Fig. 1. Rolling horizon scheme with a look-ahead window T. Fig. 4. Definitions of time metrics. For individual elevators, capacity limits are formulated a: linear inequality constraints Fig. 5. The decomposition and coordination approach. elevator subproblems, (4) is transformed into an additive form in terms of elevators by using (1) Fig. 6. Illustration of the local search. By collecting all the terms related to elevator 7 from (6), the subproblem 7 is obtained as Fig. 7. Illustration of the three-passage heuristics. Fig. 8. Stagewise costs in stage k for different cases. With (23), the problem is converted to a separable form, anc our decomposition and coordination approach can be applied. A Lagrangian function is first obtained by relaxing the assignmen' constraints (1) with nonnegative multipliers {;}, and the nev egress time constraints (23) with nonnegative multipliers {j.; } ELEVATOR CONFIGURATIONS RESULTS FOR VARIOUS TRAFFIC PATTERNS RESULTS FROM DIFFERENT METHODS FOR SUBPROBLEM SOLVING Furthermore, for each time window, interfloor has the smallest average service time among the three traffic patterns, followed by down-peak. This is reasonable since for interfloor traffic, an elevator can serve passengers on its way up and down, and for down-peak traffic on its way down, but for up-peak traffic only at the lobby. RESULTS FOR VARIOUS TRAFFIC PATTERNS WITH DIFFERENT WINDOW LENGTHS RESULTS FROM BATCH OPTIMIZATION AND 1D INCREMENTAL OPTIMIZATION TABLE VI RESULTS FOR UP-PEAK WITH DIFFERENT WINDOW LENGTHS rescheduling points over a set of passengers. For incremental optimization, however, the assignment decision is optimized upon the arrival of each passenger assuming that no future traffic information is available. To compare these two opti- mization schemes, a window length of zero is used with a rescheduling interval of 7.5 s, and results are summarized in Table VI. With incremental optimization, the average service time is 61.8 s with a standard deviation of 0.83. Both metrics are higher than those obtained by using batch optimization, al- though the average CPU time of 2.2 s is less than 5.7 s obtained before. This indicates that incremental optimization cannot take the advantage of optimizing over a set of passengers. RESULTS FOR LOW INTENSITY INTERFLOOR TRAFFIC Fig. 9. Passenger arrival rates for each interval. RESULTS FOR VARIOUS TRAFFIC PATTERNS WITH VARYING ARRIVAL RATES Fig. 10. Average service times for up-peak with different window lengths. TABLE IX RESULTS FROM THE UP-PEAK DISPATCHING POLICY AND OUR METHOD TABLE X RESULTS FOR EVACUATION WITH ELEVATORS AND WITHOUT ELEVATORS TABLE XI
