This paper considers an extension of the usual scalar valued shortest path problem to one in which there are m objective functions and for which we wish to find the efficient set of policies and paths. The paper discusses some of the deficiencies involved in the use of weighting factor methods to determine the efficient sets with respect to the original pure paths and policies. The main result is that if we wish to find which original paths are efficient with respect to the convex extensions of such sets, then they may be obtained exactly by the weighting factor method with minimisation restricted to the original pure policies, and such optimising policies will be jointly efficient in the convex extension, and hence in the original pure set. The paper also discusses linear programming and dynamic programming computational aspects.
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