Consecutive Optimizers for a Partitioning Problem with Applications to Optimal Inventory Groupings for Joint Replenishment

We consider several subclasses of the problem of grouping n items (indexed 1, 2, .., n) into m subsets so as to minimize the function g(S 1 , .. , S,). In general, these problems are very difficult to solve to optimality, even for the case m = 2. We provide several sufficient conditions on g(') that guarantee that there is an optimum partition in which each subset consists of consecutive integers (or else the partition S,,-, S,, satisfies a more general condition called semiconsecutiveness"). Moreover, by restricting attention to 'consecu-tive" (or serniconsecutive ") partitions, we can solve the partition problem in polynomial time for small values of m. If, in addition, g is symmetric, then the partition problem is solvable in purely polynomial time. We apply these results to generalizations of a problem in inventory groupings considered by the authors in a previous paper. We also relate the results to the Neyman-Pearson lemma in statistical hypothesis testing and to a graph partitioning problem of Barnes and Hoffman. al arc .-and tbi I b ar+l an br+i-bn (1) For b, = 0, we consider adb, to be +cc or-according to a > 0 or a, < 0. If ai = bi = 0, al/b 1 is defined arbitrarily so that inequality (1) holds. As usual, we let a and b denote the vectors whose coordinates are a, and bi, respectively.