Network DEA: A Modified Non-radial Approach

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1 Network DEA: A Modified Non-radial Aroach Victor John M. Cantor Deartment of Industrial and Systems Engineering National University of Singaore (NUS), Singaore, Singaore Tel: (+65) , victorjohn.cantor@u.nus.edu Kim Leng Poh Deartment of Industrial and Systems Engineering National University of Singaore (NUS), Singaore, Singaore Tel: (+65) , ohkimleng@nus.edu.sg Abstract. DEA is a non-arametric and linear rogramming based technique that attemts to maximize a decision making unit s (DMUs) relative efficiency, exressed as a ratio of oututs to inuts, by comaring a articular unit s efficiency with the erformance of a grou of similar DMUs that are delivering the same service. The traditional DEA models treat DMUs as black boxes whose internal structure is ignored. Recently, network DEA models have been introduced that treats the internal structure of a DMU as a network system. The increased interest in network DEA also roduced different tye of model formulations including the slacks-based measure network DEA (SBM-NDEA). SBM is a non-radial aroach suitable for measuring efficiencies when inuts and oututs may change non-roortionally, which is a shar contrast as comared to traditional DEA models that measures inut and outut changes roortionally. However, just like other DEA models, SBM-NDEA has its assumtions that limit its alicability and discriminative ower in efficiency measurement. The roosed model differs from existing SBM-NDEA aroaches in that it considers the exogenous inuts and oututs at the system level instead of at the rocess level and takes into account the resence of intermediate roducts in the model s objective function. Keywords: Data enveloment analysis; efficiency; non-radial; slacks-based measure 1. INTRODUCTION DEA is an otimization methodology that measures the relative efficiency of a set of DMUs with multile inuts and oututs to identify best ractices (or efficient) frontier (Charnes et al., 1978). This indicates that DEA rovides not only efficiency scores for DMUs, but also identify frontier rojections for inefficient units onto an efficient frontier. Originally, the DEA methodology considers a DMU as a black box, that is, as a single rocess that consumes inuts and roduces oututs. In other words, the internal structure of a DMU is ignored in the efficiency assessment (see, for examle, Beasley, 1990; Luo et al., 2012; Premachandra et al., 2009; Sarkis, 2000; Sherman and Gold, 1985). A major drawback of treating DMUs as black boxes is that it rohibits the identification of inefficiencies borne out of the sub rocesses that make u the internal structure of the DMU, and thus limiting the amount of information that can be gained to imrove the overall system efficiency (Chen et al., 2016; Färe and Grosskof, 2000). Disregarding the resence of sub rocesses may obtain misleading results, for instance, an overall system may be regarded as efficient even though all of its sub rocesses are not (Kao and Hwang, 2008). More significantly, there are cases in which all sub rocesses of a DMU have erformance that are worse than those of another DMU, and yet the former still has the better system erformance (Kao and Hwang, 2010). These findings suggest that the internal structure of DMUs, treating it as a network system, is required to roduce accurate results when measuring efficiencies. Most roduction systems have network structure in its oerations, where the end-to-end rocess of a DMU is divided into sub rocesses, which are then linked together by intermediate roducts (Fukuyama and Mirdehghan, 2012; Lewis and Sexton, 2004). Considering this, Färe and Grosskof (2000) concetualized the network DEA, which gained interest among researchers roducing numerous research studies thereafter. Some created models under

2 secified conditions to measure efficiencies (Fukuyama and Weber, 2010; Kao, 2014a; Tone and Tsutsui, 2009, 2010), some examine roerties ossessed by certain models (Avkiran and McCrystal, 2012; Chen et al., 2009; Chen et al., 2010), and others aly existing models to solve real world roblems (Avkiran, 2009, 2015; Chen and Yan, 2011; Lozano et al., 2013; Matthews, 2013; Yu, 2010). Kao (2014b) conducted a comrehensive review about network DEA and found out that most of the existing studies concerning network DEA are based from radial DEA models. A radial model, in which the standard DEA models fall into, measures the relative efficiency of DMUs under the assumtion of how much it can roortionally increase (decrease) all of its oututs (inuts) given its inuts (oututs) under technological constraint (Charnes et al., 1978; Farrell, 1957). However, radial models and so the standard DEA models were recognized to have two rimary limitations. First, there is a ossibility that a DMU may be measured against and referenced into a weakly efficient oint in the roduction ossibility set. These weakly efficient oints have ositive slacks with resect to strongly Pareto-efficient oints on the efficient frontier. Second, a significant roortion of DMUs can be regarded as efficient (Dyson et al., 2001). If a total ordering among the efficient DMUs is desired, one may imose an exogenously determined reference structure for inuts or oututs. For examle, restrictions on the dual multiliers or referential weights (see the review of Angulo-Meza and Lins, 2002). Otherwise, one may also use the suer-efficiency model (Anderson and Petersen, 1993), which does not require a rior weight assignment in the DEA models with weight restrictions. Tone (2001) develoed the SBM model to address the issue of referencing non-pareto-efficient targets by eliminating slacks-related biases in the efficiency measurement. Tone s SBM model comutes efficiency scores as a function of inut slacks and outut shortfalls as comared to the radial efficiency measures of standard DEA models, in which efficiency is determined based on either equi-roortional inut-contraction or oututexansion. One noticeable advantage of SBM is that it is guaranteed to identify Pareto-efficient reference oint for the evaluated DMU (Tone, 2001; Tone and Tsutsui, 2009). Therefore SBM resolves the slacks issue found to exist in the radial DEA models. The significance of analyzing the internal structure of DMUs in different network structures and the attractiveness of the SBM model both contributed to a new research front in DEA, which Tone and Tsutsui (2009) coined as SBM- NDEA aroach. The SBM-NDEA aroach refers to a non-radial DEA model using slacks-based measure where the resence of intermediate roducts in a network system is considered formally. Several studies have already used the aroach for real world alications such as in banking and finance (Lozano, 2016; Lu et al., 2014), transortation (Yu, 2010; Zhao et al., 2011), telecommunications (Moreno et al., 2013), and government (Amatatsu et al., 2012). However, the efficiency assessment of all of these studies is based on efficiency scores comuted as a ratio of the average inut reduction and outut increases of the different rocesses in a DMU. Such formulation indicates that the only way to increase system efficiency is through making each rocess efficient as ossible without increasing (decreasing) the inuts (oututs) of each of the rocesses. Alternatively, one can consider a system ersective instead of a rocess ersective in taking into account DMU s exogenous inuts and oututs. Inut slacks and outut shortfalls measured at the system level rovides the oortunity to the different rocesses to increase some inuts or decrease some oututs if it is deemed beneficial to the entire system. Such oortunity does not exist when the slacks and shortfalls are measured at the rocess level. Lozano (2015) emhasized that what is more imortant is to maintain the total inut consumtion and outut roduction, irresective of their allocation to the different rocesses. Therefore, taking the system ersective may uncover more sources inefficiencies and rovide more ambitious targets through effective allocation of resources within the DMU than a rocess ersective. 2. EXISTING SBM-NDEA APPROACH The following SBM-NDEA aroach is from Tone and Tsutsui (2009). Suose there exist n structurally homogenous DMUs (j = 1 n), i.e. all of them consists the same number of rocesses P ( =1 P) and for all the DMUs the inuts and oututs of each rocess are the same. Let m and k be the set of exogenous inuts consumed and of oututs roduced by rocess, resectively. The nonoriented weighted network SBM model under free link activities is as follows: ε 0 = min ω ω (1 1 m (1 + 1 k s i i m ) x i0 + s r i k ) y r0 s. t. λ j x ij j = x i0 s i, i = 1,, m λ j j y rj = y r0 + s + r, r = 1,, k

3 λ (,t) j j z dj = λ t (,t) j j z dj, d = 1,, D λ j 0 j, s i 0 i, s r + 0 r (1) where ε 0 is the overall efficiency of DMU 0; x ij is the observed amount of inut i consumed by rocess of DMU j; y rj is the observed amount of outut r roduced by rocess of DMU j; z (,t) dj is the observed amount of intermediate roduct d generated from rocess to rocess t. D is the total number of linking intermediate roducts. In addition, s i and s + r are inut and outut slacks of rocess, resectively while λ j is the intensity variable of rocess of DMU j. It is assumed here that an intermediate roduct d cannot be consumed and roduced simultaneously by a rocess. That is, if P in (d) is the set of rocesses that generate the intermediate roduct d while P out (d) is the set of rocesses that consumed the intermediate roduct d, then P in (d) P out (d) = for all d. Also, without a loss of generality, we assume that the intermediate roducts are comletely generated and consumed within the own DMU. The objective function denotes the fraction of the weighted average inut reduction of the different rocesses to the weighted average outut exansion of the different rocesses. The numerator is always less than or equal to one while the denominator is always greater than or equal to one. The first constraint measures the otential inut reduction for each rocess. The non-negativity of the slack variables s i means that no increase is considered in any of the rocesses. The second constraint comutes the otential outut exansion for each rocess. The nonnegativity of the shortfall variables s + r forbids that any rocess reduces any outut. The third constraint corresonds to the global balance equations for the intermediate roducts, i.e. the amount consumed by the different rocess must be equal the amount roduced. This reresents to the free links case introduced by Tone and Tsutsui (2009) where linking activities are freely determined (discretionary) while keeing continuity between inut and outut. The fixed links case where linking activities are ket unchanged (non-discretionary) would substitute the third constraint in model (1) by λ (,t) (,t) j j z dj = z dj, d = 1,, D λ t (,t) (,t) j j z dj = z dj, d = 1,, D (2) Note that model (1) corresonds to the CRS case. Adding j λ j = 1 in the constraints will make the model assume VRS. As mentioned, model (1) follows nonoriented case but it can be easily transformed into inutoriented and outut-oriented cases by the changing the objective function to ε 0 = min ω and (ξ 0 ) 1 = min ω resectively. s i (1 1 m i m ) (3) x i0 s r + (1 + 1 k i k ), (4) 3. PROPOSED NON-RADIAL NETOWRK DEA APPROACH The full list of the notations used in the roosed nonradial network DEA aroach can be found in the Aendix. Our aroach is insired in network SBM aroach of Lozano (2015) and in the relational network DEA aroach of Kao and Hwang (2008) and Kao (2009). The roosed non-radial DEA aroach can be formulated with an objective function as ξ 0 = min 1 1 I +M [ω s i i ( i I + t f i M x )] z i0 f O +N [μ s + k k ( i O + t + g y r0 z g0 subject to y r0 i N )] (5) λ j x j ij = x i0 s i i I (6) λ j y rj = y r0 + s+ j r r O (7) λ out (g) j j z fj λ in (f) j j z gj f, g (8) λ j 0 j (9) s i, s k +, t f, t g + 0 i, k, f, g (10) The objective function (5) reresents the fraction of the average total inut reduction to the average total outut exansion. The numerator is always less than or equal to one while the denominator is greater than or equal to one. Constraints (6) calculate the otential reduction that can be achieved for each inut. Likewise, constraints (7) comute the otential exansion in the total amount roduced of each outut. Constraints (8) are the relaxed version of the corresonding free links SBM-NDEA constraints in model (1). The equality sign in model (1) corresonds to letting the shadow rices of the intermediate roducts free whereas the inequality character in constraints (8) follows the

4 assumtion of non-negative shadow rices. Relaxing the constraint s has the advantag Labor (Labor1) Generation Process Labor (Labor2) Electronic Power Generated (EPGenerated) Transmission Process e of enlarging the feasibility Electronic Power Sold (EPSold) Labor (Labor3) Electronic Power Sent (EPSent) Distribution Process Electronic Power Sold (EPSold3) Figure 1: Tone and Tsutsui (2009) electric utility comanies. region of the otimization model, thus increasing the discriminating ower of the aroach. Further, in contrast to the SBM-NDEA model (1), the slacks in model (5-10) are comuted at the system level, that is, with resect to the total consumtion of inuts and oututs. Such view indicates that what is more vital is the total consumtion and outut roduction, irresective of their allocation to the different rocesses. 2.1 Identifying Characteristics of the Non-radial DEA Aroach The roosed aroach differs from existing SBM- NDEA aroaches articularly with that of Tone and Tsutsui (2009) in terms of the following reasons. First, Tone and Tsutsui s SBM-NDEA aroach imose nonnegative slacks in each rocess while the roosed nonradial network aroach comute the slacks in the whole system and that global slack is the one that should be nonnegative. If an inut or an outut is consumed or roduced by just one rocess then there is no difference. However, if an inut/outut is consumed/roduced in more than one rocess then the roosed non-radial network aroach take a system view that considers that what matters is the total inut/outut consumtion/roduction. With the relaxation of the intermediate roducts balance constraints, the roosed aroach involved a larger feasibility region and thus has more discriminating ower than Tone and Tsutsui s SBM-NDEA aroach. Second, Tone and Tsutsui s objective function is a ratio in which the numerator is the weighted average inut reduction of the different rocesses and the denominator is analogously a weighted average of the outut increases of the different rocesses. Whereas, the roosed aroach s objective function is a ratio in which the numerator is the (average) global inut reduction and the denominator the (average) global outut reduction. It must be noted again that the adotion of this objective function is in arallel with the SBM logic only that at the global (i.e. system) level. Lastly, Tone and Tsutsui s objective function does not consider slacks of intermediate roducts in the determination of the overall efficiency and rocess efficiency because the model emloy the equality constraints λ (,t) j j z dj = λ t (,t) j j z dj, d for intermediate roducts in model (1). However, as Fukuyama and Mirdehghan (2012) asserted ignoring slacks in intermediate roducts is inaroriate when one is after the efficiency status of DMUs and their rocesses. Therefore, the roosed aroach included slacks variables t f, t+ g associated with the intermediate roducts in the objective function. 4. NUMERICAL ILLUSTRATION The dataset from Tone and Tsutsui (2009) will be used to illustrate the alication of the roosed non-radial network DEA aroach. It consists of 10 DMUs corresonding to electric utility comanies (see Figure 1). Each electric utility comany consists of three rocesses connected in series: generation, transmission, and distribution. Each rocess has an exogenous inut referred to as Labor that corresonds to the number of emloyees. There are two oututs that corresond to the Electric Power Sold to large customers (exogenous outut of Transmission rocess) and to small customers (exogenous outut of Distribution rocess). Lastly, there are intermediate roducts: Electric Power Generated (which is roduced by Generation rocess and consumed by Transmission rocess) and Electric Power Sent (which is roduced by Transmission rocess and consumed by Distribution rocess). Table 1 resents the data for the inuts, oututs, and intermediate roducts of the 10 DMUs.

5 The roosed non-radial network DEA model was built and run for analysis using the otimization modeling software called LINGO It must be noted that the DEA models built for this secific analysis alone follows the inut-oriented case where the linear rogramming model is configured so as to determine how much the inut use of a DMU could reduce if used efficiently in order to achieve the same outut level. More so, a variable returns-to-scale Table 1: Tone and Tsutsui s (2009) samle data set for electric utility comanies. DMU Labor1 Labor2 Labor3 EPSold2 EPSold3 EPGenerated EPSent A B C D E F G H I J Average was assumed. These assumtions were made in to order to be able to comare the results of the roosed model with the results of both the black box and SBM-NDEA models ublished in Tone and Tsutsui s (2009) work. Three DEA models (i.e., black box model, SBM- NDEA model, and roosed model) were comared and the resulting efficiency scores are shown in Table 2. It can be seen that the efficiency scores of the black box model tend to be higher than those of the SBM-NDEA and roosed models. Eight out of the 10 DMUs were considered efficient when using the black box model whereas no DMUs were considered as overall efficient when using the SBM-NDEA and roosed models. Such results are not surrising as black box models do not consider the internal structure of DMUs unlike the network DEA models which rovides the latter that advantage of determining more sources of inefficiencies in the sub rocesses. Thus, it is aarent that when one uses a network DEA model it would hardly rovide an efficient DMU, which the roosed model adhered to. This observation also indicates that the black box model is inferior in terms of the discriminate ower to that of the SBM-NDEA and the roosed models. Figure 2 clearly shows that the discriminate ower of the black box model is inferior to that of the SBM-NDEA and roosed models. In addition, the figure shows that the ranks of the scores of the three models are not always corresonding. For examle, DMU B is scored worse in the black box model, while better in both the SBM-NDEA and roosed models. However, when comaring SBM-NDEA and roosed models it can be observed that ranks of scores of the former is always higher than the latter, indicating higher discriminate ower of the latter. These observations in the rankings between the SBM-NDEA and roosed models can be attributed to the roosed model s inclusion Table 2: Efficiency scores for the three DEA models DMU Black Box SBM-NDEA Proosed A B C D E F G A B C D E F G H I J Black Box SBM-NDEA Proosed H I J Figure 2: Comarisons of scores among the three DEA models.

6 of intermediate slacks in the objective function and comuting the slacks in the system level. The consistencies in the rankings between the SBM-NDEA and the roosed Table 3: Process efficiencies of SBM-NDEA and roosed models. model may suggest that both models work in the same way excet that the latter has more discriminate ower than the former. More so, from these results one can safely conclude that the black-box model is inferior in roviding accurate DMU SBM-NDEA Model Proosed Model Process 1 Process 2 Process 3 Process 1 Process 2 Process 3 A B C D E F G H I J efficiency measurement as oosed to the SBM-NDEA and roosed models. Table 3 resents the rocess efficiencies derived from both the SBM-NDEA and the roosed models. It must be noted that the overall efficiency scores reorted in Table 2 could be derived using the rocess efficiencies indicated in Table 3 with the corresonding inut weights for each rocess, ω 1 = ω 3 = 0.4 and ω 2 = 0.2. Given that the intermediate roducts constraints of the roosed model are the relaxed version of the SBM-NDEA, then it is logical that former s rocess efficiencies are lower than or equal to those reorted using the latter model. The differences are, however, small with the largest difference found in rocess 1 of DMU G amounting to (difference between and 0.712). More so, it can be observed from the resulting rocess efficiencies why none of the DMUs is overall efficient. That is because the overall efficiency is a weighted average of the three rocesses, which based from Table 3 no DMU had all three rocesses fully efficient. The identifying difference between the results of SBM-NDEA and the roosed model is that the latter rovides lower rocess efficiencies as comared to the former. Such results are attributable to the combination of relaxing the intermediate roducts constraints, considering intermediate roduct slacks in the objective function, and comuting these slacks at the system level in the roosed model. 5. CONCLUSION A modified non-radial aroach is roosed for network DEA. Particularly, the roosed aroach differs from Tone and Tsutsui s (2009) existing SBM-NDEA aroach for the following reasons. The inut slacks and outut shortfalls of the roosed aroach are measured at the system level thus giving freedom to the different rocesses to increase some inuts or decrease some oututs if that is deemed beneficial to the entire system. That is something not allowed when the inut slacks and outut shortfalls are measured at the rocess level just as in the case of Tone and Tsutsui s SBM-NDEA model. It has been observed from the existing SBM-NDEA models that all adhered to a rocess ersective in comuting for the efficiency of DMUs, that is, the inut slacks and outut shortfalls are comuted in each rocess. The adotion of rocess ersective restricts the different rocesses to increase/decrease some inuts/oututs and thus limits the overall imrovement that a DMU can achieve. With that in mind, it may be beneficial if a system ersective will be adoted in the efficiency measurement. That is, it gives freedom to the different rocesses to increase/decrease some inuts/oututs if that is advantageous to the whole system. Also, by taking the system ersective it may uncover more sources inefficiencies and rovide more ambitious targets through effective allocation of resources within the DMU than a rocess ersective. Furthermore, aligned with the adotion of the system ersective in efficiency measurement, the objective function of the roosed aroach considers the ratio of the average global inut reduction to the average global outut reduction. This is in contrast with Tone and Tsutsui s (2009) SBM-NDEA aroach where the objective function is a ratio of the weighted average inut reduction of the different rocesses to the weighted average of the outut increases of the different rocesses. Lastly, the roosed

7 aroach considers the slacks of intermediate roducts in determining the overall and rocess efficiency scores that are not consider reviously. The resence of intermediate roducts is an essential art in considering the internal structure of DMUs as it rovides evidence on how rocesses are interconnected to each other, which then justifies the need for it to be accounted directly in the efficiency measurement. Future research can further relax the assumtion that all sub rocesses utilize exactly the same set of inuts to generate the same set of oututs. In many real-world settings, the homogeneity of inut and outut factors may not hold. Consider, for instance, the case of different roduction lines in a manufacturing factory, or differently functioning business units oerating within an organization. One may ot to extend revious studies into a more general scenario considering intermediate measures and no matter what tye of inut and outut measures are consumed and roduced, resectively. The general scenarios that can considered with resect to inut and outut measures are: (a) all sub rocesses within any DMU have disjoint outut sets, (b) some outut measures are shared among different sub rocesses, (c) all inuts can be slit u in terms of roortions across sub rocesses, and (d) some inuts cannot be readily slit u and distributed to various sub rocesses. Aendix A. Notations The following are the notations used in the roosed non-radial network DEA aroach: n P I x ij O y rj M z fj N z gj out (g) in (f) s i + s i t f Number of DMUs Number of rocesses Set of exogenous inuts of rocess Observed amount of exogenous inut i consumed by rocess of DMU j Set of exogenous oututs of rocess Observed amount of exogenous outut r roduced by rocess of DMU j Set of endogenous inuts of rocess Observed amount of endogenous inut f roduced by other rocesses Set of endogenous oututs of rocess Observed amount of endogenous outut g utilized by other rocesses Set of rocesses that generate the endogenous outut g Set of rocesses that consume the endogenous inut f Slack of exogenous inut i Shortfall of exogenous outut r Slack of endogenous inut f t+ g Shortfall of endogenous outut g λ j Intensity variable used for rocess of DMU j when comuting linear combinations of the observed DMUs 0 DMU being evaluated/rojected REFERENCES Amatatsu, H., Ueda, T., and Amatatsu, Y. (2012). Efficiency and returns-to-scale of local governments. Journal of Oerational Research Society, 63, Andersen, P. and Petersen, N.C. (1993). A rocedure for ranking efficient units in data enveloment analysis. Management Science, 39, Angulo-Meza, L., and Lins, M.P.E. (2002). Review of methods for increasing discrimination in data enveloment analysis. Annals of Oerations Research, 116, Avkiran, N.K. (2009). Oening the black box of efficiency analysis: An illustration with UAE banks. Omega, 37, Avkiran, N.K. and McCrystal, A. (2012). Sensitivity analysis of network DEA: NSBM versus NRAM. Alied Mathematics and Comutation, 218, Beasley, J.E. (1990). Comaring university deartments. Omega, 18, Charnes, A., Cooer, W.W., and Rhodes, E. (1978). Measuring the efficiency of decision making units. Euroean Journal of Oerational Research, 2, Chen, Y., Cook, W.D., Li, N., and Zhu, J. (2009). Additive efficiency decomosition in two-stage DEA. Euroean Journal of Oerational Research, 196, Chen, Y., Cook, W.D., and Zhu, J. (2010). Deriving the DEA frontier for two-stage rocesses. Euroean Journal of Oerational Research, 202, Chen, Y., Li, Y., Liang, L., Salo, A., and Wu, H. (2016). Frontier rojection and efficiency decomosition in two-stage rocesses with slacks-based measures. Euroean Journal of Oerational Research, 250, Chen, C. and Yan, H. (2011). Network DEA model for suly chain erformance evaluation. Euroean Journal of Oerational Research, 213, Cook, W.D., Zhu, J., Bi, G.B., and Yang, F. (2010). Network DEA: Additive efficiency decomosition. Euroean Journal of Oerational Research, 207, Dyson, R.G., Allen, R., Camanho, A.S., Podinovski, V.V., Sarrico, C.S., and Shale, E.A. (2001). Pitfalls and rotocols in DEA. Euroean Journal of Oerational Research, 132,

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