggested by Kao and Hwang (2008). Under the Chen et al (2009) approach, the objective function for the VRS setting becomes 8 9 ν i x io + u 1 P >< η d z do + u 2 >= ^e o = min w 1 : P + w 2 : () ½u 1 y 1o Š >: η d z do >; where w 1, w 2 are appropriate weights. These weights must have the property that they sum to unity (convex combination of the efficiency scores of the two stages), and should be chosen such as to reflect the relative importance of the two stages. In regard to the latter property, since the output-oriented model is designed to reflect the degree to which outputs need to be enhanced to reach the frontier, it appears reasonable to define w 1, w 2 as the proportions of the outputs currently generated by the respective stages. Specifically, define these weights as η d z do w 1 = P ; w 2 = u 1 y 1o P (4) η d z do + u 1 y 1o η d z do + u 1 y 1o Under this definition, the efficiency measure (objective function) for the two stages combined becomes: min P ν i x io + u 1 + P η d z do + u 2 ^e o = P (5) η d z do + u 1 y 1o Hence, the cooperative game model for deriving the overall efficiency for the two stages combined is min P ν i x io + u 1 + P η d z do + u 2 ^e o = subject to η d z do + u 1 y 1o ν i x ij + u 1 i = 1 P 1; 8j η d z dj η d z dj + u 2 u 1 y 1j 1; 8j u 1 ; η d ; ν i 0; u 1 ; u 2 unrestricted in sign ð6þ

5 1872 Journal of the Operational Research Society Vol. 66, No.11 Clearly, as in the standard DEA model, problem (6) is highly non-linear, but can be converted to a linear format using the Charnes and Cooper (1962) transformation. As a result of this transformation, problem (6) can be expressed as the linear programming problem: ^e o = min X subject to X X X υ i x io + μ 1 + X π d z do + μ 1 y 1o = 1 υ i x ij + μ 1 - X π d z dj 0; π d z dj + μ 2 - μ 1 y 1j 0; π d z do + μ 2 8j 8j υ i ; π d ; μ 1 0; μ 1 ; μ 2 unrestricted in sign ð7þ Now let us consider the problem discussed in Section 2 where the factor internships assumes the role of both an output from Stage 1 as well as well as that of an output in Stage 2. Two-stage processes with time-phased outputs One can view the conventional two-stage process illustrated in Figure 1 as one where the primary focus of the organization (eg a business school) is directed towards those variables emanating from the second stage; in the application herein the variable Jobs plays that role. Furthermore, the outputs from Stage 1 (serving as inputs to Stage 2) such as Internships, play a pure intermediate and secondary role; they are a means to an end. In the situation where certain intermediate variables are a primary focus as well, and are, therefore, on par with the outputs from Stage 2, a conventional methodology such as that described above, is not immediately applicable. To see why this is so, let us suppose that we adapt the Stage 2 model (2) to include the Internships variable z j in the denominator (in addition to it appearing in the numerator), meaning that (2) becomes (2 ) min ^e 2 o = η d z do + u 2 ½u 1 y 1o + u 2 z o Š subject to η d z dj + u 2 1; 8j u 1 y 1j + u 2 z j u 1 ; u 2 ; η d 0; u 2 unrestricted in sign ð2 Þ It is immediately evident that by setting η = u 2 and all other variables to 0, the optimal solution becomes ^e 2 o = 1, meaning that the second stages of all DMUs would be efficient. Given this erroneous result, it is clear that inserting a variable in both numerator and denominator of the efficiency ratio renders the conventional methodology inapplicable to the situation at hand. Hence, the issue becomes one of how to model variables such as Internships in an appropriate way, given that they play a dual role of both input to and output from the second stage, while simultaneously serving as an output from Stage 1. We argue that in Stage 2, whether one looks upon Internships as an input or output, the desire is to treat that variable as one we would wish to influence. Viewed as an output from Stage 2, this variable is one where more is better, hence it would be desirable to increase that variable. If it is seen as an input, influencing Jobs, then from a technical efficiency perspective, less is better, meaning that a decrease in Internships would enhance the organization s efficiency score. In either case, it seems appropriate to model this variable in a way that allows the optimization process to indicate how much one needs to change Internships to improve efficiency. Since we are using an output-oriented methodology, treating Internships as an output automatically places it in the denominator of the second-stage efficiency ratio, meaning that it is treated as being discretionary. In its role as an input to that stage, we propose to look upon Internships as being discretionary as well, meaning that it is necessary to move it to the output (denominator) side of the efficiency ratio, but with a negative sign. This is related to the recommended treatment of non-discretionary variables as advanced by Banker and Morey (1986). Thus, viewing this from the perspective of () above, we argue that the overall efficiency ratio for the two stages should be written as: P min w 1 : ν i x io + u 1 P w 2 : 2 η d z do + u 2 ^e o = P + (8) 2 ½u 1 y 1o + gz o - hz o Š η d z do + η z o The term η z o in the Stage 1 component of (8) captures the role played by Internships as an output from that stage. In Stage 2, note that the denominator u 1 y 1o + gz o hz o permits Internships to assume both an output status represented by gz o and a discretionary input status given by hz o. It appears that a model analogous to (7) can now reasonably be used to represent the process pictured in Figure 2, albeit with a certain caveat, namely the formulation of the appropriate connection among the variables η,gand h (or more correctly their linear programming equivalents δ, γ, β, as given in (9) below). First, it is noted that the above formulation is equivalent to that where the expression gz o hz o is replaced by ωz o where the variable ω would be unrestricted in sign. If this was done, and if in the optimization ω is positive, this implies that the dominant role played by Internships is that of an output. Alternatively, if a negative ω arises, this would imply that the role of Internships as an input is dominant. Furthermore,

6 Sonia Aviles-Sacoto et al Two-stage network DEA 187 because we will view this connection between the two variables in a linear programming (LP) format, it will happen that at the optimum, one of the two multipliers g or h will automatically be zero, if we ignore for the moment the connection among η,g and h. This is because in the LP version of the optimization model, the column vectors for the two variables g and h are linearly dependent, hence both cannot be in the same basis together, meaning that both cannot be positive. This means that from a mathematical perspective, the role of Internships may be viewed as that of either an input or an output (as opposed to z o assuming both roles). Following this line of logic, we argue that whichever role internships z o assumes in Stage 2, its multiplier in that stage should take the same value as it takes as an output from Stage 1. In terms of the notation above, this means the variable η should equal either g or h, whichever is positive. With these ideas in place, and utilizing the Charnes and Cooper (1962) transformation from fractional to liner form, we propose the following integer linear programming model for determining the overall efficiency of the two-stage process. ^e o = min X υ i x io + μ 1 + X2 π d z do + μ 2 (9a) Subject to X X 2 X 2 d ==1 π d z do + δz o + μy 1o + γz o - βz o = 1 υ i x ij + μ 1 - X2 ð9bþ π d z dj - δz j 0; 8j (9c) π d z dj + μ 2 - μy 1j - γz j + βz j 0; 8j (9d) δ - γ - β = 0 γ - Me 0 β - Mf 0 e + f 1 υ i ; π d ; μ; δ; γ; β 0; μ 1 ; μ 2 (9e) (9f) (9g) (9h) unrestricted in sign ; e; f binary ð9iþ Some explanation is in order. In model (9) the variables δ, γ, β are the LP version of the multipliers η, g and h. Repeating the above argument, because we want the weight or importance attached to z j to be the same in both Stages 1 and 2, as in Kao and Hwang (2008), we need to guarantee that δ is equal to whichever of γ or β is positive. If we impose the condition δ = γ + β, as per (9e), it would appear that the requirement would now be met, except for the fact that the above claim about the LP structure forcing either γ or β to zero, is no longer valid. Specifically, the imposition of (9e) renders the column vectors for γ and β linearly independent. To bring about the requisite either/or status of the internships variable, we proceed as follows. Introduce two binary variables e and f to control the functioning of δ, γ, β. Specifically, let M denote a large positive number, and impose the constraint (9f), which allows γ to be positive only when e = 1. Similarly, constraint (9g) restricts β to be 0, except when f = 1. Now constraint (9h) insures that at most one of the binary variables will be positive, meaning that at least one of the two multipliers γ, β will be forced to zero. Constraint (9e) then restricts δ to assume the same value as whichever of the two multipliers γ, β is positive. We point out that the idea of dual role variables in DEA is not new, as the concept was originally put forward by Beasley (1995) in studying teaching and research efficiencies in UK universities. There, research was included as a factor both on the input and output sides of the ratio model. Beasley s model was later revisited by Cook et al (2006) who corrected certain irregularities in his methodology. Having obtained the overall efficiency from model (9), it is appropriate to use a type of leader-follower (or pre-emptive) methodology to arrive at individual efficiency scores for the two stages (see Cook et al, 2010). This involves designating one of the stages as being of highest priority, and then deriving an efficiency score for that stage while maintaining the optimal score for the combined process. Having determined these two scores, the score for the remaining lower priority stage will be self-evident. If Stage 1 is chosen, for example, as having preemptive priority (the leader), the efficiency score for that stage can be determined from model (10). X X 2 X ^e 1 X o = min υ i x io + μ 1 Subject to X 2 υ i x ij + μ 1 - X2 π d z do + δz o = 1 (10a) ð10bþ π d z dj - δz j 0; 8j (10c) π d z dj + μ 2 - μy 1j - γz o + βz o 0; 8j (10d) υ i x io + μ 1 + X2 π d z do + μ 2 - ^e o X 2 d ==1 π d z do + δz o + μy 1 o + γz o - βz o Þ 0 ð10eþ δ - γ - β = 0 γ - Me 0 β - Mf 0 (10f) (10g) (10h)

7 DMU Percentage of students in top 25% oftheirclass Table 1 Data Stage 1 Stage 2 Inputs Outputs Inputs Outputs x1 x2 x z1 z2 I z1 z2 I J I Percentage Academic Percentage of Percentage Percentage Percentage of applicants rating accepted receiving receiving students who rejected applicants internships internships get jobs enroled Percentage receiving institutional scholarships Percentage of accepted applicants enroled Percentage receiving institutional scholarships Percentage receiving internships 1874 Journal of the Operational Research Society Vol. 66, No.11

8 Sonia Aviles-Sacoto et al Two-stage network DEA 1875 e + f 1 υ i ; π d ; μ; δ; γ; β 0 ; μ 1 ; μ 2 unrestricted in sign ; e; f binary (10i) ð10jþ Note that constraint (10e) is imposed to insure that multipliers are chosen in a manner that maintains the overall efficiency of the two-stage process at the optimal value derived from (9). To derive the appropriate rating for the follower stage (eg, Stage 2) one needs the weights w 1, w 2 as discussed above (see (4)). In the present application, these are given by P 2 ^π d z do + ^δz o ^w 1 = P ; 2 ^π d z do + ^δz o + ^μ 1 y 1o + ^γz o - ^βz o ^μ 1 y 1o + ^γz o - ^βz o ^w 2 = P : 2 ^π d z do + ^δz o + ^μ 1 y 1o + ^γz o - ^βz o The notation bπ; etc denotes the optimal variables arising from the solution of (9). Then, the Stage 2 efficiency score is provided by ^e 2 o = ^e o - ^w 1^e 1 o (11) ^w 2 Theorem.1: The overall efficiency score of a DMU, as derived in model (9), will be efficient if and only if each of the scores for Stages 1 and 2 are efficient. Proof: Because the weights ^w 1 ; ^w 2 sum to unity, and because the overall efficiency score arising from (9) is a weighted average (using ^w 1 ; ^w 2 ) of the Stages 1 and 2 scores computed from (10) and (11), neither of which exceeds unity, then the overall score will equal 1 if and only if the two- stage scores are 1aswell. It is noted that if Stage 2 is adopted as the leader, a model similar to (10) would be solved, the results of which are discussed in the next section. Only the objective function and first constraint in the (Stage 2 as leader) problem will differ from (10a) and (10b). The remaining constraints are the same. For brevity we omit the formalities here. Before proceeding we once again emphasize that the development above can easily be generalized to allow for multiple dual-role intermediate variables. We omit the details. In the following section we discuss the application of the models of this section to the data on 7 schools of business. 4. Evaluating efficiencies of business schools Table 1 provides data on schools of business in a set 7 universities in the study group; DMU #7 is ITESM. We point DMU Overall efficiency Table 2 Overall efficiency Binary variable d Output versus input Input Output Output Output Output Output Output Output Output Output Input Input Output Output Output Output Output Output Output Input Output Input Output Output Output Output Output Output Output Output Input Input Output Output Input Input Input out that the number of DMUs in this data set is somewhat smaller than in the earlier study by Avilés-Sacoto (2012), (7 schools versus 41 used earlier) due to some key data missingfromfouroftheschools. Before carrying out the analysis, limits were set for the weights w 1, w 2, the purpose of which is to prevent the optimization process from assigning an unfairly large or unfairly small portion of the weighted outputs to one stage or the other. For purposes of the analysis, a nominal lower limit of 10% on each of the weights was imposed. Table 2 records the overall DMU efficiency for each of the 7 universities. Shown as well is the input versus output status of the dual role variable Internships. It is noted that 10 of the DMUs treat Internships as an input meaning that current Job percentages are less than they should be, in comparison to the number of Internships created. For the remaining 27 DMUs that treat Internships as outputs, both Jobs and Internships should be proportionally enhanced. Finally, the table records the values for the weights w 1, w 2. w1 w2

9 1876 Journal of the Operational Research Society Vol. 66, No.11 DMU Table Overall efficiency Two separate leader follower analyses were carried out. Table provides the efficiency scores for each case. Specifically, Columns and 4 display the efficiency scores for Stages 1 and 2 when Stage 1 is taken as leader and Stage 2 as follower. Similarly, Columns 5 and 6 record the scores when Stage 2 is taken as the leader. It can be argued that in the current setting it makes sense to use the results from Columns 5 and 6 in that it is the outcome(s) from the second stage that are of primary importance to the organization. 5. Conclusions Efficiency Scores for two separate leader-follower analyses Stage 1: Leader Stage 1 score Stage 2 score Stage 2: Leader Stage 1 score Stage 2 score In the conventional two-stage process it is generally the outputs from the second or final stage which are the primary focus of management. The intermediate outputs (those from the first stage) generally serve the sole purpose of being the inputs that generate, or at least influence those primary outputs. In this paper we have viewed the evaluation of a set of 7 schools of business from the perspective of a two-stage serial process. This particular application is one where the process differs from the conventional one in that management s primaryfocusisnot only on those outputs emanating from the second stage, but rather is directed at a larger set of outputs augmented by members from the set of intermediate (first stage) output group. The percentage of students who earn Internships is such a factor. It can be viewed both as an input to the second stage, since it influences the output Jobs, and at the same time as a factor that is on par with Jobs. A model for handling this type of process is developed herein. Another example 1 arises in the high voltage electricity transmission service, where network regulation is at issue. The obvious candidate for the final (Stage 2) output y, isthe revenue resulting from the flow of electricity. The dual role intermediate variable z might reasonably be the network reliability or capacity. An important issue then is what types of grids choose to emphasize grid capacity as Stage 1 output and a Stage 2 input and what types of grid choose grid capacity as a Stage 1 output, a Stage 2 input and also a Stage 2 output. An obvious answer might be that the former are grids facing fixed price incentive regulations, while the latter are those facing cost plus regulations, with associated Averch-Johnson effects. The Averch Johnson effect is the tendency of regulated companies to engage in excessive amounts of capital accumulation in order to expand the volume of their profits. In this application, the value of the delta multiplier would appear to indicate the behavioural response of the regulated grid. In the European setting there is an intense policy discussion concerning the different behavioural responses of investor-owned grids versus state-owned grids (which are often encouraged to build super-reliable grids irrespective of the amount of electricity going through). This is also germane to the issue of whether regulatory benchmarking should regard grid capacity as a desirable output in itself again EU regulators seem seriously spit on the issue. The ideas presented here clearly have application in other areas as well, such as financial services. A number of studies utilizing two-stage models have been done in the banking industry. If one considers deposits and total customers served as intermediate outputs from Stage 1, and which influence profit generation in the second stage, it can be argued that total customers served is a factor that might play a dual role like that played by Internships in the above application. Another area of application is supply chain efficiency measurement. While conventional multistage processes have been used to model supply chains, the ideas presented herein have not been explored in that setting. One can envision situations where 1 Suggested by a reviewer.

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