Figure 5 Illustration of slack’s duration for POI p;
Related Figures (15)
Figure 1: Illustration of a tourist route (blue dashed line) from s, to t, via POIs p;, p; and px. Green dashed arrows indicate available multimodal transfer options among POIs. Figure 2: Arrival time walking and using public transportation Unlike CSCRoutes, the algorithms introduced in this article, handle time dependent travel times among different locations/POIs. Therefore, they employ different insertion steps than CSCRoutes. Namely, the insertion steps take into account the time dependency of the travel Figure 3: traveling time from u to v as a function of the departure time from u where wait; denotes the waiting time at POI p, following a visit at p;. Figure 4 illustrates an example of inserting p,, between p; and p; shifting the visit at p; later on time. Figure 4: Illustration of p; insertion between p; and p;. A pseudo code implementation of the function shift(k, i, 7,r) follows 2, which calculates the insertion cost shift,’ in route r. The function returns oo if te insertion ol pr is infeasible. Apart from inserting a new POI in a route, our algorithms encourage replacing an already included POI by a higher profit POI. Considering that in route r, the POIs p;,p),p; are con- secutive and px is not included in the solution, pz, can replace p; if the arrival at p, takes place within its time window and the time windows of p; and the POIs following p; are not violated. Extending the terminology introduced previously, let shift,’ ; be the difference in the insertion cost from replacing POI p; by px. Then, 3.3 The Time Dependent Slack CSCRoutes (TD_S/CSCR) algorithm The TD_S@CSCR algorithm modifies the insertion step of TD_CSCR i.e., it takes an alter- native approach for determining the POI p,; that will be selected for insertion in a route r. Specifically, while TD_CSCR algorithm’s criterion for selecting the inserted POI in a route is based on the insertion cost, TD_S@€CSCR involves a more global criterion taking into considera- tion the effect of this insertion in the whole route. Figure 6: Illustration of a solution to the TTDP. A typical solution to the TTDP is illustrated in Figure 6. The tourist destination area is partitioned in nine square regions and a central location (representative) is calculated for each region (indicated by green circles). The start/end locations of the route (s1 and ty, respectively) are determined at the user query time and are indicated by the black squares. The representatives 8, (of the area where s; belongs to) and t) (of the area where t, belongs to) are visited in the beginning and in the end of the trip, respectively. POIs pi, pi2, pig and pyi4 are visited in between. Solid and dashed lines denote walking and transit transfers, respectively. Figure 7: (a) POIs locations and (b) hotels locations, in the Athens metropolitan area. Table 1: Experimental results when employing the standard profit criterion. As a general remark applied to all algorithms, the increase of the overall collected profit with the increase of the number of routes is sublinear, since the average POI profits is higher when considering low numbers of routes (i.e. for short stays, tourists tend to visit the do-not- miss POIs). Furthermore, all implemented algorithms perform remarkably well with respect to execution time since they derive solutions in less than 0.5 sec in all cases (considering topologies of 113 POIs and up to four routes). Figure 8: Illustration of a tourist route starting/ending at a hotel; solid lines indicate walking transfers while dashed lines indicate public transit transfers. Table 2: Comparison of cluster-based vs. non-cluster-based approaches. Table 3: Improvement from including the i-quality metric i-quality metric favors the insertion of POIs which belong to clusters with several high-profit POIs, not yet included into a route). Although the table reports only marginal improvement in the case of using the 7-quality metric, this is mostly due to the particular characteristics of the utilized datasets. In particular, the POI clusters of the metropolitan area of Athens are crowded in a relatively restricted geographical area, which allows to easily accessing (by foot or public transit) the most highly profitable POIs one after the other, irrespective of their cluster assignment. This results in deriving almost identical solutions when not incorporating the 7- quality metric, especially for relatively short time budgets. The effect of the i-quality metric is expected to become more evident in scenarios featuring relatively distant clusters and prolonged time budgets. In the sequel we present experimental results compiled for our TDTOPTW algorithms, using an alternative criterion for selecting the best found solution. Specifically, instead of selecting the solution with the highest aggregate profit over all routes, we select as best found solution the one
