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When entering a track from the other side, the opposite applies. Constraint 10: OPL . forall (trs in FromTrainTrack, p1 in PositionInd: p1 . trs:RsPosition:Position) ShadowAct[trs:TrainNr] Provides(ShuntAway[trs] (trs:Phase B )) TrackPositionInd[trs:Track,p1]; This constraint completes the description of the initial model. In addition to the model, we have developed a small search strategy that is presented in the next section. 14.3.2.5. Search Strategy. Although the used software has a built-in mechanism to nd solutions, this does not guarantee that a particular solution will be found within a reasonable computation time. If the default strategy would be elaborated by some speci c problem-solving knowledge, the performance could be improved. The idea is to rst schedule all activities moving toward and from one particular shunting area (ShuntAreas). This is because there is no interdependency between the assignment problem and the shunting areas. Second, we will try to combine all activities of the same rolling stock types (RsTypes). Finally, we will try to schedule per track and per position. The aim is to store units of the same kind on the same track. This is a heuristic approach applied by the planners as explained in Section 14.1.1. We have developed the following search strategy: OPL.search{ tryall (z in ShuntAreas) tryall (m in RsTypes) tryall (s in Tracks:TrackZone[s] z) tryall (p in RsPosition:p:RsType m){ tryall (trs in FromTrainTrack:trs:Track s and trs:RsPosition p) ShuntAway[trs] 1; tryall (trs in ToTrainTrack: trs:Track s and trs:RsPosition p) ShuntBack[trs] 1; }; }; This strategy tries to assign the value 1 to the elements of the arrays ShuntAway and ShuntBack, representing the choice of assigning a track, a position and a track

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time of the assignment step 4 is obviously linear in J Since the running time of such a heuristic is obviously extremely low, one may try this heuristic for several values of the parameter d and retain the best result If this very simple generating heuristic is applied to the instance of the simple job scheduling problem described in Section 82 with the parameter d 0, then it produces the schedule with Makespan 16 which is represented in Figure 87 If this heuristic is applied to this instance with the parameter d 1, then the result is a schedule with Makespan 15 A simple improvement heuristic inspired by the schedule in Figure 87 is the rule: Assign the last job to the machine with the lowest workload This rule would also result in a schedule with Makespan 15.

(A.12)

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where N(t) is the number of the round trip at the time t counted from the beginning of the Q-switch pulse. But since the whole spectrum is shifted by the same amount vB, the frequency differences of the longitudinal modes remain the same. Therefore it can be shown by a straightforward calculation that under the assumption of such a shifting mechanism the intensity still has the form of Eq. (27). Consequently, the calculated pulse shapes of Eq. (26) are very similar to the measured pulse shapes if the correct amplitudes and phases are applied (see Figs. 3.29 and 3.30). As has been explained, there is an identical phase angle for the longitudinal modes spaced by the Brillouin frequency if start and SBS resonator are tuned to the Brillouin frequency. If there are beat frequencies with a different spectral distance, their corresponding elds should have any arbitrary phase angle since there is no locking mechanism for them. Therefore, in the calculation the phases of the elds with an even index (it is even because LSBS 2LB) and a spectral distance of nB were chosen identical and the phase angles of the remaining elds with uneven index were chosen randomly. It turns out that every observed pulse shape can be calculated in an optimal way following one of the two assumptions for two different cases. Either there exists only one strong amplitude at one frequency vk in a non-length-tuned SBS resonator or in

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Improvement heuristics usually try to exchange certain parts of an obtained solution, as in the well-known 2-Opt and 3-Opt heuristics for the Traveling Salesman Problem A more sophisticated improvement heuristic in this area is the Variable-Opt method developed by Lin and Kernighan (1973) An improvement heuristic in the same vein for improving a solution of the job scheduling problem described in Section 82 could be the following: 1 For each job j1 assigned to machine 1 do 2 For each job j2 assigned to machine 2 do If switching the assignments of jobs j1 and j2 reduces the Makespan Then switch the assignments of jobs j1 and j2 3 Repeat step 1 until there are no further improvements If this improvement heuristic is applied to the schedule that was produced by the generating heuristic described above with the parameter d 0 (see Figure 8.

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