Ranking Two-Stage Production Units in Data Envelopment Analysis

Transcription

1 Asia-Pacific Journal of Operational Research Vol. 33, No. 1 (2016) (19 pages) c World Scientific Publishing Co. & Operational Research Society of Singapore DOI: /S Raning Two-Stage Production Units in Data Envelopment Analysis Roza Azizi Department of Mathematics Karaj Branch, Islamic Azad University Karaj, Iran roza.azizi@iau.ac.ir Reza Kazemi Matin Department of Mathematics Karaj Branch, Islamic Azad University Karaj, Iran rmatin@iau.ac.ir Received 11 July 2014 Revised 8 April 2015 Accepted 11 August 2015 Published 11 February 2016 In this paper, we propose a new approach to ran two-stage production systems in data envelopment analysis based on performance of the stages. To do this, a method is devised to compare the stages performance of a special two-stage unit with the corresponding stages performance of other units. The relative performance of two-stage production units is investigated under new definitions of wea and strong relations and a new raning criterion is introduced as the result. The most important features of our method is the ability to achieve raning intervals for two-stage production units based on the introduced relations over all feasible weights, as well as the ability to generate accurate information about sources of inefficiency of two-stage production units. Keywords: Data envelopment analysis; two-stage production systems; wea and strong relation; raning interval; inefficiency sources. 1. Introduction Data envelopment analysis (DEA) is a nonparametric approach for measuring the efficiency of decision-maing units (DMUs), which convert multiple inputs into multiple outputs. Although many DMUs use inputs to produce outputs at some stages such as bans, hospitals, etc., classical DEA models handle a DMU as a whole system and do not consider the internal operations of the components. Thus, decision maers (DMs) have little information about the factors that cause the system Corresponding author

2 R. Azizi & R. K. Matin to be inefficient, and conventional models such as CCR, which was introduced by Charnes et al. (1978), may determine a two-stage production unit is efficient when it is not. In recent years, many studies have focused on two-stage production units. Färe and Grossopf (1996) introduced two-stage production units for analysis. Wang et al. (1997) applied two independent models to each stage of a two-stage production unit. Seiford and Zhu (1999) examined two-stage production units to measure the efficiency scores of U.S. commercial bans. Zhu (2000) measured the efficiency of the 500 best two-stage companies. Lewis and Sexton (2004) analyzed the performance of Major League Baseball as a two-stage production unit. Chen and Zhu (2004) claimed that Wang et al. (1997) did not consider the relevance of the two stages, so they solved the problem with an assumption in which intermediate products are unnown variables. Kao and Hwang (2008) developed a relational model to measure the efficiency of a two-stage production unit in which the product of twostage efficiencies is equal to the total system efficiency. Coo et al. (2010) reviewed DEA models for two-stage production units. Raning is another issue in DEA that has been investigated. The motivation for studying this topic was discrimination of efficient units. Most raning papers are about systems without sub-processes. Andersen and Petersen (1993) introduced the AP model, named the super efficiency technique, to ran efficient units. Sexton et al. (1986) and Doyle and Green (1994) used a cross-efficiency matrix to ran units. Adler et al. (2002) reviewed some developed raning models in DEA framewor. Salo and Puna (2011) introduced a raning interval for a DMU based on its efficiency measure compared with other DMUs that considers all feasible input/output weights in raning not only the self-appraised optimal DEA weights. Several techniques are also developed for raning networ production systems. For example, Liu et al. (2009) and Liu and Lu (2010) derived information from input/output combinations to discriminate systems with a networ structure. Liu and Lu (2012) also used Chen and Zhu s (2004) method and proposed a model for raning two-stage production units. But, to the best of our nowledge, there is no direct study for the complete raning of two-stage production systems in DEA framewor. To deal with this issue, we apply Kao and Hwang s (2008) proposed model for evaluating the efficiency scores of two-stage production units. This model uses the relation of the manufacturing stages (two stages that are part of the entire two-stage production unit) and presents efficiency decomposition, which is useful for our idea to determine the best and worst ran of two-stage production units. As the result, the raning interval bounded by the best and worst rans shows all rans that can be achieved by any two-stage unit for any given set of feasible weights. The new models could be used to identify the units that either outperform or perform worse than an under evaluated unit. In addition, we can use the achieved raning interval to determine the stability of a unit s performance. Since the models

3 Raning Two-Stage Production Units in Data Envelopment Analysis are designed to deal with all feasible weights, the smallest gap between the best and worst rans of a unit shows that the unit has the most stable performance of all units compared. In addition, the models help find which stage of a DMU has more effect as an inefficient source of the DMU compared with others to determine procedures for improvement. To achieve our goal, we introduce two new concepts, wea and strong relations, which are used to introduce a new raning method based on the efficiency scores of the stages. The proposed raning method generates detailed information about sources of inefficiency in two-stage production systems in addition to determine the stages that mae a unit less efficient than other units, which help decision maers to decide which stages must be improved in order to improve the system s overall efficiency. Similar to Salo and Puna (2011), the new raning method considers all feasible weights in raning two-stage units instead of the self-appraised DEA optimal weights. The rest of this paper is organized as follows. In Sec. 2, we provide a brief review of Kao and Hwang s (2008) model, introduce new definitions of wea and strong relations, and present our new models for raning two-stage production units. In Sec. 3, examples are included. Section 4 is devoted to the extension of raning models in the presence of input prices. Conclusions are given in Sec Two-Stage Production Units and Raning Interval 2.1. Efficiency measure of two-stage production units In DEA, each observed DMU l (l =1,...,n) is specified by some non-negative inputs and outputs. Throughout this paper, the input vectors and output vectors are denoted by x l =(x 1l,...,x ml )andy l =(y 1l,...,y sl ), respectively. With these notations, Charnes et al. (1978) presented the CCR model under the CRS assumption to measure the efficiency score of DMU which could be stated as model (1): E =Maxuy /vx s.t uy l /vx l 1, l =1,...,n v, u 0. Now, suppose all production units are composed of two stages as depicted in Fig. 1. These two-stage production units consume inputs to produce intermediate products z l =(z 1l,...,z dl in the first stage, which are then used to produce final outputs in the second stage. The conventional DEA models do not tae intermediate products into account for estimating the efficiency score of two-stage production units. Similar to model (1), models (2) and (3) are used to compute the CCR efficiency measure of the first and second stages of DMU, respectively. As a result, we may determine the CCR efficiency scores of all the units and associated stages (1)

4 R. Azizi & R. K. Matin Fig. 1. Two-stage production units. independently. E 1 =Maxwz /vx s.t wz l /vx l 1, l =1,...,n w, u 0. E 2 =Maxuy /wz s.t uy l /wz l 1, l =1,...,n (3) w, u 0. To calculate the overall efficiency of a two-stage production unit, Kao and Hwang (2008) embedded the operations of the two stages, determined by the constraints of models (2) and (3) into model (1). Thus, they were able to tae into consideration the series relation of the two stages within the whole production unit, and proposed their model as follows: E =Maxuy /vx s.t uy l /vx l 1, l =1,...,n wz l /vx l 1, l =1,...,n uy l /wz l 1, l =1,...,n w, v 0. In this model, u, v and w are non-negative vectors of input weights, output weights and intermediate product weights, respectively. A special feature of this model is the use of same weights (w) for the associated multipliers of intermediate products, with both considering z as outputs or inputs in the two-stage systems. With this assumption, the decomposition relationship, E = E 1 E2, is established. That is, the efficiency of the two-stage production unit is the product of the efficiencies of the two stages. Note that aggregating the last two constraints of the model (4) leads to the first constraint, so the first set of constraints is redundant and can be omitted from the model. Since the decomposition E = E 1 E2 is not unique, Kao and Hwang (2008) proposed model (5) for measuring the largest efficiency value of the first stage while retaining the same score on the whole (2) (4)

5 Raning Two-Stage Production Units in Data Envelopment Analysis production unit. Then, they suggested using E 2 = E E 1 score of the second stage. to calculate the efficiency E 1 =Maxwz /vx s.t uy /vx = E wz l /vx l 1, uy l /wz l 1, u, w, v 0. l =1,...,n l =1,...,n (5) In the next part, the proposed model of Kao and Hwang (2008) is used to present a raning method based on the best and the worst efficiency scores of two-stage production units Raning interval in two-stage production units The first stage, the second stage, and the whole production unit can be raned for all feasible input, intermediate product, and output weights. However, the achieved raning position is dependent upon the given set of feasible weights. To overcome this issue, we suggest determining the best and worst ran of any observed two-stage production units and considering all feasible weights in the evaluation. The weights can be assumed to be restricted by applying cone ratio (Charnes et al., 1989) or assurance region (Thompson et al., 1986) methods to impose the preference information of decision maers on inputs, outputs and intermediate products. In the remainder of this paper, we assume the weights belong to (U, V, W ), which is presented by the cone ratio method, but as previously mentioned, the assurance region method can be applied as well. To introduce our new raning method, it is necessary to analyze the computed efficiency scoresfor two-stageproduction units accurately by considering all possible relations between the stage performances. Based on the computed efficiency scores, if DMU l has a better performance than DMU,wehaveE l (u, w, v) >E (u, w, v). Using the presented decomposition of Kao and Hwang (2008) for efficiency scores, the above expression is trivially true just in the following cases: (1) El 1 >E 1 & E2 l >E 2. (2) El 1 >E 1 & E2 l E 2 while E l >E. (3) El 1 E 1 & E2 l >E 2 while E l >E. In the following definitions, we consider different possible situations between a pair of two-stage units to provide accurate information about inefficiency sources

6 R. Azizi & R. K. Matin Definition 1. DMU l is said to have a wea relation to DMU if there exists a set of feasible weights such that E 1 l >E 1 & E2 l >E 2 or E 1 l >E 1 & E 2 l E 2 while E l >E or E 1 l E 1 & E2 l >E 2 while E l >E. Definition 2. DMU l is said to have a strong relation to DMU if there exists a set of feasible weights such that E 1 l >E 1 & E2 l >E 2. Next, in order to determine the best raning of a special DMU (th observed twostage production unit), we loo for the number of DMUs that have better performance than DMU. Similarly, the worst raning of DMU based on the performance of its stages is suggested as the number of DMUs that do not have worse performance than DMU. To this end, consider the following sets: El 1 >E 1 & E2 l >E 2 or WR < = DMU l,l {1,...,n} El 1 >E 1 & E2 l E 2 while E l >E, or El 1 E 1 & E2 l >E 2 while E l >E E l 1 E 1 & E2 l E 2 or WR = DMU l,l {1,...,n}\{} El 1 E 1 & E2 l E 2 while E l E, or El 1 E 1 & E2 l E 2 while E l E SR < = {DMU l,l {1,...,n} E 1 l >E 1 & E 2 l >E 2 }, SR = {DMU l,l {1,...,n}\{} E 1 l E 1 & E 2 l E 2 }. WR < is the set of DMUs which have at least one stage with a strictly higher efficiency score than the efficiency score of the corresponding stage of DMU, while the overall efficiency score of the DMUs is strictly higher than that of DMU.It is clear that only in the cases of WR < the inequality E l >E holds true. Thus, we can define WR < as a set of DMUs showing units with a strictly higher overall efficiency score than DMU.WR shows units which have at least one stage with an efficiency score that is not worse than the efficiency score of the corresponding stage of DMU, while the overall efficiency score of the units is not worse than the

7 Raning Two-Stage Production Units in Data Envelopment Analysis corresponding score of DMU. In other words, similar to WR <, WR indicates the units with an overall efficiency score that is not worse than that of DMU. Similarly, SR < (SR ) determines the units with efficiency scores of both stages are strictly higher (not lower) than corresponding stages of DMU. By means of the above introduced sets, we define the following rans for DMU : wr < =1+ WR<, wr =1+ WR, sr< =1+ SR<, sr =1+ SR, in which WR <, WR, SR<,and SR are used to show the cardinality of their corresponding sets. For example, if there is a set of feasible weights, such as (U, W, V )forwhich the efficiency scores of both stages of DMU become strictly higher than efficiency scores of similar stages of all other DMUs, then wr <,wr,sr<,andsr will be equal to one, because their corresponding sets are empty. Note that wr <,wr,sr<, and sr are different for DMU when using different sets of weights, because different sets of weights lead to different sets: WR <, WR,SR<,andSR. This means that DMU may have a different wea or strong ran for two different sets of weights. Thus, in order to evaluate the best and the worst wea ran for DMU, and to analyze all set of feasible weights, it is necessary to compute the minimum of wr < and the maximum of wr, respectively. Thus, the wea raning interval is determined as [min wr <, max wr ], considering all set of feasible weights. Similarly, to evaluate the best and worst strong ran for DMU, and to analyze all set of feasible weights, it is necessary to compute the minimum of sr < and the maximum of sr, respectively. Thus, the strong raning interval is determined as [min sr <, max sr ], considering all set of feasible weights. Wea raning of DMU can be one of the bounds or any natural number within the range [min wr <, max wr ]. Note that the best and worst wea rans of DMU are shown by the lower and upper bounds of the interval, respectively, which are independent of the choice of the feasible set of weights. The same explanation can be presented for the strong ran of DMU. According to the above descriptions and following the proposed technique in Salo and Puna (2011), we present models (6) and (7) to determine efficiency raning intervals for DMU as the under evaluation unit. In order to improve the overall performance of two-stage production systems, the new proposed models and the subsequent results enable us to determine which stage of DMU is involved in its inefficient performance. Consider the following models: min (g l + f l + h l ) l s.t wz l vx l Mg l, l uy l wz l Mf l, l

8 R. Azizi & R. K. Matin uy l vx l Mh l, l uy =1 vx =1 wz =1 g l,f l,h l {0, 1}, l (u, w, v) (U, W, V ) (6) max (g l + f l + h l ) l s.t wz l + vx l M(1 g l ), l uy l + wz l M(1 f l ), uy l + vx l M(1 h l ), uy =1 vx =1 wz =1 g l,f l,h l {0, 1}, l l l (u, w, v) (U, W, V ). (7) In these models, the weight sets are considered to be closed and bounded by the constraints uy = vx = wz = 1. The first, second and third constraints are related to the first and the second stages and to the whole production unit, respectively. M is a sufficiently large positive constant. The objective function of models (6) and (7) is to determine the optimal values of g l,f l,andh l. Next, to evaluate the best and the worst rans of DMU, we introduce d l and b l as follows: (i) If g l =1andf l =1(l, l =1,...,n)setd l =1. If g l =1,f l =0andh l =1(l, l =1,...,n)setd l =1. If g l =0,f l =1andh l =1(l, l =1,...,n)setd l =1. Otherwise, set d l =0. (ii) If g l =1andf l =1(l, l =1,...,n)setb l =1,ifnotsetb l =0. By means of the following theorems, we access the wea and strong raning interval in efficiency evaluation of observed two-stage production units. The proofs are presented in the appendix

9 Raning Two-Stage Production Units in Data Envelopment Analysis Theorem 1. Thebest/worstwearanofDMU optimal solution of (6)/(7). is 1+ l d l, based on the Theorem 2. The best/worst strong ran of DMU is 1+ l b l, based on the optimal solution of (6)/(7). Now, consider the optimal solutions of model (6). Theorem 1 counts the number of units with a strictly higher efficiency score than DMU, while the efficiency score of at least one of the stages is strictly higher than that of DMU.Theorem2 counts the number of units with a strictly higher efficiency score in both stages in comparison to the corresponding stages of DMU. Similarly, we can analyze optimal solutions of model (7). The next theorem shows that in the case of one input, one intermediate product, and one output, the worst wea/strong raning can be computed by the best wea/strong raning, so running one of the models (6) or (7) is enough to determine the raning intervals. Theorem 3. In the case of single input, single intermediate product, and a single output, allowing that the best wea and strong ran of DMU is e and t, respectively, and supposing there are m DMUs which have the same efficiency score as DMU, while the efficiency score of the similar stages are not equal, then we have: (i) The worst wea ran of DMU is equal to e+m plus the number of DMUs which have the same efficiency score as DMU in both of the corresponding stages. (ii) The worst strong ran of DMU is equal to t plus the number of DMUs, which have the same efficiency score as DMU in one of the corresponding stages, while the efficiency score of another stage is greater than or equal to that of DMU. In the next section, we illustrate the proposed approach with some numerical examples. 3. Illustrative Examples To compute the raning intervals for two-stage production units and illustrate the results, we will consider two numerical examples. Example 1. We apply our approach to a data set in the insurance industry used in Kao and Hwang (2008). In this data set, 24 non-life insurance companies in Taiwan are introduced as two-stage production units. Two inputs, two intermediate products, and two outputs are used in this evaluation. The inputs are operation expenses and insurance expenses, the intermediate products are direct written premiums and reinsurance premiums, and the outputs are underwriting profit and investment profit. In our evaluation, the weights are constrained to be non-negative, and no preference for inputs, outputs, and intermediate products are assumed

10 R. Azizi & R. K. Matin Table 1. Efficiency score, the best and worst wea and strong raning of 24 insurance DMUs. DMU Efficiency min max min max score wr < wr sr < sr The second column of Table 1 shows the efficiency score of the 24 units obtained by model (4) and is followed by the best wea, the worst wea, the best strong, and the worst strong raning of the units, respectively. In this evaluation, the best wea ran of production units DMU 2, DMU 5, DMU 12, and DMU 22 is 1, which means that there is no production unit which has better performance than these units, considering non-negative weights. The best strong ran of production units DMU 1, DMU 2, DMU 3, DMU 5, DMU 12, DMU 15, DMU 17, DMU 19, DMU 20,andDMU 22 is also 1, which shows there are no production units with higher efficiency score than these units in both stages for all feasible weights. Consider a specific unit lie DMU 4, the best wea ran of this unit is 8, which means that for a set of feasible weights, there are 7 production units with a higher efficiency score than DMU 4, while the efficiency score of at least one of the stages is higher than the corresponding stages of DMU 4. For another instance, the worst wea and strong ran of DMU 1 is 16 and 6, respectively, which shows that the maximum number of production units with at least the same efficiency score as production unit 1 is 15, and the maximum number

11 Raning Two-Stage Production Units in Data Envelopment Analysis of production units with efficiency scores of both stages are at least the same as the corresponding stages of production unit 1 is 5. The interval achieved for wea raning of DMU 1 is [2, 16], which means, considering all set of feasible weights, DMU 1 may ran as the 2nd DMU, the 16th DMU, or any place between 2 and 16 among all 24 DMUs based on wea relation. In other words, there is no set of feasible weights for which DMU 1 rans as the 1st or as the 17th to 24th based on wea relation. To analyze the result of our models more accurately, we are presenting the optimal solution of the model (6) for DMU 1. The optimal solution, easily obtained by running the model (6), is ({l; g l =1} =19, {l; f l =1} =3, 5, 12, {l; h l =1} =5). This means DMU 19 is the only DMU, which has better performance than DMU 1 in comparison of their respective first stages. DMU 3, DMU 5,andDMU 12 are the units which have better performance than DMU 1 in their second stage. DMU 5 is the only unit with a better efficiency score than DMU 1. With respect to the mentioned results, the best wea ran of DMU 1 is 2, because there is just one unit, DMU 5, with a higher efficiency score than DMU 1, while the efficiency score of at least one of its stages is strictly higher than the corresponding stages of DMU 1.Usingthese results, to upgrade the best wea ran of DMU 1 from 2 to 1, DM should improve performance of the second stage of this unit. The best strong ran of DMU 1 is 1, because {l; g l =1} and {l; f l =1} are disjointed and there is no unit with a strictly higher efficiency score than DMU 1 in either of its stages. The optimal set of weights of the model (6) for DMU 1 is achieved as (v 1,v 2,w 1, w 2,u 1,u 2 ) = ( E-06, E-06, E-06, E-06, E-07, E-05). Note that the constraints uy = vx = wz =1of model (6) lead to the following efficiency scores for the two-stage production unit 1 and the corresponding stages, respectively: E 1 = uy 1 /vx 1 =1, E 1 1 = wz 1/vx 1 =1, E 2 1 = uy 1/wz 1 =1. Using the optimal weights of the model (6) for DMU 1 and the mentioned formulas for the whole unit and the stages, the efficiency score of each unit can be determined. For example, the efficiency score of DMU 2 using the achieved weights is uy 2 /vx 2 = Thus, its wea raning is worse than the wea raning of DMU 1 because its efficiency score is less than that of DMU 1, or the wea raning of DMU 20 is worse than the wea raning of DMU 1 due to its efficiency score which is uy 20 /vx 20 = based on the optimal weights of the model (6) for DMU 1. Also, it is concluded that the use of this set of weights reveals that DMU 2 has better performance than DMU 20. As shown in Table 2, there is just one DMU which has better performance than DMU 1 for the computed optimal set of weights based on wea relation, which is DMU 5 (the efficiency score of DMU 5 with the computed optimal weights for DMU 1 in the model (6) is uy 5 /vx 5 = , while the efficiency score of DMU 1 is 1). In order to mae a comparison between DMU 1 andotherdmusbasedonstrong relation, considering the optimal weights of the model (6), we need to calculate the

12 R. Azizi & R. K. Matin Table 2. Data of five hypothetical DMUs, and their system and stages efficiency. DMU Input Intermediate Output Efficiency of Efficiency Efficiency of product whole unit of first stage second stage efficiency scores of the stages of other DMUs using the achieved optimal weights. If there exists a DMU which has a strictly higher efficiency score than DMU 1 in its both stages, its best strong ran will be better than DMU 1, which is not included in this example. The optimal weights of the model (6) are too small due to the constraints uy = vx = wz = 1 and large amount of inputs, intermediate products, and outputs. We note that large or small amounts of weights have no effect on the results of the models. To illustrate, consider the data set of Example 1 when all elements are divided by a million. The optimal weights of the model (6) for DMU 1 are achieved in the same way as the former weights. The only difference is that all of the weights are multiplied by a million. As the result, the sets g l,f l,andh l are the same as the ones achieved before. Hence, using the new set of data, similar rans are achieved for DMU 1 compared to the results of Table 1. Similarly, we can obtain the efficiency score of all DMUs using the set of the weights which determines the worst wea ran and the worst strong ran. Thus, the best wea and strong rans refer to the weights which are achieved by running model (6) and the worst wea and strong rans refer to the weights which are achieved by running model (7). In the next example, the results of Theorem 3 are analyzed. Example 2. Table 2 shows data for five hypothetical DMUs with one input, one intermediate product, and one output. The last three columns show the efficiency scores of two-stage production units, and the efficiency scores of the first and second stages, which are obtained by model (4), model (5) and E 2 = E, respectively. The E 1 efficiency scores are rounded to two decimal places. Table 3 reports the best wea, the worst wea, the best strong, and the worst strong ran of DMUs, respectively in the new raning method. The weights are constrained to be non-negative. Note that the computed best and worst rans are equal for DMU 2 and DMU 5. That is the same result which is presented in Theorem 3. Also, the best and the worst strong rans of DMU 1 are equal, but this is not the case for its wea raning, because there is one DMU, which has the same efficiency score as DMU 1.Wehave the similar results for DMU 4.DMU 3 and DMU 4 have the same efficiency score in their second stage, while DMU 4 has strictly higher efficiency score than DMU 3 in

13 Raning Two-Stage Production Units in Data Envelopment Analysis Table 3. The best and the worst wea and strong raning of 5 DMUs. DMU minwr < maxwr minsr < maxsr its first stage. So, as Theorem 3 states, the best and the worst wea rans of DMU 3 are the same, but the worst strong ran of DMU 3 is equal to its best ran plus one. 4. Price Existence In the previous sections, we presented a new complete raning method for the twostage production systems, based on computing the best and worst raning of the observed units. As it is described, we extended the idea of Salo and Puna (2011) to analyze relative performance of two-stage production units by introducing the new definitions of wea and strong relations. Another possible extension of this idea is providing the best and worst raning of production units in cost (profit) efficiency analysis, when the input (output) prices are available. Here, we briefly discuss the idea of computing raning intervals for cost efficiency evaluation when the units have single-stage. Possible extension of cost efficiency raning intervals in the case of two-stage production systems is straightforward. To obtain the raning intervals in the presence of input prices, we need to modify the Salo and Puna (2011) models for analyzing the cost efficiency scores in which producing the output vector y at the minimum cost is considered. Suppose that the input price vector for all units is c, then the actual cost for DMU is cx and the minimum cost of producing the target output y is min c x; x L(y ), where L(y) shows the input set in model (1). See Färe and x Grossopf (1985; 1994) for more details. Now, to determine the best and the worst cost ran of a system, by taing the dual model of min cx; x L(y ); we present the following models: x min 1+ z l z,u l s.t uy l cx l Mz l, for l uy =1, z l {0, 1}, for l u U (8)

14 R. Azizi & R. K. Matin Table 4. Cost efficiency score, the data set, and the best and the worst cost raning of five DMUs. DMU Cost Input 1 Input 2 Output 1 Output 2 Best Worst efficiency cost ran cost ran c max 1+ z l z,u l s.t uy l + cx l M(1 z l ), for l uy =1, z l {0, 1}, for l u U. (9) Here c is cx. The best/worst cost ran of DMU is then computed by 1 + l z l, based on the optimal solution of (8)/(9). We use the following example to illustrate the proposed cost-raning method. Example 3. Assume that there are five DMUs, which consume two inputs to produce two outputs. The input prices are 1.2 and 0.8, respectively. The weights are considered to be non-negative. Table 4 depicts cost efficiency scores, the data set, and the best and the worst cost raning. The two last columns show the best and worst cost raning of DMUs, respectively. The results show that DMU 2 has the best cost performance due to its cost raning interval, which is [1, 1], and DMU 5 has the worst cost performance due to its worst cost ran, which is 5, while its best cost ran is worse than that of other DMUs. 5. Conclusion We have introduced a new raning interval method for two-stage production units based on the efficiency score of the stages. In introducing the raning intervals, all feasible weights are considered, not just self-appraised optimal weights. Two new definitions, wea and strong relations, are given, and, using the proposed model of Kao and Hwang (2008), the best and the worst raning situations of two-stage production units are computed. The proposed raning method compares performance of any two-stage unit with the other units without using self-appraised optimal weights of the units. In addition, the proposed method provides accurate

15 Raning Two-Stage Production Units in Data Envelopment Analysis information about sources of inefficiency in two-stage production units by checing the performance situation of the stages. Also, cost-raning models are suggested when input prices are available. The results were illustrated by some numerical examples. Appendix ProofofTheorem1.Let the best raning of DMU is achieved at (u, w, v) (U, W, V ). So, there exists D = WR < such that El 1 >E 1 & E2 l >E 2 or El 1 >E 1 & E2 l E 2 while E l >E or El 1 E 1 & E2 l >E 2 while E l >E for l D. Let û = u uy, ˆv = v vx, ŵ = w wx,then(û, ˆv, ŵ) (U, V, W )andûy =1, ˆvx =1, ŵx = 1. Now consider the following sets: H = {l (1,...,n) E l >E } G = {l (1,...,n) E 1 l >E 1 } F = {l (1,...,n) E 2 l >E 2 }. Let g l =1,f l =1andh l =1(l ) forl G, l F and l H, respectively, and g l =0,f l =0andh l =0(l ) forl/ G, l / F and l/ H, respectively. So, for any l/ G, we have the following inequality for the first stage: El 1 (w, v) E 1 (w, v) 1 E1 l (w, v) E1 l (ŵ, ˆv) E 1 = (w, v) E 1 = (ŵ, ˆv) ŵz l ˆvx l ŵz ˆvx = ŵz l ˆvx l ŵz l ˆvx l ŵz l ˆvx l 0. For l G, multiplying g l =1byM leads us to the first set of constraint. For any l/ F, we have the following inequality for the second stage: E 2 l (u, w) E 2 E2 l (u, w) E2 l (û, ŵ) (u, w) 1 E 2 = (u, w) E 2 = (û, ŵ) ûy l ŵz l ûy l ŵz l 0. For l F, multiplying f l =1byM leads to the second constraint. ûy l ŵz l ûy ŵz = ûy l ŵz l

16 R. Azizi & R. K. Matin For any l/ H, we have the following inequality for the whole system: El (u, w, v) E(u, w, v) 1 E l (u, w, v) E (u, w, v) = E ŵzl l (û, ŵ, ˆv) E (û, ŵ, ˆv) = ˆvx l ŵz ˆvx = ŵx l ˆvx l ŵz 1 ˆvx l ŵz l ˆvx l 0. For l H, multiplying h l =1byM leads us to the third constraint. In the cases of g l =1&f l =1, or g l =1,f l =0&h l =1, or g l =0,f l =1&h l =1, it can be concluded that l D(l ). In the mentioned cases, we set d l =1. Therefore, 1 + l d l =1+ D =1+ WR < =minwr < = the best wea raning of DMU. Now, let the worst raning of DMU is achieved at (u, w, v) (U, W, V ). So, there exists D = WR such that El 1 E 1 & E2 l E 2 or El 1 E 1 & E2 l E 2 while E l E or El 1 E 1 & E2 l E 2 while E l E for l D. Let û = u uy, ˆv = v vx, ŵ = w wx.then(û, ˆv, ŵ) (U, V, W )andûy =1, ˆvx =1, wz ˆ =1.Andlet, H = {l (1,...,n)\ E l E } G = {l (1,...,n)\ E 1 l E 1 } F = {l (1,...,n)\ E 2 l E 2 }. Let g l =1,f l =1andh l =1(l ) forl G, l F and l H, respectively, and g l =0,f l =0andh l =0(l ) forl/ G, l / F and l/ H, respectively. So, for any l G, we have the following inequality for the first stage: (w, v) El 1 (w, v) 1 E1 (w, v) El 1 (w, v) = E1 (ŵ, ˆv) El 1 (ŵ, ˆv) = E 1 ŵz ˆvx ŵz l ŵz l ˆvx l ˆvx l = 1 ŵz l ˆvx l 1 ŵz l ˆvx l ŵz l + ˆvx l 0. For l/ G, multiplying (1 g l )=1byM leads us to the first constraint

17 Raning Two-Stage Production Units in Data Envelopment Analysis For any l F, we have the following inequality for the second stage: E 2 ûy ŵz ûy l (u, w) E2 l (u, w) 1 E2 (u, w) El 2 (u, w) = E2 (û, ŵ) El 2 (û, ŵ) = = 1 ŵz l ûy l ŵz l ûy l ŵz l 1 ûy l ŵz l ûy l + ŵz l 0. For l/ F, multiplying (1 f l )=1byM leads us to the second constraint. For any l H, we have the following inequality for the whole system: E (u, w, v) E l (u, w, v) 1 E (u, w, v) El (u, w, v) = E (û, ŵ, ˆv) = (û, ŵ, ˆv) E l ûy ˆvx ûy l ûy l ˆvx l ˆvx l = 1 ûy l ˆvx l 1 ûy l ˆvx l ûy l + ˆvx l 0. For l/ H, multiplying (1 h l )=1byM leads us to the third constraint. In the cases of g l =1&f l =1,org l =1,f l =0&h l =1,org l =0,f l =1&h l =1, l D(l ) can be concluded. In each of the mentioned case, we set d l =1, therefore 1+ l d l =1+ D =1+ WR =maxwr = the worst wea raning of DMU. Proof of Theorem 2. The procedure of proof in this theorem is similar to that of Theorem 1. Proof of Theorem 3. When there are just one input, intermediate product, and output, a unique set of weights will be achieved by running (6) and (7) due to the constraints uy = vx = wz = 1. Here, we prove the theorem for wea raning, and the similar proof can be presented for strong raning. Without loss of the generality, assume there is no DMU which has the same efficiency score as DMU with different efficiency scoresforthesimilarstages. By evaluating the optimal amount of model (6), and definition of d l,wenow El 1 >E 1 & E2 l >E 2 or d l =1 for l D = l {1,...,n El 1 >E 1 & E2 l E 2 while E l >E. (A.1) or El 1 E 1 & E2 l >E 2 while E l >E

18 R. Azizi & R. K. Matin By evaluating the optimal amount of model (7), and definition of d l,wehave d l =1forl D E l 1 E 1 & E2 l E 2 or = l {1,...,n}\{} El 1 E 1 & E2 l E 2 while E l E. (A.2) or El 1 E 1 & E2 l E 2 while E l E Since u, v and w are equal in the both models, and according to (a) and (b), the only difference between the best and the worst wea raning is the number of DMUs, which have the same efficiency score as DMU in the both stages. References Adler, N, L Friedman and Z Sinuany-Stern (2002). Review of raning methods in the data envelopment analysis context. European Journal of Operational Research, 140, Andersen, P and NC Petersen (1993). A procedure for raning efficient units in data envelopment analysis. Management Science, 39, Charnes, A, WW Cooper and E Rhodes (1978). Measuring the efficiency of decision maing units. European Journal of Operational Research, 2, Charnes, A, WW Cooper, QL Wei and ZM Huang (1989). Cone ratio data envelopment analysis and multi-objective programming. International Journal of Systems Science, 20, Chen, Y and J Zhu (2004). Measuring information technology s indirect impact on firm performance. Information Technology & Management Journal, 5, Coo, WD, L Liang and J Zhu (2010). Measuring performance of two-stage networ structures by DEA: A review and future perspective. Omega, 38, Doyle, JR and R Green (1994). Efficiency and cross-efficiency in data envelopment analysis: Derivatives, meanings and uses. Journal of the Operational Research Society, 45, Färe, R and S Grossopf (1996). Productivity and intermediate products: A frontier approach. Economic Letters, 50, Färe, R, S Grossopf and CAK Lovell (1985). Measurement of Efficiency of Production. Boston: Kluwer-Nijhoff Publishing Co., Inc. Färe, R, S Grossopf and CAK Lovell (1994). Production Frontiers. Cambridge: Cambridge University Press. Kao, C and SN Hwang (2008). Efficiency decomposition in two-stage data envelopment analysis: An application to non-life insurance companies in Taiwan. European Journal of Operational Research, 85, Lewis, HF and TR Sexton (2004). Networ DEA: Efficiency analysis of organizations with complex internal structure. Computers & Operations Research, 31, Liu, JS and WM Lu (2010). DEA and raning with the networ based approach: A case of R&D performance. Omega, 38, Liu, JS, WM Lu, C Yang and M Chuang (2009). A networ-based approach for increasing discrimination in data envelopment analysis. Journal of the Operational Research Society, 60,

19 Raning Two-Stage Production Units in Data Envelopment Analysis Liu, JS and WM Lu (2012). Networ-based method for raning of efficient units in twostage DEA models. Journal of the Operational Research Society, 63, Salo, A and A Puna (2011). Raning intervals and dominance relations for ratio-based efficiency analysis. Management Science, 57, Seiford, LM and J Zhu (1999). Profitability and maretability of the top 55 U.S. commercial bans. Management Science, 45, Sexton, TR, RH Silman and AJ Hogan (1986). Data envelopment analysis: Critique and extensions. Measuring Efficiency: An Assessment of Data Envelopment Analysis, RH Silman (ed.), pp San Francisco: Jossey-Bass. Thompson, RG, FD Singleton Jr, RM Thrall and BA Smith (1986). Comparative site evaluations for locating a high-energy physics lab in Texas. Interfaces, 16, Wang, CH, R Gopal and S Zionts (1997). Use of data envelopment analysis in assessing information technology impact on firm performance. Annals of Operations Research, 73, Zhu, J (2000). Multi-factor performance measure model with an application to Fortune 500 companies. European Journal of Operational Research, 123, Biography Roza Azizi (MS) is a Lecturer in Applied Mathematics and Operation Research Group in Islamic Azad University in Iran. Her research interests lie in the broad area of performance management with special emphasis on the quantitative methods of performance measurement, and especially those based on the broad set of methods nown as Data Envelopment Analysis (DEA). Reza Kazemi Matin, is an Associate Professor of Operational Research at Islamic Azad University, Karaj Branch. His areas of research interests include mathematical modeling, performance measurement and management, efficiency and productivity analysis. Dr. Kazemi Matin has published in such journals as Omega, European Journal of Operational Research, Agricultural Economics, Operations Research Society of Japan, Measurement and Applied Mathematical Modelling. He is a member of Iranian Operations Research Society