### Table 3 Results for the unconstrained e$cient frontier

1998

"... In PAGE 25: ... These values mean that (excluding initialisation) each heuristic evaluates exactly 1000N solutions using Algorithm 1 for each value of j. The results for the unconstrained e$cient frontier are shown in Table3 . In that table we show, for each of our quot;ve data sets and each of our three heuristics: (a) the median percentage error, (b) the mean percentage error, (c) the total computer time in seconds.... In PAGE 25: ... Note here that all the computer times presented in this paper exclude the time needed to calculate the error measures. It is clear from Table3 that our GA heuristic is best able to approximate the UEF with an average mean percentage error of 0.0114%, the SA heuristic next best with an average mean percentage error of 0.... In PAGE 26: ... It can be seen from Table 4 that over our quot;ve test data sets no one of our heuristic algorithms is uniformly dominant. Although the GA heuristic performs better than the SA heuristic, which in turn performs better than the TS heuristic, the di!erences are not nearly as marked as they were for the UEF ( Table3 ). For some data sets there are considerable di!erences in the percentage error measures, indicating that the algorithms give signi quot;cantly di!erent results.... ..."

### Table 4 Results for the cardinality constrained e$cient frontier

1998

"... In PAGE 26: ...he results for our heuristic algorithms with K quot;10 and ei quot;0.01, di quot;1(i quot;1,...,N) are shown in Table4 . In that table we show, for each of our quot;ve data sets and each of our three heuristics: (a) the median percentage error, (b) the mean percentage error, (c) the number of (undominated) e$cient points, (d) the total computer time in seconds.... In PAGE 26: ... Note that for the column labelled lt; we did not eliminate from lt;(j) any dominated solutions. It can be seen from Table4 that over our quot;ve test data sets no one of our heuristic algorithms is uniformly dominant. Although the GA heuristic performs better than the SA heuristic, which in turn performs better than the TS heuristic, the di!erences are not nearly as marked as they were for the UEF (Table 3).... In PAGE 27: ...4. Discussion There are a number of points which can be made with respect to Table4 and these are discussed in this section. T.... In PAGE 28: ...6% 32.3% In Table4 we have shown results for quot;ve data sets, one particular value of K and one set of values for ei and di. Plainly for di!erent data sets/values the results will be di!erent.... In PAGE 28: ...or ei and di. Plainly for di!erent data sets/values the results will be di!erent. However we believe that our key point, namely that it is important to use a number of heuristics and to pool their results, is established. As stated above, the percentage errors given in Table4 are overestimates of the errors associated with each heuristic as they are derived from the UEF, which dominates the CCEF. One point that is important however is the distribution of these e$cient points along this frontier.... In PAGE 29: ... This is illustrated in Fig. 9 for the DAX data set (which has the highest mean (pooled) error in Table4 ) using the pooled results for all three heuristics. In that quot;gure we have plotted the curves for K quot;2, 3, 4 and 5 (ei quot;0.... ..."

### Table 1. Progressive refinement: a set of BSP-tree frontiers.

2006

"... In PAGE 9: ...16:33 WSPC/Guidelines ijcga Parallel Solid Modeling Using BSP Dataflow 9 box of the split representation. The refinement process of the frontier of a BSP-tree is de- scribed in Table1 . Such a progressive splitting is efficient when the volume of fuzzy cells gradually tends to zero.... In PAGE 9: ...Adaptive refinement The refinement algorithm in Table1 is made adaptive after adding a step before line 8, projecting the bounding ball of c(n) on the view plane. If either the projection is outside the viewport or its radius is below a threshold, the node is set to IN, since it is useless to detail it more.... ..."

### Table 3: Properties of the frontier search

2000

"... In PAGE 18: ... To verify this result on a reliable bases monte-carlo simulation would be necessary. Table3 summarizes the properties of DEA, stochastic frontier and the fron- tier search algorithm introduced here. 6 Conclusion This paper introduced a genetic programming based methodology to determine production frontiers.... ..."

### Table 3. Results for Game 1 * represents the most balanced solution on the non-dominated frontier + represents a solution on the non-dominated frontier italics represents a non-dominated solution among those found

"... In PAGE 11: ... They contain the payoffs for both players as well as indicating the points on the non-dominated frontier. Game 1 is reported in Table3 , Game 2 in Table 4 and Game 3 in Table 5. Across all the games, the strategies of maximin, maximax and weighted sum all record instances of finding the most balanced non-dominated solution.... ..."

### Table 3. Results for Game 1 * represents the most balanced solution on the non-dominated frontier + represents a solution on the non-dominated frontier italics represents a non-dominated solution among those found

"... In PAGE 11: ... They contain the payofis for both players as well as indicating the points on the non-dominated frontier. Game 1 is reported in Table3 , Game 2 in Table 4 and Game 3 in Table 5. Across all the games, the strategies of maximin, maximax and weighted sum all record instances of flnding the most balanced non-dominated solution.... ..."

### Table 1: Computation of head, tail and frontier strings

"... In PAGE 5: ....1 Let the sample set be S+ = {ab, aabb, aaabbb}. The even-linear skeletons are constructed for the sample set S+ and using these even-linear skeletons a node set ND(S+) is constructed: ND(S+)={N11,N21,N22,N31,N32,N33}. Head strings, tail strings and frontier strings for each of the node in the ND(S+)arecomputedandareshowninthe Table1 . UsinginformationintheTable1,aninitialpartition... ..."

### Table 8 DEA2 frontier- Indices of efficiency

"... In PAGE 18: ...9843) and now passes over to be declared efficient and also forms part of the reference group for another 23 units. Table8 shows the new indices of efficiency. In order to grade the efficient units, we have used a method that has frequently been applied in the DEA literature.... ..."

### TABLE 5 CONVENTIONAL ALGORITHMS ON THE TRANSLOG COST FRONTIER MODEL

1994

Cited by 42

### Table 2 Mean statistics of the stochastic frontier shadow cost function estimation

"... In PAGE 17: ...772, which implies that the one sided error term U dominates the symmetric error V. In Table2 , mean statistics of the estimated stochastic frontier shadow cost function are shown for two periods, 1980-84 and 1985-93, and for the whole period, 1980-93.... ..."