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The paper presents a mathematical investigation into bus scheduling. The passenger arrival rate is supposed to be given; the problem is to determine the bus departure rate as a function of time. The primary objective is to minimize the number of buses that is needed. A secondary criterion is the minimization of the passenger waiting time, for which a calculus of variation technique is used. Although the paper deals mainly with a single busroute, it is also shown how the theory can be extended to the case of a pair of linked busroutes. The practical implications are illustrated by an example. The fleetsize formula that is used here is thought to be applicable to many transportation systems. [ABSTRACT FROM AUTHOR]

In a previous paper, APPELGREN (1969), a decomposition algorithm for a class of vessel scheduling problems was presented. In some problems, the algorithm gives fractional solutions that cannot be interpreted as feasible schedules. This paper treats two integer programming methods that can be used to resolve these cases. The cutting plane method that was first tested was abandoned because it was not able to solve all the test problems. The second method is a branch-and- bound algorithm, where the branching is performed on one of the `essential' fractional variables and where the bounds are obtained by the decomposition algorithm. All fractional problems that have been found by simulation or in regular use of the algorithm have been solved, mostly with one branching only. There are fundamental difficulties in combining these integer programming methods with the DANTZIG-WOLFE decomposition, since the constraints generated in the master program have to be taken into account in the solution of the subprograms. The success in this case is due to the simple structure of the master LP problem. [ABSTRACT FROM AUTHOR]

The article focuses on models of fleet routing and scheduling problems for transportation systems, which are formulated as integer linear programs with 0-1 variables. The topics discussed in this paper are concerned with important operational problems confronting the management of the competitive air transportation system. Some of these problems are significant for other transportation systems, including those with public-service rather than profit objectives. The problems are those connected with fleet utilization, fleet size, routing, and scheduling. Work that is related to some of these problems has been carried out by operations researchers and has been limited because, when optimal solutions were sought, computation times were excessive. The principal method of attack has been dynamic programming, which usually involves a very large state space for any real-life transportation system. The basic models are of the discrete-time rather than of the continuous-time variety. This property permits the formulation of the optimization problems as integer linear programs. The aim is to formulate a realistic model and to obtain its optimal solution