One-warehouse Multi-retailer Problem Research Articles

A heuristic algorithm to solve the one‐warehouse multiretailer problem with an emission constraint

AbstractIn this paper, we consider the one‐warehouse multiretailer problem with a global carbon emission cap constraint (OWMR‐EC). This constraint aims at limiting the carbon emissions related to the production, setup, and inventory‐holding operations. We develop a penalized relaxation (PR) method to heuristically solve the considered problem, both with and without the possibility of having initial inventory. This heuristic uses in itself another heuristic that we propose to solve the standard one‐warehouse multiretailer problem (OWMR). Our PR method is tested on numerous instances adapted from the literature. Our results indicate that the penalized method is able to find between 87.4% and 89.8% of feasible solutions for this NP‐hard problem, with an average optimality gap of 2.1% and 2.2% depending on the algorithms we use to solve the different subproblems involved in the method. The results show that our method is highly effective in terms of run‐time and solution quality, when a feasible solution is found. Furthermore, the results indicate that the heuristic for the standard OWMR is also very effective. We further perform a sensitivity analysis on the optimal solutions of the OWMR‐EC to better understand the implications of the carbon emission cap constraint. The sensitivity analysis indicates that the marginal cost of reducing carbon emissions increases as the emission cap decreases. The analysis also shows that the correlation between the cost and emission parameters has an important impact on the potential to further lower the emissions, compared to the emission of the minimum cost solution.

A comparison of formulations for a three-level lot sizing and replenishment problem with a distribution structure

We address a three-level lot sizing and replenishment problem with a distribution structure (3LSPD), which is an extension of the one-warehouse multi-retailer problem (OWMR). We consider one production plant that produces one type of item over a discrete and finite planning horizon. The items produced are used to replenish warehouses and then retailers using direct shipments. Each retailer is linked to a unique warehouse and there are no transfers between warehouses nor between retailers. We also assume that transportation is uncapacitated. However, we consider the possibility of imposing production capacity constraints at the production plant level. The objective is to minimize the sum of the fixed production and replenishment costs and of the variable inventory holding costs at all three levels. We compare 13 different MIP formulations to solve the problem. All of these formulations are adapted from existing MIP formulations found in the one-warehouse multi-retailer literature, but most formulations are new in the context of the 3LSPD. We run experiments on both balanced and unbalanced networks. Our results indicate that the multi-commodity formulation is well suited for uncapacititated instances and that the echelon stock reformulations are better for capacitated instances. They also show that the richer formulations are not necessarily the best ones and that the unbalanced instances are harder to solve.

Approximation algorithms for k-echelon extensions of the one warehouse multi-retailer problem

In this paper, we consider k-echelon extensions of the deterministic one warehouse multi-retailer problem. We give constant factor approximation algorithms for some of these extensions when k is fixed. We focus first on the case without backorders and we give a \((2k-1)\)-approximation algorithm under general assumptions on the evolution of the holding costs as products move toward the final customers. We then improve this result to a k-approximation when the holding costs are monotonically non-increasing or non-decreasing (which is a natural situation in practice). Finally we address problems with backorders: we give a 3-approximation for the one-warehouse multi-retailer problem with backlog and a k-approximation algorithm for the k-level Joint Replenishment Problem with backlog (a variant where inventory can only be kept at the final retailers). Ours results are the first constant approximation algorithms for those problems. In addition, we demonstrate the potential of our approach on a practical case. Our preliminary experiments show that the average optimality gap is around 15%.

Heuristics Based on Genetic Algorithms for the Capacitated Multi Vehicle Production Distribution Problem

In this paper, we consider the integration of production, inventory and distribution decisions in a supply chain composed of one production facility supplying several retailers located in the same region. The supplier is far from the retailers compared to the distance between retailers. Thus, the traveling cost of each vehicle from the supplier to the region is assumed to be fixed and there is a fixed delivery (service) cost for each visited retailer. The objective is to minimize the sum of the costs at the production facility and at the retailers. The problem is more general than the One-Warehouse Multi-Retailer problem and is a special case of the Production Routing Problem. Five heuristics based on a Genetic Algorithm are proposed to solve the problem. In particular, three of them include the resolution of a Mixed Integer Program as subproblem to generate new individuals in the population. The results show that the heuristics can find optimal solutions for small and medium size instances. On large instances, the gaps obtained by the heuristics in less than 300 s are better than the ones obtained by a standard solver in two hours.

Fast Approximation Algorithms for the One-Warehouse Multi-Retailer Problem Under General Cost Structures and Capacity Constraints

We consider a well-studied multi-echelon (deterministic) inventory control problem, known in the literature as the one-warehouse multi-retailer (OWMR) problem. We propose a simple and fast 2-approximation algorithm for this NP-hard problem, by recombining the solutions of single-echelon relaxations at the warehouse and at the retailers. We then show that our approach remains valid under quite general assumptions on the cost structures and under capacity constraints at some retailers. In particular, we present the first approximation algorithms for the OWMR problem with nonlinear holding costs, truckload discount on procurement costs, or with capacity constraints at some retailers. In all cases, the procedure is purely combinatorial and can be implemented to run in low polynomial time.

On reformulations for the one-warehouse multi-retailer problem

We consider the one-warehouse multi-retailer problem (OWMR) and perform both theoretical and computational studies of different formulations for the problem. Several formulations that were not yet directly applied to the OWMR are considered, namely an echelon stock formulation strengthened with valid inequalities, a two-level lot-sizing based formulation strengthened with valid inequalities, a multicommodity formulation and a dynamic programming based formulation. These are also compared with a strengthened echelon stock formulation and the previously best performing formulations available in the literature, a transportation and a shortest path formulations, which were studied in a previous work (Solyali and Sural in Ann Oper Res 196(1):517–541, 2012). The formulations were ordered according to the provided linear relaxation bounds, assuming there is no available stock at the beginning of the planning horizon. Experimental results using a commercial MIP solver indicate that a partial version of the two-level lot-sizing based formulation strengthened with valid inequalities and the multicommodity formulation outperform the others, especially as the sizes of the instances become larger. Besides, these two new best performing formulations allow the solver to prove optimality of instances for which not even the linear relaxation could be solved using the previously best performing formulations.

A constant approximation for the one-warehouse multi-retailer problem with backorder

Abstract We consider two deterministic inventory models : the One-Warehouse Multi-Retailer problem and its special case the Joint Replenishment Problem. Both models have been studied extensively in their classical version, where demand has to be satisfied on time, but there are very few results in the case where backlog is allowed. This work aims to develop a simple and fast method to provide an approximate solution for both models. More precisely, our algorithm has a worst case guarantee of three for the One-Warehouse Multi-Retailer problem with backlog. To the best of our knowledge, it is the first constant approximation for such a general and complex model. We also introduce a new lower bound based on a decomposition into single-echelon systems of these problems and improve the approximation guarantee to two in the special case of the Joint Replenishment model.

The one-warehouse multi-retailer problem: reformulation, classification, and computational results

We consider the one-warehouse multi-retailer problem where a warehouse replenishes multiple retailers with deterministic dynamic demands over a horizon. The problem is to determine when and how much to order to the warehouse and retailers such that the total system-wide costs are minimized. We propose a new (combined transportation and shortest path based) integer programming reformulation for the problem in addition to the echelon stock and transportation based formulations in the literature. We analyze the strength of the LP relaxations of three formulations and show that the new formulation is stronger than others. We also show that the new and transportation based formulations are equivalent for the joint replenishment problem, where the warehouse is a crossdocking facility. We extend all formulations to the case with initial inventory at the warehouse and reveal the relation among their LP relaxations. We present our computational experiments with all formulations over a set of randomly generated test instances.

The one-warehouse multiretailer problem with an order-up-to level inventory policy

We consider a two-level system in which a warehouse manages the inventories of multiple retailers. Each retailer employs an order-up-to level inventory policy over T periods and faces an external demand which is dynamic and known. A retailer's inventory should be raised to its maximum limit when replenished. The problem is to jointly decide on replenishment times and quantities of warehouse and retailers so as to minimize the total costs in the system. Unlike the case in the single level lot-sizing problem, we cannot assume that the initial inventory will be zero without loss of generality. We propose a strong mixed integer program formulation for the problem with zero and nonzero initial inventories at the warehouse. The strong formulation for the zero initial inventory case has only T binary variables and represents the convex hull of the feasible region of the problem when there is only one retailer. Computational results with a state-of-the art solver reveal that our formulations are very effective in solving large-size instances to optimality. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010

A Constant Approximation Algorithm for the One-Warehouse Multiretailer Problem

Deterministic inventory theory provides streamlined optimization models that attempt to capture trade-offs in managing the flow of goods through a supply chain. We will consider two well-studied deterministic inventory models, called the one-warehouse multiretailer (OWMR) problem and its special case the joint replenishment problem (JRP), and give approximation algorithms with worst-case performance guarantees. That is, for each instance of the problem, our algorithm produces a solution with cost that is guaranteed to be at most 1.8 times the optimal cost; this is called a 1.8-approximation algorithm. Our results are based on an LP-rounding approach; we provide the first constant approximation algorithm for the OWMR problem and improve the previous results for the JRP.