(t+1) to the t-frontier and (t+1)-frontier, indicating the location of DMUo whether the current performance of all DMUs is better then before; the latter, R2, is the relative locations of DMUo
in time t to the t-frontier and (t+1)-frontier, indicating the location of DMUo whether the future performance of all DMUs will be better than now.
If R1>1, it indicates DMUo is right in the current period that entire performance is better than the last period; If R1<1, it indicates DMUo is right in the current period that entire performance is worse than the last period; If R1=1, the performances over two periods even.
On the contrary, If R2 >1, it indicates DMUo is right in the current period that entire performance will be better than now; If R2 <1, it indicates DMUo is right in the current period that entire performance will be worse than now; If R2=1, the performances over two periods even.
Figure 1 Frontier shift
As depicted in Figure 1, a company’s performance in period t could be the six possible locations, A1t~ At6. The oblique line that connects the origin and the intersection of the two frontiers is the tradeoff on the strategy changes. A1t, At2, and At3 locate on the upper part and inside the t-frontier, between the two frontiers, and outside the (t+1)-frontier respectively. The distances of At2 and At3 to the t- and (t+1)-frontiers respectively are the measurement of super-efficiencies. Similarly, At4, At5, and At6 locate on the lower part and inside the (t+1)-frontier, between the two frontiers, and outside the t-frontier respectively. The distances of A t6 and A 5t to the t- and (t+1)-frontiers respectively are the measurement of super-efficiencies. It is noticeable that the locations of the six points A1t+1~A6t+1 have similar occasions.
(t +1)-frontier
·A1t t-frontier
·At4
·At5
。At4+1
。A1t+1
。At2+1
。At2
。At3
。A3t+1
。A5t+1
。At6+1
。At6
Input 1
Input 2 Tradeoff on the two
inputs shifts (strategy changes)
express the efficiency measurement of each point by the ratio of distances; for instance, by drawing a line that connects the origin and point A1t+1. The line intersects with the t-frontier and (t+1)-frontier at points α1 and β1, respectively. The ratio of Dt(xot+1,yto+1) to
O , respectively. Thus,
)
β . Similarly, drawing a line connects the origin and point A1t. The line
intersects with the t-frontier and (t+1)-frontier at points γ1 and δ1, respectively. Tables 1 and 2 depict the models employed to measure the two distances. The signs of R1 and R2 in the last columns are visible from Figure 1.
In Figure 1, a downward frontier shift (towards the origin) from period t to (t+1) represents a positive shift. The converse situation (away from the origin) represents a negative shift. For a company, from period t to (t+1), the four possible frontier shifts are as follows in (a)~(d). The 36 possible movements are depicted in Table 3.
Table 1 The computation of ratio R1
t+1 Dt(xto+1, yot+1) Dt+1(xot+1,yot+1) R1=
At4+1 Use (M3)
Table 2 The computation of ratio R2
t Dt(xot,yot) Dt+1(xto,yot) R2=
Table 3 The four possible frontier shifts for a company between two periods To period (t+1)
At6
(a) If R2>1 and R1>1,
then the FSo must be larger then 1.0, indicating the DMUo has a positive shift and the technology of DMUo progresses. As shown in Figure 1, the points of period t, A1t, At2, and At3 in the upper part could be one of the points at period (t+1) in the upper part, A1t+1, A2t+1, and A3t+1.
(b) If R2<1 and R1<1,
then the FSo must be less then 1.0, indicating the DMUo has a negative shift and the technology of DMUo declines. As shown in Figure 1, the points of period t, At4, At5, and At6 in the lower part could be one of the points at period (t+1) in the lower part, A4t+1, A5t+1, and A6t+1.
(c) If R2<1 and R1>1,
then FSo may be larger or less then 1.0. But, certainly we can conclude DMUo moves from a negative shift facet towards a positive shift facet. Also, there is a change in the tradeoff between the two inputs. Furthermore, FSo <1 indicates that the change resulting from the positive shift facet is less than that of the negative shift facet; and, on average, the technology of DMUo declines. In contrast, FSo >1 indicates that the change resulting from the positive shift facet is lager than that of the negative shift facet; and, on average, the technology of DMU progresses. FS =1 indicates that, on average, the technology of DMU remains the
same. As shown in Figure 1, the points of period t, At4, At5, and At6 in the lower part could be one of the points at period (t+1) in the upper part, A1t+1, At2+1, and A3t+1.
(d) If R2>1 and R1<1,
then FSo may be greater or less then 1.0. But, we can certainly conclude DMUo moves from a positive shift facet towards a negative shift facet. Also, there is a change in the tradeoff between the two inputs. Furthermore, FSo <1 indicates that the change resulting from the positive shift facet is less than that of the negative shift facet; and, on average, the technology of DMUo declines. In contrast, FSo >1 indicates that the change resulting from the positive shift facet is lager than that of the negative shift facet; and, on average, the technology of DMUo progresses. FSo =1 indicates that on average the technology of DMUo remains the same.
As shown in Figure 1, the points of period t, A1t, At2, and At3 in the upper part could be one of the points at period (t+1) in the lower part, A4t+1, A5t+1, and A6t+1.
3.1 Definition of TECo
Note that Mto+1= TECo× FSo and TECo =Dt+1(xot+1,yot+1) Dt(xot,yot) if (i) TECo >1, indicating Dt+1(xot+1,yot+1)>Dt(xot,yot). This implies that DMUo in time (t+1) is closer to the frontier in time t, (ii) TECo <1 implies DMUo in time (t+1) is further away from the frontier in (t+1) than DMUo in time t to the frontier in t, and (iii) TECo=1 implies DMUo in time (t+1) is as close to the (t+1)-frontier as DMUo in time t to the t-frontier.