Consecutive optimizors for a partitioning problem with applications to optimal inventory groupings for joint replenishment

Bibliography: p. 20.A.K. Chakravarty, J.B. Orlin and U.G. Rothblum

Constrained partitioning problems

AbstractWe consider partitioning problems subject to the constraint that the subsets in the partition are independent sets or bases of given matroids. We derive conditions for the functions F and [fnof] such that an optimal partition (S∗1, S∗2,…, S∗k) which minimizes F([fnof](S1),…, [fnof](S k)) has certain order properties. These order properties allow to determine optimal partitions by Greedy-like algorithms. In particular balancing partitioning problems can be solved in this way

Convex Combinatorial Optimization

We introduce the convex combinatorial optimization problem, a far reaching generalization of the standard linear combinatorial optimization problem. We show that it is strongly polynomial time solvable over any edge-guaranteed family, and discuss several applications

Stochastic Transportation-Inventory Network Design Problem

In this paper, we study the stochastic transportation-inventory network design problem involving one supplier and multiple retailers. Each retailer faces some uncertain demand. Due to this uncertainty, some amount of safety stock must be maintained to achieve suitable service levels. However, risk-pooling benefits may be achieved by allowing some retailers to serve as distribution centers (and therefore inventory storage locations) for other retailers. The problem is to determine which retailers should serve as distribution centers and how to allocate the other retailers to the distribution centers. Shen et al. (2000) and Daskin et al. (2001) formulated this problem as a set-covering integer-programming model. The pricing subproblem that arises from the column generation algorithm gives rise to a new class of submodular function minimization problem. They only provided efficient algorithms for two special cases, and assort to ellipsoid method to solve the general pricing problem, which run in O(n⁷ log(n)) time, where n is the number of retailers. In this paper, we show that by exploiting the special structures of the pricing problem, we can solve it in O(n² log n) time. Our approach implicitly utilizes the fact that the set of all lines in 2-D plane has low VC-dimension. Computational results show that moderate size transportation-inventory network design problem can be solved efficiently via this approach.Singapore-MIT Alliance (SMA

Stochastic joint replenishment problems: periodic review policies

Operations Managers of manufacturing systems, distribution systems, and supply chains address lot sizing and scheduling problems as part of their duties. These problems are concerned with decisions related to the size of orders and their schedule. In general, products share or compete for common resources and thus require coordination of their replenishment decisions whether replenishment involves manufacturing operations or not. This research is concerned with joint replenishment problems (JRPs) which are part of multi-item lot sizing and scheduling problems in manufacturing and distribution systems in single echelon/stage systems. The principal purpose of this research is to develop three new periodic review policies for stochastic joint replenishment problem. It also highlights the lack of research on joint replenishment problems with different demand classes (DSJRP). Therefore, periodic review policy is developed for this problem where the inventory system faces different demand classes that are deterministic demand and stochastic demand. Heuristic Algorithms have been developed to obtain (near) optimal parameters for the three policies as well as a heuristic algorithm has been developed for DSJRP. Numerical tests against literature benchmarks have been presented

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 'consecutive" (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. C · lg · · ·II CIL ·I D···1C- ·------- · 111-ET a, , a and b,-, b be real numbers ordered so that for some integer 0 r n, b, *.., b, are negative, b,+,.., b are nonnegative and al ar-- c.-- and tbi I b ar+l an br+i- bn For b, = 0, we consider adb, to be +cc or- according to a> 0 or a, < 0. If ai = bi = 0, al/b1 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. Subject clasification: 334 partitioning items into subgroups, 625 optimal inventory groupings