Mandell Bellmore

Pathology of Traveling-Salesman Subtour-Elimination Algorithms

The traveling-salesman problem has been the target of a substantial number of computational algorithms over the last two decades. Reported computational experience with these algorithms varies widely; authors, however, have generally failed to explain this variation adequately, or to offer predictive theories for their approaches. This paper a develops an underlying theory for the problem, b predicts pathological performance of some existing techniques, and c presents two algorithms, based upon the theory, with predictable polynomial growth in expected computation time and resistence to pathological problems.

An Algorithm to Determine the Reliability of a Complex System

The method most often suggested for determining the reliability of a system is to construct a reliability network, enumerate from the network all mutually exclusive working states of the system, calculate the probability of occurrence of each working state, and sum these probabilities. For a complex system this is not a practical method for there is a very large number of working states. Esary and Proschan suggest a lower bound approximation to reliability that requires the enumeration of a much smaller set of system states. These states are called minimal cuts. An algorithm is presented to determine the set of minimal cuts and thus calculate a lower bound to system reliability. The algorithm is intended for digital-computer implementation and computational times are provided.

Generalized Penalty-Function Concepts in Mathematical Optimization

Given a mathematical program, this paper constructs an alternate problem with its feasibility region a superset of the original mathematical program. The objective function of this new problem is constructed so that a penalty is imposed for solutions outside the original feasibility region. One attempts to choose an objective function that makes the optimal solutions to the new problem the same as the optimal solutions to the original mathematical program.

On Multi-Commodity Maximal Dynamic Flows

Ford and Fulkerson have shown that a single-commodity maximal dynamic flow can be obtained by solving a transshipment problem associated with the static network and thereby finding the maximal temporally repeated dynamic flow. This flow is known to be an optimal dynamic flow. However, this result cannot be extended to the multi-commodity maximal dynamic flow problem. This paper shows that, for sufficiently large number of time periods, the difference between the multi-commodity maximal dynamic flow and the temporally repeated multi-commodity flow is bounded by a constant. In addition, it calculates this bound.