9.5 Advanced Pareto Dominance
9.5.2 Tests with Dominance
In this section, we will test the influence of the different definitions of dominance. We will look at the combination of criteria, and of some of the rows in rule set (A). Additionally, we will examine different wage sets applying rule set (AW) for our standard customer, a businessman or a handicapped person traveling in a wheelchair.
9.5.2.1 Pareto Vs. Relaxed Pareto Vs. Advanced Pareto
We compared the profiles Pareto (P), relaxed Pareto (R), and advanced Pareto (A) (see Table9.10). Relaxing the criterion travel time from (P) to (R) increases the runtime by 15% and the number of all significant operations by nearly 9% and results in 50% more connections. As expected, advanced Pareto is much faster, it requires only one sixth or seventh of the runtime, respectively. Besides, it determines less connections. The other profiles calculate 4 (relaxed Pareto) or nearly 3 (Pareto) times as many optima in the end, but filtered according to advanced Pareto dominance, advanced is (of course) the best. The quality for (R) is only worse in 13.5% of the queries, whereas (P) is worse in 72.12%.
Since (P) disregards alternatives created by the relaxations, its poor quality was to be excepted. The fifth equation protects connections that achieve desired wages if we combine more than one criterion and thus creates new optima. This and the differences between the relaxations in (R) and (A) explain the discrepancies between those two versions.
Optima Runtime PQ extracts Labels created
Profile avg avg ratio avg ratio avg ratio
in ms in % in % in %
Pareto 22.48 2,466.08 100.00 192,178 100.00 305,426 100.00 relaxed 33.57 2,857.06 115.85 209,140 108.83 331,947 108.68
Fadvanced 8.45 412.14 16.71 57,926 30.14 90,921 29.77
Table 9.10: The influence of using profiles Pareto (P), relaxed Pareto (R), or advanced Pareto (A) for dominance testing. Runtime, number of significant operations and the average number of solutions.
Criteria Optima Runtime PQ extracts Labels created
# time price rel ic avg avg ratio avg ratio avg ratio
in ms in % in k in % in k in %
1F √ √ √ √
8.45 412.1 100.0 57.93 100.0 90.92 100.0
2 √ √ √
8.20 507.3 123.1 72.33 124.9 106.14 116.7
3 √ √ √
6.49 219.8 53.3 34.87 60.2 54.74 60.2
4 √ √
5.07 143.9 34.9 28.47 49.2 42.35 46.6
5 √ √ √
4.67 121.6 29.5 20.61 35.6 33.15 36.5
6 √ √
4.39 138.4 33.6 25.73 44.4 39.66 43.6
7 √ √
3.82 71.4 17.3 11.01 19.0 17.72 19.5
8 √
3.11 71.6 17.4 11.04 19.1 18.17 20.0 Table 9.11: The influence of toggling the criteria ticket cost (price), reliability (rel), and number of interchanges (ic). Runtime, number of significant operations, and the average number of solutions.
9.5.2.2 Influence of the Criteria
In this section we want to measure the share of computational complexity of each of the criteria. To this end we either activated or deactivated the criteria price, reliability of interchanges and number of interchanges (see Table9.11). Of course, it is not reasonable to deactivate criterion travel time. We observe that the number of advanced Pareto optima decreases as we deactivate criteria.
Number of interchanges ic
Toggling the criterioniconly (odd rows and their subsequent rows), the number of optima is less affected, than toggling any other of the criteria. The biggest difference is together with time and price (from #3 to #4), where we get 1.42 optima less without interchanges, in all other combinations the difference is only 0.71, 0.28, or 0.25 optima.
Interestingly,timeandprice is the only combination, where enablingic increases the runtime (by about 55%), for all other combinations the additional criterionic speeds up the search. This is due to the fact that travel time and reliability of interchanges are correlated with the number of interchanges, whereas the price is not.
Additionally,ic is severely limited to a range of usually no more than 2 to 5 different values for the optimal results to a given query. There are no queries for which one optimal connection is a direct one and another one has 7 interchanges.
Reliability of interchanges rel and ticket cost price
The combinationstime+price (#4) andtime+rel (#6) have nearly identical runtime, so price andrel appear to be equally complicated. However, disabling onlyrel (#3) in the standard version speeds up the search by 46%, whereas disablingprice(#5) nets us over 70% due to the correlation withic that onlyrel exhibits.
There are not as many additional optima forrel (e.g. 0.85 from #7 to #5 or 1.96 from
#3 to #1) as forprice(e.g. 2.67 from #7 to #3 or 3.78 from #5 to #1). However, some of the latter originate again from the missing correlation with criterionic, as seen above (e.g. 1.42 additional optima from #4 to #3).
9.5 Advanced Pareto Dominance 141 9.5.2.3 Other Criteria
We introduced the search for special offers and the criterion sleeping time in night trains in Chapter 6. Here, we want to answer the question: How does the performance in the respective sections relate to that observed in this section? In Section9.1and Section9.2, we used relaxed Pareto dominance with three or four criteria, therefore the runtimes have to be compared to our relaxed Pareto experiment (cf. Table9.10in the previous section).
Comparison to the Search for Special Offers In Section 6.2 we discussed the search for special tariffs, discounts or fix prices. The tests were run using three real criteria (time, number of interchanges, and price) and artificial ones (extendibility to a connection eligible for a certain special offer). The results were presented in Section9.1.
The approx. 170k priority queue extractions and a runtime of 1.7 seconds are about 80%
of the priority queue operations and 60% of the runtime of the relaxed Pareto reference version with our four criteria.
Comparison to the Search for Night Trains The search for night trains introduces a different fourth criterion in Section6.4. The required runtime as observed in Section9.2 was comparable, albeit slightly faster, as the criterion sleeping time is easier to handle than the criterion reliability of interchanges. There are much less connections “protected”
during dominance due to their sleeping time, than there are differences in the reliability score high enough to warrant considering more alternatives.
9.5.2.4 Influence of the “Rows”
In rule set (A) we have five rows (r= 5). Instead of measuring the impact of the criteria as in the previous section, we now want to have a closer look at the influence of the
“rows” in advanced Pareto dominance. We give runtime and the numbers of significant operations without row 5 (w/o 5), without rows 3+4 (w/o 3+4) and for rows 1 or 5 only in Table9.12. Note that we have to change to the relaxation termreltime instead of zero in the first row, if row 5 with the relaxation term rel5 is disabled.
We observe that the variants w/o 5 and w/o 3+4 have a worse quality for about half of the queries, butw/o 3+4 misses double the amount of optimal connections. Not surprisingly after that observation, w/o 3+4 is also much faster.
Runtime PQ extracts Labels created Quality loss
Rows avg ratio avg ratio avg ratio conn query
in % in % in % in % in %
allF 413.53 100.0 57,926 100.0 90,921 100.0 - -w/o 5 350.64 84.8 52,169 90.1 82,029 90.2 15.6 49.4 w/o 3+4 232.28 56.2 37,375 64.5 58,662 64.5 28.2 54.6 only 5 143.98 34.8 26,689 46.1 40,843 44.9 64.5 79.2 only 1 71.6 17.4 11,037 19.1 18,168 20.0 69.7 83.6 Table 9.12: The influence of the rows in advanced Pareto dominance (A). Runtime, number of significant operations, the average number of solutions and quality loss in connections (conn) and queries (query).
Basic Business Handicapped profile customer person
wage unit W B H
δic min / ic 30 10 120
δsec min / % -1 -0.5 -2
δcost min /e 15 2.4 15
δcost e/ h 4 25 4
Table 9.13: Differing wages for our profiles standard (W), business customer (B) and handicapped person (H) in advanced Pareto dominance (AW).
In variants only 1 and only 5 about two thirds of optimal connections are missing and for for fifths of the queries we have worse quality. Both variants share similar scores but found different connections. Optimizing the weighted sum in only 5 is about twice as demanding as optimizing only the first row with travel time and the other relaxation term.
9.5.2.5 Advanced Pareto Dominance with Different Wages
Next, we will examine different wage profiles for rule set (AW). Besides our standard profile (W) we will look at profiles suited for a business customer (B) or a handicapped person (H).
Profile Businessman We assume that the proposed wages W are suitable to find attractive connections for most passengers. However, if we know more about a customer in advance, say a businessman with a much higher priority on the travel time than on anything else, we may raise the importance of time in our wages, for example changing to the business customer profileB in Table9.13. Now we will spend at most 10 minutes to save one interchange instead of 30 minutes.
Using the advanced Pareto dominance profile (AW) with either of the two profiles and filtering the resulting connections according to profile B, the quality of the result set is identical. Of course, the result sets are not identical in the first place, but there is no advanced Pareto optimal connection in one set that is not also present in the other with respect to profileB. On the other hand, Whereas the runtime decreases significantly for the business customer profile, as the search does not have to provide that many cheaper or other alternatives (cf. Table9.14). With the standard wage profile more than double the runtime is required. A search with wage profileBis not more complicated than a two
Runtime PQ extracts Labels created
Profile avg ratio avg ratio avg ratio
in ms in % in % in %
business B 156.97 100.00 25,111 100.00 39,977 100.00 standard W 387.06 246.59 53,688 213.80 84,528 211.44
Table 9.14: Searching for connections for customer businessman using different wage profiles and rule set (AW). Runtime and number of significant operations.
9.6 Goal-Direction and Domination by Terminal 143