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Several modifications to the Dantzig and Fulkerson tanker scheduling problem are described. The original formulation addressed the determination of the minimal number of homogeneous tankers needed to meet a fixed schedule of deliveries. An earlier modification considered an insufficient number of vehicles to make all deliveries and a set of utilities associated with each vehicle delivery. The maximum utility solution to this problem can be found using the Fulkerson out-of kilter algorithm. The paper further extends the vehicle constrained problem to allow a span of feasible delivery times in lieu of a fixed schedule. This extension results in nonintegral valued flows in the network, precluding the application of network algorithms. The concept of a generalized bundle constraint is presented and a column-generation linear programming solution technique is described. The paper considers extensions to a fleet of vehicles with differing characteristics. A solution technique for the multicommodity flow problem is described. (Author), Prepared in cooperation with Johns Hopkins Univ., Baltimore, Md.

A set of edges D (called an isolation set) is said to isolate a set of nodes R from an undirected network if every chain between the nodes in R contains at least one edge from the set D. Associated with each edge e of the network is a positive cost c(e). The isolation problem is concerned with finding an isolation set such that the sum of its edge costs is a minimum. The paper formulates the problem of determining the minimal cost isolation as a 0-1 integer linear programming problem. An algorithm is presented which applies a branch and bound enumerative scheme to a decomposed linear program whose dual sub-problems are minimal cost network flow problems. Computational results are given. The problem is also formulated as a special quadratic assignment problem and an algorithm is presented that finds a local optimal solution. This local solution is used for an initial bound. (Author), Prepared in cooperation with Johns Hopkins Univ., Baltimore, Md. Presented at the National ORSA meeting (35th), Denver, Colorado, 17 Jun 69.