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Programming Relaxation (programming + relaxation)

Distribution by Scientific Domains
Mathematics and Statistics100%

Selected Abstracts

Approximation algorithms for general one-warehouse multi-retailer systems

NAVAL RESEARCH LOGISTICS: AN INTERNATIONAL JOURNAL, Issue 7 2009
Zuo-Jun Max Shen
Abstract Logistical planning problems are complicated in practice because planners have to deal with the challenges of demand planning and supply replenishment, while taking into account the issues of (i) inventory perishability and storage charges, (ii) management of backlog and/or lost sales, and (iii) cost saving opportunities due to economies of scale in order replenishment and transportation. It is therefore not surprising that many logistical planning problems are computationally difficult, and finding a good solution to these problems necessitates the development of many ad hoc algorithmic procedures to address various features of the planning problems. In this article, we identify simple conditions and structural properties associated with these logistical planning problems in which the warehouse is managed as a cross-docking facility. Despite the nonlinear cost structures in the problems, we show that a solution that is within ,-optimality can be obtained by solving a related piece-wise linear concave cost multi-commodity network flow problem. An immediate consequence of this result is that certain classes of logistical planning problems can be approximated by a factor of (1 + ,) in polynomial time. This significantly improves upon the results found in literature for these classes of problems. We also show that the piece-wise linear concave cost network flow problem can be approximated to within a logarithmic factor via a large scale linear programming relaxation. We use polymatroidal constraints to capture the piece-wise concavity feature of the cost functions. This gives rise to a unified and generic LP-based approach for a large class of complicated logistical planning problems. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009 [source]

Models and heuristics for a minimum arborescence problem

NETWORKS: AN INTERNATIONAL JOURNAL, Issue 1 2008
Christophe Duhamel
Abstract The Minimum Arborescence problem (MAP) consists of finding a minimum cost arborescence in a directed graph. This problem is NP-Hard and is a generalization of two well-known problems: the Minimum Spanning Arborescence Problem (MSAP) and the Directed Node Weighted Steiner Tree Problem (DNWSTP). We start the model presentation in this paper by describing four models for the MSAP (including two new ones, using so called "connectivity" constraints which forbid disconnected components) and we then describe the changes induced on the polyhedral structure of the problem by the removal of the spanning property. Only two (the two new ones) of the four models for the MSAP remain valid when the spanning property is removed. We also describe a multicommodity flow reformulation for the MAP that differs from well-known multicommodity flow reformulations in the sense that the flow conservation constraints at source and destination are replaced by inequalities. We show that the linear programming relaxation of this formulation is equivalent to the linear programming relaxation of the best of the two previous valid formulations and we also propose two Lagrangean relaxations based on the multicommodity flow reformulation. From the upper bound perspective, we describe a constructive heuristic as well as a local search procedure involving the concept of key path developed earlier for the Steiner Tree Problem. Numerical experiments taken from instances with up to 400 nodes are used to evaluate the proposed methods. © 2007 Wiley Periodicals, Inc. NETWORKS, 2008 [source]

The ring grooming problem

NETWORKS: AN INTERNATIONAL JOURNAL, Issue 3 2004
Timothy Y. Chow
Abstract The problem of minimizing the number of bidirectional SONET rings required to support a given traffic demand has been studied by several researchers. Here we study the related ring-grooming problem of minimizing the number of add/drop locations instead of the number of rings; in a number of situations this is a better approximation to the true equipment cost. Our main result is a new lower bound for the case of uniform traffic. This allows us to prove that a certain simple algorithm for uniform traffic is, in fact, a constant-factor approximation algorithm, and it also demonstrates that known lower bounds for the general problem,in particular, the linear programming relaxation,are not within a constant factor of the optimum. We also show that our results for uniform traffic extend readily to the more practically important case of quasi-uniform traffic. Finally, we show that if the number of nodes on the ring is fixed, then ring grooming is solvable in polynomial time; however, whether ring grooming is fixed-parameter tractable is still an open question. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 44(3), 194,202 2004 [source]

Analysis of algorithms for two-stage flowshops with multi-processor task flexibility

NAVAL RESEARCH LOGISTICS: AN INTERNATIONAL JOURNAL, Issue 1 2004
George L. Vairaktarakis
Abstract In this article we introduce a 2-machine flowshop with processing flexibility. Two processing modes are available for each task: namely, processing by the designated processor, and processing simultaneously by both processors. The objective studied is makespan minimization. This production environment is encountered in repetitive manufacturing shops equipped with processors that have the flexibility to execute orders either individually or in coordination. In the latter case, the product designer exploits processing synergies between two processors so as to execute a particular task much faster than a dedicated processor. This type of flowshop environment is also encountered in labor-intensive assembly lines where products moving downstream can be processed either in the designated assembly stations or by pulling together the work teams of adjacent stations. This scheduling problem requires determining the mode of operation of each task, and the subsequent scheduling that preserves the flowshop constraints. We show that the problem is ordinary NP-complete and obtain an optimal solution using a dynamic programming algorithm with considerable computational requirements for medium and large problems. Then, we present a number of dynamic programming relaxations and analyze their worst-case error performance. Finally, we present a polynomial time heuristic with worst-case error performance comparable to that of the dynamic programming relaxations. © 2003 Wiley Periodicals, Inc. Naval Research Logistics, 2004. [source]