A joint measurement of efficiency and effectiveness for non-storable commodities: Integrated data envelopment analysis approaches

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Decision Support

A joint measurement of efﬁciency and effectiveness for non-storable commodities:

Integrated data envelopment analysis approaches

Yu-Chiun Chiou

a

, Lawrence W. Lan

b,*

, Barbara T.H. Yen

a a

Institute of Trafﬁc and Transportation, National Chiao Tung University, 4F, 118 Sec.1, Chung-Hsiao W. Rd., Taipei 10012, Taiwan, ROC

b

Department of Global Marketing and Logistics, MingDao University, 369 Wen-Hua Rd., Peetow, Changhua 52345 Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 28 March 2008 Accepted 9 March 2009 Available online 20 March 2009 Keywords:

Integrated data envelopment analysis Non-storable commodities

Service effectiveness Technical efﬁciency Technical effectiveness

a b s t r a c t

Efficiency and effectiveness for non-storable commodities represent two distinct dimensions and a joint measurement of both is necessary to fully capture the overall performance. This paper proposes two novel integrated data envelopment analysis (IDEA) approaches, the integrated Charnes, Cooper and Rhodes (ICCR) and integrated Banker, Charnes and Cooper (IBCC) models, to jointly analyze the overall performance of non-storable commodities under constant and variable returns to scale technologies. The core logic of the proposed models is simultaneously determining the virtual multipliers associated with inputs, outputs, and consumption by additive specifications for technical efficiency and service effectiveness terms with equal weights. We show that both ICCR and IBCC models possess the essential properties of rationality, uniqueness, and benchmarking power. A case analysis also demonstrates that the proposed novel IDEA approaches have higher benchmarking power than the conventional separate DEA approaches. More generalized specifications of IDEA models with unequal weights are also elaborated.

1. Introduction

Data envelopment analysis (DEA) is a technique for measuring the relative efficiency of decision making units (DMUs) which produce similar products. Measures of both technical efficiency (a transformation of factors to production) and service effectiveness (consumption of production) for storable commodities are essentially the same because the commodities, once produced, can be stockpiled until con-sumed. Nothing will be lost throughout the transformation from production to consumption if one assumes that all the stockpiles are even-tually sold, there is no storage cost, and there is no loss incurred. Namely, conventional measures for storable commodities assume perfect sale and no storage cost effectiveness. However, technical efficiency and service effectiveness for non-storable commodities, such as trans-port services, represent two distinct measurements because one can never store the surplus service during periods of low demand (off-peak hours) for use during periods of high demand ((off-peak hours). When such non-storable commodities are produced and a portion of which are not concurrently consumed, the technical effectiveness (a joint effect of both technical efficiency and service effectiveness) would be less than the technical efficiency. To explain this concept,Fielding (1987)first introduced three performance measures for a pub-lic transit system by defining technical efficiency as the ratio of production to factors, service effectiveness as the ratio of consumption to production, and technical effectiveness as the ratio of consumption to factors as depicted inFig. 1. As shown inFig. 1, once the transport production (e.g. seat-miles) is transformed from such factors as labor, vehicle, and fuel, seat-miles must be consumed immediately by the passengers; otherwise they are exhausted and wasted. Thus, both technical efficiency and service effectiveness should be jointly evaluated to account for the portion of seat-miles not utilized in practice. The technical effectiveness depends not only on how well the production (seat-miles) is transformed from the factors but also on how well the consumption (passenger-miles) is transformed from the production. Any poor performance of transport services can be attributed to either poor technical efficiency or poor service effectiveness or a combi-nation of both. Without separation of technical efficiency and service effectiveness measurements, it is difficult to scrutinize the sources of poor performance. In other words, to assess the system performance for non-storable commodities, it would become more informative if one could have jointly analyzed the efficiency and effectiveness measurements.

* Corresponding author. Tel.: +886 4 887 6660x7500; fax: +886 4 887 9013. E-mail address:[email protected](L.W. Lan).

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Over the past three decades, various DEA models have been widely used to evaluate the technical efficiency or technical effectiveness of decision making units (DMUs) in different organizations or industries. In transport performance evaluation, numerous applications of DEA have also been found in various fields, including airline (e.g.Schefczyk, 1993; Charnes et al., 1996; Sengupta, 1999; Alder and Golany, 2001), airport (e.g.Salazar de La Cruz, 1999; Joseph, 2000; Martin and Roman, 2001; Adler and Berechman, 2001), maritime (e.g.Tongzon, 2001; Cullinane et al., 2006), transit (e.g.Nolan, 1996; Kerstens, 1996; Viton, 1998; Cowie and Asenova, 1999; Odeck and Alkadi, 2001; Nolan et al., 2002; Karlaftis, 2003, 2004; Boame, 2004; Sheth et al., 2007; Margari et al., 2007), and railway (e.g.Oum and Yu, 1994; Cowie, 1999). Most of these works, however, merely evaluated the performance from the perspective of technical efficiency or technical effectiveness.

In order to completely and fairly evaluate the relative performance of non-storable transport services, several recent works have em-ployed various DEA approaches to evaluating the efﬁciency and effectiveness. In general, they can be divided into four categories: sep-arate DEA model (hereinafter, SDEA; e.g.Karlaftis, 2004; Chiou and Chen, 2006), separate two-stage DEA model (hereinafter, STDEA; e.g.

Rousseau and Rousseau, 1997; Lan and Lin, 2003; Keh et al., 2006), network DEA model (hereinafter, NDEA; e.g.Yu and Lin, 2008; Yu, 2008; Kao, 2009), and integrated two-stage DEA model (hereinafter, ITDEA; e.g.Kao and Hwang, 2008; Chen et al., in press; Chen et al., 2009). The SDEA employs independent DEA models to measure technical efficiency, service effectiveness, and technical effectiveness separately. Hence, paradoxical improvement strategies were usually generated based on the results of these independent DEA models. To overcome this shortcoming, the STDEA uses an input-oriented DEA model to evaluate the technical efficiency and an output-oriented DEA model to assess the service effectiveness, holding the output level unchanged. Although the STDEA model will not generate con-flicting improvement strategies, it suggests the organization be divided into two independent departments: production and sale, such that the performance of one department is not interrelated with that of the other department. This is of course not exactly true from the organizational perspective. The lack of interrelated performance among different departments may be solved by the NDEA or ITDEA modeling. However, due to the complexity of the modeling, the scale economy and slack values for each DMU are hard to compute by the NDEA model, proposed byYu and Lin (2008)andYu (2008), which is only applicable to the case of constant returns to scale. The ITDEA model proposed byChen et al. (in press)can be applied to both technologies of constant and variable returns to scale, and the scale economy and slack values can easily be computed as well. However, for the ease of transforming the objective function into a lin-ear form, the ITDEA model sets rather restricted weights proportional to the relative contributions of inputs, outputs and consumption in association with their corresponding virtue multipliers. This would lead to difficulties provided that the organization would value the weights differently across the departments. Strictly speaking, the weights should represent the relative importance of efficiency and effectiveness valued by the evaluator or the decision maker, and they should remain unchanged in evaluating all DMUs. To further rec-tify this shortcoming, this paper develops integrated DEA (IDEA) models which jointly evaluate the non-storable commodities’ efficiency and effectiveness. In fact, the pioneering IDEA concept was proposed and tested in our early work (Chiou et al., 2007). The present study further extends the IDEA models to generalized IDEA models. Moreover, important underlying properties of the proposed models are also proven. In this paper, we also demonstrate the applicability and superiority of our proposed IDEA models and generalized IDEA models.

The rest of the paper is organized as follows. The formulation of the IDEA models under constant and variable returns to scale contexts is proposed in Section2. The essential properties—rationality, uniqueness, benchmarking power—are proven in Section3. To demonstrate the applicability of the proposed IDEA models and to compare the benchmarking power with the conventional SDEA models, a case analysis is presented in Section4. Generalized IDEA approaches with unequal weights are further elaborated and analyzed in Section5. Finally, con-cluding remarks and suggestions for future studies are addressed.

2. Model formulation

DEA is a method for measuring the relative efﬁciency of DMUs that perform similar tasks. A DEA model under constant returns to scale (CRS) context was developed byCharnes et al. (1978, CCR model hereinafter). A DEA model under variable returns to scale (VRS) context was later developed by Banker et al. (1984, BCC model hereinafter)based on the CCR model by adding the convexity constraint. To simul-taneously measure the efﬁciency and effectiveness for non-storable commodities with avoidance of the above mentioned shortcomings, this paper proposes two integrated DEA approaches under CRS and VRS contexts, which are termed as integrated CCR (ICCR) model and integrated BCC (IBCC) model, respectively. The formulation of the proposed ICCR and IBCC models is given below.

Factors • Labor • Vehicle • Operating network • Fuel Production • Vehicle miles • Seat-miles • Vehicle journey Consumption • Passenger -miles • Passengers • Revenue Technical efficiency measurement Service effectiveness measurement Technical effectiveness mesurement

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2.1. Integrated CCR model

The proposed integrated CCR model [ICCR] aims to maximize the technical efﬁciency and service effectiveness by simultaneously solv-ing for virtual multipliers correspondsolv-ing to factor, production, and consumption variables under CRS assumptions. The model is formulated as follow: ½ICCR Max u;v;w Hk¼ PR r¼1urykr PJ j¼1

v

jxkj ! þ PS s¼1wszks PR r¼1urykr ! ð1Þ s:t: X R r¼1 uryir6 XJ j¼1

v

jxij; i ¼ 1; 2; . . . ; I; ð2Þ XS s¼1 wszis6 XR r¼1 uryir; i ¼ 1; 2; . . . ; I; ð3Þ

v

jP0; j ¼ 1; 2; . . . ; J; ð4Þ wsP0; s ¼ 1; 2; . . . ; S; ð5Þ urP0; r ¼ 1; 2; . . . ; R; ð6Þ

where Hk2 ½0; 2 represents the overall efficiency score of DMU k. If Hkequals to two, the DMU is defined relatively efficient; otherwise the

DMU is relatively inefﬁcient. xkjrepresents the jth input of DMU k. ykrdenotes the rth output of DMU k. zksrepresents the sth consumption of

the DMU k. The variables

v

j;ur;wsare corresponding virtual multipliers of the jth input, the rth output, and the sth consumption. I; J; S; R are

the number of DMUs, inputs, outputs, and consumption, respectively.

Adding slack to each constraint, [ICCR] can then be reformulated as [ICCR-S]:

½ICCR-S Max u;v;w Hk¼ PR r¼1urykr PJ j¼1

v

jxkj ! þ PS s¼1wszks PR r¼1urykr ! ð7Þ s:t: X R r¼1 uryir¼ XJ j¼1

v

jxij sij; i ¼ 1; 2; . . . ; I; ð8Þ XS s¼1 wsðzisþ sisÞ ¼ XR r¼1 uryir; i ¼ 1; 2; . . . ; I; ð9Þ

v

jP0; j ¼ 1; 2; . . . ; J; ð10Þ wsP0; s ¼ 1; 2; . . . ; S; ð11Þ urP0; r ¼ 1; 2; . . . ; R: ð12Þ

For a given production ðyirÞ, [ICCR-S] model can be expressed as: ½ICCR-S Max u;v;w Hk¼ XR r¼1 urykr ! XR r¼1 urykr ! þ X S s¼1 wszks ! XJ j¼1

v

jxkj ! ð13Þ s:t: X J j¼1

v

jxkj ! XR r¼1 urykr ! ¼ 1; ð14Þ XR r¼1 uryir¼ XJ j¼1

v

jxij sij; i ¼ 1; 2; . . . ; I; ð15Þ XS s¼1 wsðzisþ sisÞ ¼ XR r¼1 uryir; i ¼ 1; 2; . . . ; I; ð16Þ

v

With the optimal virtual multipliers, the technical efﬁciency, service effectiveness and technical effectiveness of DMU k can be, respectively, calculated as follows: hk¼ PR r¼1urykr PJ j¼1

v

jxkj ! ; gk¼ PS s¼1wszks PR r¼1urykr ! ; and ok¼ Ps s¼1wszks PJ j¼1

v

jxkj ! :

Based upon the optimal slack values, we can determine the amount of inputs to be curtailed or the amount of consumption to be added or promoted to achieve efﬁciency.

2.2. Integrated BCC model

The above ICCR model can be easily extended to an integrated BCC model [IBCC] by simply adding the convexity constraint, which is expressed as:

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½IBCC Max u;v;w PR r¼1urykr u0 PJ j¼1

v

jxkj ! þ PS s¼1wszks u1 PR r¼1urykr u0 ! ð20Þ s:t: X R r¼1 uryir u06 XJ j¼1

v

j xij sij ; i ¼ 1; 2; . . . ; I; ð21Þ XS s¼1 wsðzisþ sisÞ u16 XR r¼1 uryir u0; i ¼ 1; 2; . . . ; I; ð22Þ

v

3. Properties of the proposed models

In what follows we prove the ICCR and IBCC models exhibiting three essential properties: rationality, uniqueness and benchmarking power.

3.1. Rationality property 3.1.1. Rationality of ICCR

According toCharnes et al. (1978), the efficiency of any DMU is obtained as the maximum of a ratio of weighted outputs, subject to the condition that the similar ratio for every DMU be less than or equal to unity. Since the proposed ICCR model is to maximize two aspects of efficiency values, the overall efficiency value should be less than or equal to two. The measure of the efficiency of any DMU can also be obtained in a similar way. Mathematically,

½ICCR0 Max u;v;w PR r¼1urykr PJ j¼1

v

jxkj ! þ PS s¼1wslks PR r¼1urykr ! ð26Þ s:t: PR r¼1uryir PJ j¼1

v

jxij 61; i ¼ 1; 2; . . . ; I; _ð27Þ PS s¼1wslis PR r¼1uryir 6_1; _{i ¼ 1; 2; . . . ; I;} _ð28Þ

v

jP0; j ¼ 1; 2; . . . ; J; ð29Þ wsP0; s ¼ 1; 2; . . . ; S; ð30Þ urP0; r ¼ 1; 2; . . . ; R: ð31Þ Let E0r¼ xR xr and E 00

r¼llRrrespectively represent the technical efﬁciency (ratios of inputs at a given output) and service effectiveness (ratios of consumption at a given output), where xRis the minimum input that can produce the given output and xris the actual input being rated

from the same output. Likewise, lRis the maximum consumption that can be generated from the given output and lris the actual

consump-tion being rated from the same output. The overall efﬁciency can be calculated as Er¼ E0rþ E00r ¼xxRrþ

lr

lR. Essentially, 0 6 Er 6_2. Alternately, we can also derive the overall efﬁciency, Er, of our proposed ICCR as follows. For any given output y,

Max u;v;w hr¼ uyr

v

xrþ wlr uyr ð32Þ s:t: uyR

v

xR 61; _ð33Þ uyr

v

xr 61; _ð34Þ wlR uyR 61; _ð35Þ wlr uyr 6_1; _ð36Þ u;

v

;w P 0: ð37Þ

Let u_;

_v

_;_w _{represent the optimal triplet of corresponding values. As x}

R6xr;lRPlr and yR¼ yr¼ y imply uyr¼ uyR¼

v

xR and

u_y

r¼ uyR¼ wlR, we then have the following results and relationships: Technical efficiency ¼u _y r

v

_x r ¼u _y R

v

_x r ¼

v

_x R

v

_x r ¼ E0 r; Service effectiveness ¼w _l r u_y r ¼w _l r u_y R ¼w _l r w_l R ¼ E00 r: Thus, ho¼u _y r v_x_rþw _l_r u_y r¼ E 0 rþ E 00 r¼ Er.

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No matter which alternatives are adopted, the efficiency scores determined by the ICCR model are proven to possess an essential prop-erty of rationality 0 6 Er62, because the optimal values of ICCR model have satisfied the definition of efficiency.

3.1.2. Rationality of IBCC

The proposed IBCC model can be expressed as follow:

½IBCC0 Max u;v;w PR r¼1urykr u0 PJ j¼1

v

jxkj ! þ PS s¼1wslks u1 PR r¼1urykr u0 ! ð38Þ s:t: PR r¼1uryir u0 PJ j¼1

v

jxij 6_1; _{i ¼ 1; 2; . . . ; I;} _ð39Þ PS s¼1wslis u1 PR r¼1uryir u0 6_1; _{i ¼ 1; 2; . . . ; I;} _ð40Þ

v

Similar to the measure of efﬁciency in the ICCR model, we can derive the overall efﬁciency from the IBCC model as follows:

Max u;v hr¼ uyr u0

v

xr þwlr u1 uyr u0 ð44Þ s:t: uyR u0

v

xR 6_1; _ð45Þ uyr u0

v

xr 6_1; _ð46Þ wlR u1 uyR u0 6_1; _ð47Þ wlr u1 uyr u0 61; _ð48Þ u;

v

;w P 0: ð49Þ Let u_;

_v

_;_w_;_u

0;u1 represent the optimal set of corresponding values. As xR6x_r;lRPlr and yR¼ yr¼ y imply

u_y

r u0¼ uyR u0¼

v

xRand uyr u0¼ uyR u0¼ wlR u1, we can then obtain the following relationships: Technical efficiency ¼u _y r u0

v

_x r ¼ u_y R u0

v

_x r ¼

v

_x R

v

_x r¼ E 0 r; Service effectiveness ¼w _l r u1 u_y r u0 ¼w _l r u1 u_y R u0 ¼w _l r u1 w_l R u1 ¼w _l rwu1 w _l R_wu1 ¼lr u1 w lRuw1 ; 0 < Service effectiveness ¼lr u1 w lRu_w1

<1; where u1is a scale variable:

When u1>0, the result of lr>lrwu1can be obtained, suggesting that the inputs for DMU r should be reduced to reach its optimal scale. When u1¼ 0, the result of lr¼ lrwu1can be obtained, suggesting that DMU r is already at its optimal scale. When u1<0, the result of

lr<lr_wu1can be obtained, suggesting that DMU r needs to expand to reach its optimal scale.

In conclusion, the efﬁciency scores determined by the IBCC model have been proven to exhibit an essential property of rationality. Addi-tionally, the IBCC model can further indicate the improving direction of each DMU to reach its optimal scale.

3.2. Uniqueness property 3.2.1. Uniqueness of ICCR

To show the uniqueness of the joint efﬁciency measurement of the ICCR model, we have to prove that the virtual multipliers of u, v, and w determined by the ICCR model are a global optimum, not a local optimum. Technically, we examine the concavity or convexity of the objective function as well as of the feasible region for this nonlinear programming problem. Concavity and convexity establish necessary conditions for optimality and the Karush–Kuhn–Tucker (KKT) conditions establish sufﬁcient conditions.

For simplicity, without loss of generality, the mathematical model of [ICCR-S] is examined under the case of a single variable for each of the three stages of the model: input, output and service. Since all the constraints in [ICCR-S] are linear, the feasible set deﬁned by these constraints is deﬁnitely convex. The bordered Hessian matrix of objective function of [ICCR-S] can be derived as:

H ¼ 0 0 y1_lu2 0 2x1_y

_v

3_u _x1_y

_v

2 y1_lu2 x1_y

_v

2 _2y1_lu3 w ;

where the signs of the ﬁrst, second and third leading principal minors of H are Hj 1j 6 0; Hj 2j P 0 and Hj 3j 6 0, indicating that the bordered

Hessian is negative semi-deﬁnite and the objective function is a concave function. In other words, the sufﬁcient conditions for a global max-imum are proven.

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3.2.2. Uniqueness of IBCC

For simplicity and without loss of generality, the mathematical model of [IBCC-S] is also examined under the case of single variable in three aspects of input, output and service. Likewise, the bordered Hessian matrix of objective function of [IBCC-S] can be derived as:

H ¼ 0 0 0 l uy uð 0Þ2 yl uy uð 0Þ2 0 2x1

_v

3 _{uy u} 0 ð Þ 0 xð

v

xÞ2 xyð

v

xÞ2 0 0 0 uy uð 0Þ2 y uy uð 0Þ2 l uy uð 0Þ2 xð

v

xÞ2 uy uð 0Þ2 2 wl uð 1Þ uy uð 0Þ3 2y wl uð 1Þ uy uð 0Þ3 yl uy uð 0Þ2 xyð

v

xÞ2 y uy uð 0Þ2 2y wl uð 1Þ uy uð 0Þ3 2y2ðwl u1Þ uy uð 0Þ3 ;

where, the signs of principal minors of H are Hj 1j 6 0; Hj 2j P 0; Hj 3j 6 0; Hj 4j P 0 and Hj 5j 6 0, indicating that the bordered Hessian is

neg-ative semi-deﬁnite and the objective function is a concave function. In other words, the sufﬁcient conditions for a global maximum are proven.

3.3. Benchmarking power property

The benchmarking power of DEA models, in this study, is defined as ‘‘the fewer number of efficient DMUs, the higher the benchmarking power of the model.” To show the benchmarking power of the IDEA model, we have to prove that the performance score evaluated by the IDEA model is lower than or equal to that evaluated by the conventional DEA model. On the other hand, if the IDEA model rates a DMU as overall efficient, the conventional DEA model should also rate the DMU as both ‘‘technically efficient” and ‘‘service effective.”

Let u

T;

v

Tand uS;wSrepresent the optimal set of virtual multipliers determined by the conventional DEA models in aspects of technical

efﬁciency and service effectiveness, respectively. Assuming that DMU R is evaluated as technical efﬁciency and service effectiveness by the conventional DEA models, implying that hTR¼

u TyR v TxR¼ 1 and oSR¼ w SlR u

SyR¼ 1. Two cases are discussed. First, if u

T¼ uS, then DMU R will be also

evaluated as overall efﬁcient by optimally setting

v

I ¼

v

T;uI ¼ uT¼ uS, and wI ¼ wS, then HR¼ u IyR v IxRþ w IlR u IyR¼ u TyR v TxRþ w SlR u SyR¼ 2. If u T–uS, due to

the uniqueness property of the proposed IDEA and conventional DEA model,uTyR v TxR> u IyR v TxR, if u I –uTand w SlR u SyR> w SlR u IyR, if w

I–wS. Thus, for the case

of

v

I ¼

v

T;uI ¼ uT;wI ¼ wS;HR¼ u TyR v TxRþ w SlR u

TyR<2 and for the case of

v

I ¼

v

T;uI ¼ uS;wI ¼ wS;HR¼ u SyR v TxRþ w SlR u

SyR<2. It can be concluded that the proposed IDEA model exhibits higher benchmarking power than the conventional DEA models.

4. Application

The main contribution of this study is to develop the novel IDEA approaches and to prove the theoretical properties exhibited in the proposed ICCR and IBCC models. To further demonstrate the applicability and superiority of our proposed IDEA models and to compare the benchmarking power with the conventional SDEA models (more speciﬁcally, SCCR and SBCC models associated with CRS and VRS tech-nologies), a real case analysis from Taiwanese intercity bus companies is conducted.

4.1. Data

Currently, there are 39 intercity bus companies in Taiwan. We take these bus companies as our case analysis. Potential variables of two factor variables (number of buses and operating network), two production variables (number of bus runs and bus-km) and four consump-tion variables (operating revenue, number of passengers, passenger-km and average number of on-board passengers per run) are consid-ered; all of these data are available from the annual report published by Ministry of Transportation and Communications.Table 1presents the descriptive data of these variables. To select important and relevant variables, regression analyses are further conducted by respectively regressing production variables on factor variables and consumption variables on production variables, respectively. The results are pre-sented inTable 2. Note that all the explanatory variables have shown positive and signiﬁcant effects on at least one of the associated depen-dent variables, suggesting the appropriateness of the above variables selected.

4.2. Efﬁciency scores

The optimal virtual multipliers corresponding to all variables are determined by the proposed IDEA approaches, ICCR and IBCC models, which jointly measure the overall efﬁciency scores of each bus company under CRS and VRS respectively, and by the SDEA approaches,

Table 1

Summary of descriptive data for the case of 39 Taiwanese intercity bus companies in 2007.

Item Factor variable Production variable Consumption variable

Number of buses Operating network (km) Number of bus runs

Vehicle-km Operating revenue (NT) Number of passengers Passenger-km Average number of on-board passengers per run Median 24 92 37,526 1,852,906 40,062,094 441,581 19,311,871 13.30 Std. Dev. 208 1,666 243,305 39,782,472 700,333,679 4,530,141 588,420,076 4.44 Max 1,083 7,810 1,266,527 126,078,237 3,574,792,434 26,330,194 2,217,682,256 17.59 Min 4 65 3,866 297,875 2,272,480 26,056 2,114,183 2.33

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SCCR and SBCC models, which separately measure the efﬁciency scores of each company under CRS and VRS respectively.Table 3compares the efﬁciency scores under CRS by ICCR and SCCR models; whereasTable 4compares the scores under VRS by IBCC and SBCC models.

Table 3

Scores of overall and individual efﬁciencies for each company under constant returns to scale.

DMU ICCR model SCCR model

Overall efficiency Technical efficiency Service effectiveness Technical effectiveness Technical efficiency Service effectiveness Technical effectiveness

1 1.464 0.572 0.892 0.510 0.573 0.916 0.579 2 1.555 0.570 0.985 0.561 0.699 1.000* 0.993 3 1.410 0.767 0.643 0.493 0.799 0.809 0.587 4 1.531 0.830 0.701 0.582 0.830 0.726 0.671 5 1.174 0.393 0.781 0.307 0.414 0.818 0.440 6 0.823 0.153 0.670 0.102 0.574 0.704 0.570 7 1.597 0.636 0.961 0.611 0.673 1.000* 1.000* 8 1.045 0.105 0.940 0.099 0.569 1.000* 0.328 9 0.679 0.266 0.461 0.121 0.285 0.456 0.175 10 1.976 1.000* 0.976 0.976 1.000* 0.976 1.000* 11 1.666 0.754 0.912 0.688 0.754 1.000* 1.000* 12 1.644 0.683 0.961 0.656 0.696 1.000* _0.891 13 1.933 1.000* 0.933 0.933 1.000* 1.000* 1.000* 14 1.634 0.634 1.000* 0.634 0.668 1.000* 0.884 15 1.518 0.771 0.747 0.576 0.772 0.851 1.000* 16 1.627 0.761 0.866 0.659 0.761 0.869 0.695 17 1.157 0.505 0.652 0.329 0.539 0.808 0.574 18 1.423 0.764 0.659 0.504 0.764 0.659 0.685 19 1.832 1.000* 0.832 0.832 1.000* 0.832 1.000* 20 1.836 0.890 0.946 0.842 0.902 1.000* 0.971 21 1.362 0.623 0.738 0.460 0.651 0.820 0.642 22 1.776 0.985 0.791 0.779 1.000* 0.948 0.925 23 2.000* 1.000* 1.000* 1.000* 1.000* 1.000* 1.000* 24 1.613 0.655 0.957 0.628 0.656 1.000* 0.917 25 1.258 0.469 0.789 0.370 0.502 0.798 0.592 26 1.261 0.570 0.691 0.394 0.627 0.693 0.747 27 1.281 0.478 0.803 0.384 0.478 0.804 0.471 28 0.765 0.212 0.553 0.117 0.213 0.653 0.260 29 1.569 1.000* 0.569 0.569 1.000* 0.738 0.983 30 1.393 0.625 0.768 0.480 0.668 0.826 0.720 31 1.240 0.605 0.635 0.384 0.634 0.712 0.609 32 1.151 0.375 0.776 0.291 0.407 0.781 0.411 33 1.443 0.443 1.000* 0.443 0.446 1.000* 0.922 34 2.000* 1.000* 1.000* 1.000* 1.000* 1.000* 1.000* 35 1.456 0.456 1.000* 0.456 0.456 1.000* 0.909 36 1.274 0.683 0.591 0.403 0.684 0.757 0.691 37 0.836 0.192 0.644 0.124 0.192 0.768 0.251 38 0.973 0.531 0.442 0.235 0.531 0.506 0.395 39 1.467 0.468 0.999 0.468 0.468 1.000* 0.909 Note:*

Denotes the DMU achieving corresponding efﬁciency (effectiveness). Table 2

Regression results from technical efﬁciency and service effectiveness perspectives.

Perspective Dependent variable Independent variable

Number of bus Operating network Number of bus run (in thousand) Bus-km (in thousand)

Technical efﬁciency Number of bus runs (in thousand) 0.04 0.24

(0.00) (4.72)

R2_{¼ 0:92}

Bus-km (in thousand) 266.91 10.02

(18.07) (5.42)

R2

¼ 0:99

Service effectiveness Operating revenue (in thousands) 521.24 14.64

(5.71) (26.25)

R2¼ 0:99

Number of passengers (in thousand) 25.46 0.05

(36.29) (11.73)

R2

¼ 0:99

Passenger-km (in thousand) 678.53 10.93

(9.73) (25.62)

R2_{¼ 0:99}

Average number of on-board passengers per run 0.01 0.00

(3.15) (1.27)

R2

¼ 0:35 Note: t values in parentheses.

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Note fromTable 3that only two bus companies (DMU 23 and DMU 34) are benchmarked as overall efficient by the proposed ICCR mod-el. In contrast, three bus companies (DMU 13, DMU 23 and DMU 34) are evaluated as overall efficient by the SCCR models. Namely, the proposed ICCR model has higher benchmarking power than the conventional SCCR models. By definition, the overall efficient score of the proposed ICCR model is equal to the sum of scores of technical efficiency and service effectiveness. However, the SCCR models do not possess this essential relationship. Also note fromTable 4that four companies are benchmarked as overall efficient by the proposed IBCC model, whereas nine companies have been assessed as overall efficient by the SBCC models. Once again, our proposed IBCC model demonstrates superior benchmarking power over the conventional SBCC model. In sum, the proposed IDEA approach is superior to con-ventional SDEA approach in terms of the benchmarking power.

Using the proposed IBCC model, we further examine the signs of u0ðu1Þ to identify the scale property for technical efﬁciency (service

effectiveness). The DMU is characterized with increasing returns to scale (IRS) if u

0<0ðu1<0Þ. If u0>0ðu1>0Þ, the DMU is decreasing

returns to scale (DRS). If u

0¼ 0ðu1¼ 0Þ, the DMU is constant returns to scale (CRS). The results are summarized inTable 5. Note that

most DMUs are characterized with IRS in their production or consumption, implying enlarging the scale may be required for most of the Taiwanese intercity bus companies to become more technical efﬁciency and service effective.

4.3. Slack analysis

To propose improvement strategies for the inefficient companies, slack values for each of the factor and consumption variables are com-puted according to [ICCR-S] models. The results are reported inTable 6. Except for two efficient companies (DMU23 and DMU34), most of the inefficient companies require either reducing factor amounts or raising consumption amounts. Taking DMU 9 as an example, decreas-ing 6.46% of buses, 9.08% of operatdecreas-ing network, or increasdecreas-ing operatdecreas-ing revenue by 11.05% would achieve efficiency frontier. Note that con-tradictory improvement suggestions are likely to emerge on the basis of slack analysis, provided that SDEA approaches are employed for the same case analysis.

Table 4

Scores of overall and individual efﬁciencies for each company under variable returns to scale.

DMU IBCC model SBCC model

Overall efficiency Technical efficiency Service effectiveness Technical effectiveness Technical efficiency Service effectiveness Technical effectiveness

1 1.490 0.579 0.911 0.528 0.579 0.916 0.599 2 1.230 1.000* 0.230 0.230 1.000* 1.000* 1.000* 3 1.132 0.533 0.599 0.319 0.835 0.823 0.589 4 1.535 0.826 0.709 0.586 0.831 0.731 0.707 5 1.315 0.489 0.826 0.404 0.489 0.831 0.475 6 0.977 0.248 0.729 0.180 0.803 0.729 0.719 7 1.880 1.000* 0.880 0.880 1.000* 1.000* 1.000* 8 0.903 0.138 0.765 0.106 0.579 1.000* 0.332 9 0.712 0.261 0.451 0.118 0.430 0.705 0.365 10 1.977 1.000* 0.977 0.977 1.000* 0.977 1.000* 11 1.751 0.754 0.997 0.752 0.754 1.000* 1.000* 12 1.710 0.710 1.000* 0.710 0.748 1.000* 0.893 13 1.439 1.000* 0.439 0.439 1.000* 1.000* 1.000* 14 1.998 1.000* 0.998 0.998 1.000* 1.000* 1.000* 15 1.539 0.771 0.769 0.592 1.000* 0.859 1.000* 16 1.652 0.798 0.853 0.681 0.798 0.874 0.696 17 1.204 0.204 1.000* 0.204 1.000* 1.000* 1.000* 18 1.355 0.690 0.665 0.459 0.850 0.665 0.715 19 1.836 1.000* 0.836 0.836 1.000* 0.836 1.000* 20 1.691 0.699 0.992 0.693 0.904 1.000* 0.973 21 1.446 0.610 0.836 0.510 0.673 0.836 0.643 22 1.823 0.996 0.828 0.824 1.000* 0.951 0.951 23 2.000* _1.000* _1.000* _1.000* _1.000* _1.000* _1.000* 24 1.659 0.659 1.000* 0.659 0.659 1.000* 0.917 25 1.313 0.512 0.801 0.410 0.519 0.801 0.594 26 1.389 0.686 0.703 0.483 0.697 0.703 0.763 27 1.282 0.478 0.804 0.385 0.482 0.807 0.475 28 1.572 0.571 1.000* 0.571 0.571 1.000* 0.571 29 1.768 1.000* 0.768 0.768 1.000* 0.768 1.000* 30 1.786 0.887 0.899 0.797 1.000* 1.000* 0.895 31 1.361 0.683 0.678 0.463 0.725 0.786 0.644 32 1.016 0.220 0.796 0.175 0.415 0.796 0.419 33 1.547 0.547 1.000* 0.547 0.575 1.000* 0.946 34 2.000* 1.000* 1.000* 1.000* 1.000* 1.000* 1.000* 35 2.000* 1.000* 1.000* 1.000* 1.000* 1.000* 1.000* 36 1.608 0.818 0.790 0.646 0.823 0.790 0.816 37 0.985 0.188 0.797 0.150 0.260 0.797 0.266 38 1.000 0.444 0.556 0.247 0.554 0.556 0.504 39 2.000* 1.000* 1.000* 1.000* 1.000* 1.000* 1.000* Note:*

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5. Generalized IDEA approaches 5.1. Models

Both of the abovementioned two novel IDEA approaches have adopted an additive form of technical efﬁciency and service effectiveness terms with equal weights. More generalized speciﬁcations of the proposed IDEA approaches can be reformulated by introducing unequal weights for both terms. The generalized ICCR model [GICCR] can thus be formulated as:

½GICCR Max u;v;w Hk¼

a

PR r¼1urykr PJ j¼1

v

jxkj ! þ ð1

a

Þ PS s¼1wqlkq PR r¼1urykr ! ð50Þ s:t: X R r¼1 uryir¼ XJ j¼1

v

j xij sij ; i ¼ 1; 2; . . . ; I; ð51Þ XS s¼1 wsðlisþ sisÞ ¼ XR r¼1 uryir; i ¼ 1; 2; . . . ; I; ð52Þ

v

jP0; j ¼ 1; 2; . . . ; J; ð53Þ wsP0; s ¼ 1; 2; . . . ; S; ð54Þ urP0; r ¼ 1; 2; . . . ; R; ð55Þ

where

a

is the weight for technical efﬁciency and ð1

aÞ is the weight for service effectiveness. If, for instance, the decision maker wishes to

place more emphasis on service effectiveness, then

a

should be set less than 0.5.

Similarly, the generalized IBCC model [GIBCC] can be formulated by introducing

a

as the weight for technical efﬁciency and ð1

aÞ as

the weight for service effectiveness.

Table 5

Returns to scale for each company.

DMU Technical efﬁciency Service effectiveness

u 0 RTS u1 RTS 1 0.101 IRS 0.123 IRS 2 2.120 DRS 2.760 DRS 3 0.035 IRS 0.004 DRS 4 0.075 IRS 0.088 IRS 5 0.484 IRS 0.522 IRS 6 0.568 IRS 0.617 IRS 7 0.740 IRS 0.804 IRS 8 0.022 IRS 0.011 DRS 9 0.996 IRS 1.000 IRS 10 0.383 IRS 2.940 DRS 11 0.002 DRS 2.352 DRS 12 0.191 IRS 0.170 IRS 13 0.154 DRS 0.159 DRS 14 0.107 DRS 4.315 DRS 15 1.000 IRS 1.000 IRS 16 0.107 DRS 0.103 DRS 17 0.999 IRS 1.000 IRS 18 0.360 IRS 0.390 IRS 19 0.139 IRS 0.154 IRS 20 0.063 IRS 0.093 IRS 21 0.389 IRS 0.425 IRS 22 0.110 IRS 0.133 IRS 23 0.000 CRS 0.000 CRS 24 0.065 DRS 0.270 DRS 25 0.149 IRS 0.124 IRS 26 0.264 IRS 0.350 IRS 27 0.001 DRS 0.001 IRS 28 0.999 IRS 1.000 IRS 29 0.787 IRS 0.816 IRS 30 0.632 IRS 0.672 IRS 31 0.159 IRS 0.200 IRS 32 0.967 IRS 0.981 IRS 33 0.249 IRS 0.307 IRS 34 0.000 CRS 0.000 CRS 35 1.000 IRS 1.000 IRS 36 0.622 IRS 0.676 IRS 37 1.000 IRS 1.000 IRS 38 1.000 IRS 1.000 IRS 39 3.924 DRS 0.704 DRS

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½GIBCC Max u;v;w

a

PR r¼1urykr u0 PJ j¼1

v

jxkj ! þ ð1

a

Þ PS s¼1wslks u1 PR r¼1urykr u0 ! ð56Þ s:t: PR r¼1uryir u0 PJ j¼1

v

jxij 6_1; _{i ¼ 1; 2; . . . ; I;} _ð57Þ PS s¼1wslis u1 PR r¼1uryir u0 6_1; _{i ¼ 1; 2; . . . ; I;} _ð58Þ

v

jP0; j ¼ 1; 2; . . . ; J; ð59Þ wsP0; s ¼ 1; 2; . . . ; S; ð60Þ urP0; r ¼ 1; 2; . . . ; R: ð61Þ Table 6

Slack values of factors and consumption under constant returns to scale (in percentage).

DMU Factor variable Consumption variable

Number of buses Operating network Operating revenue Number of passengers Passenger-km Average number of passengers on board per run

1 0.00 8.01 1.23 1.23 1.23 0.00 2 0.00 1.23 1.23 0.00 0.00 1.23 3 27.97 1.30 1.23 1.23 1.23 28.38 4 1.23 42.00 1.23 4.95 1.23 1.23 5 2.11 68.93 0.00 0.00 0.00 4.75 6 1.94 0.00 0.00 0.00 0.00 6.03 7 12.35 1.23 1.23 1.23 1.23 1.23 8 19.72 0.00 0.00 1.23 0.00 1.43 9 6.46 9.08 0.00 0.00 0.00 11.05 10 1.24 17.08 1.24 0.00 0.00 1.69 11 15.13 3.67 23.77 0.00 1.23 0.00 12 4.50 9.10 8.67 0.00 0.00 0.00 13 16.91 7.05 1.00 7.05 0.00 1.17 14 0.01 47.70 0.00 10.81 4.31 0.00 15 4.27 0.00 0.00 0.00 0.00 5.55 16 43.44 0.00 18.35 0.00 0.00 1.59 17 11.53 0.00 1.23 0.00 43.02 0.00 18 30.13 75.25 0.00 0.00 0.00 11.46 19 26.82 6.25 0.00 26.51 0.00 0.00 20 17.73 0.00 0.50 0.00 0.00 0.00 21 0.00 12.58 0.00 0.00 0.00 26.24 22 8.15 0.00 0.00 0.00 0.00 1.23 23 0.00 0.00 0.00 0.00 0.00 0.00 24 0.00 27.42 0.00 35.89 0.00 0.00 25 12.94 0.00 21.21 0.00 0.00 0.00 26 2.86 0.00 1.76 0.00 0.00 0.00 27 59.77 0.00 0.00 0.00 0.00 12.04 28 0.00 11.47 0.00 0.00 0.00 1.20 29 3.10 0.00 0.00 0.00 0.00 0.22 30 4.04 6.00 1.24 1.21 0.00 0.00 31 0.00 9.46 0.00 0.00 23.43 1.27 32 24.93 0.00 0.00 0.00 0.00 0.89 33 24.16 0.59 0.00 0.00 0.00 0.10 34 0.00 0.00 0.00 0.00 0.00 0.00 35 2.35 0.00 0.00 0.00 0.00 6.20 36 2.94 0.00 0.00 0.00 1.23 7.33 37 23.76 31.25 0.00 31.25 0.00 0.00 38 1.65 0.00 0.00 0.00 0.00 2.32 39 1.58 0.00 1.23 0.00 0.00 0.00 Table 7

Technical efﬁciency and service effectiveness for DMU 6 under various weights.

a Technical efﬁciency Service effectiveness

0.1 0.128 0.702 0.2 0.130 0.701 0.3 0.148 0.681 0.4 0.153 0.670 0.5 0.153 0.670 0.6 0.159 0.637 0.7 0.160 0.619 0.8 0.160 0.619 0.9 0.160 0.619

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Table 8

Slack values of factors and consumption for DMU6 under various weights (in percentage).

a Factor variable Consumption variable

Number of buses Operating network Operating revenue Number of passengers Passenger-km Average number of passengers on board per run

0.1 7.53 0.00 0.00 0.00 0.00 6.14 0.2 2.31 0.00 0.00 1.23 0.00 4.88 0.3 2.29 0.04 2.79 1.29 1.99 4.89 0.4 2.12 0.04 2.37 1.10 1.69 5.17 0.5 1.94 0.00 0.00 0.00 0.00 6.03 0.6 9.38 0.00 0.00 1.29 0.00 4.87 0.7 5.12 0.00 0.00 1.23 0.00 1.52 0.8 5.13 0.00 0.00 1.23 0.00 6.27 0.9 1.90 0.00 0.00 0.00 0.00 5.23 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 α Score Technical efficiency Service effectiveness

Fig. 2. Technical efﬁciency and service effectiveness for DMU 6 under various weights.

Table 9

Technical efﬁciency for each DMU under various weights.

DMU a(weight) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.490 0.542 0.546 0.572 0.572 0.572 0.572 0.572 0.572 2 0.570 0.570 0.569 0.570 0.570 0.570 0.570 0.570 0.570 3 0.537 0.538 0.538 0.572 0.767 0.799 0.799 0.799 0.799 4 0.751 0.777 0.805 0.830 0.830 0.830 0.830 0.830 0.830 5 0.259 0.377 0.379 0.393 0.393 0.393 0.411 0.411 0.414 6 0.128 0.130 0.148 0.153 0.153 0.159 0.160 0.160 0.160 7 0.584 0.584 0.584 0.584 0.636 0.636 0.673 0.673 0.673 8 0.105 0.105 0.105 0.105 0.105 0.516 0.517 0.550 0.568 9 0.266 0.266 0.266 0.266 0.266 0.275 0.275 0.285 0.285 10 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 11 0.637 0.637 0.637 0.637 0.754 0.754 0.754 0.754 0.754 12 0.683 0.683 0.683 0.683 0.683 0.683 0.683 0.683 0.683 13 0.315 0.929 0.951 0.951 1.000 1.000 1.000 1.000 1.000 14 0.634 0.634 0.634 0.634 0.634 0.634 0.634 0.634 0.642 15 0.520 0.630 0.771 0.771 0.771 0.771 0.771 0.772 0.772 16 0.748 0.761 0.761 0.761 0.761 0.761 0.761 0.761 0.761 17 0.499 0.499 0.505 0.505 0.505 0.505 0.505 0.505 0.505 18 0.763 0.763 0.763 0.763 0.764 0.764 0.764 0.764 0.764 19 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 0.628 20 0.633 0.843 0.855 0.890 0.890 0.890 0.892 0.902 0.902 21 0.381 0.567 0.567 0.567 0.623 0.651 0.651 0.651 0.651 22 0.651 0.651 0.900 0.984 0.985 1.000 1.000 1.000 1.000 23 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 24 0.634 0.634 0.634 0.634 0.655 0.655 0.656 0.656 0.656 25 0.462 0.462 0.462 0.469 0.469 0.469 0.502 0.502 0.502 26 0.570 0.570 0.570 0.570 0.570 0.627 0.627 0.627 0.627 27 0.478 0.478 0.478 0.478 0.478 0.478 0.478 0.478 0.478 28 0.163 0.163 0.163 0.163 0.212 0.212 0.213 0.213 0.213 29 0.706 0.706 0.799 1.000 1.000 1.000 1.000 1.000 1.000 30 0.247 0.604 0.604 0.625 0.625 0.625 0.646 0.646 0.663 31 0.222 0.564 0.587 0.605 0.605 0.605 0.605 0.613 0.626 32 0.370 0.370 0.370 0.375 0.375 0.407 0.407 0.407 0.407 33 0.443 0.443 0.443 0.443 0.443 0.443 0.443 0.443 0.443 34 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 35 0.456 0.456 0.456 0.456 0.456 0.456 0.456 0.456 0.456 36 0.477 0.477 0.477 0.543 0.683 0.684 0.684 0.684 0.684 37 0.150 0.150 0.192 0.192 0.192 0.192 0.192 0.192 0.192 38 0.383 0.431 0.447 0.531 0.531 0.531 0.531 0.531 0.531 39 0.462 0.468 0.468 0.468 0.468 0.468 0.468 0.468 0.468

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5.2. Case analysis

Various weights are attempted to DMU 6, for instance, for the same case analysis as above and the detailed results are presented inTable 7andFig. 2. These results show that technical efﬁciency increases and service effectiveness decreases as

a

increases. Moreover, slack values for factor and consumption variables associated with various weights are also detailed inTable 8. Looking into the slack values under the weights

a

¼ 0:1; 0:5 and 0.9, for example, one can see the improvement pressure to reduce the number of buses and to increase the average number of on-board passengers per run so as to achieve both efﬁciency and effectiveness are relieved as

a

increases. It indicates that DMU 6 is an efficiency-emphasis company—the preference of efficiency over effectiveness will lessen the improvement pressure for this com-pany. Such a case analysis demonstrates that change in weights can not only alter the performance measures but also influence the improvement strategies.

The technical efficiency and service effectiveness for each of 39 bus companies under various weights are further detailed inTables 9 and 10, respectively. Note fromTable 9that six DMUs are evaluated as technical efficient by ICCR model ða¼ 0:5Þ, but only four remain efficient by GICCR model (aranging from 0.1 to 0.9). Also note fromTable 10that five DMUs are evaluated as service effective by ICCR model ða¼ 0:5Þ and these DMUs remain effective by GICCR model (aranging from 0.1 to 0.9). It is interesting to note that only two com-panies, DMU 23 and DMU 34, are robustly overall efficiency because these two companies are originally benchmarked as both technical efficient and service effective by ICCR model (Table 3) and they remain technical efficient and service effective by GICCR model.

6. Concluding remarks

To more correctly evaluate the overall performance and to fully capture the insights of lacking efficiency or effectiveness for non-stor-able commodities, it is imperative to measure the efficiency and effectiveness simultaneously because both terms represent distinct as-pects of performance. As transport services are typically non-storable commodities, conventional measurement of technical efficiency or technical effectiveness only represents one aspect of the performance. The managers may also need to know the service effectiveness to understand how much consumption (passenger-miles or ton-miles) would be generated from the output (vehicle-miles). This paper pro-poses two novel integrated data envelopment analysis (IDEA) approaches, including the ICCR and IBCC models, to jointly measure the over-all performance for non-storable commodities from two aspects: technical efficiency and service effectiveness. We prove that the proposed

Table 10

Service effectiveness for each DMU under various weights.

DMU a(weight) 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.915 0.906 0.904 0.892 0.892 0.892 0.892 0.892 0.890 2 0.984 0.986 0.984 0.986 0.985 0.985 0.985 0.984 0.985 3 0.810 0.809 0.809 0.788 0.643 0.610 0.610 0.610 0.610 4 0.726 0.721 0.713 0.701 0.701 0.701 0.701 0.701 0.701 5 0.818 0.790 0.789 0.781 0.781 0.781 0.745 0.745 0.725 6 0.702 0.701 0.681 0.670 0.670 0.637 0.619 0.619 0.619 7 1.000 1.000 1.000 1.000 0.961 0.961 0.898 0.898 0.900 8 0.940 0.940 0.940 0.940 0.940 0.283 0.282 0.420 0.304 9 0.046 0.046 0.461 0.461 0.461 0.451 0.451 0.425 0.425 10 0.976 0.976 0.976 0.976 0.976 0.976 0.976 0.977 0.976 11 1.000 1.000 1.000 1.000 0.912 0.912 0.912 0.912 0.912 12 0.961 0.961 0.961 0.961 0.961 0.961 0.961 0.961 0.955 13 0.788 0.954 0.948 0.948 0.933 0.933 0.933 0.933 0.933 14 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 15 0.817 0.792 0.747 0.747 0.747 0.747 0.747 0.742 0.742 16 0.869 0.866 0.866 0.866 0.866 0.866 0.866 0.866 0.866 17 0.701 0.701 0.652 0.652 0.652 0.652 0.652 0.652 0.652 18 0.659 0.659 0.659 0.659 0.659 0.658 0.661 0.658 0.658 19 0.833 0.833 0.832 0.832 0.832 0.834 0.832 0.835 0.882 20 0.999 0.966 0.962 0.946 0.946 0.946 0.946 0.912 0.912 21 0.822 0.788 0.787 0.788 0.738 0.707 0.707 0.707 0.707 22 0.950 0.949 0.843 0.792 0.791 0.773 0.773 0.772 0.772 23 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 24 0.907 0.907 0.907 0.907 0.957 0.957 0.961 0.961 0.961 25 0.793 0.793 0.793 0.789 0.789 0.790 0.720 0.720 0.720 26 0.691 0.691 0.691 0.691 0.691 0.620 0.620 0.620 0.620 27 0.803 0.803 0.803 0.803 0.803 0.803 0.803 0.803 0.803 28 0.588 0.588 0.588 0.588 0.553 0.553 0.550 0.551 0.550 29 0.736 0.736 0.699 0.569 0.569 0.569 0.569 0.569 0.569 30 0.846 0.779 0.779 0.768 0.768 0.768 0.726 0.726 0.595 31 0.713 0.655 0.646 0.635 0.635 0.635 0.635 0.606 0.518 32 0.779 0.779 0.779 0.776 0.776 0.735 0.735 0.734 0.734 33 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 34 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 35 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 36 0.761 0.761 0.761 0.722 0.591 0.589 0.589 0.589 0.589 37 0.647 0.647 0.644 0.644 0.644 0.644 0.644 0.644 0.644 38 0.499 0.493 0.488 0.442 0.442 0.442 0.442 0.442 0.442 39 1.000 0.999 0.999 0.999 0.999 0.999 0.999 0.999 0.999

(13)

ICCR and IBCC models possess the essential properties of rationality, uniqueness, and benchmarking power. In addition to this theoretical contribution to the DEA literature, we also demonstrate that the proposed IDEA approaches have revealed higher benchmarking power than the conventional separate DEA approaches for practical applications. We therefore recommended our proposed IDEA approaches (with equal weights) or generalized IDEA approaches (with unequal weights) be used for non-storable commodities’ overall efﬁciency measurement.

Some directions for future studies can be identified. The IDEA models are specified in an additive form in the present paper, other spec-ification forms of IDEA models or even multi-objective IDEA models deserve further exploration. The present paper only demonstrates the overall efficiency measurement for bus transit services with two departments—production (technical efficiency) and sale (service effective-ness). It is a challenging issue to extend our proposed IDEA models to evaluate the overall performance for an enterprise with more than two departments vertically and/or horizontally interrelated, e.g., the supply chain systems within an enterprise, the postal mail pickup, processing and delivery, the air-express courier’s ground operation (pickup/delivery and processing) and air transport and hubs, among others. Although the proposed IDEA models have been proven and demonstrated with higher benchmarking power than the conventional SDEA models, further comparisons with other DEA models aiming at enhancement of benchmarking power, such as super-efficiency and cross-efficiency models (e.g.Banker et al., 1989; Chen, 2002), are worthy of investigation.

Acknowledgement

The authors are indebted to three anonymous referees who have provided very constructive comments and insightful suggestions to improve the quality of this paper.

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