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.
The Solution of a Railway Locomotive Scheduling Problem
A method is described for finding a minimum cost set of schedules for railway locomotives to work a given set of trains. The times at which the trains start may be fixed or variable. A heuristic method, based on a linear programming model, is described. This gives good integer solutions to the problem.
Limitations on the interpretation of biotite substitutions from chemical analyses of natural samples
Biotites, as well as other phases, are sufficiently complex chemically that there are numerous sets of linearly independent exchange components with which to describe the composition of a naturally occurring crystal. Even after complete chemical analysis, this leads to ambiguities in the correlation of exchange components with valid substitution mechanisms in complex phases. The best means of identifuing valid substitutions are by the observation of simple endmember compositions, the investigation of compositional changes in zoned crystals or petrogenetic sequences, and by direct experimental synthesis.