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 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

Strongly Polynomial Primal-Dual Algorithms for Concave Cost Combinatorial Optimization Problems

We introduce an algorithm design technique for a class of combinatorial optimization problems with concave costs. This technique yields a strongly polynomial primal-dual algorithm for a concave cost problem whenever such an algorithm exists for the fixed-charge counterpart of the problem. For many practical concave cost problems, the fixed-charge counterpart is a well-studied combinatorial optimization problem. Our technique preserves constant factor approximation ratios, as well as ratios that depend only on certain problem parameters, and exact algorithms yield exact algorithms. Using our technique, we obtain a new 1.61-approximation algorithm for the concave cost facility location problem. For inventory problems, we obtain a new exact algorithm for the economic lot-sizing problem with general concave ordering costs, and a 4-approximation algorithm for the joint replenishment problem with general concave individual ordering costs

On the inventory routing problem with stationary stochastic demand rate

One of the most significant paradigm shifts of present business management is that individual businesses no longer participate as solely independent entities, but rather as supply chains (Lambert and Cooper, 2000). Therefore, the management of multiple relationships across the supply chain such as flow of materials, information, and finances is being referred to as supply chain management (SCM). SCM involves coordinating and integrating these multiple relationships within and among companies, so that it can improve the global performance of the supply chain. In this dissertation, we discuss the issue of integrating the two processes in the supply chain related, respectively, to inventory management and routing policies. The challenging problem of coordinating the inventory management and transportation planning decisions in the same time, is known as the inventory routing problem (IRP). The IRP is one of the challenging optimization problems in logis-tics and supply chain management. It aims at optimally integrating inventory control and vehicle routing operations in a supply network. In general, IRP arises as an underlying optimization problem in situations involving simultaneous optimization of inventory and distribution decisions. Its main goal is to determine an optimal distribution policy, consisting of a set of vehicle routes, delivery quantities and delivery times that minimizes the total inventory holding and transportation costs. This is a typical logistical optimization problem that arises in supply chains implementing a vendor managed inventory (VMI) policy. VMI is an agreement between a supplier and his regular retailers according to which retailers agree to the alternative that the supplier decides the timing and size of the deliveries. This agreement grants the supplier the full authority to manage inventories at his retailers'. This allows the supplier to act proactively and take responsibility for the inventory management of his regular retailers, instead of reacting to the orders placed by these retailers. In practice, implementing policies such as VMI has proven to considerably improve the overall performance of the supply network, see for example Lee and Seungjin (2008), Andersson et al. (2010) and Coelho et al. (2014). This dissertation focuses mainly on the single-warehouse, multiple-retailer (SWMR) system, in which a supplier serves a set of retailers from a single warehouse. In the first situation, we assume that all retailers face a deterministic, constant demand rate and in the second condition, we assume that all retailers consume the product at a stochastic stationary rate. The primary objective is to decide when and how many units to be delivered from the supplier to the warehouse and from the warehouse to retailers so as to minimize total transportation and inventory holding costs over the finite horizon without any shortages

Online Algorithms for Multi-Level Aggregation

In the Multi-Level Aggregation Problem (MLAP), requests arrive at the nodes of an edge-weighted tree T, and have to be served eventually. A service is defined as a subtree X of T that contains its root. This subtree X serves all requests that are pending in the nodes of X, and the cost of this service is equal to the total weight of X. Each request also incurs waiting cost between its arrival and service times. The objective is to minimize the total waiting cost of all requests plus the total cost of all service subtrees. MLAP is a generalization of some well-studied optimization problems; for example, for trees of depth 1, MLAP is equivalent to the TCP Acknowledgment Problem, while for trees of depth 2, it is equivalent to the Joint Replenishment Problem. Aggregation problem for trees of arbitrary depth arise in multicasting, sensor networks, communication in organization hierarchies, and in supply-chain management. The instances of MLAP associated with these applications are naturally online, in the sense that aggregation decisions need to be made without information about future requests. Constant-competitive online algorithms are known for MLAP with one or two levels. However, it has been open whether there exist constant competitive online algorithms for trees of depth more than 2. Addressing this open problem, we give the first constant competitive online algorithm for networks of arbitrary (fixed) number of levels. The competitive ratio is O(D^4 2^D), where D is the depth of T. The algorithm works for arbitrary waiting cost functions, including the variant with deadlines. We also show several additional lower and upper bound results for some special cases of MLAP, including the Single-Phase variant and the case when the tree is a path

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

Policies for inventory/distribution systems: The effect of centralization vs. decentralization

This paper concerns with a multi-echelon inventory/distribution system considering one-warehouse and N-retailers. The retailers are replenished from the warehouse. We assume that the demand rate at each retailer is known. The problem consists of determining the optimal reorder policy which minimizes the overall cost, that is, the sum of the holding and replenishment costs. Shortages are not allowed and lead times are negligible. We study two situations: when the retailers make decisions independently and when the retailers are branches of the same firm. Solution methods to determine near-optimal policies in both cases are provided. Computational results on several randomly generated problems are reported