Directional distance function

The main difference between the conventional model and the network model is that in this model the allocation of shared input x_cj^PC, x_dj^PCC between the two processes is not

given a priori like the other inputsx_aj^PH,x_bj^PU,x_ej^C, and provides a possibility to use the outputs of the production process as intermediate inputs in the consumption process. The conventional model does not provide for such allocation and processes, since it can be viewed as an aggregation of this network model that obscures the subprocess, such as the model constructed for measuring technical efficiency in multimode bus transit by Viton (1997).

8.3.2 Directional distance function

The most general of distance is the directional distance function, which is defined on the output set P(x) by Standard DEA or Farrell (1957) technical efficiency measures are closely related to the

distance function. Shephard’s (1970) input and output distance functions, are defined as

{

: ( / )

}

function expands (reduces) the outputs (inputs) proportionally as much as is feasible. The reciprocal of the output distance function is known as the Farrell output measure of

technical efficiency. The input and output distance functions are special cases of the directional distance function Eq. (8.2).

If the input oriented DEA model is chosen, then the directional distance function )

. Similarly, if the output oriented DEA is taken, then 1

When all of the three parts as mentioned in section 8.3.1 have different efficiency scores θ_k^H,θ_k^U,θ_k^C, this study introduces the three functions θ_k^H(X^I,Y^I), )θ_k^U(X^II,Y^II

and )θ_k^C(X^III,Y^III which will provide a measure of how efficient a firm k is at HB, UB and consumption process, respectively. If no input was associated with both HB and UB, or among HB, UB and consumption process, then the overall process doesn’t have the characteristics of the consumed services occur concurrently with produced services in the real world. The efficiency score of each part could then be calculated by ignoring all inputs associated with any one of the three parts and by running the model only for each part as follows:

As for the evaluation framework of the transit firm mentioned above, the evaluation concepts include the production and consumption processes. The actual overall process is generally not modeled explicitly; rather, one simply specifies what enters the box and what exits. The conventional models for DEA performance measurement are based on thinking about production/consumption as a “black box”(Färe and Grosskopf, 2000). This in fact does not suit the transit performance measurement application. Therefore this study

modified and applied the network DEA model introduced by Färe and Grosskopt (1996) and the multiactivity DEA model developed by Mar Molinero (1996) to model both production and consumption activities in the transit firms. This offers a possibility to integrate consumption and production into a transit performance evaluation framework.

Moreover, HB and UB are considered as two major production activities of a transit firm.

Diez-Ticio and Mancebon(2002) advocate the use of multiactivity DEA technique as a more powerful shared inputs treating method. Thus allocation of shared input resources into the activities of the production and consumption processes is allowed. Furthermore, the network DEA performance evaluation framework is an appropriate concept which has an insight in the overall performance profile of the transit firm. That is to say, the combination of the production and consumption processes into one overall technology will provide insights into the inter-related effects between the two major processes of a transit firm. The network DEA basis of assessment provides the opportunity of exploring a simultaneous input minimization for production technique, and an output maximization for marketing strategy for the overall performance, which extends the typical cost effectiveness measure from the literature.

For the illustration of the network performance measurement, this study chooses to evaluate firm k relative to the network technology (8.1) by means of a directional distance function. This measure is as follows:

)

(

1

)

, 1, , ,

The non-linear programming network DEA model in Eq (8.3), which is subject to the constraints, can be explained by each process as follows:

8.3.2.1 Production process constraints

The multiactivity DEA Model (Mar Molinero, 1996) can be applied to the determination of HB and UB efficiency at a set of transit firms in Taiwan. Some inputs will

contribute only to the HB or the UB, while some other inputs will contribute to both the HB and the UB. The efficiency model in this process has an input contraction orientation and seeks to estimate the cost efficiency (1-θ_k^H) and (1−θ_k^U) of transit firm k. The assessment is pursued under a constant returns to scale assumption, while the objective is incorporated regarding the cost minimization characteristics of transit production, which is consistent with the concept proposed by Talley and Anderson (1981). This results in the following constraints: Allocation of shared inputs to HB and UB:

( ) ( )

x , are quantities of inputs a and outputs f associated only with the HB activity of transit firm j.

PU gj PU bj y

x , are quantities of inputs b and outputs g associated only with the UB activity of transit firm j.

PC

xcj is quantities of inputs c associated with HB and UB at transit firm j.

λU H,

λ are positive constants associated with the HB production process and the UB production process.

µc is the proportion of the joint inputs c associated with the HB.

U k H k ,θ

θ are the maximum proportion inputs that can be reduced in the HB and UB activity, respectively, by transit firm k.

8.3.2.2 Consumption process constraints

The assessment of the service effectiveness of transit firms is based on an input-output set in which some inputs Y_f^PH,Y_g^PU are outputs of the production process, the characteristics and attributes are considered as intermediate inputs together with inputs

PCC

Xd , such as management labor associated with HB, UB and the consumption process, and inputs X associated only with consumption in a consumption process, while the _e^C

output Y_l^C,Y_o^C are the passenger-kms and passengers, which are outputs of HB and UB activities of the production process consumed by passengers, respectively. The solution of this process yields θ_k^C which conveys information on the necessary adjustments to individual consumption outputs of each transit firm, based on the assumption of maximization of the ridership. This results in the following constraints:

Consumption process:

,

Allocation of shared inputs to HB, UB and the consumption process:

( )

xej is the quantity of inputs e associated only with the consumption activity of transit firm j

PCC

xdj is the quantity of inputs d associated with HB, UB and consumption activity of transit firm j

2 1, _d

d α

α are the proportion of joint inputs d associated with HB and UB, respectively

C oj C lj y

y , are the quantities of outputs l, o associated only with the consumption activity of transit firm j

C

θ is the maximum proportion consumed outputs that can be expanded by transit firm k. k

8.3.2.3 Environmental constraints

To further consider the effect of the environmental factors on the performance of transit firms, the environmental variables as were included non-discretionary inputs by imposing a restriction of the following form:

The effect of environmental factors on the consumption process:

,

where ^eCwj is quantities of environmental factors w associated only with the consumption activities of transit firm j.

The objective is to jointly maximize θ , _k^H θ and _k^U θ . The solution of Eq.(8.3) _k^C yields an overall measure of transit firm k which includes the HB cost efficiency (1-θ_k^H),

UB cost efficiency (1-θ^U_k ) , service effectiveness (1+θ_k^C) , and cost effectiveness )

(1+θ_k . The objective function takes the form:

Max θ^k =w_k^Hθ^kH +w^U_kθ^kU +w^C_kθ^kC

As is the case in the above equation, the coefficients w_k^H,w^U_k ,w^C_k are associated with the priorities given to the two activities and processes, respectively. In order to emphasize the relative importance of each activity or process, the ws can be normalized so that they add up to one, e.g., w_k^H +w^U_k +w_k^C =1. The assessed transit firm k will be termed efficient

if and only if θ_k^H =θ_k^U =θ_k^C =0 in the optimal solution of Eq. (8.3~8.15). Since the model incorporates a process in the assessment, the results are useful for distinguishing cost efficiency, service effectiveness and cost effectiveness in consideration of the inter-related effects among these measures. In addition, the results are also useful for setting improvement targets for inefficient activities of the transit firm.

8.4 The data

In this study, drivers, vehicles, fuel and network length are used as specific inputs for both highway bus service and urban bus service, respectively; technical staff (mechanics) are used as shared inputs for both HB and UB; management staff as shared input for HB, UB and the consumption process; sales staff as a preassigned specific input for the consumption process; and population density and car ownership are used as two major environmental variables for the consumption process. The multiactivity DEA model will be applied to overcome the shared inputs issue. As for the output measure, vehicle-kms and

frequencies of service are used as intermediate produced outputs for production process as well as for HB and UB respectively; and passenger-kms and passengers are used as final (consumed) output for consumption process as well as for HB and UB respectively.

The indicator data to be used in the measurement of cost efficiency, service effectiveness and cost effectiveness in Taiwanese motorbus transit is a sample of 24 firms, all long established operators and located all over the island in 2001. All these DMUs operated both HB and UB services. A system which provided only either HB or UB would therefore be excluded. All data used in the modified network model were obtained from the annual statistical reports published by the National Federation of Bus Passenger Transportation of the Republic of China for 2002.

A modification of the network DEA model introduced by Färe and Grosskopf (1996) is proposed, and the multiactivity DEA model developed by Mar Molinero (1996) as well as Fielding’s theory is also incorporated into this all-in-one model to investigate the multimode transit performance. The modified network model represents two production and one consumption nodes, as well as intermediate products between the production node and consumption node in the multimode bus transit technology. The modified network DEA model is applied to estimate the efficiency or effectiveness of the multimode bus transit systems, in which the shared inputs used could be allocated to each node (activity), and the optimal proportion of shared inputs used may vary from one firm to another. Thus the optimal proportions in the modified model are derived from the data instead of being fixed in advance. (This differs from Löthgren and Tambour (1999), where a network model with its allocation data was based on budget data in terms of percentage of total labor hours).

Then Fielding’s three important indicators (1987) corresponding to the characteristics of the production process coinciding with the consumption process in the transit system are presented. In addition, the transit performance is thought to be sensitive to the environment in which the system operates, and hence environmental factors that affect performance must be identified and taken into account (Giuliano, 1981).

In the modified model, inputs x_aj^PH ∈R₊^ma such as drivers are used in the production node of HB to produce intermediate output y^PH_fj ∈R₊^sf such as vehicle-kms. This intermediate output y^PH_fj is used as input together with x and _e^C x_e^C∈R₊^mz such as sales

staff in the consumption node to produce final output y_l^C ∈R₊^zl such as passenger-kms.

The same method can be applied to the urban bus service. The network relationship among netput is illustrated in Figure 8.3. The production technology of the multimode bus transit is represented using proxies for inputs and outputs of each of the two modes, that is, eight specific inputs (four for HB and the other four for UB), two shared inputs and two environmental variables; two intermediate outputs and two final outputs. The following set of variables, labeled according to the relationships in Figure 8.3, are used in the empirical application for each mode.

Inputs for highway bus service ( x_a^PH ): The four specific inputs are given by DRIVER

=

PH

xd (the number of transportation labor used by this mode in providing service), HICLEx_v^PH =VΕ (the fleet sizes, which are taken to be the total number of vehicles operated in maximum service by this mode), x^PH_f =FUEL (the number of liters of fuel by mode) and x_l^PH =NWLTH (network length by mode).

Shared inputs ( x_d^PCC,x_c^PC ): These are given by x_m^PCC =MGT (the number of

management labor used by two modes and the consumption node), and x_t^PC =MEC (the number of mechanics used by two modes). The data includes a preassigned specific input (x ), sales staff, _e^C x_s^C =SALE for each firm in terms of the number of sales labor devoted to the consumption node in the network model. The allocation of these data are based on the resulting data being derived from the application of the multiactivity DEA model, which is

capable of objectively assigning a share to the different activities/processes which will allow for the independent treatment of each of these different activities/processes. This information allows a separation of the shared inputs, which is necessary for an implementation of the modified network model.

Inputs for urban bus service (x ): In the same manner as the highway bus service, the _b^PU three individual inputs for urban service are given by x_d^PU =DRIVER (the amount of

transportation labor used by this mode in providing service), x_v^PU =VEHICLE (the fleet sizes, which are taken to be the total number of vehicles operated in maximum service by mode), x^PU_f =FUEL (the number of liters of fuel by mode) and x_l^PU =NWLTH(network length by mode).

Outputs for highway bus service ( y^PH_f ): The intermediate output is given by

hPH

y =VEHKM (vehicle-kms) and the final output（y ） is given by _h^C y^C_h =PASSKM (passenger-kms).

Outputs for urban bus service (y_g^PU): In the smae manner as the highway ubs service, the intermediate output is given by y_u^PU=FREQ (frequencies of service) and the final output（y_o^C）is given by y_u^C=PASS (passengers).

Environmental variables (e ): Following Levaggi (1994) and Chu et al. (1992), two _w^C environmental factors are considered in this paper. There is a set of “environmental factors”

including CARe_c^C = , e^C_p =POP (the quantities of car ownership and population density influencing the consumption process), to describe the situation in which the DMU finds itself.

Table 8.1 presents 17 netput used in this study to capture the cost efficiency, service effectiveness and cost effectiveness. Table 8.2 shows the summary statistics.

Table 8.1 Inputs and Outputs Measures Used in the Model

Specific inputs Shared inputs

Intermediate

Production process of highway bus service

Drivers: x_d^PH=DRIVER Vehicles: x_v^PH=VEHICLE Fuel: x^PH_f =FUEL

Network

Length:x_l^PH=NWLTH

Managements: x_m^PCC =MGT (Shared inputs to HB, UB and consumption process)

Production process of urban bus service

Drivers: x_d^PU=DRIVER Vehicles: x_v^PU=VEHICLE Fuel:

x

^PU_f =FUEL

Network Length:

x

_l^PU

=NWLTH

Mechanics: x_t^PC=MEC (Shared inputs to HB and UB)

Frequency:

Table 8.2 Variables and Descriptive Statistics

Mean Stdev Maximum Minimum

Individual inputs Process of HB service

DRIVER 158.1 137.0 451.0 8.0

VEHICLE 121.0 107.8 387.0 8.0

FUEL 3,084,916.9 2,431,013.4 8,154,152.0 281,700.0

NWLTH 1,248.2 1,134.0 3,765.9 31.4 Process of UB service

DRIVER 137.2 175.1 633.0 3.0

VEHICLE 120.3 145.2 557.0 3.0

FUEL 3,386,796.7 4,857,287.8 16,362,765.0 44,381.0

NWLTH 248.5 228.9 1,006.5 18.0

Shared inputs

MEC 39.6 32.3 117.0 3.0

MGT 31.3 39.6 193.0 1.0

Intermediate outputs

VEHKM 8,533,146.4 6,956,059.8 25,378,595.3 775,231.7

FREQ 790,040.5 1,131,977.4 4,115,662.0 14,540.0 Final outputs

PASSKM 103,807,107.6 109,260,433.4 417,903,218.0 5,189,557.0

PASS 20,446,874.1 30,453,096.6 105,091,579.0 76,818.0 Environmental variables

CAR 222.7 17.3 249.7 191.4

POP 4,016.2 3,252.5 11,864.0 73.0

8.5 Results and discussions

There are a number of results can be found with regard to the all-in-one network model, and separate the conventional DEA model for measuring performance. In this section the results obtained are summarized in Table 8.3 will be comment. Recall that if the value of

the cost efficiency ( )

( 2

values less than 1 denote ‘inefficient’. On the other hand, if the value of the service effectiveness (1+θ_k^C) or cost effectiveness

(

1+θ_k

)

equals to unity, it denotes ‘effective’, whereas values greater than 1 denote ‘ineffective’. For each transit firm eight performance measures were calculated. The three basic measures that were obtained by the network model, θ_k^H, θ_k^U and θ_k^C as well as the other two induced measures ) HB and UB activity cost efficiency, and in the fourth column the service effectiveness are evaluated on the basis of their ability to share common inputs among different activities, and determine simultaneously the efficiency and effectiveness. This is a credible concept, and from this viewpoint the results make clear that among the 24 transit firms measured, only 7, those ofTable 8.3 Efficiency or effectiveness scores of the network model Ta-you, Fu-ho, Kuang-hua, Tamshui, Chungli, Fengyuan and Pingtung can be considered as cost efficient, which is shown in the third column of Table 8.3. If highway bus services are concentrated on, as shown in the first column, then the bus services of 8 of the transit firms demonstrated a productive behavior superior to the rest. Regarding the urban bus services, a maximum level of efficiency was achieved by 10 transit firms, with DMUs that were efficient in each of the two services corresponding in only seven cases, i.e., Ta-you, Fu-ho, Kuang-hua, Tamshui, Chungli, Fengyuan, Pintung.

Table 8.3 Efficiency or Effectiveness Scores of the Network Model

Ta-you 1.000 1.000 1.000 1.000 1.000

Fu-ho 1.000 1.000 1.000 1.000 1.000

Chung-hsing 0.692 0.899 0.796 1.153 1.179

Chih-nan 0.964 0.748 0.856 1.000 1.072 Kuang-hua 1.000 1.000 1.000 1.000 1.000 Tamshui 1.000 1.000 1.000 1.000 1.000

Hsin-ho 0.865 1.000 0.933 1.118 1.093

Taipei 0.996 0.949 0.973 1.000 1.014

Sanchung 0.786 1.000 0.893 1.000 1.053

Capital 0.966 0.986 0.976 1.000 1.012

Hsintien 0.995 0.998 0.997 1.000 1.002

Hualien 0.941 0.870 0.906 1.000 1.047

Taoyuan 0.822 1.000 0.911 1.000 1.045

Chungli 1.000 1.000 1.000 1.000 1.000

Hsinchu 0.929 0.975 0.952 1.000 1.024 Fengyuan 1.000 1.000 1.000 1.000 1.000 Chu-yeh 0.753 0.747 0.750 1.000 1.125 Taichung 0.760 0.550 0.655 1.187 1.266

Jen-you 0.837 0.449 0.643 1.000 1.179

Changhua 0.887 0.507 0.697 1.000 1.152

Chiayi 0.857 0.680 0.769 1.113 1.172

Tainan 0.903 0.491 0.697 2.227 1.765

Kaohsiung 1.000 0.514 0.757 1.000 1.122 Pingtung 1.000 1.000 1.000 1.000 1.000 Maximum 1.000 1.000 1.000 2.227 1.765 Minimum 0.692 0.449 0.643 1.000 1.000

Mean 0.915 0.848 0.882 1.075 1.097

Median 0.953 0.981 0.922 1.000 1.046

Stdev 0.097 0.204 0.126 0.252 0.162

Note: (1) )

Regarding average efficiency, it is worth noting that this clearly differs between the two activities, with the highway bus service showing a higher average rate of efficiency.

The priority given by transit firms to this type of service, because of the greater returns in profit for operating them, could be one of the reasons for explaining this phenomenon.

With regard to the service effectiveness measure, 19 out of 24 are categorized effective transit firms, while only 5 are categorized as ineffective. When the mean service effectiveness score is greater than 1 (1.075) it denotes ‘ineffective’ for the sample as a whole. As to the cost effectiveness measure, it is found that only six ranked as effective while the other 18 ranked as ineffective firms. The average cost effectiveness was also greater than 1 (1.097) indicating ‘ineffective’ for the sample.

These network results are very much different when the transit firm performance is judged on the basis of separate measures of cost efficiency, service effectiveness and cost effectiveness, terms which are described in previous literatures following the interpretations proposed by Fielding (1987). There is one further point that deserves discussion. It would be reasonable to compare the rates obtained from the network model with those derived from a conventional DEA model. In the latter no regard is given to the possible technological differences of the various activities engaged in by multimode transit firms, integrating them into one single measurement model, nor are the cost efficiency, service effectiveness and cost effectiveness determined simultaneously. The results of the comparison are set out in Table 8.4.

In order to provide statistically robust findings concerning the firms’ performance, the paired difference experiments are applied. This experiment is conducted to verify whether the sample firms of the two kinds of models were drawn from the same performance populations for the three measures, respectively. The significance of the t-values is set as a two-tailed test at 0.025 acceptance level. As shown in the last column of Table 8.4, the test of significance yielded a t-value of 2.894, which shows a statistically significant difference

Table 8.4 Descriptive Statistics of the Conventional and Network Models’ Performance Scores and the Results of Test of Significance

Network model Conventional model Test of significance

Number of

Note: (1) Paired difference experiments are used to test the same mean between two groups. The null hypothesis is that there is no significant difference in the efficiency or effectiveness scores between

Note: (1) Paired difference experiments are used to test the same mean between two groups. The null hypothesis is that there is no significant difference in the efficiency or effectiveness scores between

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