5. Case study
5.4. Weight analysis for generalized IDEA model
The above IDEA model has adopted an additive form of technical efficiency and service effectiveness, which implicitly assumes equal weights for both terms. A more generalized IDEA model with various weights can be expressed as follows:
If the weights of production and marketing department are allowed to be
endogenous, some DMUs might reach efficiency by totally ignoring the performance of production department or marketing department, which might not be very reasonable in practice. Thus, this study set the weights as exogenous parameter (α ).
Where, α is the weight of technical efficiency, which is subjectively given by the decision maker. (1-α) is the weight of service effectiveness. If the decision maker concerns more about the technical efficiency than the service effectiveness, then α can be set lager than 0.5, vice versa. Taking DMU 5 as an example, the technical efficiency and service effectiveness with various weight combinations are computed and the technical efficiency will increase and the service effectiveness will decrease as α gets larger.
Obviously, the generalized IDEA model can provide the decision-maker with wider spectrum of information than only the equal-weight information.
In this section, we will discuss influence of weight change for each DMU.
First, we will demonstrate the result of DMU 1. From Table 8 and Figure 4, we could realize that only when α equal to 0.1 and 0.2 weight change will change efficiency score. When α larger than 0.3 weight change would no effect on efficiency score no matter for technical efficiency or service effectiveness. That’s because technical efficiency is reach optimal value when α equal to 0.3. No matter how the weight changes technical efficiency will adjust by itself in order to reach optimal value.
For relative efficient DMU such DMU 1, weight changes wouldn’t increase efficiency score by large scale. For this kind DMUs, they need to improve their efficiency by adjust their input, output and/or consumption variables rather than change their weight only. We could know that weight changing isn’t so important for efficient DMUs than for inefficient DMUs.
Table 8 The technical efficiency and service effectiveness of DMU 1 with various weight combinations
α Technical efficiency Service effectiveness
0.1 0.88245 0.85085
0.2 0.91033 0.84733
0.3 1.00000 0.82141
0.4 1.00000 0.82141
0.5 1.00000 0.82141
0.6 1.00000 0.82141
0.7 1.00000 0.82141
0.8 1.00000 0.82141
0.9 1.00000 0.82141
Fig 4 The shapes of technical efficiency and service effectiveness of DMU 1 with various weight combinations
The result of DMU 2 is showing in Table 9 and Figure 5. DMU 2 is relative
0.00 0.20 0.40 0.60 0.80 1.00 1.20
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
α
Score
Technical efficiency Service effectiveness
efficiency in technical efficiency. When α get larger, technical efficiency will increase slowly. However the score change cause by weight change isn’t significant. The reason cause this result is the same as DMU 1.
If DMU 2 wishes to increase its efficiency score, it needs to adjust is input, output and/or consumption variables rather than change its weight only.
Table 9 The technical efficiency and service effectiveness of DMU 2 with various weight combinations
α Technical efficiency Service effectiveness
0.1 0.98480 0.79051
0.2 0.98480 0.79051
0.3 0.99268 0.78836
0.4 0.99268 0.78836
0.5 0.99268 0.78836
0.6 1.00000 0.77920
0.7 1.00000 0.77920
0.8 1.00000 0.77920
0.9 1.00000 0.77920
Fig 5 The shapes of technical efficiency and service effectiveness of DMU 2 with various weight combinations
The result for DMU 3 is displaying in Table 10 and Figure 6. They show
0.00 0.20 0.40 0.60 0.80 1.00 1.20
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
α
Score
Technical efficiency Service effectiveness
that weight changing causes significant improvement in performance. That’s because DMU 3 is a relative inefficiency DMU. When the weight changes, the efficiency scores will have apparent change. If DMU 3 wants to improve its efficiency score, it can either modify weight combinations or adjust input, output and/or consumption variables.
From this example, we could know that weight change is more useful for relative inefficiency DMU. However once efficiency score of technical efficiency or service effectiveness is close to unity under certain weight combination, the weight change is no more useful in improving efficiency score. In this case, when α =0.4, no matter how the weight change the efficiency score isn’t change at all. Unless the weight combination become extreme such as α=0.9. Then the sector which gets most source will have higher efficiency score such as technical efficiency.
Table 10 The technical efficiency and service effectiveness of DMU 3 with various weight combinations
α Technical efficiency Service effectiveness
0.1 0.78654 0.85611
0.2 0.83382 0.84874
0.3 0.93082 0.82127
0.4 0.94107 0.81624
0.5 0.94107 0.81624
0.6 0.94107 0.81624
0.7 0.94107 0.81624
0.8 0.94107 0.81624
0.9 0.94355 0.80406
Fig 6 The shapes of technical efficiency and service effectiveness of DMU 3 with various weight combinations
Table 11 and Figure 7 are demonstrated the result of DMU 4. DMU 4 is a relative inefficient DMU as well. Its efficiency score in sensitive about weight change. It can improve it efficiency performance through weight change and/or adjusting input, output and/or consumption variables.
The result of DMU 5 is demonstrating in Table 12 and Figure 10. DMU 5 will have the same pattern as DMU 4 because DMU 5 is an inefficient DMU as well.
Table 11 The technical efficiency and service effectiveness of DMU 4 with various weight combinations
α Technical efficiency Service effectiveness
0.1 0.87147 0.78877
0.2 0.87147 0.78877
0.3 0.87147 0.78877
0.4 0.87147 0.78877
0.5 0.89702 0.77083
0.6 0.97748 0.68851
0.7 0.98773 0.67224
0.8 0.99026 0.66504
0.9 0.99026 0.66504
0.70 0.75 0.80 0.85 0.90 0.95 1.00
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
α
Score
Technical efficiency Service effectiveness
Fig 7 The shapes of technical efficiency and service effectiveness of DMU 4 with various weight combinations
Table 12 The technical efficiency and service effectiveness of DMU 5 with various weight combinations
α Technical efficiency Service effectiveness
0.1 0.940981 0.85118
0.2 0.967532 0.84522
0.3 0.967532 0.84522
0.4 0.967532 0.84522
0.5 0.967532 0.84522
0.6 0.972326 0.83987
0.7 0.972327 0.83987
0.8 0.995025 0.75819
0.9 0.996021 0.75031
0.00 0.20 0.40 0.60 0.80 1.00 1.20
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
α
Score
Technical efficiency Service effectiveness
0.7 0.75 0.8 0.85 0.9 0.95 1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
α
Scores
Technical efficiency Service effectiveness
Fig 8 The shapes of technical efficiency and service effectiveness of DMU 5 with various weight combinations
The result of weight change of DMU 6 is showing in Table 13 and Figure 9.
DMU 6 is a relative efficient DMU. Its service effectiveness reaches unity which means its sale department is benchmark. No matter how the weight changes it will adjust by itself unless the weight is in extreme position such as α =0.9. In this situation, production department get most source so technical efficiency will perform better and service effectiveness will become worse. If DMU 6 wants to improve its performance, it better focus on adjusting its input, output and/or consumption variables.
Table 13 The technical efficiency and service effectiveness of DMU 6 with various weight combinations
α Technical efficiency Service effectiveness
0.1 0.95003 1.00000
0.2 0.95003 1.00000
0.3 0.95003 1.00000
0.4 0.95003 1.00000
0.5 0.95003 1.00000
0.6 0.95003 1.00000
0.7 0.95003 1.00000
0.8 0.95003 1.00000
0.9 0.95276 0.98852
Fig 9 The shapes of technical efficiency and service effectiveness of DMU 6 with various weight combinations
Table 14 and Figure 10 demonstrate the outcomes of DMU 7. Like DMU 6, DMU 7 is a relative efficient DMU. However it performs better in technical efficiency. Only when weight combination become extreme such as α =0.1, service effectiveness will perform better. Otherwise technical efficiency will always be benchmark. Talking about performance improvement about DMU 7, it could focus on modify its input, output and/or consumption variables rather than find optimal weight combination.
Table 14 The technical efficiency and service effectiveness of DMU 7 with various weight combinations
α Technical efficiency Service effectiveness
0.1 0.96007 0.81723
0.2 1.00000 0.81024
0.3 1.00000 0.81024
0.4 1.00000 0.81024
0.5 1.00000 0.81024
0.6 1.00000 0.81024
0.7 1.00000 0.81024
0.8 1.00000 0.81024
0.9 1.00000 0.81024
0.00 0.20 0.40 0.60 0.80 1.00 1.20
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
α
Score
Technical efficiency Service effectiveness
Fig 10 The shapes of technical efficiency and service effectiveness of DMU 7 with various weight combinations
Both the scores of technical efficiency and service effectiveness of different weight combination for DMU 8 and DMU 10 are demonstrating in Table 15, Table 16, Figure 11 and Figure 12. Although DMU 8 and DMU 9 looks like relative efficient DMUs in the efficient score, they are not benchmarks in both production and sale department.
They are still sensitive about weight change especially at some extreme weight combination such as α=0.1 or 0.2. When weight combination is in extreme level, weight change will have certain influence on scores. If the weight combination is in average level such as α =0.5 or 0.6, efficiency score will has little response about weight change.
For this kind DMUs, they can improve their performance scores both through weight change and adjusting their input, output and/or consumption variables. However the main improvement approach of this kind should focus on adjusting their variables.
0.00 0.20 0.40 0.60 0.80 1.00 1.20
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
α
Score
Technical efficiency Service effectiveness
Table 15 The technical efficiency and service effectiveness of DMU 8 with various weight combinations
α Technical efficiency Service effectiveness
0.1 0.90691 0.94448
0.2 0.97387 0.93167
0.3 0.97387 0.93167
0.4 0.97387 0.93167
0.5 0.99002 0.91646
0.6 0.99002 0.91646
0.7 0.99002 0.91646
0.8 0.99002 0.91646
0.9 0.99002 0.91646
Fig 11 The shapes of technical efficiency and service effectiveness of DMU 8 with various weight combinations
0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
α
Score
Technical efficiency Service effectiveness
Table 16 The technical efficiency and service effectiveness of DMU 10 with various weight combinations
α Technical efficiency Service effectiveness
0.1 0.87781 0.97950
0.2 0.90262 0.97615
0.3 0.90262 0.97615
0.4 0.90262 0.97615
0.5 0.90262 0.97615
0.6 0.90262 0.97615
0.7 0.90262 0.97615
0.8 0.90262 0.97615
0.9 0.90262 0.97615
Fig 12 The shapes of technical efficiency and service effectiveness of DMU 10 with various weight combinations
The result of DMU 9 and DMU 11 is shown in Table 17, Table 18, Figure 13, and Figure 14. Both DMU 9 and DMU 11 reach overall efficiency. Their efficiency score no matter technical efficiency or service effectiveness are all equal to unit. For these kinds DMU, different weight combinations have insignificant effect.
0.82 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 1.00
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
α
Score
Technical efficiency Service effectiveness
Because both produce and sale sector of these kinds DMUs will adjust by themselves, weight change will have no power in improving efficiency scores.
Once DMU reaches overall efficiency, weight change is not an important issue it should concern.
Table 17 The technical efficiency and service effectiveness of DMU 9 with various weight combinations
α Technical efficiency Service effectiveness
0.1 1.00000 1.00000
0.2 1.00000 1.00000
0.3 1.00000 1.00000
0.4 1.00000 1.00000
0.5 1.00000 1.00000
0.6 1.00000 1.00000
0.7 1.00000 1.00000
0.8 1.00000 1.00000
0.9 1.00000 1.00000
Fig 13 The shapes of technical efficiency and service effectiveness of DMU 9 with various weight combinations
0.00 0.20 0.40 0.60 0.80 1.00 1.20
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
α
Score
Technical efficiency Service effectiveness
Table 18 The technical efficiency and service effectiveness of DMU 11 with various weight combinations
α Technical efficiency Service effectiveness
0.1 1.00000 1.00000
0.2 1.00000 1.00000
0.3 1.00000 1.00000
0.4 1.00000 1.00000
0.5 1.00000 1.00000
0.6 1.00000 1.00000
0.7 1.00000 1.00000
0.8 1.00000 1.00000
0.9 1.00000 1.00000
Fig 14 The shapes of technical efficiency and service effectiveness of DMU 11 with various weight combinations
Table 19 and Figure 15 are displaying the result of different weight combinations for DMU 12. There is an interesting shape on Figure 15. The efficiency scores is fixed in two value. When α is lying between 0.1 and 0.5 and between 0.6 and 0.9.
First, when α is between 0.1 and 0.5, service effectiveness is unity. Sale department will adjust by itself until the source become less and less. Which means production department get most source such as α is lying 0.5 and
0.00 0.20 0.40 0.60 0.80 1.00 1.20
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
α
Score
Technical efficiency Service effectiveness
between 0.6 and 0.9. When α is large enough, the technical efficiency will be performance better.
The DMU like DMU 12, which is relative inefficiency in one department and efficiency in another department, can improve its performance by finding optimal weight combinations and modifying its variable values. Both these method can help to better its performance.
Table 19 The technical efficiency and service effectiveness of DMU 12 with various weight combinations
α Technical efficiency Service effectiveness
0.1 0.89205 1.00000
0.2 0.89205 1.00000
0.3 0.89205 1.00000
0.4 0.89205 1.00000
0.5 0.89205 1.00000
0.6 1.00000 0.85259
0.7 1.00000 0.85259
0.8 1.00000 0.85259
0.9 1.00000 0.85259
Fig 15 The shapes of technical efficiency and service effectiveness of DMU 12 with various weight combinations
0.75 0.80 0.85 0.90 0.95 1.00 1.05
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
α
Score
Technical efficiency Service effectiveness
Result of DMU 13, DMU 14 and DMU 15 are demonstrating in Table 20, Table 21 and Table 22. The shapes of these results are displaying in Figure 16, Figure 17 and Figure 18. From these results, we could know that all these DMU are relative efficiency. Only weight combinations are in extreme level, the efficiency scores will be different and the difference is only in a small scale.
If these kinds DMUs hope to raise its efficiency scores, they better to concentrate on adjusting their variable values.
Table 20 The technical efficiency and service effectiveness of DMU 13 with various weight combinations
α Technical efficiency Service effectiveness
0.1 0.96949 0.84731
0.2 0.96949 0.84731
0.3 0.96949 0.84731
0.4 0.96949 0.84731
0.5 0.96949 0.84731
0.6 0.96949 0.84731
0.7 0.96949 0.84731
0.8 0.96949 0.84731
0.9 0.98929 0.72021
Fig 16 The shapes of technical efficiency and service effectiveness of DMU 13 with various weight combinations
0.00 0.20 0.40 0.60 0.80 1.00 1.20
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
α
Score
Technical efficiency Service effectiveness
Table 21 The technical efficiency and service effectiveness of DMU 14 with various weight combinations
α Technical efficiency Service effectiveness
0.1 0.97151 0.93701
0.2 1.00000 0.93016
0.3 1.00000 0.93016
0.4 1.00000 0.93016
0.5 1.00000 0.93016
0.6 1.00000 0.93016
0.7 1.00000 0.93016
0.8 1.00000 0.93016
0.9 1.00000 0.93016
Fig 17 The shapes of technical efficiency and service effectiveness of DMU 14 with various weight combinations
0.88 0.90 0.92 0.94 0.96 0.98 1.00 1.02
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
α
Score
Technical efficiency Service effectiveness
Table 22 The technical efficiency and service effectiveness of DMU 15 with various weight combinations
α Technical efficiency Service effectiveness
0.1 1.00000 0.96504
0.2 1.00000 0.96504
0.3 1.00000 0.96504
0.4 1.00000 0.96504
0.5 1.00000 0.96504
0.6 1.00000 0.96504
0.7 1.00000 0.96504
0.8 1.00000 0.96504
0.9 1.00000 0.96504
Fig 18 The shapes of technical efficiency and service effectiveness of DMU 15 with various weight combinations
5.5. Overall weight analysis
From above analysis, we could know that change weight will influence the performance of DMU. Then, Table 23 and Figure 19 will demonstrate the technical efficiency of each DMU with various weight combinations and Table 24 and Figure 20 is the result of service effectiveness. Obviously, the generalized integrated DEA model can provide the decision-maker with wider spectrum of information than only the equal-weight information.
0.94 0.95 0.96 0.97 0.98 0.99 1.00 1.01
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
α
Score
Technical efficiency Service effectiveness
Table 23 Technical efficiency of each DMU with various weight combinations α (Weight)
Route 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 0.8824 0.9103 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2 0.9848 0.9848 0.9927 0.9927 0.9927 1.0000 1.0000 1.0000 1.0000 3 0.7865 0.8338 0.9308 0.9411 0.9411 0.9411 0.9411 0.9411 0.9436 4 0.8715 0.8715 0.8715 0.8715 0.8970 0.9775 0.9877 0.9903 0.9903 5 0.9410 0.9675 0.9675 0.9675 0.9675 0.9723 0.9723 0.9950 0.9960 6 0.9500 0.9500 0.9500 0.9500 0.9500 0.9500 0.9500 0.9500 0.9528 7 0.9601 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 8 0.9069 0.9739 0.9739 0.9739 0.9900 0.9900 0.9900 0.9900 0.9900 9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 10 0.8778 0.9026 0.9026 0.9026 0.9026 0.9026 0.9026 0.9026 0.9026 11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 12 0.8920 0.8920 0.8920 0.8920 0.8920 1.0000 1.0000 1.0000 1.0000 13 0.9695 0.9695 0.9695 0.9695 0.9695 0.9695 0.9695 0.9695 0.9893 14 0.9715 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 15 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
Fig 19 The shapes of technical efficiency of Each DMU with various weight combinations
Cost efficiency
0.7500 0.8000 0.8500 0.9000 0.9500 1.0000 1.0500
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Weights
Efficiency value
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Table 24 Service effectiveness of each DMU with various weight combinations α (Weight)
Route 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1 0.8508 0.8473 0.8214 0.8214 0.8214 0.8214 0.8214 0.8214 0.8214 2 0.7905 0.7905 0.7884 0.7884 0.7884 0.7792 0.7792 0.7792 0.7792 3 0.8561 0.8487 0.8213 0.8162 0.8162 0.8162 0.8162 0.8162 0.8041 4 0.7888 0.7888 0.7888 0.7888 0.7708 0.6885 0.6722 0.6650 0.6650 5 0.8512 0.8452 0.8452 0.8452 0.8452 0.8399 0.8399 0.7582 0.7503 6 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.9885 7 0.8172 0.8102 0.8102 0.8102 0.8102 0.8102 0.8102 0.8102 0.8102 8 0.9445 0.9317 0.9317 0.9317 0.9165 0.9165 0.9165 0.9165 0.9165 9 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 10 0.9795 0.9761 0.9761 0.9761 0.9761 0.9761 0.9761 0.9761 0.9761 11 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 12 1.0000 1.0000 1.0000 1.0000 1.0000 0.8526 0.8526 0.8526 0.8526 13 0.8473 0.8473 0.8473 0.8473 0.8473 0.8473 0.8473 0.8473 0.7202 14 0.9370 0.9302 0.9302 0.9302 0.9302 0.9302 0.9302 0.9302 0.9302 15 0.9650 0.9650 0.9650 0.9650 0.9650 0.9650 0.9650 0.9650 0.9650
Fig 20 The shapes of Service effectiveness of Each DMU with various weight combinations
Service Effectiveness
0.7500 0.8000 0.8500 0.9000 0.9500 1.0000 1.0500
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Weights
Efficiency value
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
From Table 23, Table 24, Figure 19 and Figure 20, we can realize that efficient DMUs, such DMU 9 and DMU 11, has low sensitivity about weight change because no matter how the weight change the efficient DMU will adjust by itself. Only the inefficient DMUs, such as DMU 5, will have high sensitivity about weight change.
Through weight analysis, each DMU would have its own suggestion on improvement approach. Different DMU would have different features, and weight analysis provides another way in analyzing each DMU. Each DMU can find best method in improving its performance.
From above two figures, we could know that the changes of the efficiency score no matter technical efficiency or service effectiveness are not very huge. If DMUs try to improve its performance, they better focus on adjusting the variable.
6. Conclusions and suggestions
6.1. Conclusions
This paper develops integrated DEA (IDEA) models, under constant-returns-to-scale (ICCR model) and variable-returns-to-scale (IBCC model) technologies, to measure the overall efficiency and separate efficiencies for non-storable commodities, from various aspects of technical efficiency, service effectiveness, and technical effectiveness. Some major findings can be concluded as follows:
(1) The proposed IDEA model, either ICCR or IBCC, is proven with rationality and uniqueness properties. The property of rationality suggests that the scores obtained from this integrated model are efficient values rather than meaningless figures. The property of uniqueness guarantees that the efficiency scores obtained from this model are global maximum rather than local maximum.
(2) Our proposed IDEA models can be employed to measure the overall efficiency of non-storable commodities such as transportation services.
The applicability of the proposed IDEA model has been demonstrated by a case study, from which the IDEA model has revealed higher discrimination power than the conventional separated DEA models.
(3) Compared with conventional separated DEA model, the proposed IDEA model can explain for non-storable commodities more explicitly. Because the IDEA model can jointly account for the production and sale departments of non-storable commodities, it is superior to conventional DEA models.
6.2. Suggestions
Some directions for future studies can be identified as follows.
(1) The weight analysis of the proposed IDEA model is worthy to make a further study because the weight in this study is an exogenous variable, not an endogenous variable. One could add the weight variable into the integrated DEA model and let the model decide the optimal weight for each department.
(2) An additive form of proposed IDEA model is derived in this paper, other forms of IDEA models or even multi-objective IDEA models deserve further exploration.
(3) The present paper only demonstrates the overall efficiency measure for two departments -- production (technical efficiency) and sale (service effectiveness). The proposed IDEA model can easily be extended to evaluate the overall performance of an organization with more than two departments that are vertically and/or horizontally coordinated, e.g., the supply chain managing of a firm, the mails processing of the post office, among others.
(4) More applications to other non-storable cases with the proposed IDEA model and more comparisons with other types of DEA models are also worthy of further study.
References
(1) Adler, N. and Berechman, J. (2001) “Measuring airport quality from the airlines’ viewpoint: an application of data envelopment analysis,” Transport Policy, Vol.8, pp.171-181.
(2) Allen, R. and Thanassoulis, E. (2004) “Improving envelopment in data envelopment analysis,” European Journal of Operational Research, Vol. 154, pp.363–379.
(3) Appa, G. and Williams, H.P. (2006) “A new framework for the solution of DEA models,” European Journal of Operational Research, Vol.172, pp.604–615.
(4) Banker, R.D., Charnes, A., and Cooper, W.W. (1984) “Some models for estimating technical and scale inefficiencies in data envelopment analysis,”
Management Science, Vol. 30, pp.1078-1092.
(5) Banker, R.D., Cooper, W.W., Seiford, L.M., Thrall, R.M., and Zhu J. (2004)
“Returns to scale in different DEA models,” European Journal of Operational Research, Vol.154, pp.345–362.
(6) Charnes, A., Cooper, W.W., and Rhodes, E. (1978) “Measuring the efficiency of decision-marking units,” European Journal of Operational Research, Vol.2, pp.429–444.
(7) Cherchye, L., Kuosmanen, T., and Post, T. (2001) “Alternative treatments of congestion in DEA: a rejoinder to Cooper, Gu, and Li,” European Journal of Operational Research, Vol.132, pp.75-80.
(8) Chiou, Y.C. and Chen, Y.H. (2006) “Route-based performance evaluation of Taiwanese domestic airlines using data envelopment analysis,” Transportation Research Part E, Vol. 42, pp.116–127.
(9) Coelli, T. and Perelman, S. (1999) “A comparison of parametric and non-parametric distance functions: with application to European railways,”
European Journal of Operational Research, Vol.117, pp.326-339.
(10) Cooper, W.W., Gu, B.,and Li, S. (2001) “Comparisons and evaluations of alternative approaches to the treatments of congestion DEA,” European Journal of Operational Research, Vol.132, pp.62-74.
(11) Cowie, J. and Asenova, D. (1999) “Organisation form, scale effects and efficiency in the British bus industry,” Transportation, Vol.26, pp.231–248.
(12) Cullinane, K., Wang, T.F., Song, D.W. and Ji, P. (2006) “The technical efficiency of container ports: Comparing data envelopment analysis and
stochastic frontier analysis,” Transportation Research Part A, Vol. 40, pp.354–374.
(13) El-Mahgary, S., Lahdlma, R. (1995) “Data envelopment analysis: visualizing the result,” European Journal of Operational Research, Vol.85, pp.700-710.
(14) Fielding, G.J., Timlynn, T.B. and Brenner, M.E. (1984) “Performance evaluation for bus transit,” Transportation Research, Vol. 19, pp.73-82.
(15) Fielding, G.J.,Glauthier, R.E. and Lave, C.A. (1978) “Performance indicators for transit management,” Transportation, Vol. 7, pp.365-379.
(16) Fukuyama, H. (2000) “Returns to scale and scale elasticity in data envelopment analysis,” European Journal of Operational Research, Vol. 125, pp.93-112.
(17) Karlaftis, M.G. (2003) “Investigating transit production and performance: A programming approach,” Transportation Research Part A, Vol.37, pp.225–240.
(18) Karlaftis, M.G. (2004) “A DEA approach for evaluating the efficiency and effectiveness of urban transit systems,” European Journal of Operational Research, Vol.152, pp.354–364.
(19) Lan, L.W. and Lin, T.J. (2003) “Technical efficiency and service effectiveness for railways industry: DEA approaches,” Journal of the Eastern Asia Society for Transportation Studies, Vol.5, pp.2932-2947.
(20) Lan, L.W. and Lin, T.J. (2005) “Measuring railway performance with adjustment of environmental effects, data noise and slacks,” Transportmetrica, Vol. 1, No. 2 , pp.161-189.
(21) Lan, L.W. and Lin, T.J. (2006) “Performance measurement for railway transport: stochastic distance function with inefficiency and ineffectiveness effects,” Journal of Transport Economics and Policy, Vol.40, Part 3, pp.383-408.
(22) Lin, T.J. (2004) Productive Efficiency, Service Effectiveness, Productivity and Sales Force Measurements for Rail Transport Industry, Ph.D. dissertation, Institute of Traffic and Transportation, National Chiao Tung University.
(23) Norm, M. and Stoker, B. (1991) “Data Envelopment Analysis. The assessment of performance.”
(24) Odeck, J. and Alkadi, A. (2001) “Evaluating efficiency in the Norwegian bus industry using data envelopment analysis,” Transportation, Vol.28, pp.211–232.
(25) Peck, M.W., JR, Scheraga, C.A. and Boisjoly, R.P. (1998) “Assessing the relative efficiency of aircraft maintenance technologies: An application of data envelopment analysis,” Transportation Research, Vol.32, pp.261-269.
(26) Pels, E., Nijkamp, P. and Rietveld, P. (2001) “Relative efficiency of European airports,” Transportation Policy, Vol. 8, pp.183-192.
(27) Thanassoulis, E. (2001) “Introduction to the theory and application of data envelopment analysis.”
(28) Tongzon, J. (2001) “Efficiency measurement of selected Australian and other international ports using data envelopment analysis,” Transportation Research Part A, Vol. 35, pp.113-128.
(29) Tzeng, G.H. and Chiang, C.I. (2000) “A new efficiency measure for DEA:
Efficiency achievement measure established on fuzzy multiple objective programming,” Journal of Management, Vol.17, pp.369-388.
(30) Viton, P.A. (1998) “Changes in multi-mode bus transit efficiency, 1988–1992,”
Transportation, Vol.25, pp.1–21.
(31) Yun, Y.B., Nakayama, H., and Tanino, T. (2004) “A generalized model for data envelopment analysis,” European Journal of Operational Research, Vol.157, pp.87–105.