Better Approximation Bounds for the Joint Replenishment Problem

The Joint Replenishment Problem (JRP) deals with optimizing shipments of goods from a supplier to retailers through a shared warehouse. Each shipment involves transporting goods from the supplier to the warehouse, at a fixed cost C, followed by a redistribution of these goods from the warehouse to the retailers that ordered them, where transporting goods to a retailer $\rho$ has a fixed cost $c_\rho$. In addition, retailers incur waiting costs for each order. The objective is to minimize the overall cost of satisfying all orders, namely the sum of all shipping and waiting costs. JRP has been well studied in Operations Research and, more recently, in the area of approximation algorithms. For arbitrary waiting cost functions, the best known approximation ratio is 1.8. This ratio can be reduced to 1.574 for the JRP-D model, where there is no cost for waiting but orders have deadlines. As for hardness results, it is known that the problem is APX-hard and that the natural linear program for JRP has integrality gap at least 1.245. Both results hold even for JRP-D. In the online scenario, the best lower and upper bounds on the competitive ratio are 2.64 and 3, respectively. The lower bound of 2.64 applies even to the restricted version of JRP, denoted JRP-L, where the waiting cost function is linear. We provide several new approximation results for JRP. In the offline case, we give an algorithm with ratio 1.791, breaking the barrier of 1.8. In the online case, we show a lower bound of 2.754 on the competitive ratio for JRP-L (and thus JRP as well), improving the previous bound of 2.64. We also study the online version of JRP-D, for which we prove that the optimal competitive ratio is 2

Approximation algorithms and hardness results for the joint replenishment Problepm with constant demands

19th Annual European Symposium, Saarbrücken, Germany, September 5-9, 2011. ProceedingsIn the Joint Replenishment Problem (JRP), the goal is to coordinate the replenishments of a collection of goods over time so that continuous demands are satisfied with minimum overall ordering and holding costs. We consider the case when demand rates are constant. Our main contribution is the first hardness result for any variant of JRP with constant demands. When replenishments per commodity are required to be periodic and the time horizon is infinite (which corresponds to the so-called general integer model with correction factor), we show that finding an optimal replenishment policy is at least as hard as integer factorization. This result provides the first theoretical evidence that the JRP with constant demands may have no polynomial-time algorithm and that relaxations and heuristics are called for. We then show that a simple modification of an algorithm by Wildeman et al. (1997) for the JRP gives a fully polynomial-time approximation scheme for the general integer model (without correction factor). We also extend their algorithm to the finite horizon case, achieving an approximation guarantee asymptotically equal to √9/8

Approximation Algorithms for the Joint Replenishment Problem with Deadlines

The Joint Replenishment Problem (JRP) is a fundamental optimization problem in supply-chain management, concerned with optimizing the flow of goods from a supplier to retailers. Over time, in response to demands at the retailers, the supplier ships orders, via a warehouse, to the retailers. The objective is to schedule these orders to minimize the sum of ordering costs and retailers' waiting costs. We study the approximability of JRP-D, the version of JRP with deadlines, where instead of waiting costs the retailers impose strict deadlines. We study the integrality gap of the standard linear-program (LP) relaxation, giving a lower bound of 1.207, a stronger, computer-assisted lower bound of 1.245, as well as an upper bound and approximation ratio of 1.574. The best previous upper bound and approximation ratio was 1.667; no lower bound was previously published. For the special case when all demand periods are of equal length we give an upper bound of 1.5, a lower bound of 1.2, and show APX-hardness

The submodular joint replenishment problem

The joint replenishment problem is a fundamental model in supply chain management theory that has applications in inventory management, logistics, and maintenance scheduling. In this problem, there are multiple item types, each having a given time-dependent sequence of demands that need to be satisfied. In order to satisfy demand, orders of the item types must be placed in advance of the due dates for each demand. Every time an order of item types is placed, there is an associated joint setup cost depending on the subset of item types ordered. This ordering cost can be due to machine, transportation, or labor costs, for example. In addition, there is a cost to holding inventory for demand that has yet to be served. The overall goal is to minimize the total ordering costs plus inventory holding costs. In this paper, the cost of an order, also known as a joint setup cost, is a monotonically increasing, submodular function over the item types. For this general problem, we show that a greedy approach provides an approximation guarantee that is logarithmic in the number of demands. Then we consider three special cases of submodular functions which we call the laminar, tree, and cardinality cases, each of which can model real world scenarios that previously have not been captured. For each of these cases, we provide a constant factor approximation algorithm. Specifically, we show that the laminar case can be solved optimally in polynomial time via a dynamic programming approach. For the tree and cardinality cases, we provide two different linear programming based approximation algorithms that provide guarantees of three and five, respectively.National Science Foundation (U.S.) (CAREER Grant CMMI-0846554)United States. Air Force Office of Scientific Research (Award FA9550-11-1-0150)SMA GrantSolomon Buchsbaum AT&T Research Fun

Algorithms and hardness results for the jump number problem, the joint replenishment problem, and the optimal clustering of frequency-constrained maintenance jobs

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2012.Cataloged from PDF version of thesis.Includes bibliographical references (p. 107-110).In the first part of this thesis we present a new, geometric interpretation of the jump number problem on 2-dimensional 2-colorable (2D2C) partial order. We show that the jump number of a 2D2C poset is equivalent to the maximum cardinality of an independent set in a properly defined collection of rectangles in the plane. We then model the geometric problem as a linear program. Even though the underlying polytope may not be integral, we show that one can always find an integral optimal solution. Inspired by this result and by previous work of A. Frank, T. Jordan and L. Vegh [13, 14, 15] on set-pairs, we derive an efficient combinatorial algorithm to find the maximum independent set and its dual, the minimum hitting set, in polynomial time. The combinatorial algorithm solves the jump number problem on convex posets (a subclass of 2D2C posets) significantly faster than current methods. If n is the number of nodes in the partial order, our algorithm runs in 0((n log n)2.5) time, while previous algorithms ran in at least 0(n9 ) time. In the second part, we present a novel connection between certain sequencing problems that involve the coordination of activities and the problem of factorizing integer numbers. We use this connection to derive hardness results for three different problems: -- The Joint Replenishment Problem with General Integer Policies. -- The Joint Replenishment Problem with Correction Factor. -- The Problem of Optimal Clustering of Frequency-Constrained Maintenance Jobs. Our hardness results do not follow from a standard type of reduction (e.g., we do not prove NP-hardness), and imply that no polynomial-time algorithm exists for the problems above, unless Integer Factorization is solvable in polynomial time..by Claudio Telha Cornejo.Ph.D

Online Energy Storage Management: an Algorithmic Approach

Motivated by the importance of energy storage networks in smart grids, we provide an algorithmic study of the online energy storage management problem in a network setting, the first to the best of our knowledge. Given online power supplies, either entirely renewable supplies or those in combination with traditional supplies, we want to route power from the supplies to demands using storage units subject to a decay factor. Our goal is to maximize the total utility of satisfied demands less the total production cost of routed power. We model renewable supplies with the zero production cost function and traditional supplies with convex production cost functions. For two natural storage unit settings, private and public, we design poly-logarithmic competitive algorithms in the network flow model using the dual fitting and online primal dual methods for convex problems. Furthermore, we show strong hardness results for more general settings of the problem. Our techniques may be of independent interest in other routing and storage management problems

Algoritmos de aproximação para problemas de alocação de instalações e outros problemas de cadeia de fornecimento

Orientadores: Flávio Keidi Miyazawa, Maxim SviridenkoTese (doutorado) - Universidade Estadual de Campinas, Instituto de ComputaçãoResumo: O resumo poderá ser visualizado no texto completo da tese digitalAbstract: The abstract is available with the full electronic documentDoutoradoCiência da ComputaçãoDoutor em Ciência da Computaçã