LOCOMOTIVE SCHEDULING UNDER UNCERTAIN DEMAND
Each day, railroads face the problem of allocating power to trains. Often, power requirements for each train are not known with certainty, and the fleet of locomotives may not be homogeneous. To deal with both of these complications, we formulate a multicommodity flow problem with convex objective function on a time-space network. The convex objective allows us to minimize expected cost under uncertainty by penalizing trip arcs likely to have too little power. Our solution heuristic sends locomotives down the shortest paths (based on marginal arc costs) in the time-space network and then attempts to improve interchanges of locomotives around cycles. Two lower bounds are also developed by relaxing the multicommodity aspect of the problem. In 19 test problems, ranging in size from 15 to 404 arcs, the heuristic performed well, with short running times and costs averaging within 3% of the best of the two lower bounds developed.
Scheduling Locomotives and Car Transfers in Freight Transport
We present a new model for a strategic locomotive scheduling problem arising at the Deutsche Bahn AG. The model is based on a multicommodity minimum cost flow formulation that is also used for public bus scheduling problems. However, several new aspects have to be additionally taken into account such as cyclic departures of the trains, time windows on starting and arrival times, network load-dependent travel times, and a transfer of cars between trains. The model is formulated as an integer linear programming problem (ILP). The formulation is improved by preprocessing and additional cutting planes. Solutions are obtained using a randomized greedy heuristic in combination with commercial ILP solvers. Computational results are presented for several real-world test instances.
Knocking in an Internal-Combustion Engine
The question remains open of the relation between the phenomena of knocking in the engine and the explosion wave. The solution of this problem is the object of this paper. The tests were conducted on an aircraft engine with a pyrex glass window in the cylinder head. Photographs were then taken of various combinations of fuels and conditions.