# Efficiency Bounds for Two-Stage Production Systems

Authors: Xiao Shi
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

Traditional data envelopment analysis (DEA) models find the most desirable weights for each decision-making unit (DMU) in order to estimate the highest efficiency score as possible. These efficiency scores are then used for ranking the DMUs. The main drawback is that the efficiency scores based on weights obtained from the standard DEA models ignore other feasible weights; this is due to the fact that DEA may have multiple solutions for each DMU. To overcome this problem, Salo and Punkka (2011) deemed each DMU as a “Black Box” and developed models to obtain the efficiency bounds for each DMU over sets of all its feasible weights. In many real world applications, there are DMUs that have a two-stage production system. In this paper, we extend the Salo and Punkka’s (2011) model to a more common and practical case considering the two-stage production structure. The proposed approach calculates each DMU’s efficiency bounds for the overall system as well as efficiency bounds for each subsystem/substage. An application for nonlife insurance companies has been discussed to illustrate the applicability of the proposed approach and show the usefulness of this method.


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## Body

## 1. Introduction

Data Envelopment Analysis (DEA), first developed by Charnes et al. [1], has been proven as an effective tool for performance evaluation and benchmarking. This technique makes no assumptions on the production function and imposes no subjective weights on multiple inputs and multiple outputs. DEA has been widely applied in many areas [2]. The DEA technique allows a DMU to choose the most favorable weights to achieve the best possible relative efficiency. However, the standard efficiency scores do not consider all the possible weights as they only consider the weights most favorable to each DMU. The main issue that has been ignored in the past literature is that the efficiency score of a DMU relative to other DMUs can change over different weights when applying the DEA models. Hence, it is important to consider all possible weights to evaluate each DMU.To overcome this problem, Salo and Punkka [3] have proposed a procedure to obtain the efficiency bounds by taking into account all possible weights (see also [4]). That is to say, the efficiency bounds show how the DMUs’ efficiency ratios relate to each other for all feasible weights, rather than for those weights only for which the data envelopment analysis (DEA) efficiency score of some DMU is maximized. They have introduced an efficiency bound for all possible efficiency scores that is determined by the lower and upper bounds of the efficiency scores. The efficiency bounds show how much more efficient a given DMU can be relative to some other DMU or a subset of other DMUs. For this purpose, Green et al. [5] developed a new model by putting the CCR model into a mixed-binary linear programming framework to obtain the efficiency bounds in data envelopment analysis. Entani and Tanaka [6] proposed the interval DEA model to obtain an efficiency interval consisting of evaluations from both the optimistic and pessimistic viewpoints. Wang and Yang [7] proposed a pair of bounded DEA models to measure the overall performances of a group of decision-making units (DMUs), which were characterized by interval efficiencies. To overcome the problem of these models incapable of determining an efficiency interval for any DMU when there is a zero value for each output, Azizi and Wang [8] proposed a pair of improved bounded DEA models to overcome the drawback. All these methods for obtaining the efficiency bounds treated each DMU as a “Black Box”. Thus, they ignored the internal structure of the production system.However, as discussed in many DEA studies, in many real application, DMUs have a two-stage structure, i.e., outputs from the first stage become the inputs to the second stage. Outputs from the first stage are referred to as intermediate measures. Seiford and Zhu [9] use the standard DEA approach to measure the profitability and marketability of US commercial banks which does not address potential conflicts between the two stages arising from the intermediate measures. For example, the second stage may have to reduce its inputs (intermediate measures) in order to achieve an ‘efficient’ status, which imply a reduction in the first stage outputs. In a survey by Cook et al. [10], they pointed out that the approaches of modeling DMUs with a two-stage production process can be categorized as four types, i.e., standard DEA methodology, efficiency decomposition methodology, network DEA, and game-theoretic approaches. The standard DEA methodology simply uses the standard DEA model, i.e., two separate DEA models to calculate the efficiencies of two stages (e.g., Seiford and Zhu [9]; Zhu [11]; and Sexton and Lewis [12]); the efficiency decomposition methodology is that given the efficiency scores of stage 1 and stage 2, the overall efficiency could be defined as the product or the arithmetic mean of two substages’ efficiencies (e.g., Kao and Hwang [13]; Chen et al. [14], and Chen et al. [15]); the network DEA approach extends the two-stage process to more general situation (e.g., Tone and Tsutsui [16]; Tone and Tsutsui [17]; Izadikhah et al. [18]); game-theoretic approaches introduce game theory to the efficiency evaluation of two-stage structure (e.g., Liang et al. [19]; Zha and Liang [20]; Li et al. [21]; Guo and Zhu [22]; and Izadikhah et al. [18]). Except for the standard DEA approach, all other approaches attempt to correct for the above-referenced conflict issue. And, two-stage DEA has been extensively applied to many areas, such as hotels ([23, 24]; Huang et al. [25]), R&D departments (Li et al. [21]; Liu and Lu, [26]), information technology (Shao and Lin [27]; Chen and Zhu [28]; and Kao and Hwang [29]), insurance companies (Yang [30]; Kao and Hwang [13]), industry (Wu et al. [31]; Chen et al. [32]; and Li et al. [33]), and banks (Paradi et al. [34]; Huang et al. [23, 24]; Wang et al. [35]; and Zhu et al. [36]).In this paper, we develop a method to obtain the efficiency bounds for the classic two-stage production systems as discussed by Seiford and Zhu [9], Chen and Zhu [28], and Kao and Hwang [13]. That is, the first subsystem uses inputs to produce outputs that then become the inputs to the second subsystem to produce the final outputs. The proposed model calculates each DMU’s efficiency bounds for the overall system as well as two subsystems. Unlike conventional efficiency scores, the results show how the DMUs’ efficiency ratios for the overall system and two subsystems relate to each other for all feasible weights. We believe that this process provides more accurate information for decision makers by identifying the best (and/or worst) DMUs in the overall system and both subsystems over all feasible weights. Besides, the proposed approach provides information regarding the sensitivity of the DMU’s efficiency bounds for the overall system and both two subsystems over sets of all feasible weights.The remainder of this paper is organized as follows. In the next section, the procedure of efficiency bounds by Salo and Punkka [3] has been reviewed briefly. Then, in Section 3, a method is developed to obtain the efficiency bounds considering the two-stage production systems. This is followed by illustration example in Section 4. An application is also given in this section to show the usefulness of the proposed procedure. Finally, conclusions and direction for future research are given in Section 5.


## 2. Efficiency Bounds by Salo and Punkka [3]

Assume that there aren DMUs denoted as DMUjj=1,…,n. Each DMU uses inputs xiji=1,…,m to produce outputs yrjr=1,…,s. Based on the definition of Charnes et al. [1], the efficiency of DMUk is calculated by the CCR multiplier form as follows:(1)EkCCRu,v=max⁡∑r=1suryrk∑i=1mvixik(2)s.t.∑r=1suryrj-∑i=1mvixij≤0∀j(3)ur≥0∀rvi≥0∀iThe model could be transformed into linear programming model by use of the Charnes-Cooper transformation [37]. ur∗r=1,…,s and vi∗i=1,…,m are the optimal solution of model (2) and the associated optimal output and input weights. DMUk is termed efficient if and only if the optimal objective is equal to one and the optimal weight vectors are all larger than zero; i.e., EkCCRu∗,v∗=1 and ur∗>0,vi∗>0. For any feasible weights urr=1,…,s and vii=1,…,m, Salo and Punkka [3] defined efficiency dominance between DMUs based on the efficiency scores of model (2).Definition 1.
D M U k dominates DMUl (denoted by DMUk≻DMUl) if and only if (4)Eku,v≥Elu,v∀u,v∈Su,SvEku,v>Elu,vforsomeu,v∈Su,Sv
Thus, ifDMUk≻DMUl, the efficiency ratio of DMUk is at least as high as that of DMUl for all feasible weights, and moreover, there exist some weights for which its efficiency is strictly higher.The dominance relation in Definition1 could be calculated based on the following pairwise efficiency ratio:(5)Dk,lu,v=Eku,vElu,v.However, the relative efficiency ratio (5) is nonlinear in weights u,v. Salo and Punkka [3] proposed the following model to maximize and minimize the ratio through linear programming.Theorem 2.
The optimum of the maximization (minimization problem)(6)maxu,vminu,v∑r=1suryrks.t.∑r=1suryrl∑i=1mvixil∑i=1mvixik=1ur,vi≥0,∀r,iis the maximum of (minimum) Dk,lu,v for all weight scenarios, i.e., D_k,l and D¯k,l.The optimization problems in model (6) obtain the upper and lower bounds on how efficient DMUk can be relative to DMUl across feasible weights. It is worth noting that if the benchmark set L contains all DMUs, then D_k,l(or D¯k,l) is equal to the CCR-DEA score. If DMUk is not contained in the benchmark set L, D_k,l(or D¯k,l) is the super efficiency of DMUk relative to this set of DMUs (see, e.g., [38]). For example, if D_k,l=1.2, the efficiency score of DMUk is at least 20% higher than that of DMUl. And, if D¯k,l=1.5, the efficiency score of DMUk can be at most 50% higher than that of DMUl. If the minimum D_k,l is greater than one, DMUk dominates DMUl. If the minimum D_k,l is less than one, the dominance does not hold. If the minimum is equal to one, we could judge the dominance relation by further maximizing the linear program of model (6). If the resulting maximum D¯k,l is greater than one, the dominance holds, but if not, then DMUk and DMUl have the same efficiency score (2) for all feasible weights.Based on model (6), the lower efficiency bound of DMUk, which is the efficiency of DMUk relative to the most efficient DMUs in the benchmark group for different inputs/outputs weights. Thus, the lower bound of DMUk' efficiency scores denoted by D_k,L¯ is minl∈LD_k,l, i.e., D_k,L¯=minl∈LD_k,l.The following proposition shows how to find the upper bound of efficiency scores, denoted asD¯k,L¯, which is relative to the most efficient DMUs in the benchmark group for DMUk.Theorem 3.
D ¯ k , L ¯ is the optimum of the maximization problem(7)maxu,v∑r=1suryrks.t.∑r=1suryrl≤∑i=1mvixil,l∈L∑i=1mvixik=1ur,vi≥0,∀r,i.
The optimal value of model (7) is the upper bound of DMUk’ efficiency scores over sets of all feasible weights. The upper bound of DMUk’ efficiency scores defined as the maximum of values of how efficient DMUk is relative to the most efficient DMUs in the benchmark group for different input/output weights. Based on models (6) and (7), the efficiency bounds D_k,L¯,D¯k,L¯ of DMUk can be computed. These models treat the production systems as “Black Box”.The approach can be generalized to systems composed of two subsystems connected in series. In the next section, we will discuss how to calculate the efficiency bounds for each DMU with a two-stage production system.


## 3. Efficiency Bounds for Two-Stage Production Systems

Suppose the operation of a DMU can be divided into two subsystems or processes, as depicted in Figure1. For DMUk, subsystem 1 applies inputs xiki=1,...,m to produce the intermediate products zdkd=1,...,D. All these intermediate products are then used by subsystem 2 to produce the final outputs yrkr=1,...,s. Based on the definition of Kao and Huang [13], DMUk’s efficiency scores for the overall system and two subsystems are defined as(8)Ek=∑ruryrk∑ivixik(9)Ek1=∑dwd1zdk∑ivixik(10)Ek2=∑ruryrk∑dwd2zdkwhere urr=1,...,s and vii=1,...,m are the output weights and input weights, respectively. Accordingly, wd1d=1,...,D and wd2d=1,...,D are the weights attached to the intermediate measures for subsystem 1 and subsystem 2, respectively. Similar to Kao and Hwang [13] and Liang et al. [19], we assume that the weights attached to the intermediate outputs in both subsystem 1 and subsystem 2 are the same, i.e., wd1=wd2. This assumption represents the serial relationship between the two subsystems [14]. If we solve the two-stage DEA without this assumption, then our method is identical to independently employing the model for each subsystem. Therefore, this paper assumes wd1=wd2=wd.Figure 1
Two-stage production system.


### 3.1. Efficiency Bounds of a DMU for the Overall System

As discussed in Section2, choosing different weights may lead to different efficiency scores for a “Black Box” DMU. Similarly, choosing different weights may result in different efficiency scores for a DMU with a two-stage production system.Proposition 4.
The optimum of the maximization (or minimization) problem(11)maxu,v,wminu,v,w∑r=1suryrks.t.∑r=1suryrl=∑d=1Dwdzdl∑d=1Dwdzdl=∑i=1mvixil∑i=1mvixik=1ur,vi,wd≥0,∀r,i,dis the maximum (or minimum) of Dk,lu,w,v for all weight scenarios, i.e., D¯k,lu,w,v (or D_k,lu,w,v). The proof of this proposition is given in the Appendix.
In the following model, the optimum of the maximization problem, i.e.,D¯k,L¯, is the upper efficiency bound of DMUk considering each DMU' two-stage production structure.Proposition 5.
D ¯ k , L ¯ is the optimum of the maximization problem(12)maxu,v∑r=1suryrks.t.∑r=1suryrl≤∑d=1Dwdzdl,l∈L∑d=1Dwdzdj≤∑i=1mvixil,l∈L∑i=1mvixik=1ur,vi≥0,∀r,i.
By solving model (12), the upper efficiency bound D¯k,L¯ could be obtained. The proof of this proposition is given in the Appendix.


### 3.2. Efficiency Bounds of a DMU for Both Subsystems

In this section, we discuss the efficiency bounds of DMUs for two subsystems.Proposition 6.
The optimum of the maximization (minimization problem)(13)minv,w∑d=1Dwdzdks.t.∑d=1Dwdzdj=∑i=1mvixil∑i=1mvixik=1wd,vi≥0,∀d,iis the minimum of Dk,l1w,v for subsystem 1 for all weight scenarios.Based on model (13), the lower bound of DMUk's efficiency scores for subsystem 1, which is the efficiency of DMUk relative to the most efficient DMUs in the benchmark group for different inputs/outputs weights. Thus, the lower bound of DMUk’s efficiency scores for subsystem 1 denoted by D_k,L¯1 is minl∈LD_k,l1, i.e., D_k,L¯1=minl∈LD_k,l1. The proof of this proposition is similar to the proof of Theorem 3 in Salo and Punkka [3].In the following model, the optimum of the maximization problem, i.e.,D¯k,L¯1, is the upper bound of DMUk' efficiency score for subsystem one.Proposition 7.
D ¯ k , L ¯ 1 is the optimum of the maximization problem(14)maxu,v∑d=1Dwdzdks.t.∑d=1Dwdzdj≤∑i=1mvixil,l∈L∑i=1mvixik=1wd,vi≥0,∀d,i.
The optimal value of model (14) may be greater than 1 or less than one or equal to 1. If the optimal value is greater than 1, then DMUk dominates DMUl. The proof of this proposition is similar to the proof of Theorem 4 in Salo and Punkka [3].Similarly, the lower and upper bound ofDMUk' efficiency scores for subsystem 2 are calculated by the following two linear programs.Proposition 8.
The optimum of the minimization problem(15)minu,w∑r=1suryrks.t.∑r=1suryrl=∑d=1Dwdzdl∑d=1Dwdzdk=1ur,vi,wd≥0,∀r,i,dis the minimum of Dk,l2u,v for all weight scenarios.Based on model (15), the lower bound of DMUk's efficiency scores for subsystem 2, which is the efficiency of DMUk relative to the most efficient DMUs in the benchmark group for different inputs/outputs weights. Thus, the lower bound of DMUk's efficiency scores for subsystem 2 denoted by D_k,L¯2 is minl∈LD_k,l2, i.e., D_k,L¯2=minl∈LD_k,l2. The proof of this proposition is similar to the proof of Theorem 3 in Salo and Punkka [3].Proposition 9.
D _ k , L ¯ 2 is the optimum of the maximization problem(16)maxu,v∑r=1suryrks.t.∑r=1suryrl≤∑d=1Dwdzdl,l∈L∑d=1Dwdzdk=1ur,vi≥0,∀r,i.
Thus, the efficiency boundsD_k,L¯2,D¯k,L¯2 of DMUk for subsystem 2 could be obtained by model (15) and model (16).


## 3.1. Efficiency Bounds of a DMU for the Overall System

As discussed in Section2, choosing different weights may lead to different efficiency scores for a “Black Box” DMU. Similarly, choosing different weights may result in different efficiency scores for a DMU with a two-stage production system.Proposition 4.
The optimum of the maximization (or minimization) problem(11)maxu,v,wminu,v,w∑r=1suryrks.t.∑r=1suryrl=∑d=1Dwdzdl∑d=1Dwdzdl=∑i=1mvixil∑i=1mvixik=1ur,vi,wd≥0,∀r,i,dis the maximum (or minimum) of Dk,lu,w,v for all weight scenarios, i.e., D¯k,lu,w,v (or D_k,lu,w,v). The proof of this proposition is given in the Appendix.
In the following model, the optimum of the maximization problem, i.e.,D¯k,L¯, is the upper efficiency bound of DMUk considering each DMU' two-stage production structure.Proposition 5.
D ¯ k , L ¯ is the optimum of the maximization problem(12)maxu,v∑r=1suryrks.t.∑r=1suryrl≤∑d=1Dwdzdl,l∈L∑d=1Dwdzdj≤∑i=1mvixil,l∈L∑i=1mvixik=1ur,vi≥0,∀r,i.
By solving model (12), the upper efficiency bound D¯k,L¯ could be obtained. The proof of this proposition is given in the Appendix.


## 3.2. Efficiency Bounds of a DMU for Both Subsystems

In this section, we discuss the efficiency bounds of DMUs for two subsystems.Proposition 6.
The optimum of the maximization (minimization problem)(13)minv,w∑d=1Dwdzdks.t.∑d=1Dwdzdj=∑i=1mvixil∑i=1mvixik=1wd,vi≥0,∀d,iis the minimum of Dk,l1w,v for subsystem 1 for all weight scenarios.Based on model (13), the lower bound of DMUk's efficiency scores for subsystem 1, which is the efficiency of DMUk relative to the most efficient DMUs in the benchmark group for different inputs/outputs weights. Thus, the lower bound of DMUk’s efficiency scores for subsystem 1 denoted by D_k,L¯1 is minl∈LD_k,l1, i.e., D_k,L¯1=minl∈LD_k,l1. The proof of this proposition is similar to the proof of Theorem 3 in Salo and Punkka [3].In the following model, the optimum of the maximization problem, i.e.,D¯k,L¯1, is the upper bound of DMUk' efficiency score for subsystem one.Proposition 7.
D ¯ k , L ¯ 1 is the optimum of the maximization problem(14)maxu,v∑d=1Dwdzdks.t.∑d=1Dwdzdj≤∑i=1mvixil,l∈L∑i=1mvixik=1wd,vi≥0,∀d,i.
The optimal value of model (14) may be greater than 1 or less than one or equal to 1. If the optimal value is greater than 1, then DMUk dominates DMUl. The proof of this proposition is similar to the proof of Theorem 4 in Salo and Punkka [3].Similarly, the lower and upper bound ofDMUk' efficiency scores for subsystem 2 are calculated by the following two linear programs.Proposition 8.
The optimum of the minimization problem(15)minu,w∑r=1suryrks.t.∑r=1suryrl=∑d=1Dwdzdl∑d=1Dwdzdk=1ur,vi,wd≥0,∀r,i,dis the minimum of Dk,l2u,v for all weight scenarios.Based on model (15), the lower bound of DMUk's efficiency scores for subsystem 2, which is the efficiency of DMUk relative to the most efficient DMUs in the benchmark group for different inputs/outputs weights. Thus, the lower bound of DMUk's efficiency scores for subsystem 2 denoted by D_k,L¯2 is minl∈LD_k,l2, i.e., D_k,L¯2=minl∈LD_k,l2. The proof of this proposition is similar to the proof of Theorem 3 in Salo and Punkka [3].Proposition 9.
D _ k , L ¯ 2 is the optimum of the maximization problem(16)maxu,v∑r=1suryrks.t.∑r=1suryrl≤∑d=1Dwdzdl,l∈L∑d=1Dwdzdk=1ur,vi≥0,∀r,i.
Thus, the efficiency boundsD_k,L¯2,D¯k,L¯2 of DMUk for subsystem 2 could be obtained by model (15) and model (16).


## 4. Empirical Illustration

To illustrate the proposed approach of efficiency bounds for two-stage production systems, we use the following example. In this section, we take the data set of 24 nonlife insurance companies from [13]. These nonlife insurance companies’ whole production system has a typical two-stage structure. The production system is divided into two subsystems: premium acquisition and profit generation. These companies are evaluated by using two inputs, two intermediates, and two outputs. Table 1 reports the efficiency bounds of each DMU based on our approach and Salo and Punkka [3]’s approach. Column 5 and column 6 in Table 1 report the DMUs’ efficiency bounds treating the production system as a “Black Box”, which are represented graphically blue in Figure 2. Column 3 and column 4 in Table 1 report the DMUs’ efficiency bounds when considering the inner production structure, which are represented graphically red in Figure 2.Table 1
Ranking intervals for each nonlife insurance company.
DMU NO. Our approach Salo & Punkka [3]’s model D _ k , L ¯ u , w , v D ¯ k , L ¯ u , w , v D _ k , L ¯ u , v D ¯ k , L ¯ u , v Taiwan Fire 1 0.1253 0.6992 0.1626 0.9840 Chung Kuo 2 0.1565 0.6248 0.1161 1.0000 Tai Ping 3 0.0374 0.6900 0.0485 0.9884 China Mariners 4 0.0317 0.3042 0.0535 0.4882 Fubon 5 0.6035 0.7670 0.2282 1.0000 Zurich 6 0.1057 0.3897 0.127 0.5938 Taian 7 0.1118 0.2766 0.1984 0.4698 Ming Tai 8 0.1587 0.2752 0.2004 0.4148 Central 9 0.0673 0.2233 0.1296 0.3270 The First 10 0.1413 0.4660 0.1672 0.7807 Kuo Hua 11 0.0047 0.1639 0.0154 0.2826 Union 12 0.2005 0.7596 0.1182 1.0000 Shingkong 13 0.0568 0.2078 0.1167 0.3527 South China 14 0.0847 0.2886 0.1812 0.4696 Cathay Century 15 0.1415 0.6138 0.299 0.9793 Allianz President 16 0.0504 0.3202 0.1372 0.4717 Newa 17 0.0948 0.3600 0.2454 0.6349 AIU 18 0.0418 0.2588 0.1616 0.4271 North America 19 0.0119 0.4112 0.184 0.8220 Federal 20 0.0068 0.5465 0.302 0.9351 Royal&Sun Alliance 21 0.0017 0.2008 0.1202 0.3328 Asia 22 0.0011 0.5895 0.285 1.0000 AXA 23 0.0008 0.4203 0.0011 0.5990 Mitsui Sumitomo 24 0.0043 0.1348 0.0517 0.2571Figure 2
Efficiency bounds for each nonlife insurance company considering and without considering the two-stage production system.From Figure2, we can find that if we do not consider the two-stage production structure, the best DMUs are Chung Kuo (DMU 2), Fubon (DMU 5), Union (DMU 12), and Asia (DMU 22) as they have the best efficiency score of 1. Though these DMUs may have the best efficiency score of 1, they have wide efficiency bounds. Among these five DMUs, Asia (DMU 22) has the best performance as it has the narrowest efficiency bounds as well as the best efficiency score of 1. The worst DMUs are Tai Ping (DMU 3), China Mariners (DMU 4), Kuo Hua (DMU 11), AXA (DMU 23), and Mitsui Sumitomo (DMU 24) as their lower efficiency scores are less than 0.1. Among these five DMUs, the upper efficiency bound of Mitsui Sumitomo (DMU 24) is 0.2571, which is smaller than that of other four DMUs. Thus, Mitsui Sumitomo (DMU 24) is the worst DMU.The best performers (or worst performers) based on our approach may not be the same as that of Salo and Punkka [3]’s approach. The red bar char in Figure 2 reports the DMUs’ efficiency bounds when the two-stage structure is considered. Figure 2 shows that Taiwan Fire (DMU 1) may be the best DNU as its upper efficiency has the largest efficiency score of 0.6992. AXA (DMU 23) may be the worst DMU as its lower efficiency has the least efficiency score of 0.0008.Besides, we can compare some DMUs over sets of all feasible weights. For example, Chung Kuo (DMU 2) has a best efficiency score of 0.6248 and a worst efficiency score of 0.1565, while Mitsui Sumitomo (DMU 24) has a best efficiency score of 0.1348 and a worst efficiency score of 0.0043. That is,D_24,L¯<D_1,L¯ and D¯24,L¯<D_2,L¯<D¯2,L¯. Hence, Chung Kuo (DMU 2) always performs better than Mitsui Sumitomo (DMU 24) regardless of the choice of the weights.When the two-stage structure is considered, the efficiency bounds may be narrower based on our approach than that of Salo and Punkka [3]’s approach. For example, as shown in Column 3 and 4 in Table 1, it could be found that Taian (DMU 7) has the best efficiency score of 0.2766 and the worst efficiency of 0.1118. However, as shown in Column 5 and 6 in Table 1, it has the best efficiency score of 0.4698 and the worst efficiency score of 0.1984 over all feasible weights. Thus, the efficiency bounds based on Salo and Punkka [3]’s approach are wider.As for two subsystems, Table2 shows the subsystem’s efficiency bounds of each insurance company. The third and fourth columns in Table 2 are the lower and upper efficiency scores for subsystem 1. The fifth and sixth columns in Table 2 are the lower and upper efficiency scores for subsystem 2.Table 2
Efficiency bounds for two subsystems.
DMU NO. Efficiency bounds for sub-system 1 Efficiency bounds for sub-system 2 D _ k , L ¯ 1 D ¯ k , L ¯ 1 D _ k , L ¯ 2 D ¯ k , L ¯ 2 Taiwan Fire 1 0.2183 0.9926 0.1253 0.7134 Chung Kuo 2 0.4619 0.9985 0.1565 0.6275 Tai Ping 3 0.1427 0.6900 0.0374 1.0000 China Mariners 4 0.0947 0.7243 0.0317 0.4323 Fubon 5 0.4468 0.8375 1.0000 1.0000 Zurich 6 0.2426 0.9637 0.1057 0.4057 Taian 7 0.1639 0.7521 0.1118 0.5378 Ming Tai 8 0.2891 0.7256 0.1587 0.5113 Central 9 0.1392 1.0000 0.0673 0.2920 The First 10 0.1285 0.8615 0.1413 0.6736 Kuo Hua 11 0.1639 0.7405 0.0047 0.3267 Union 12 0.2849 1.0000 0.2005 0.7596 Shingkong 13 0.2067 0.8107 0.0568 0.5435 South China 14 0.1186 0.7246 0.0847 0.5178 Cathay Century 15 0.1910 1.0000 0.1415 0.7047 Allianz President 16 0.1026 0.9072 0.0504 0.3847 Newa 17 0.0873 0.7233 0.0948 1.0000 AIU 18 0.2536 0.7935 0.0418 0.3737 North America 19 0.1231 1.0000 0.0119 0.4158 Federal 20 0.0336 0.9332 0.0068 0.9014 Royal&Sun Alliance 21 0.0103 0.7505 0.0017 0.2795 Asia 22 0.0037 0.5895 0.0011 1.0000 AXA 23 0.0127 0.8501 0.0003 0.5599 Mitsui Sumitomo 24 0.0607 1.0000 0.0043 0.3351It shows that Central (DMU 9), Union (DMU 12), Cathay Century (DMU 15), North America (DMU 19), and Mitsui Sumitomo (DMU 24) are the best performers in subsystem 1. As Union (DMU 12) has the narrowest efficiency bound, it is the best DMU for subsystem 1. The source of inefficiency could also be identified; for example, Fubon (DMU 5) has the efficiency bound of0.6035,0.7670. It can be seen that it performs well in subsystem 2 as the efficiency bound of subsystem 2 is 1,1, while it does not perform well in subsystem 1 as the efficiency bound of subsystem 1 is 0.4468,0.8375. Therefore, the reason why Fubon (DMU 5)’s overall efficiency is so low is its bad performance in subsystem 1. As for the subsystem 2, Tai Ping (DMU 3), Fubon (DMU 5), Newa (DMU 17), and Asia (DMU 22) are the best DMUs as their best efficiency scores are all 1. AXA (DMU 23) and Mitsui Sumitomo (DMU 24) may be the worst DMU as its worst efficiency score is the smallest.


## 5. Conclusions and Direction for Future Research

In previous DEA literature, each DMU is evaluated by using the most favorable weights. However, it ignores other feasible weights. To overcome this problem, Salo and Punkka [3] deemed each DMU as a “Black Box” and developed a series of models to obtain the efficiency bounds over sets of all feasible weights. In this paper, we expand their method by considering the internal structure of the DMUs. We extend their method to compute efficiency bounds for a two-stage production system and illustrate the method by revisiting reported DEA studies. Thus, the “Black Box” is opened, and more accurate information on the efficiency bounds for the overall system and both subsystems is provided to the decision maker. Unlike conventional efficiency scores, the results show how the DMUs’ efficiency ratios for the overall system and two subsystems relate to each other for all feasible weights. The efficiency measure used in this paper is radial; some nonradial measures have also been proposed in the literature, such as the slack-based measure [16, 17]. Obtaining the efficiency bounds for two-stage production systems based on nonradial DEA is another interesting avenue to explore in the future.



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*Source: 2917537-2018-07-26.xml*