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Fire hazard analysis often includes the use of mathematical models of the egress of the individual occupants of a structure which is involved in a fire. In this paper, we introduce some mathematical optimization models which produce as output at least one path for each occupant which is optimal with respect to some measurement. Network based models of increasing levels of complexity and realism are demonstrated by means of a sequence of examples. These examples include single attribute constant costs; single varying costs, two attribute constant costs, and finally two attribute time-varying costs imposed on network links. Dynamic programming functional equations which are based on the principle of optimality are presented. A multi-attribute analysis is proposed to evaluate a building design with respect to evacuation paths.