One-warehouse Multi-retailer Systems Research Articles

Inventory management for one warehouse multi-retailer systems with carbon emission costs

In this paper, we study inventory management for the one warehouse multi-retailer (OWMR) system with considering carbon emissions, in which we determine the reorder intervals for the warehouse and the retailers while minimizing the inventory ordering and holding cost and the carbon emission cost. We consider the carbon emissions from the holding and replenishing inventory activities in the OWMR system, and calculate the carbon emissions based on the frequency of the inventory replenishment at the warehouse and the retailers. We focus on stationary and integer-ratio policies, and formulate the problem as an optimization model with a convex objective function and a set of integer-ratio constraints. We relax the integer-ratio constraints, and introduce an algorithm to solve the corresponding relaxed problem to optimality in O(nlogn) times, where n represents the number of retailers. Based on the optimal solution of the relaxed problem, we build an integer-ratio policy, i.e., power-of-two (POT) policy, with 94% optimality for the primal problem. The computational results show that with considering the carbon emission cost, the reorder intervals of the warehouse and the retailers both increase, and that the integrated model proposed in this paper has more superiorities in terms of reducing the system-wide cost and the carbon emissions.

A Power-of-Two Ordering Policy for One-Warehouse Multiretailer Systems with Stochastic Demand

We study a two-echelon supply chain with one warehouse and N (nonidentical) retailers facing stochastic demand. An easy-to-implement inventory policy, the so-called power-of-two (POT) policy, is proposed to manage inventory for the system. To maintain a certain service level, safety stocks are kept at the warehouse and each retailer outlet to buffer random demand. Our analysis highlights the important role of the warehouse safety stock level, which, in addition to the length of the warehouse order interval, significantly affects the lengths of the retailers' order intervals. By combining the length of the warehouse order interval with the warehouse safety stock level, we introduce a plane partition method and develop a polynomial time algorithm to find a POT policy for arbitrary target service levels. The long-run average cost of the proposed POT policy is guaranteed to be no more than 1.26 times the optimal POT policy cost. We also show that our proposed policy can be computed in O(N3).

A numerical study on the effect of the balance assumption in one-warehouse multi-retailer inventory systems

One-warehouse multi-retailer systems under periodic review have been studied extensively in the literature. The optimal policy has not been characterized yet. It would require solving a multi-dimensional dynamic program, which is hard due to the curse of dimensionality. In order to let the dynamic program decompose, researchers often make the so-called balance assumption. All available heuristics for periodic review distribution systems are based on some form of this assumption. For these heuristics, often further approximate steps are applied. We investigate the pure effect of the balance assumption in this paper. The balance assumption is the relaxation of a set of constraints in the original dynamic program and yields a lower bound model, which we solve exactly. This gives us a lower bound for the optimal cost of the original model. An upper bound for the true optimal cost is obtained by simulating the optimal policy for the relaxed problem with a slightly modified allocation rule. This modified policy is referred to as the LB heuristic policy. We use the relative gap between the upper and lower bound as a measure to assess the impact of the balance assumption. Based on extensive testing, we identify when the gap is small, and when not. For those instances with small gaps, both the lower bound is tight and the performance of the LB heuristic policy is close to the optimal. We also identify many practically relevant settings under which the balance assumption yields large gaps. For these instances, either the lower bound is poor or the LB heuristic policy is far from optimal, or both. In any case, it implies that more research is needed to develop better lower bounds and/or better heuristics for these instances.

Approximation algorithms for general one‐warehouse multi‐retailer systems

AbstractLogistical 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 ε‐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 + ε) 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

Two-stage inventory models with a bi-modal transportation cost

This paper discusses inventory models over an infinite planning horizon with constant demand rate and two modes of transportation. These transportation options include truckloads and a less than truckload carrier. An optimal algorithm is derived for a one-warehouse one-retailer system. A power-of-two heuristic algorithm is also proposed for a one-warehouse multi-retailer system. Computational results are provided to show that, on the average, the heuristic algorithm is at least 94% effective.

A new algorithm for one-warehouse multi-retailer systems under stationary nested policy

This study aims at optimally coordinating inventory among all the partners in a supply chain system with a central warehouse and multiple local retailers so as to minimize the average total costs. After reviewing the literature, we found that no study proposes an efficient solution approach that guarantees to solve the optimal solution for the one-warehouse multi-retailer (OWMR) lot-sizing problem. The solution approaches in the literature share a common problem, namely, they do not have insights into the characteristics of the optimal-cost curve. Therefore, this study first focuses on performing a full theoretical analysis on the characteristics of the optimal-cost curve. Then, by utilizing our theoretical results, we derive an effective search algorithm that is able to solve the optimal solution for the OWMR lot-sizing problem under stationary nested policy. Based on our random experiments, we demonstrate that our search algorithm out-performs the other heuristics.

A two-echelon inventory system with transportation capacity constraint

Consider a one-warehouse multi-retailer system under constant and deterministic demand, which is subjected to transportation capacity for every delivery period. To search for the best stationary zero inventory ordering (ZIO) policy, or the best power-of-two policy, or the best nested policy, the problem is formulated as a 0–1 integer linear program in which the objective function comprises of a fixed transportation cost whenever a delivery is made and the inventory costs for both the warehouse and retailers. To overcome the transportation capacity limitation, we extend the policies to allow for staggering deliveries. It is shown that with transportation capacity constraint the non-staggering policy can have its effectiveness close to 0% from the best staggering policy and the power-of-two policy with staggering allowed can have its effectiveness close to 0% from the optimal policy. Nevertheless in general, the power-of-two policy fairs well on a number of randomly generated problems. To solve the large distribution network problem, an efficient heuristic based on the power-of-two policy with staggering of deliveries is suggested.

Inventory Cost Effect of Consolidating Several One-Warehouse Multiretailer Systems

Consolidation of warehouses is a new trend in global logistics management, and the reduction in order processing and inventory costs is often cited as one of the main motivations. In this note we show that when retailers face constant demand rates and their ordering costs are independent of the warehouse that services them, consolidated systems are rarely suboptimal and always lead to close-to-optimal inventory replenishment costs. In particular, we prove that using two (one) properly selected warehouses, the systemwide inventory replenishment cost is in the worst case at most 2% (14.75%) more than the optimal.

Effectiveness of (R, Q) policies in one-warehouse multiretailer systems with deterministic demand and backlogging

Consider a distribution system with a central warehouse and multiple retailers. Customer demand arrives at each of the retailers continuously at a constant rate. The retailers replenish their inventories from the warehouse which in turn orders from an outside supplier with unlimited stock. There are economies of scale in replenishing the inventories at both the warehouse and the retail level. Stockouts at the retailers are backlogged. The system incurs holding and backorder costs. The objective is to minimize the long-run average total cost in the system. This paper studies the cost effectiveness of (R, Q) policies in the above system. Under an (R, Q) policy, each facility orders a fixed quantity Q from its supplier every time its inventory position reaches a reorder point R. It is shown that (R, Q) policies are at least 76% effective. Numerical examples are provided to further illustrate the cost effectiveness of (R, Q) policies. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 422–439, 2000

One-Warehouse Multiretailer Systems with Centralized Stock Information

We consider a distribution system with a central warehouse and multiple retailers. The warehouse orders from an outside supplier and replenishes the retailers which in turn satisfy customer demand. The retailers are nonidentical, and their demand processes are independent compound Poisson. There are economies of scale in inventory replenishment, which is controlled by an echelon-stock, batch-transfer policy. For the special case with simple Poisson demand, we develop an exact method for computing the long-run average holding and backorder costs of the system. Based on this exact method, we provide approximations for compound Poisson demand. Numerical examples are used to illustrate the accuracy of the approximations. We also present a numerical comparison between the average costs of a heuristic, echelon-stock policy and an existing lower bound on the average costs of all feasible policies.

Lower Bounds for Multi-Echelon Stochastic Inventory Systems

We establish lower bounds on the minimum cost of managing certain production-distribution networks with setup costs at all stages and stochastic demands. These networks include serial, assembly, and one-warehouse multi-retailer systems. We obtain the bounds through novel cost-allocation schemes. We evaluate the bounds' performance for one-warehouse multi-retailer systems by comparing them with simple, heuristic policies. The bounds are quite tight for systems with a small number of retailers. We also present simplified proof of known optimality results for serial and assembly systems.

Approximation Procedures for the One-Warehouse Multi-Retailer System

Two heuristic procedures for a one-warehouse multi-retailer system are developed. Based on the accuracy desired, the first heuristic evaluates a specified number of points. The relative error is within a bound that approaches 1/(√2 ln 2) − 1 ≈ 2.014%. The complexity of the heuristic is O(n) for a fixed number of evaluations. Although our bound only approaches the one of Roundy (1985), when only a small number of points are evaluated, our method is faster. We show that the bound for our procedure and two bounds proposed by Roundy (1985) are tight. The second heuristic pertains to a class of polities called stationary interval policies. For this class of policies, we develop a fully polynomial-time approximation scheme where the relative error is within ε > 0, and the computational effort increases as a linear function of 1/√ε. Computational experiments show that these heuristics perform well in practice.

Minimizing Separable Convex Objectives on Arbitrarily Directed Trees of Variable Upper Bound Constraints

An extension of the Economic Order Quantity (EOQ) model to multi-stage production-distribution systems and the isotonic regression problem are known to be equivalent and to be solvable in O(N4) time. The following specializations of the model are solvable in O(N log N) time: joint replenishment systems, pure assembly systems, nested policies in pure distribution systems, nonnested policies in one-warehouse multi-retailer systems, nested policies in multi-item, one-warehouse, multi-retailer systems, and systems in which the underlying network is a series-parallel digraph. We show here that a generalization of this problem is solvable in O(N log N) time whenever the undirected version of the underlying network is a tree. The algorithm we propose is used as a subroutine in an algorithm solving the EOQ problem on general circuitless directed graphs, the subject of a companion paper.