A multi-objective distance friction minimization model for performance assessment through data envelopment analysis

European Journal of Operational Research 279 (2019) 132–142

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Decision Support

A multi-objective distance friction minimization model for performance assessment through data envelopment analysis Beibei Xiong a,b,c, Haoxun Chen c, Qingxian An d,∗, Jie Wu b a

School of Business Administration, Hunan University, Changsha 410082, China School of Management, University of Science and Technology of China, Hefei 230026, China c Industrial Systems Optimization Laboratory, Charles Delaunay Institute, UMR CNRS 6281, University of Technology of Troyes, Troyes 10004, France d School of Business, Central South University, Changsha 410083, China b

a r t i c l e

i n f o

Article history: Received 4 June 2018 Accepted 7 May 2019 Available online 13 May 2019 Keywords: Data envelopment analysis Distance friction minimization model Augmented ε -constraint method

a b s t r a c t The distance friction minimization (DFM) model proposed by Suzuki, Nijkamp, Rietveld, & Pels (2010) is an important data envelopment analysis (DEA) model for performance improvement. This model has been widely applied to improve the performance of government, finance and transportation sectors and energy-environment management. The DFM model is constructed to set input and output improvements based on the optimal weight vector of the traditional CCR model. However, non-unique optimal weight vectors of the CCR model may be obtained when different linear programming solvers are used. Thus, different values for the two distance friction (DF) objectives may be obtained. Furthermore, the DF values obtained using one optimal weight vector may be dominated by those obtained using another optimal vector. To address this issue, we propose an improved DFM model that considers all the possible optimal weight vectors of the CCR model to determine a set of non-dominated DF values definitively. Our DFM model is a multi-objective, quadratic and nonlinear programming model that is solvable by an augmented ε -constraint method that ensures all solutions are Pareto efficient. We demonstrate the applicability of our new model by using it to analyse China’s transportation sectors. © 2019 Elsevier B.V. All rights reserved.

1. Introduction Data envelopment analysis (DEA) is a mathematical programming-based approach used to evaluate the relative efficiencies of a group of homogenous decision-making units (DMUs) especially with multiple inputs and multiple outputs. This approach was originally proposed by Charnes, Cooper, & Rhodes (1978). Since its proposal, the theoretical developments of DEA, such as the BCC model (Banker, 1984), additive model (Charnes, Cooper, Golany, Seiford, & Stutz, 1985), slack-based measure (SBM) model (Aparicio, Ortiz, & Pastor, 2017; Tone, 2001), cross efficiency model (Liu, Song, & Yang, 2019; Sexton, Silkman, & Hogan, 1986), network DEA model (Färe & Grosskopf, 1996; Tone & Tsutsui, 2017; An, Wen, Li & Ding, 2019) and distance friction minimization (DFM) model (Suzuki, Nijkamp, Rietveld, & Pels, 2010), have shown remarkable progress. Besides, DEA has been applied in numerous areas, such as operational research, public policy, energy-environment management and regional development (Boussemart, Leleu, Shen, Vardanyan, & Zhu, 2019; Cooper, Seiford, & Tone, 2006; Emrouznejad, Jablonský, Banker, & Toloo, 2017; Galagedera, 2019; Zhou, Yang, Chen, & Zhu, 2018). ∗

Corresponding author. E-mail address: [email protected] (Q. An).

https://doi.org/10.1016/j.ejor.2019.05.007 0377-2217/© 2019 Elsevier B.V. All rights reserved.

DEA assesses the relative efficiency of a DMU by constructing a piecewise linear production frontier and then measures the distance between the observed DMU and the reference one, which serves as a benchmarking target. DEA can thus provide the efficiency analysis and benchmarking information for each evaluated DMU. A DMU is considered efficient when it is on the efficient production frontier but is considered inefficient otherwise. The reference point of each inefficient DMU can be obtained by increasing outputs, by reducing inputs or by increasing outputs and reducing inputs simultaneously. It should be noted that the selection of a suitable direction (path) for the improvement of inefficient DMU is crucial. If the improvement direction is not selected appropriately, the obtained efficiency and benchmarking information might give misleading indications about how to improve the efficiency. Directions provided by radial projection methods, such as the CCR and BCC models, are widely used to improve the efficiencies of DMUs. However, the application of radial projection methods in DEA has been criticised (e.g. Färe and Lovell, 1978). For example, the reference point that is obtained on the basis of the radial projection method is not necessarily efficient and may instead to be weakly efficient (Korhonen, Dehnokhalaji, & Nasrabadi, 2018). Moreover, the generated reference point is usually the furthest efficient projection of the evaluated DMU and may be not a

B. Xiong, H. Chen and Q. An et al. / European Journal of Operational Research 279 (2019) 132–142

representative projection of the evaluated DMU. Other popular methods, such as directional distance function method, provide directions by considering the preferences of the decision-maker. However, if the priori information for the improvement direction used by a DMU is biased, an incorrect benchmarking target for the DMU will be obtained. The distance friction minimization (DFM) model proposed by Suzuki et al. (2010) is widely used to address the above drawbacks. The DFM model improves the performance of an inefficient DMU by identifying its appropriate movement towards the benchmarking target on the efficient (production) frontier. This movement positions the benchmarking target to be as close as possible to the original inputs and outputs of the evaluated DMU. In this approach, a multi-objective quadratic programming (MOQP) problem with two distance friction functions given by a Euclidean distance metric is proposed to improve the efficiency of an inefficient DMU. This MOQP problem is solved by using the DFM Solver, which transforms a multi-objective problem into a monoobjective problem, to obtain the input/output distance values of the evaluated DMU. Such an approach provides a fresh perspective on efficiency improvement and can address input reduction and output augmentation simultaneously. The methodology and application of the DFM model have received increased attention recently. In reference to the seminal ideas of Banker and Morey (1986), Suzuki and Nijkamp (2013) extended the DFM approach to account for fixed factors or non-discretionary variables. Wanke, Barros, and Figueiredo (2014) used the DFM approach with fixed factors to assess the efficiency drives of the Brazilian motor carrier industry. Wanke and Barros (2015) applied a DFM approach to investigate the drives for output-increasing/input-saving potentials in the Brazilian railway industry for the period of 2004–2012. Suzuki and Nijkamp (2016) integrated an inflexible factor in the target-oriented DFM model to investigate the efficiency and the energy-environment-economic targets of the EU and of APEC and ASEAN countries. The book ‘Regional Performance Measurement and Improvement’ by Suzuki and Nijkamp (2017) introduced additional details, extended models, and several successful applications of DFM. DFM models not only retain the advantages of traditional DEA models but also have the advantages of setting the improvement projection on the basis of input/output distance without incorporating a priori information. DFM models are non-radial models that allow inefficient DMUs through input reduction and output augmentation to achieve efficient. However, these previous DFM models present one deficiency that may affect their successful application. For example, in the original DFM model proposed by Suzuki et al. (2010), an arbitrary optimal weight vector provided by the CCR model is selected and used directly in the MOQP model to obtain the minimum distance friction values. We will illustrate this process in Section 2.1. However, given that the optimal weight vectors of the CCR model are usually not unique (Chen, 2018; Kao & Liu, 2019), the DFM model may produce different distance friction values and different improvement directions for the same inefficient DMU if different linear programming solvers are used to solve the CCR model. Furthermore, the DF values obtained with an arbitrary optimal weight vector of the CCR model may be dominated by those obtained with another optimal weight vector. This phenomenon implies that the selection of the optimal weight vector of the CCR model in the DFM model greatly affects the DF values, input reduction and output augmentation. We will illustrate this phenomenon by an empirical example. Moreover, the weighted method will confer the following problems to the models: 1) the generation of efficient extreme solutions, 2) the sensitivity of results to the weight assigned to each objective function and 3) the production of a set of dominated objective function values for the MOQP problem (Mavrotas, 2009). To

133

overcome these deficiencies, we propose a new DFM model that consistently finds the reasonable optimal weight vector of the CCR model to ensure that the generated DF values are non-dominated. In our model, the optimal weight vector used in the MOQP (i.e. model 3 of Suzuki et al., 2010) to obtain the minimum DF values is not constant but is rather a constrained variable to guarantee that the efficiency of the inefficient DMU remains unchanged. Therefore, all of the possible optimal weight vectors of the CCR model are considered in the generation of the minimum DF values. Besides, the nonlinear constraints of our DFM model can also be transformed into linear constraints through variable substitution. Furthermore, in contrast to the weighted method, an augmented ε-constraint (AUGMECON) method is applied to solve the MOQP. The application of AUGMECON ensures that Pareto-efficient solutions are found via lexicographic optimization and that deficiencies of the weighted method are avoided. The rest of the paper is organized as follows. Section 2 describes an introduction to the DFM model. A numerical example is given in this section to illustrate the weakness of the DFM model. The new multi-objective distance friction minimization model is presented in Section 3. Section 3 also provides a discussion of the proposed application of the AUGMECON method to solve the new MOQP model. Section 4 presents the application of our improved model in a case study and provides a comparison of the results of our model with those of the DFM model. The conclusions and some directions for future research are given in Section 5. 2. Distance friction minimization (DFM) model and its weakness This section briefly introduces the DFM model proposed by Suzuki et al. (2010). It also illustrates the weaknesses of the DFM model. 2.1. DFM model by Suzuki et al. (2010) A set J DMUs where each DMU j (j = 1, … , J) uses M inputs xm j (m = 1, . . . , M ) to produce S outputs ys j (s = 1, . . . , S) is considered. vm and us are the weights given to the mth input and sth output, respectively. The DFM model proposed by Suzuki et al. (2010) comprises five steps, as follows: Step 1. Solve the CCR model, i.e. model (1), to evaluate DMUo and to obtain efficiency θo∗ and optimal weights u∗s and v∗m .

max θ =



us yso

s

s.t.

M 

vm xmo = 1,

m=1

−

M 

vm xm j +

m=1

S 

us ys j ≤ 0 ( j = 1, . . . , J ),

s=1

vm ≥ 0, us ≥ 0.

(1)

θo∗ ,

Step 2. Given the value of obtain the optimal slack values s−∗ , s+∗ by solving the following model:

max

λ,s− ,s+

ω = es− + es+

s.t. s− = θo∗ xo − X λ, s+ = Y λ − yo ,

λ ≥ 0, s− ≥ 0, s+ ≥ 0.

(2)

Models (1) and (2) state that a DMU can be categorized into three types by θo∗ , s−∗ and s+∗ , as follows: (1) When θo∗ = 1, s−∗ = s+∗ = 0, the evaluated DMUo is (strongly) efficient. The terms ‘efficient’ and ‘strongly efficient’ are synonymous when no specific explanation is provided.

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(2) When θo∗ = 1,s−∗ = 0 or s+∗ = 0, the evaluated DMUo is weakly efficient. The improvement solutions are generated by formulas

xˆo = θo∗ xo − s−∗ , yˆo = yo + s+∗ .

(3)

(3) When θo∗ = 1,s−∗ = 0 or s+∗ = 0, the evaluated DMUo is inefficient. The improvement solutions are generated by Steps 3, 4 and 5. Step 3. Obtain the input reduction distances and output augmentation distances through the following MOQP problem on the basis of the optimal solution of model (1), i.e., (v∗m , u∗s ) and the distance friction functions F r x and F r y introduced by Euclidean distance in weighted spaces:



M 

x

min F r =

x (v∗m xmo − v∗m dmo )2

m=1

 min F r y =

S 

2

y (u∗s yso − u∗s dso )

M 

x v∗m (xmo − dmo )=

m=1 S 

y u∗s (yso + dso )=

s=1

2θo∗ , 1 + θo∗

Number of outputs

Number of DMUs

1 1 1 1 2 2 2 2 3 3 3 3 4 4 4 4

1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

50 50 50 50 50 50 50 50 50 50 50 50 50 50 50 50

DMU1 DMU2

Number of DMUs with non-unique optimal weights 0 1 3 4 4 4 5 11 17 20 17 17 24 27 20 21

X1

X2

Y

2 4

2 4

6 4

2θo∗ , 1 + θo∗ Theorem 1. The optimal weights from model (1) may be nonunique.

x xmo − dmo ≥ 0 ( m = 1, . . . , M ), x dmo ≥ 0 ( m = 1, . . . , M ), y dso ≥ 0 ( s = 1, . . . , S ).

x dmo

Number of Inputs

Table 2 Input and output data of two DMUs.

s=1

s.t.

Table 1 Checking the non-uniqueness of optimal weights on randomly generated instances.

(4)

y dso

In model (4), and are the distance reduction and distance augmentation for input xmoand output yso, respectively, and are variables. v∗m and u∗s are constants obtained through Step 1. The first and second constraints refer to the target values of input reduction and output augmentation, respectively. The third constraint is a limitation of input reduction and indicates that the input value cannot be negative. x∗ , d y∗ ) with the optimal solution of model Step 4. Denote (dmo so (4) to express the friction minimization solution for an inefficient DMUo by using formula (5), x∗ x∗mo = xmo − dmo (m = 1, . . . , M ), y∗ y∗so = yso + dso ( s = 1, . . . , S )

θo∗

(5) x∗mo,

y∗so

Step 5. Set = 1 and replace xmo, yso by in model (2) to obtain the related optimal s−∗∗ , s+∗∗ . The solution for improving the efficiency of an inefficient DMUo can be now expressed by ∗ −∗∗ x∗∗ ( m = 1 , . . . , M ), mo = xmo − s ∗ +∗∗ y∗∗ ( s = 1 , . . . , S ). so = yso + s

(6)

Through the DFM model, it can be seen that the optimal solution of model (1) is critical for the rest steps. Doyle and Green (1994) indicated that “the weights, which maximize DMU k’s simple efficiency, may not be unique”, that is, the weights for CCR model may not be unique. Chen (2018) also pointed out that “the optimal input and output weights of the CCR models involved in the cross evaluation may be not unique” when he studied the cross efficiency. Based on the Balinski-Tucker’s Simplex Tableau, Tijssen and Sierksma (1998) gave a corollary about the conditions for optimal solutions of a linear program in different scenarios. Appa (2002) also indicated that some linear programs may have multiple optimal solutions. Model (1) is just the CCR model which is a linear program, thus the corollary in Tijssen and Sierksma (1998) is also suitable for CCR model. Then, we have the following theorem without proof.

Besides, In order to show that the non-uniqueness of optimal weight vector of CCR model is not rare, except for giving the above-mentioned example, we also test randomly generated instances.In this test, both the number of inputs and the number of outputs of the CCR model are selected from 1 to 4. There are totally 16 combinations of input number and output number. For each combination, the data sets of 50 DMUs are randomly generated. The values of all inputs and outputs of each DMU are randomly generated according to the uniform distribution defined on [0, 100]. The results are given in Table 1. It should be noted that the DMUs with one input and one output always have the unique optimal weight because the optimal weight is set for satisfying wx0 = 1 and uy0 = θo∗ . From this table, we can find that the optimal weight of a DMU may not be unique especially when it has the large number of inputs and outputs. This shows the non-unique optimal weights commonly exist for the CCR model especially when the number of inputs and outputs is larger. A small numerical example in the following section can also verify the non-uniqueness of the optimal weights to model (1).

2.2. Weakness of the DFM model Multiple optimal solutions to the CCR model may yield difx∗ , d y∗ ) and dissimilar minimum DF ferent optimal solutions (dmo so x ∗ y ∗ values (F r , F r ) for model (4) through the DFM method. This phenomenon can be illustrated through a numerical example, we present a small numerical example with only two DMUs. Each DMU uses two inputs X1 , X2 to produce one output Y . The data are listed in Table 2. Firstly, we use the CCR model (i.e., model (1)) to evaluate the efficiencies of DMUs 1 and 2. The efficiency of DMU 1 is 1 and that of DMU 2 is 1/3. Thus, DMU 1 is efficient, and DMU 2 is inefficient. In accordance with model (1), the following linear programming is given to obtain the optimal solutions for DMU 2:

B. Xiong, H. Chen and Q. An et al. / European Journal of Operational Research 279 (2019) 132–142 M 

Table 3 Comparison of the results obtained with the three sets of optimal weight vectors. Optimal vectors

d1x∗

d2x∗

d y∗

F r x∗

F r y∗

(0, 1/4, 1/12) (1/4, 0, 1/12) (1/12, 2/12, 1/12)

[0,4] 2 1

2 [0,4] 5/2

2 2 2

1/2 1/2 √ 1/2 2

1/3 1/3 1/3

vm xmo = 1,

m=1 M 

vm xm j −

max θ2 = 4u

y dso ≥0,

vm ≥ 0, us ≥ 0 ( m = 1, . . . , M ), ( s = 1, . . . , S ).

4v1 + 4v2 − 4u ≥ 0,

v1 , v2 ≥ 0, u ≥ 0.

(7)

(v∗1 , v∗2 , u∗ ),

The optimal weight vector of inputs and output which could be any vector that satisfies the condition 1 of{v∗1 + v∗2 = 14 , u∗ = 12 , v∗1 , v∗2 ≥ 0, u∗ ≥ 0}, could be obtained by using model (7). Thus, the optimal weight vectors are nonunique. Here, three sets of optimal weight vectors, namely, (0, 1/4, 1/12), (1/4, 0, 1/12) and (1/12, 2/12, 1/12), are selected. The optimal input reduction distance, the optimal output augmentation distance and the minimum DF values for the three sets of optimal weight vectors are calculated in accordance with Step 3 of the DFM model and are shown in Table 3. Table 3 shows that different optimal weight vectors provide different optimal solutions to model (4) in Step 3. By considering the minimum DF values as two objectives, we find that the (F r x∗ , F r y∗ ) based on the optimal solution vectors (0, 1/4, 1/12) and (1/4, 0, 1/12) are all dominated by those based on (1/12, 2/12, 1/12). This result indicates that the selection of optimal solution vectors is critical for Step 3 of the DFM model. In addition, the projection points on the efficient production frontier obtained through Steps 4 and 5 for the three optimal weight vectors are also different and are (2, 2, 6), (2, 2, 6) and (3, 3/2, 6).

 min F r x =

 min F r y =

M 

(αmo − pmo )2

m=1 S 

(βso − qso )2

s=1 S 

qso =

s=1 M 

θo∗ (1−θo∗ ) , 1 + θo∗

αmo = 1,

m=1 S 

This section will introduce an improved DFM model which is a new multi-objective distance friction minimization model. We apply the AUGMECON method to solve the new MOQP in the improved DFM model.

βso = θo∗ ,

s=1 M 

vm xm j −

m=1

3.1. Model The numerical example presented in the previous section indicates that the optimal weight vector of model (1) is not unique and that the selection of different optimal weight vectors will likely result in different minimum DF values for Step 3 of the DFM model. Moreover, the minimum DF values of model (4) may be dominated. To solve the weaknesses of the DFM model, we propose a new MOQP, which can minimize the DF values F r x and F r y under the condition that its weight vector is the optimal solution of model (1). The new MOQP model is presented below.



M 

x

min F r =

x (vm xmo − vm dmo )2

m=1



S 

2

y (us yso − us dso )

s=1 M 

x vm (xmo − dmo )=

m=1

us (yso +

y dso

2θ , 1 + θo∗ ∗ o

2θo∗ )= , 1 + θo∗

(8)

In model (8), vm and us are variables. By contrast, in model (4), vm and us are constants. θo∗ is the efficiency of DMUo obtained by solving model (1). In this model, the third to fifth constraints are used to guarantee that the efficiency of the evaluated DMUo remains unchanged during the calculation of the minimum DF values. The other parameters retain the same meanings as in model (4). This problem is a multi-objective, nonlinear programming problem. Model (8) can be transformed into the following x = p , u y = β and u d y = model by settingvm xmo = αmo, vm dmo mo s so so s so qso :

3. A new multi-objective distance friction minimization model

s=1

us yso = θo∗ ,

x x xmo − dmo ≥ 0, dmo ≥ 0,

2v1 + 2v2 − 6u ≥ 0,

S 

us ys j ≥ 0 ( j = 1, . . . , J ),

s=1

s=1

s.t. 4v1 + 4v2 = 1,

s.t.

S 

m=1 S 

min F r y =

135

S 

us ys j ≥ 0 ( j = 1, . . . , J ),

s=1

vm xmo = αmo, us yso = βso, 0 ≤ pmo ≤ α mo , qso ≥ 0, βso ≥ 0 (m = 1, . . . , M ), (s = 1, . . . , S ).

(9)

Model (9) is a multi-objective mathematical programming (MOMP) problem. Hwang and Masud (1979) and Steuer (1986) suggested that methods for solving MOMP problems could be classified into three categories, namely, prior methods, interactive methods and generation methods. Mavrotas (2009) stated that generation methods provide remarkable advantages over prior and interactive methods. The weighting method and the ε − constraint method, which can provide a representative subset of the Pareto set, are the most widely used generation methods. ∗∗ , p∗∗ , β ∗∗ , q∗∗ , v∗∗ , u∗∗ ) is a Pareto-efficient Definition 1. (αmo mo so so m s  , p , β  , q , v , u ) solution of model (9) if no other solution (αmo  mo so so m s M M 2   ∗∗ ∗∗ 2 that satisfies m=1 (αmo − pmo ) ≤ m=1 (αmo − pmo ) ,



S

s=1

 − q )2 ≤ (βso so

inequality exists.



S

s=1

∗∗ − q∗∗ )2 with at least one strict (βso so

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B. Xiong, H. Chen and Q. An et al. / European Journal of Operational Research 279 (2019) 132–142 Table 4 Payoff table obtained through the lexicographic optimization of the objective functions.

3.2. AUGMECON method Suzuki et al. (2010) used the DFM solver to solve MOMP. In contrast to the approach taken by Suzuki et al. (2010), we apply AUGMECON to solve model (9) given that the simple additive weighting method has several deficiencies. The AUGMECON method was developed on the basis of the well-known ε constraint method, which can provide an approximation of the Pareto frontier (Ehrgott, 2005; Mavrotas, 2009). We introduce this method briefly. Without loss of generality, we consider the following MOMP with minimized objectives:

min ( f1 (z ), f2 (z ), . . . , ft (z )) s.t. z ∈ H.

(10)

where z is the vector of decision variables, f1 (z ), . . . , ft (z ) are the t objectives and H is the feasible region. To ensure that the Pareto-optimal solution of the original multiobjective model is obtained, the AUGMECON method transforms t-1 objective function to equalities by explicitly incorporating the appropriate slack or surplus variables. Meanwhile, the slack or surplus variable is used as a second term (with low priority in a lexicographic manner) in the objective function. As a result, the aforementioned MOMP model can be re-established as follows:

min F r x min F r y



min z =

M 

(αmo − pmo )2 − σ

m=1

 s.t.

S 

LX UX

UY LY

s¯ r

(βso − qso )2 + s¯ = e,

s=1 M 

pmo =

m=1

s.t.

S 

f 2 (z ) + s¯2 = e2 , ...

qso =

s=1 M 

(11)

where e2 , . . . , et are the satisfaction levels that stipulate the maximum requirement on the constrained objectives. δ is an adequately small number, and rl (l = 2, . . . , t ) is the range length of the lth objective. s¯l (l = 1, . . . , t ) refers to the slack between the lth objective value and el (l = 2, . . . , t ). The lower and upper bounds of the objectives are obtained through the lexicographic optimization method. This method can generate a set of Pareto-efficient solutions by properly adjusting the number of grid points in each objective function range. The AUGMECON method has several advantages over the weighting method, as follows: (1) The AUGMECOM method can generate multiple pareto-optimal solutions even all pareto-optimal solutions, whereas the weighting method can generate a single pareto-optimal solution for the given weight. (2) In the AUGMECON method, the scaling of the objective functions does not affect the obtained results, while in the weighting method, the scaling has strong influence in the obtained results. (3) It is difficult to determine the weights in the weighting method. Different weights may lead to different solutions. (4) The AUGMECON method can control the number of generated efficient solutions by properly adjusting the number of grid points in each objective function range, but the weighting method cannot realize this so easily. In the following work, we will illustrate the application of the AUGMECON method in our improved DFM model. Firstly, we obtain the lower and upper bounds of the first and the second objective functions of model (9) through the lexicographic optimization method. That is, we optimize the first objective function (of high priority) to obtain min F r x = LX. Then, we optimize the second objective function by adding the constraint F r x = LX to retain the optimal solution of the first optimisation, thereby obtaining UY . Analogously, we optimize the second objective function (of high priority), thereby obtaining min F r y = LY . Then, we optimize the first objective function by adding the constraint F r y = LY to retain the

F ry

optimal solution of the second objective function, thereby obtaining UX. The payoff table obtained through the lexicographic optimization of the two objective functions is shown in Table 4. Then, we divide the range of the second objective function into C equal intervals (i.e. C + 1 grid points). We set e as a value from the LY to UY with the step increase of (UY − LY)/C. If C is sufficiently large, then we can approximately obtain all the Pareto-efficient set solutions to model (9) through the following single-objective programming model (12).

min ( f1 (z ) − δ × (s¯2 /r2 + ... + s¯t /rt ))

ft (z ) + s¯t = et .,

F rx

1 − θo∗ , 1 + θo∗

θo∗ (1−θo∗ ) , 1 + θo∗

αmo = 1,

m=1 S 

βso = θo∗ ,

s=1 M 

vm xm j −

m=1

S 

us ys j ≥ 0 ( j = 1, . . . , J ),

s=1

vm xmo = αmo, us yso = βso, 0 ≤ pmo ≤ α mo , qso ≥ 0, βso ≥ 0 (m = 1, . . . , M ), (s = 1, . . . , S ).

(12)

In model (12), σ is an adequately small positive number and is usually set as 0.0 0 0 0 01, r is the range of the second objective function caculated from the payoff table and is calculated as r = UY − LY. s¯ refers to the slack between the second objective value and e, and s¯ r is used to avoid scaling problems. Moreover, the second term in the objective function avoids the generation of weakly efficient ∗∗ , p∗∗ , β ∗∗ , q∗∗ , v∗∗ , u∗∗ , s¯∗∗ ) is the solutions. We assume that (αmo mo so so m s  M ∗∗ ∗∗ 2 optimal solution of model (12) and F r x∗∗ = m=1 (αmo − pmo ) , F r y∗∗ =



S

s=1

∗∗ − q∗∗ )2 . Then, we have the following theorem: (βso so

∗∗ , p∗∗ , β ∗∗ , q∗∗ , v∗∗ , u∗∗ , s¯∗∗ ) is the optimal soTheorem 2. If (αmo mo so so m s ∗∗ , p∗∗ , β ∗∗ , q∗∗ , v∗∗ , u∗∗ ) must be a lution of model (12), then (αmo mo so so m s Pareto-efficient (efficient) solution of model (9). ∗∗ , p∗∗ , β ∗∗ , q∗∗ , v∗∗ , u∗∗ , s¯∗∗ ) be the optimal soProof. Let (αmo mo so so m s ∗∗ , p∗∗ , β ∗∗ , q∗∗ , v∗∗ , u∗∗ ) is not a lution of model (12). If (αmo mo so so m s Pareto-efficient solution of model (9), then an alternative so , p , β  , q , v , u ) and that satisfies lution denoted by (αmo mo so so m s  M  2 M x x∗∗ , F r y =   ∗∗ ∗∗ 2 Fr = m=1 (αmo − pmo ) ≤ m=1 (αmo − pmo ) = F r

B. Xiong, H. Chen and Q. An et al. / European Journal of Operational Research 279 (2019) 132–142



S

  2 s=1 (βso − qso ) ≤



S

s=1

∗∗ − q∗∗ )2 = F r y∗∗ , with at least one (βso so

strict inequality exists.  , p , β  , q , v , u ) must satisfy the constraints of Thus, (αmo mo so so m s model (9), that is,

 

S 

(β  so − q so )2 = F rx



s=1 S 

(β  so − q so )2 = F ry

p mo =

m=1 S  s=1 M 

q so =

y∗∗ y∗∗ so = yso + dso (s = 1, . . . , S ).

θo∗ (1−θo∗ ) , 1 + θo∗

∗∗ −∗∗∗ x∗∗∗ (m = 1, . . . , M ), mo = xmo − s ∗∗ +∗∗∗ y∗∗∗ (s = 1, . . . , S). so = yso + s

α  mo = 1,

m=1

(15)

The difference between our improved DFM model and the DFM model is illustrated in Fig. 1.

β  so = θo∗ ,

s=1 M 

(14)

∗∗ Step 5. Set θo∗ = 1 and replace xmo,yso by x∗∗ mo ,yso in model (2) to −∗∗∗ + ∗∗∗ obtain the related optimal s ,s . The solution for improving the efficiency of an inefficient DMUo can now be expressed as:

1 − θo∗ , 1 + θo∗

m=1 S 

Step 3. Model (12) shows that we can obtain the Pareto-efficient ∗∗ , p∗∗ , β ∗∗ , q∗∗ , v∗∗ , u∗∗ ) of model (9) and the minisolution (αmo mo so so m s x∗∗ and mum DF values F r x∗∗ and F r y∗∗ . The optimal distances dmo y∗∗ x = dso of model (8) can be calculated by usingvm xmo = αmo, vm dmo y pmo, us yso = βso and us dso = qso. x∗∗ and d y∗∗ by the optimal solution of model Step 4. Denote dmo so (8). Thus, the friction minimization solutions for an inefficient DMUocan be expressed using the following formulas: x∗∗ x∗∗ mo = xmo − dmo (m = 1, . . . , M ),



s=1 M 

137

v m xm j −

S 

4. Application in the China’s transportation industry

u s ys j ≥ 0

( j = 1, . . . , J ),

s=1

v m xmo = α  mo, u s yso = β  so, 0 ≤ p mo ≤ α  mo , q so ≥ 0, β  so ≥ 0 (m = 1, . . . , M ), (s = 1, . . . , S ). Setting s¯ = e −



S

s=1



 − q )2 = e − F r y , (α  , p , β  , q , (βso so mo mo so so

vm , us , s¯ ) is a feasible solution of model (12). Given   F r x∗∗ ≥F r x , F r y∗∗ ≥ F r y with at least one strict inequalM s¯   2 ity, is strictly smaller than m=1 (αmo − pmo ) − σ r  M ∗∗ 2 s¯ ∗∗ ∗∗ m=1 (αmo − pmo ) − σ r . This condition contradicts the initial ∗∗ , p∗∗ , β ∗∗ , q∗∗ , v∗∗ , u∗∗ , s¯∗∗ ) is the optimal soassumption that (αmo mo so so m s lution of model (12). Thus, the optimal solution of model (12) can produce the Pareto-efficient solution of model (9).  On the basis of model (9), we can obtain the following theorem without proof. Theorem 3. The vector (F r x∗∗ , F r y∗∗ ) obtained from model (12) is the non-dominated vector of the two objective values of model (9). The process for the improved DFM approach is given by the following algorithm. Step 1. Solve the CCR model (i.e. model 1) to evaluate DMUo and obtain the optimal objective value θo∗ . Step 2. Given the value of θo∗ , obtain the optimal slack values s−∗ , s+∗ by solving model (2). Categorize each DMU on the basis of θo∗ ,s−∗ and s+∗ as follows: (1) When θo∗ = 1, s−∗ = s+∗ = 0: The evaluatedDMUo is efficient. (2) When θo∗ = 1,s−∗ = 0 or s+∗ = 0: The evaluated DMUo is weakly efficient. The improvement solutions are generated by formulas

xˆo = θo∗ xo − s−∗ , yˆo = yo + s+∗ .

(13)

(3) Whenθo∗ = 1,s−∗ = 0 or s+∗ = 0: The evaluated DMUo is inefficient. The improvement solutions are generated through Steps 3, 4 and 5.

Many DEA works have focused on the analysis of transportation sectors. In this paper, we analyse China’s transportation sectors on the basis of their characteristics and their relative weights. We apply the improved DFM model to identify the efficient DMUs that determine the efficient production frontier upon which inefficient DMUs will be projected (i.e. the peers that inefficient transportation sectors need to follow) and to determine the projection of the inefficient DMUs with the minimum DF values. We provide a new perspective to policymakers on the directions that inefficient transportation sectors can follow to improve their efficiency.

4.1. Input and output variables We apply our improved DFM model to examine the efficiencies of 31 transportation sectors in mainland China and offer directions for improving the efficiency of inefficient sectors. We consider capital and labour as two basic inputs in accordance with the principles of economics. In reference to previous works (e.g. Ng & Chang, 2003; Shi, Bi, & Wang, 2010; Chang et al., 2013; Wu et al., 2016), we use the amount of the fixed capital investment of the transportation sector to represent capital stock input because capital stocks statistics by industry in China are unavailable. Therefore, the amount of the fixed capital investment of the transportation sector (FCIT) (billion yuan) and the number of the employed labour of the transportation sector (NELT) (thousand persons) are considered input variables. Gross domestic product by transportation sector (GDPT) (billion yuan), passenger turnover volume (PTV) (100 million passenger-km) and freight turnover volume (FTV) (100 million ton-km) are selected as output variables. These input and output variables have been widely used in previous studies (Wu et al., 2016; Liu, Zhang, Zhu, & Chu, 2017). The datum related to all variables are available in the China Statistical Yearbook 2017 (National Bureau of Statistics of China, 2017). Table 5 shows that according to the results of model (1), Beijing, Tianjin, Hebei, Liaoning, Shanghai, Anhui, Jiangxi, Henan, Hunan and Shaanxi are strongly efficient or weakly efficient compared with other sectors. Specifically, the results of model (2) show that Shaanxi is weakly efficient, whereas all other nine provinces are strongly efficient. China’s transportation sectors have difference efficiencies.

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Fig. 1. Analytical framework of the DFM model and the improved DFM model. Table 5 Raw data for China’s transportation sectors and their CCR efficiencies.

Beijing Tianjin Hebei Shanxi Inner Mongolia Liaoning Jilin Heilongjiang Shanghai Jiangsu Zhejiang Anhui Fujian Jiangxi Shandong Henan Hubei Hunan Guangdong Guangxi Hainan Chongqing Sichuan Guizhou Yunnan Tibet Shaanxi Gansu Qinghai Ningxia Xinjiang

NELT

FCIT

GDPT

PTV

FTV

CCR efficiency

26.1 4.3 30.2 17.1 17.5 62.8 22.4 8.4 38.9 65.9 41.6 19.6 19 26.4 57.9 17.9 42.9 13 57.8 18.5 4.6 20 21.4 9.6 12.9 1 7.7 6.5 1.2 2.6 7.9

761.6 735.1 2095.3 912.6 1427.7 661.2 1170 1134.6 944.9 2551 2581.9 1628.5 2505.4 965.4 2982.2 1954.5 2833 1944.4 3032.3 1849.8 467.1 1630.7 3738 1779.9 2577.5 542.3 1584.6 1100 589.9 367.7 836.1

1060.97 725.31 2369.27 930.75 1141.97 1245.27 558.38 758.01 1237.32 2837.16 1774.37 826.9 1689.82 796.47 2725.41 1938.06 1297.48 1356.56 3209.72 855.67 199.89 848.22 1472.57 987.47 328.41 31.26 771.77 271.25 94.99 205.75 567.54

268.49 262.05 1238.12 360.56 374.97 936.09 431.27 471.08 214.42 1468.48 1074.99 1187.36 593.27 970.67 1188.92 1684.27 1232.32 1500.85 1887.45 743.83 120.36 506.29 941.6 674.86 446.06 39.78 755.67 613.39 125.11 109.72 458.06

825.43 2302.32 12,332.68 3565.46 4341.74 12,113.49 1478.52 1532.54 19,317.76 7653.78 9789.33 10,896.37 6070.59 3897.75 8884.34 7383.54 5922.87 4056.86 21,801.65 4260.41 1060.75 2968.29 2504.11 1482.28 1600.07 124.63 3444.92 2170.05 475.8 819.94 1803.88

1.0 0 0 1.0 0 0 1.0 0 0 0.859 0.731 1.0 0 0 0.429 0.684 1.0 0 0 0.891 0.615 1.0 0 0 0.664 1.0 0 0 0.764 1.0 0 0 0.481 1.0 0 0 0.937 0.507 0.463 0.481 0.502 0.772 0.325 0.359 1.0 0 0 0.886 0.997 0.615 0.676

4.2. Comparison of the efficiency improvement projections provided by the improved DFM and DFM models We apply the improved DFM model to find an optimal solution that simultaneously minimizes the two DF values of inefficient transportation sectors, such as Inner Mongolia Province. For comparison, we use the improved DFM and DFM models to identify the two minimum DF values of the 21 inefficient transportation sectors. The results are shown in Table 6. Columns 2 to 7 of Table 6 show the DF values F r x and F r y obtained through our improved DFM model with three different values of e, and Columns 8 and 9 show the minimum DF values

obtained through the DFM model. In Step 3 of the improved DFM model, we use model (12) to calculate the Pareto-efficient solutions of model (9) by dividing the range of the second objective function F r y into two equal intervals (i.e. three values of e). By increasing e from LY to UY with the step size of (UY − LY )/2, we obtain three groups of minimum DF values, which are shown in Columns 2 to 3 (F r x (1 ), F r y (1 )), Columns 4 to 5 (F r x (2 ), F r y (2 )) and Columns 6 to 7 (F r x (3 ), F r y (3 )). As e increases, the value of the first objective function F r x does not increase and that of the second objective function F r y does not decrease. Moreover, in some sectors, F r y increases although F r x does not change. Actually, in these sectors, the decrease of F r x is negligible. This decrease cannot be

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Table 6 Comparison of two DF values obtained through the improved DFM model and DFM model. Improved DFM

Shanxi Inner Mongolia Jilin Heilongjiang Jiangsu Zhejiang Fujian Shandong Hubei Guangdong Guangxi Hainan Chongqing Sichuan Guizhou Yunnan Tibet Gansu Qinghai Ningxia Xinjiang

DFM

F r x (1 )

F r y (1 )

F r x (2 )

F r y (2 )

F r x (3 )

F r y (3 )

F rx

F ry

0.74538 0.68536 0.46097 0.60399 0.76108 0.60513 0.68407 0.68598 0.53641 0.78390 0.47802 0.44756 0.48582 0.66829 0.87155 0.49020 0.52795 0.84342 0.99874 0.68548 0.66005

0.79324 0.61715 0.19660 0.39386 0.76962 0.31480 0.52985 0.66123 0.29244 0.64862 0.19836 0.17465 0.28189 0.23714 0.47598 0.11254 0.13389 0.60054 0.74510 0.30421 0.48864

0.74525 0.68261 0.46097 0.60399 0.76108 0.60460 0.68407 0.68598 0.53641 0.78390 0.47623 0.44756 0.48582 0.66829 0.87155 0.49020 0.52795 0.84342 0.99874 0.68545 0.65717

0.79484 0.62586 0.19660 0.39386 0.76962 0.31845 0.52985 0.67746 0.29244 0.64873 0.20627 0.17465 0.28189 0.23724 0.48216 0.11254 0.13389 0.60054 0.74510 0.30438 0.49122

0.74525 0.68258 0.46097 0.60399 0.76108 0.60447 0.68407 0.68598 0.53641 0.78377 0.47601 0.44756 0.48582 0.66829 0.87155 0.49020 0.52795 0.84342 0.99874 0.68545 0.65716

0.79637 0.63411 0.19660 0.39386 0.76962 0.32197 0.52985 0.67746 0.29244 0.64883 0.21258 0.17465 0.28189 0.23735 0.48699 0.11254 0.13389 0.60054 0.74510 0.30438 0.49376

0.74525 0.68258 0.46097 0.60399 0.76108 0.60447 0.68407 0.68598 0.53641 0.78377 0.47601 0.44756 0.48582 0.66829 0.87155 0.49020 0.52795 0.84342 0.99874 0.68545 0.65716

0.86641 0.760 0 0 0.58045 0.77009 0.89985 0.71344 0.70983 0.78597 0.59510 0.94775 0.66583 0.63136 0.59875 0.65550 0.83500 0.52667 0.54817 0.91126 0.99790 0.72476 0.72975

observed when we round the number to four decimal places, as shown in Table 6. For example, the decrease in F r x of Sichuan and Guizhou cannot be observed after rounding. Comparing the two minimum DF values obtained by the DFM model and the improved DFM model shows that the first minimum DF values of these sectors are almost identical. Nevertheless, the second minimum DF values of these sectors obtained through the improved model are considerably smaller than those obtained through the DFM model. Table 6 shows that the minimum DF values obtained through the DFM model are not non-dominated. The minimum DF values obtained through our improved DFM model dominate those obtained through the DFM model. For example, the set of the minimum DF values of Sichuan from the DFM model is (0.66829, 0.65550), whereas those obtained through our improved DFM model are (0.66829, 0.23714), (0.66829, 0.23724) and (0.66829, 0.23735). Thus, the solutions obtained through our improved DFM model (0.66829, 0.23735) dominate those obtained through the DFM model (0.66829, 0.65550). Moreover, the other two solutions obtained by our method are all non-dominated. In addition, multiple efficient solutions and multiple minimum DF values can be generated by our improved DFM model when the value of e is adjusted. Decision-makers can identify the most preferred solution on the basis of their preferences. If they have no preferences, then any set of non-dominated values obtained by our model can be selected because they are all non-dominated sets of DF values. x∗∗ and d y∗∗ for all the Given that showing all of the results of dmo so three sets of e is unnecessary, we assume that the decision-maker prefers a balance between the two friction distance function values. In this case, e will be set by the middle value (LY + UY )/2. The distances for the inefficient provinces are shown in Table 7. We can use Steps 4 and 5 of our improved DFM model to obtain the improvement projections of the inefficient transportation sectors, which are given in Table 8, on the basis of the optimal disx∗∗ and d y∗∗ of China’s transportation sectors. Note that the tances dmo so target for Shannxi is given because it is the only weakly efficient tranportation sector. In order to compare the improved DFM model with the DFM model, we show the detail results of the projections based on these two models for the 21 inefficient transportation sectors in Table 9.

Table 7 Distances of the inputs and outputs of transportation sectors.

Shanxi Inner Mongolia Jilin Heilongjiang Jiangsu Zhejiang Fujian Shandong Hubei Guangdong Guangxi Hainan Chongqing Sichuan Guizhou Yunnan Tibet Gansu Qinghai Ningxia Xinjiang

d1x∗∗

d2x∗∗

d1y∗∗

d2y∗∗

d3y∗∗

0.001 0.004 0.0 0 0 1.878 7.098 0.046 0.0 0 0 0.0 0 0 0.014 0.050 7.866 1.985 0.033 7.099 1.233 6.576 0.472 0.436 0.002 0.674 0.0 0 0

87.71 273.52 565.59 103.91 112.78 756.63 578.68 508.80 1139.65 125.08 327.94 126.41 725.18 0.00 0.00 0.00 0.00 0.00 0.00 0.00 195.22

69.09 164.04 271.25 231.01 178.20 576.78 341.28 307.78 0.00 151.34 187.86 18.48 318.73 608.06 144.42 0.00 0.00 0.00 154.63 0.61 117.88

0.00 0.00 0.00 0.00 0.00 31.59 0.00 0.00 451.31 0.88 161.86 0.00 0.00 210.36 58.41 286.56 25.07 59.24 0.23 0.32 11.17

0.00 0.00 0.00 0.00 0.00 350.11 0.00 0.00 0.00 10.94 1781.79 613.40 0.00 0.00 0.00 353.23 9.64 0.00 0.00 314.01 0.00

We can find from Table 9 that CCR model has the largest average change ratios of inputs (−35.06% for NELT and −40.09% for FCIT). The original DFM model and the improved DFM model have similar average change ratios in the reduction of NELT (−14.03% and −14.09%), but the improved model has a much smaller reduction of 19.44% in the FCIT whereas the original model has a reduction of 31.07%. The increments of outputs GDPT, PT V and FT V for the improved DFM model are 28.25%, 18.31% and 64.09%, respectively, whereas those increments for the original DFM model are 31.81%, 17.7% and 53.92%, respectively. By an analysis similar to that of Suzuki et al. (2010), the improved DFM model has a better average performance on projection than the original DFM model and the CCR model. The projections of the original DFM model and the improved DFM model are both on the efficient production frontier, therefore it is impossible that the reductions of all inputs and the increments of all outputs of each province obtained by the improved DFM model are smaller than those obtained by the original DFM model. For example, the DFM projection in Table 9 shows that Jiangsu should reduce its FCIT by 7.3%, together with an

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Table 8 Benchmarks for the identification of inefficient and weakly efficient transportation sectors.

Shanxi Inner Mongolia Jilin Heilongjiang Jiangsu Zhejiang Fujian Shandong Hubei Guangdong Guangxi Hainan Chongqing Sichuan Guizhou Yunnan Tibet Gansu Qinghai Ningxia Xinjiang Shannxi

NELT (10 0 0 persons)

FCIT (billion Yuan)

GDPT (billion Yuan)

PTV (100 million passenger-kilometer)

17.10 17.50 22.40 6.52 58.80 41.55 19.00 57.90 42.89 57.75 10.63 2.62 19.97 14.30 8.37 6.32 0.53 6.06 1.20 1.93 17.10 7.7

824.89 1154.18 604.41 1030.69 2438.22 1825.27 1926.72 2473.40 1693.35 2907.22 1521.86 340.69 905.52 3202.99 1779.90 1250.87 542.30 110 0.0 0 589.90 274.82 824.89 1123.25

999.84 1306.01 829.63 989.02 3015.36 2351.15 2031.10 3033.19 1297.48 3361.06 1043.53 218.37 1166.95 2080.63 1131.89 644.18 31.26 271.25 249.62 206.36 999.84 776.12

457.57 650.78 431.27 471.08 1468.48 1106.58 898.43 1270.29 1683.63 1888.33 905.69 120.36 506.29 1151.96 733.27 732.62 64.85 672.63 125.34 110.04 457.57 755.67

FTV (100 million ton-kilometer) 6139.60 7367.69 5933.97 3204.82 19,564.63 10,139.44 8520.11 22,452.15 5922.87 21,812.59 6042.20 1674.15 2968.29 7462.16 4100.80 1953.30 134.27 2170.05 475.80 1157.29 6139.60 3530.30

Table 9 CCR, DFM and improved DFM projections results for all DMUs. CCR Shanxi NELT −14.1% FCIT −14.1% GDPT 0.0% PTV 17.1% FTV 6.4% Heilongjiang NELT −31.6% FCIT −31.6% GDPT 0.0% PTV 0.0% FTV 71.3% Fujian NELT −33.6% FCIT −33.6% GDPT 0.0% PTV 13.3% FTV 1.1% Guangdong NELT −6.3% FCIT −6.3% GDPT 0.0% PTV 0.0% FTV 0.0% Chongqing NELT −51.9% FCIT −51.9% GDPT 0.0% PTV 0.0% FTV 34.3% Yunnan NELT −67.5% FCIT −71.0% GDPT 30.4% PTV 0.0% FTV 0.0% Qinghai NELT −0.3% FCIT −62.4% GDPT 28.2% PTV 0.0% FTV 0.0%

DFM

IDFM

−0.0% −9.6% 7.6% 17.8% 0.8%

−0.0% −9.6% 7.4% 26.9% 72.2%

−25.8% −6.5% 30.5% 0.0% 102.2%

−22.4% −9.2% 30.5% 0.0% 109.1%

−0.0% −23.1% 20.2% 51.4% 40.4%

−0.0% −23.1% 20.2% 51.4% 40.4%

−0.0% −4.1% 4.7% 0.0% 0.0%

−0.0% −4.1% 4.7% 0.0% 0.0%

−0.0% −44.5% 37.6% 0.0% 56.8%

−0.0% −44.5% 37.6% 0.0% 56.8%

−51.0% −63.2% 101.5% 64.2% 23.6%

−51.0% −51.5% 96.2% 64.2% 22.1%

−0.1% −62.4% 28.4% 0.2% 0.0%

−0.1% 0.0% 162.8% 0.2% 0.0%

CCR Inner Mongolia NELT −26.9% FCIT −26.9% GDPT 0.0% PTV 48.0% FTV 24.8% Jiangsu NELT −10.9% FCIT −10.9% GDPT 0.0% PTV 0.0% FTV 94.5% Shandong NELT −23.6% FCIT −23.6% GDPT 0.0% PTV 1.7% FTV 19.9% Guangxi NELT −49.3% FCIT −49.3% GDPT 0.0% PTV 0.0% FTV 0.0% Sichuan NELT −49.8% FCIT −53.9% GDPT 0.0% PTV 0.0% 82.6% FTV Tibet NELT −64.1% FCIT −89.1% GDPT 18.5% PTV 0.0% FTV 0.0% Ningxia NELT −38.5% FCIT −38.5% GDPT 0.0% PTV 0.0% FTV 0.0%

DFM

IDFM

−0.0% −19.2% 15.6% 79.4% 52.2%

−0.0% −19.2% 14.4% 73.6% 69.7%

−0.0% −7.3% 6.3% 0.0% 86.8%

−10.8% −4.4% 6.3% 0.0% 155.6%

−0.0% −17.1% 13.4% 0.5% 3.3%

−0.0% −17.1% 13.4% 0.5% 3.3%

−44.2% −28.9% 9.9% 0.0% 55.4%

−42.5% −17.7% 22.0% 21.8% 41.8%

−33.2% −37.5% 41.4% 22.3% 159.9%

−33.2% −14.3% 41.3% 22.3% 198.0%

−47.2% −85.4% 76.2% 53.2% 32.2%

−47.2% 0.0% 0.0% 63.0% 7.7%

−25.9% −32.2% 0.0% 12.9% 38.5%

−25.9% −25.3% 0.3% 0.3% 41.1%

Jilin NELT FCIT GDPT PTV FTV Zhejiang NELT FCIT GDPT PTV FTV Hubei NELT FCIT GDPT PTV FTV Hainan NELT FCIT GDPT PTV FTV Guizhou NELT FCIT GDPT PTV FTV Gansu NELT FCIT GDPT PTV FTV Xinjiang NELT FCIT GDPT PTV FTV

CCR

DFM

IDFM

−57.1% −57.1% 0.0% 0.0% 94.4%

−0.0% −48.3% 48.6% 0.0% 204.2%

−0.0% −48.3% 48.6% 0.0% 204.2%

−38.5% −38.5% 0.0% 0.0% 0.0%

−0.0% −29.3% 33.8% 0.0% 5.7%

−0.1% −29.3% 32.5% 2.9% 3.6%

−51.9% −51.9% 4.8% 0.0% 0.0%

−0.0% −40.2% 10.5% 36.6% 15.0%

−0.0% −40.2% 10.5% 36.6% 15.0%

−53.7% −53.7% 0.0% 0.0% 0.0%

−47.8% −39.6% 9.2% 22.5% 25.5%

−43.1% −27.1% 9.2% 0.0% 57.8%

−22.8% −33.7% 0.0% 0.0% 106.2%

−12.8% −23.6% 21.3% 2.4% 152.2%

−12.8% 0.0% 14.6% 8.7% 176.7%

−11.4% −11.4% 118.8% 0.0% 0.0%

−6.7% −7.1% 130.2% 8.4% 0.0%

−6.7% 0.0% 0.0% 9.7% 0.0%

−32.4% −32.4% 0.0% 0.0% 23.2%

−0.0% −23.3% 21.2% 0.0% 77.7%

−0.0% −23.3% 20.8% 2.4% 70.8%

B. Xiong, H. Chen and Q. An et al. / European Journal of Operational Research 279 (2019) 132–142 Table 10 Efficiency improvement projection results of CCR-I, DFM and the improved DFM (example of Hainan Province). DMU:

CCR-I

I/O

Data

NELT FCIT GDPT PTV FTV

4.6 467.1 199.89 120.36 1060.75

DMU:

DFM

I/O

Data

− ( 1 − θ ∗ )x0

−s−∗ s+∗

−2.5 −250.8

0 0 0 0 0.0

%

2.1 216.3 199.9 120.4 1060.8

−2.5 −250.8 0 0 0.0

−53.7% −53.7% 0.0% 0.0% 0.0%

Projection ∗ xxmo ∗ yxmo

Difference x∗ −dmo − s−∗ x∗ dmo + s+∗

%

−s−∗ s+∗

2.2 98.0 18.5 0.0 270.4

0 86.8 0 27.1 0

2.4 282.3 218.4 147.4 1331.12

−2.2 184.8 18.5 27.1 270.4

−47.8% −39.6% 9.2% 22.5% 25.5%

−s−∗ s+∗

Projection ∗ xxmo ∗ yxmo

Difference x∗ −dmo − s−∗ x∗ dmo + s+∗

%

x∗ dmo x∗ dmo

2.0 126.4 18.5 0.0 613.4

0 0 0 0 0

2.6 340.7 218.4 120.4 1674.1

−2.0 −126.4 18.5 0 613.4

−43.1% −27.1% 9.2% 0.0% 57.8%

4.6 467.1 199.89 120.36 1060.75

DMU:

Improved DFM

I/O

Data

4.6 467.1 199.89 120.36 1060.75

Difference

x∗ dmo y∗ dso

NELT FCIT GDPT PTV FTV

NELT FCIT GDPT PTV FTV

Projection

increase in GDPT of 6.3%, FTV of 86.8%, in order to become efficient. On the other hand, the results of the improved DFM show that a reduction of NELT by 10.8% and FCIT by 4.4%, together with an increase of GDPT by 6.3% and FTV by 155.6% are required to become efficient. Here, we take an example of Hainan province to compare the efficiency improvement projection of the CCR model, DFM model and the improved DFM model in more details. Table 10 shows the outcomes of Hainan as an example of the input-slack projection type. Other DMUs of this type can be found in Table 9. The uniform reduction in inputs for CCR model is 53.7%. Neither the original DFM model nor the improved DFM model has uniform reduction of inputs for the Hainan province. They both require a large reduction in NELT but a small reduction in FCIT. Compared with the original DFM model, the change ratios of DCIT and PTV are smaller for the improved DFM model, both models have the equal change ratio of GDPT, but the improved DFM model has a larger change ratio of FTV than DFM model. 5. Conclusions DEA is a non-parametric analytical methodology used for efficiency analysis and benchmarking analysis. The benchmarking information related to the elimination of inefficiency represents a unique advantage of DEA methodology over other efficiency analysis methods. Besides, the selection of the reference point, which serves as the benchmarking target, is crucial for evaluating the performance of the DMU. Therefore, finding the appropriate benchmarking target is very important, and is related to the efficiency measure as well as to the benchmarking information itself. In this paper, we construct the DFM model as an alternative to the standard CCR model for benchmarking analysis. The DFM model can avoid the use of a linear proportional improvement but still allows input reduction and output augmentation. The main ad-

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vantage of the DFM model is that it provides a benchmarking target on the efficient production frontier that is as close as possible to the input and output profiles of the evaluated DMU. However, given that the minimum DF values may be calculated on the basis of an arbitrary optimal solution of the CCR model, the final obtained DF values may not be non-dominated. We illustrate this phenomenon with a numerical example. To overcome this nondominance, we propose an improved DFM model that guarantees that the minimum DF values are obtained. We apply the AUGMECON method to produce non-dominated efficient solutions to the MOQP problem. This method can control the number of generated efficient solutions by properly adjusting the number of grid points for the second DF values. Finally, both the original DFM model and the improved DFM model are applied to evaluate the efficiencies and to conduct benchmarking analysis of China’s 31 transportation sectors, and the results obtained by the two models and the CCR model are compared. This comparison shows that the improved DFM model has a better average performance on projection than the original DFM model and the CCR model, but the new model has not always a better projection performance than that of the original DFM model in some cases. However, for the criterions we pay attention to in this study, i.e., the distance friction values, our numerical results show that these values obtained by the improved DFM model dominate those obtained by the original DFM model, which is consistent with a theorem in this paper. We offer two directions for future studies. One is to consider the undesirable outputs for our model. In this case, the number of objective functions will be increased to three. Thus, coordinating these three goals in optimization will be a crucial issue. Another is the integration of our DFM model with other types of DEA models, such as the BCC and SBM models. Acknowledgements The authors thank three anonymous reviewers for their helpful comments. The research is supported by National Natural Science Foundation of China (Nos. 71871223, 71501189, 71631008, 71571173, 71790615), Innovation-Driven Planning Foundation of Central South University (2019CX041), the ANR (French National Research Agency) project ANR-14-CE22- 0017. References An, Q., Wen, Y., Ding, T., & Li, Y. (2019). Resource sharing and payoff allocation in a three-stage system: Integrating network DEA with the Shapley value method. Omega, 85, 16–25. Aparicio, J., Ortiz, L., & Pastor, J. T. (2017). Measuring and decomposing profit inefficiency through the Slacks-Based Measure. European Journal of Operational Research, 260(2), 650–654. Appa, G. (2002). On the uniqueness of solutions to linear programs. Journal of the Operational Research Society, 53(10), 1127–1132. Banker, R. D. (1984). Estimating most productive scale size using data envelopment analysis. European Journal of Operational Research, 17(1), 35–44. Banker, R. D., & Morey, R. C. (1986). Efficiency analysis for exogenously fixed inputs and outputs. Operations Research, 34(4), 513–521. Boussemart, J. P., Leleu, H., Shen, Z. Y., Vardanyan, M., & Zhu, N. (2019). Decomposing banking performance into economic and credit risk efficiencies. European Journal of Operational Research, 227(2), 719–726. Chang, Y. T., Zhang, N., Danao, D., & Zhang, N. (2013). Environmental efficiency analysis of transportation system in China: A non-radial DEA approach. Energy policy, 58, 277–283. Charnes, A., Cooper, W. W., & Rhodes, E. (1978). Measuring the efficiency of decision making units. European Journal of Operational Research, 2(6), 429–444. Charnes, A., Cooper, W. W., Golany, B., Seiford, L. M., & Stutz, J. (1985). Foundations of data envelopment analysis and Pareto–Koopmans empirical production functions. Journal of Econometrics, 30(1–2), 91–107. Chen, H. (2018). Average lexicographic efficiency for data envelopment analysis. Omega, 74, 82–91. Cooper, W. W., Seiford, L. M., & Tone, K. (2006). Introduction to data envelopment analysis and its uses. Boston: Springer Science + Business Media Inc.. Doyle, J., & Green, R. (1994). Efficiency and cross-efficiency in DEA: Derivations, meanings and uses. Journal of the Operational Research Society, 45(5), 567–578. Ehrgott, M. (2005). Multicriteria optimization: 491. Springer Science & Business Media.

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