A Constant Approximation Algorithm for the One-Warehouse Multiretailer Problem

Levi, Retsef; Roundy, R.; Shmoys, David B.; Sviridenko, Maxim

doi:10.1287/mnsc.1070.0781

Cited by 84 publications

(68 citation statements)

References 19 publications

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“…Levi, Roundy and Shmoys [LRS06] gave a 2-approximation algorithm based on a primal-dual scheme. Using randomized rounding, Levi et al [LRSS08,LS06] (building on [LRS05]) improved the approximation ratio to 1.8; Nonner and Souza [NS09] reduced it further to 5/3. These results use a natural linear-program (LP) relaxation, which we use too.…”

Section: I} a Solution (Also Called A Schedule) Is A Set Of Ordementioning

confidence: 99%

Approximation Algorithms for the Joint Replenishment Problem with Deadlines

Bieńkowski

Byrka

Chrobák

et al. 2013

Automata, Languages, and Programming

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The Joint Replenishment Problem (JRP) is a fundamental optimization problem in supplychain 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, computerassisted 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.

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Section: I} a Solution (Also Called A Schedule) Is A Set Of Ordementioning

confidence: 99%

Approximation Algorithms for the Joint Replenishment Problem with Deadlines

Bieńkowski

Byrka

Chrobák

et al. 2013

Automata, Languages, and Programming

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“…To benchmark the quality and speed of the solutions, we also solve Problem CP for each of the instances generated with CPLEX MIP solver with a time limit of 1800 s. Furthermore, to investigate possibly stronger formulations, we also solve the MIP version of the formulation introduced in Levi et al [15].…”

Section: Computational Resultsmentioning

confidence: 99%

“…Levi et al [15] develop constant approximation algorithms for the Single-Warehouse Multi-Retailer Problem with Fixed Charge ordering costs (see next paragraph for details) and extend them to accommodate FTL charges identical to ours. In contrast with our model, however, they restrict themselves to those solutions in which the entire demand of a retailer at a particular time period (d i t ) must travel together (i.e., in the same time period) from supplier to warehouse and then again together from warehouse to retailer.…”

Section: Introductionmentioning

confidence: 99%

Single‐warehouse multi‐retailer inventory systems with full truckload shipments

Jin¹,

Muriel

2009

Naval Research Logistics

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Abstract:We consider a multi-stage inventory system composed of a single warehouse that receives a single product from a single supplier and replenishes the inventory of n retailers through direct shipments. Fixed costs are incurred for each truck dispatched and all trucks have the same capacity limit. Costs are stationary, or more generally monotone as in Lippman (Management Sci 16, 1969, 118-138). Demands for the n retailers over a planning horizon of T periods are given. The objective is to find the shipment quantities over the planning horizon to satisfy all demands at minimum system-wide inventory and transportation costs without backlogging. Using the structural properties of optimal solutions, we develop (1) an O(T 2 ) algorithm for the single-stage dynamic lot sizing problem; (2) an O(T 3 ) algorithm for the case of a single-warehouse single-retailer system; and (3) a nested shortest-path algorithm for the single-warehouse multi-retailer problem that runs in polynomial time for a given number of retailers. To overcome the computational burden when the number of retailers is large, we propose aggregated and disaggregated Lagrangian decomposition methods that make use of the structural properties and the efficient single-stage algorithm. Computational experiments show the effectiveness of these algorithms and the gains associated with coordinated versus decentralized systems. Finally, we show that the decentralized solution is asymptotically optimal.

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“…Several heuristics for the non-stationary additive JRP have been proposed with varying degrees of theoretical performance guarantees in Veinott Jr (1969), Zangwill (1969), Kao (1979), Joneja (1990), Federgruen and Tzur (1994), Levi et al (2006), and Stauffer et al (2011). The current best approximation algorithms for the additive JRP with non-stationary demand are due to Levi et al (2008) and Bienkowski et al (2013), which have approximation ratios of 1.80 and 1.8791, respectively. Becchetti et al (2006) and Khanna et al (2002) have considered special cases of the tree JRP with assumptions on the holding cost structure.…”

Section: Literature Reviewmentioning

confidence: 99%

“…The cost of the root node is K r and the cost of each leaf i is K i . In the remainder of this section, we advance the ideas of Levi et al (2008) to describe a 3-approximation algorithm for the tree-JRP via LP rounding.…”

Section: Tree Joint Replenishment Problemmentioning

confidence: 99%

The submodular joint replenishment problem

et al. 2015

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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

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A Constant Approximation Algorithm for the One-Warehouse Multiretailer Problem

Cited by 84 publications

References 19 publications

Approximation Algorithms for the Joint Replenishment Problem with Deadlines

Approximation Algorithms for the Joint Replenishment Problem with Deadlines

Single‐warehouse multi‐retailer inventory systems with full truckload shipments

The submodular joint replenishment problem

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