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The Traveling Salesman Problem involves nding the shortest route between a number of cities. This route must visit each of the cities exactly once and end in the same city as it started. As easy as it is to describe, this problem is notoriously diicult to solve. It is widely believed that there is no eecient algorithm that can solve it accurately. On the… (More)

New observations are made about two lower bound schemes for single-machine min-sum scheduling problems. We find that the strongest bound of those provided by transportation problem relaxations can be computed by solving a linear program. We show the equivalence of this strongest bound and the bound provided by the LP relaxation of the time-indexed integer… (More)

While the process of IMRT planning involves optimization of the dose distribution, the procedure for selecting the beam inputs for this process continues to be largely trial-and-error. We have developed an integer programming (IP) optimization method to optimize beam orientation using mean organ-at-risk (MOD) data from single-beam plans. Two test cases were… (More)

—Large-scale discrete optimization problems are difficult to solve, especially when different kinds of real constraints are considered. Conventionally, standard mathematical programming is a general approach for discrete optimization, but may suffer from the unacceptable long solution time in applications. On the other hand, some heuristics/metaheuristics… (More)

Dynamic programming, branch-and-bound, and constraint programming are the standard solution principles for nding optimal solutions to machine scheduling problems. We propose a new hybrid optimization framework that integrates all three methodologies. The hybrid framework leads to powerful solution procedures. We demonstrate our approach through the… (More)

In recent articles we presented a general methodology for finite optimization. The new method, the Nested Partitions (NP) method, combines partitioning, random sampling, a selection of a promising index, and backtracking to create a Markov chain that converges to a global optimum. In this paper we demonstrate, through examples, how the NP method can be… (More)