This paper presents a model for determining how a central government can most efficiently allocate resources among other levels of government. The model explicitly includes the fact that lower levels of government can make independent decisions once they have been given resources by the central government. A key feature of the model is the mathematical formulation of the central government's objective of distributing resources efficiently, while at the same time being as fair as possible to all those receiving allocations. An algorithm for solving the model is presented along with a numerical example.

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The generalized network problem and the closely related restricted dyadic problem are two special model types that occur frequently in applications of linear programming. Although they are next in order after pure network or distribution problems with respect to ease of computation, the jump in degree of difficulty is such that, in the most general problem, there exist no algorithms for them comparable in speed or efficiency to those for pure network or distribution problems. There are, however, numerous examples in which some additional special structure leads one to anticipate the existence of algorithms that compare favorably with the efficiency of those for the corresponding pure cases. Also, these more special structures may be encountered as part of larger or more complicated models. In this paper we designate by topological properties two special structures that permit evolution of efficient algorithms. These follow by extensions of methods of Charnes and Cooper and of Dijkstra for the corresponding pure network problems. We obtain easily implemented algorithms that provide an optimum in one “pass” through the network. The proofs provided for these extended theorems differ in character from those provided (or not provided) in the more special “pure” problem algorithms published.

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Users of linear programming computer codes have realized the necessity of evaluating the capacity, effectiveness, and accuracy of the solutions provided by such codes. Large scale linear programming codes at most installations are assumed to be generating correct solutions without ever having been *'bench-marked*' by test problems with known solutions. The reason for tbis failure to adequately test tbe codes is tbat rarely are tbere large problems with known solutions readily available. This paper presents a tbeoretical justification and an illustrative implementation of a method for generating linear programming test problems with known solutions. The method permits tbe generation of test problems tbat are of arbitrary size and have a wide range of numerical characteristics.