CHAPTER 8 The Case Study 4
8.2 The framework of performance evaluation
8.3.1 Network production possibility set
The starting point of all above-mentioned proposals is that when one DMU jointly carries out various activities and processes which can not be assumed to be technologically identical, is to separate these activities and processes into different technologies in a network model. The application of the traditional DEA model is to evaluate transit performance which assumes that the DMU is equally efficient in all its activities, and to assess cost efficiency, service effectiveness and cost effectiveness by using three separate DEA models. The main problem is that what is by nature concurrent is treated in a consecutive manner, which could lead to a significant degree of distortion in the interpretation of the results.
In order to solve this problem, this study proposes a modified network DEA model introduced by Fare and Grosskopf (2000), in order to represent a production and consumption process in the transit technology. Consider, for the sake of simplicity, a two-activity transit firm with only two different services which outputs are the intermediate inputs of the consumption process in the model.
Consider a set of j=
{
1,...,n}
transit firms which use input quantities ), , , ,
(XajPH XbjPH XcjPC XdjPCC XejC
X = , a={1,...,ma} , b=
{
1,...,mb}
, c={
1,...,mc}
,{
md}
d = 1,..., , e=
{
1,...,me}
to produce intermediate output quantities YP =(YfjPH,YgjPU),{
sf}
f = 1,..., , g=
{
1,...,sg}
, and final output quantities YC =(YljC,YojC), l ={
1,...,zl}
,{
zo}
o= 1,..., , where inputs, which are associated only with HB, will be given the superscript PH. Inputs which are associated only with UB will be given the superscript PU, inputs which contribute to both highway bus service and urban bus service will be given the superscript PC, inputs which are associated only with the consumption process will be given the superscript C, whereas inputs which contribute to both highway bus service and urban bus service as well as the consumption process will be given the superscript PCC. For example, XaPH =(xdPH,xPHf ,xvPH,xlPH) are inputs associated solely with HB, such as drivers, fuel, vehicles and network length, but XcPC =(xtPC) is an input associated in part with HB and in part with UB, such as mechanics. YfPH =(yhPH) representing produced outputs of HB service, such as vehicle-kms, is an output which is solely associated with HB, whereas YgPU =(yuPU) are the results of UB service, such as frequencies of service. YP has the characteristics of intermediate products which are intermediate to the production system and are consumed by the consumption system together with specific inputs
) ( sC
C
e x
X = of the consumption process, such as sales staff deliver to the consumption process. The products of the consumption process, YC =(YlC,YoC), which is the final output matrix with outputs Y and lC Y , outputs oC YlC =(yhC) are the result of the consumed HB service, such as passenger-kms, outputs YoC =(yuC) are the result of consumed UB service, such as passengers. The network model is depicted in Figure 8.3.
PCC
Shared inputs for HB, UB and consumption Cost effectiveness (Overall performance)
Figure 8.3 Performance Dimensions of Multi-mode Transit System
In the situation where there are inputs associated with both activities or among activity HB, activity UB and the consumption process, then one will assume that these shared inputs can be apportioned between HB and UB, or among activity HB, activity UB and consumption process. In this way, each joint input will contribute to the determination of the cost efficiency of the HB, and of the cost efficiency of the UB in the production process, as well as the determination of the service effectiveness of the consumption process.
Assume that the proportion of the shared inputs assigned to each one of the said activities are µc and 1−µc, and the proportion of the shared inputs assigned to each one of the said activities and the consumption process are α1d, αd2 and 1−α1d −αd2. Then the model for the network DMUs will have the HB, UB and consumption process production possibility set APH,APU,AC, defined as follows:
If A is the smallest set satisfying the convexity, constant returns to scale, free PH disposability, and minimum extrapolation postulates (Tsai and Mar Molinero, 2002), subject to the condition that each of the input-output observations (XI,YI)∈APH, then the
Similarly, the UB input set PU(YII) for each Y can be defined as II
The network model gives some insights into how inputs may be shared by different processes and inter-related effects between different processes. Here, xcjPC can be allocated
between the two activities, xdjPCC can be allocated among the two activities and the other process, and the others are pre-assigned to a specific process for each DMU j. In this case, the network production possibility set is shown below in (8.1).
⎩⎨
,
The main difference between the conventional model and the network model is that in this model the allocation of shared input xcjPC, xdjPCC between the two processes is not
given a priori like the other inputsxajPH,xbjPU,xejC, 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).