Chapter 3 Methodology
3.3 Directional distance function
3.3 Directional distance function
Conventional DEA models can only consider output expansion or input contraction, but not both. When the technology set is characterized by variable returns to scale (VRS), both output-oriented and input-oriented technical efficiencies are not equal generally. If we evaluate efficiency of tourist hotel by the output-oriented approach, we may not fully characterize operational management of hotels since it cannot distinguish between quasi-fixed inputs and variable inputs.
Other the other hand, it may overestimate the ability of adjustment of hotel management if the input-oriented model ignores the existence of quasi-fixed inputs (Ouellette and Vierstraete, 2004). Furthermore, the objective of tourist hotels is to expand outputs rather than to contract inputs. Hence, it is inappropriate to evaluate hotels’ efficiency ignored output expansion. The directional distance function, capable of expanding outputs and contracting inputs simultaneously, can fulfill the request of this study.
Färe and Grosskopf (2005) define the directional distance function as follows:
{ }
variable input vector, quasi-fixed input vector, and output vector, respectively;
( )
largest feasible contraction of input vector x
in the −gx
direction. Note that we treat quasi-fixed input vector k
as fixed. This specification can not only characterize the property of quasi-fixed inputs in the operational management of
hotels, but also satisfy the request of output expansion. The value of the directional distance function D( )⋅ =βˆ
. The efficient DMU is corresponding to ˆ 0β = . In other words, the technology frontier is constructed by those DMU associated with
β = . Hence, the larger the value ˆβ , the farer the DMU from the frontier. ˆ 0
3.4 Meta-frontier
Due to different national cultures, operational philosophy, managerial mode, and etc., domestic and foreign-owned hotels apparently belong to different operating systems and thus the assumption of convexity may not be valid. The meta-frontier approach allows each group to have its own group-frontier. The meta-frontier is defined as a common boundary that envelops the group frontiers. The technology set associated with meta-frontier could be convex or non-convex. We will illustrate by figure 3.2.
Assume that there are two groups, A and B. The frontier of group A is the line segment connecting A1, A2, G and A3. Similarly, the frontier of groups B consists of points B1, G, B2 and B3. If the technology set is non-convex, the relevant meta-frontier is the line segment connecting A1, A2, G, B2, and B3. It is apparent that group-frontiers exhaust the meta-frontier; in other words, each part of meta-frontier belongs to at least one of group-frontiers. If the technology set is convex, the relevant meta-frontier is the line segment connecting A1, A2, B2, and B3.
The convexity allows that the input-output combinations beyond the boundaries of group-frontiers such as the dot line connecting A2 and B2. It may imply that existing technology can upgrade through spillover and/or mutually learning among groups for a considerable period. In this sense, the non-convex meta-frontier is
suitable to analyze efficiency in the short run, while the convex meta-frontier may be appropriate for the analysis of the long run. Furthermore, we follow the basic assumption that the quasi-fixed inputs cannot be adjusted in the short run, but they are variable in the long run.
Figure 3.2 Illustration of the meta-frontier
Before drawing the meta-frontier, we assume that the technology is constant within a given time period. The meta-technology is regarded as true technology, the group-technologies are considered as revealed technology. In the convex case, the production possibility set (true technology) which constructed by the technology integration of groups is greater then the case of non-convex. Moreover, the strategies of firms are usually focused on the adjustment of current operating condition in the short run, but the strategies of firms are focused on the overall planning in the long run.
We now describe how to incorporate the directional distance function in the meta-frontier approach. Let Tm be the meta-technology set that envelopes the G
A1 A2
G A3
B1 B2
B3
frontier of group A frontier of group B
X Y
0
group frontiers such that Tm =T1∪T2∪ ∪... TG where Tg is the technology set of group g, g=1,2, , .… G The directional distance function relative to the meta-technology set can be expressed as:
{ }
This study applies the following approach to calculate the direction distance function for the non-convex meta-technology set: (1) calculate the direction distance function of each DMU based on the efficient frontier of group g, say ˆ
βg, g = 1, 2,.., G; (2) the relevant value of the direction distance function ˆβ for each DMU is the m
maximum of
{
β βˆ ˆ1, , ,2 … βˆG}
; i.e. βˆm ={
β βˆ ˆ1, , ,2 … βˆG}
. For G = 2, the linear programming of DMU j under VRS can be written as:( )
slack and the n-th input slack, respectively; ε is a small non-Archimedean quantity,
usually being 10−6. The first constraint labeled (4b) seeks largest contraction of the n-th variable input in the direction gnx. The constraints in (4c) search for largest expansion of p-th output in the direction g . Expression (4d) holds the quasi-fixed py inputs to fixed in the short run. Constraints (4e) to (4g) ensure the technology is VRS.
The non-convex meta-technology set is only suitable for the analysis of the short run. We employ the convex meta-technology set to analyze the efficiency in the long run. In addition, we assume that the quasi-fixed inputs can be adjusted in the long run, so all inputs are variable. The corresponding linear programming is:
( )
group-technologies regard as revealed technology. In other words, we measure the group-efficiency based on the revealed technology, while evaluate the meta-efficiency based on the true technology. Hence, each DMU can generate two directionalFigure 3.3 Illustration of meta-efficiency and technology gap degree of outputs expanded and inputs reduced from ( ,x y)
to the group frontier in
. The technology gap (TG) represents the difference between group frontier and meta-frontier.
the point Aˆ
(
x−βˆg yx, +βˆgy)
along the direction (−g gx, )y . Similarly, the directional distance function D ⋅m( )projects the point A to the meta-frontier at the point A* (x−βˆmg yx, +βˆm( ) )⋅ gy along the direction (−g gx, )y . We can image that TG can translate the point Aˆ on the group-frontier to the meta-frontier at the point
(
(x−βˆgx)−TGgx, (y+βˆgy)+TGgy)
along the direction (−g gx, )y . However, both points A* and(
(x−βˆgx)−TGgx, (y+βˆgy)+TGgy)
are coincide.Hence, we obtain ˆβm = +βˆ TG.