Optimal outbound dispatch policies: Modeling inventory and cargo capacity

In this paper, we present an optimization model for coordinating inventory and transportation decisions at an outbound distribution warehouse that serves a group of customers located in a given market area. For the practical problems which motivated this paper, the warehouse is operated by a third party logistics provider. However, the models developed here may be applicable in a more general context where outbound distribution is managed by another supply chain member, e.g., a manufacturer. We consider the case where the aggregate demand of the market area is constant and known per period (e.g., per day). Under an immediate delivery policy, an outbound shipment is released each time a demand is realized (e.g., on a daily basis). On the other hand, if these shipments are consolidated over time, then larger (hence more economical) outbound freight quantities can be dispatched. In this case, the physical inventory requirements at the third party warehouse (TPW) are determined by the consolidated freight quantities. Thus, stock replenishment and outbound shipment release policies should be coordinated. By optimizing inventory and freight consolidation decisions simultaneously, we compute the parameters of an integrated inventory/outbound transportation policy. These parameters determine: (i) how often to dispatch a truck so that transportation scale economies are realized and timely delivery requirements are met, and (ii) how often, and in what quantities, the stock should be replenished at the TPW. We prove that the optimal shipment release timing policy is nonstationary, and we present algorithms for computing the policy parameters for both the uncapacitated and finite cargo capacity problems. The model presented in this study is considerably different from the existing inventory/transportation models in the literature. The classical inventory literature assumes that demands should be satisfied as they arrive so that outbound shipment costs are sunk costs, or else these costs are covered by the customer. Hence, the classical literature does not model outbound transportation costs. However, if a freight consolidation policy is in place then the outbound transportation costs can no longer be ignored in optimization. Relying on this observation, this paper models outbound transportation costs, freight consolidation decisions, and cargo capacity constraints explicitly. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 531–556, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10030     

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

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(T²) algorithm for the single‐stage dynamic lot sizing problem; (2) an O(T³) 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. © 2009 Wiley Periodicals, Inc. Naval Research Logistics 2009     

Fixed‐interval joint‐replenishment policies for distribution systems with multiple retailers and stochastic demand

We consider a distribution system consisting of a central warehouse and a group of retailers facing independent stochastic demand. The retailers replenish from the warehouse, and the warehouse from an outside supplier with ample supply. Time is continuous. Most previous studies on inventory control policies for this system have considered stock‐based batch‐ordering policies. We develop a time‐based joint‐replenishment policy in this study. Let the warehouse set up a basic replenishment interval. The retailers are replenished through the warehouse in intervals that are integer multiples of the basic replenishment interval. No inventory is carried at the warehouse. We provide an exact evaluation of the long‐term average system costs under the assumption that stock can be balanced among the retailers. The structural properties of the inventory system are characterized. We show that, although it is well known that stock‐based inventory control policies dominate time‐based inventory control policies at a single facility, this dominance does not hold for distribution systems with multiple retailers and stochastic demand. This is because the latter can provide a more efficient mechanism to streamline inventory flow and pool retailer demand, even though the former may be able to use more updated stock information to optimize system performance. The findings of the study provide insights about the key factors that drive the performance of a multiechelon inventory control system. © 2013 Wiley Periodicals, Inc. Naval Research Logistics 60: 637–651, 2013     

An exact analysis of a joint production‐inventory problem in two‐echelon inventory systems

We consider a two‐echelon inventory system with a manufacturer operating from a warehouse supplying multiple distribution centers (DCs) that satisfy the demand originating from multiple sources. The manufacturer has a finite production capacity and production times are stochastic. Demand from each source follows an independent Poisson process. We assume that the transportation times between the warehouse and DCs may be positive which may require keeping inventory at both the warehouse and DCs. Inventory in both echelons is managed using the base‐stock policy. Each demand source can procure the product from one or more DCs, each incurring a different fulfilment cost. The objective is to determine the optimal base‐stock levels at the warehouse and DCs as well as the assignment of the demand sources to the DCs so that the sum of inventory holding, backlog, and transportation costs is minimized. We obtain a simple equation for finding the optimal base‐stock level at each DC and an upper bound for the optimal base‐stock level at the warehouse. We demonstrate several managerial insights including that the demand from each source is optimally fulfilled entirely from a single distribution center, and as the system's utilization approaches 1, the optimal base‐stock level increases in the transportation time at a rate equal to the demand rate arriving at the DC. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Capacitated warehouse location model with risk pooling

In this article, we introduce the capacitated warehouse location model with risk pooling (CLMRP), which captures the interdependence between capacity issues and the inventory management at the warehouses. The CLMRP models a logistics system in which a single plant ships one type of product to a set of retailers, each with an uncertain demand. Warehouses serve as the direct intermediary between the plant and the retailers for the shipment of the product and also retain safety stock to provide appropriate service levels to the retailers. The CLMRP minimizes the sum of the fixed facility location, transportation, and inventory carrying costs. The model simultaneously determines warehouse locations, shipment sizes from the plant to the warehouses, the working inventory, and safety stock levels at the warehouses and the assignment of retailers to the warehouses. The costs at each warehouse exhibit initially economies of scale and then an exponential increase due to the capacity limitations. We show that this problem can be formulated as a nonlinear integer program in which the objective function is neither concave nor convex. A Lagrangian relaxation solution algorithm is proposed. The Lagrangian subproblem is also a nonlinear integer program. An efficient algorithm is developed for the linear relaxation of this subproblem. The Lagrangian relaxation algorithm provides near‐optimal solutions with reasonable computational requirements for large problem instances. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

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     

Managing components for assemble‐to‐order products with lead‐time‐dependent pricing: The full‐shipment model

We study a component inventory planning problem in an assemble‐to‐order environment faced by many contract manufacturers in which both quick delivery and efficient management of component inventory are crucial for the manufacturers to achieve profitability in a highly competitive market. Extending a recent study in a similar problem setting by the same authors, we analyze an optimization model for determining the optimal component stocking decision for a contract manufacturer facing an uncertain future demand, where product price depends on the delivery times. In contrast to our earlier work, this paper considers the situation where the contract manufacturer needs to deliver the full order quantity in one single shipment. This delivery requirement is appropriate for many industries, such as the garment and toy industries, where the economies of scale in transportation is essential. We develop efficient solution procedures for solving this optimization problem. We use our model results to illustrate how the different model parameters affect the optimal solution. We also compare the results under this full‐shipment model with those from our earlier work that allows for multiple partial shipments. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Centralized inventory control in a two‐level distribution system with Poisson demand

This paper introduces a new replenishment policy for inventory control in a two‐level distribution system consisting of one central warehouse and an arbitrary number of nonidentical retailers. The new policy is designed to control the replenishment process at the central warehouse, using centralized information regarding the inventory positions and demand processes of all installations in the system. The retailers on the other hand are assumed to use continuous review (R, Q) policies. A technique for exact evaluation of the expected inventory holding and backorder costs for the system is presented. Numerical results indicate that there are cases when considerable savings can be made by using the new (α₀, Q₀) policy instead of a traditional echelon‐ or installation‐stock (R, Q) policy. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 798–822, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10040     

Warehouse sizing to minimize inventory and storage costs

This paper considers a warehouse sizing problem whose objective is to minimize the total cost of ordering, holding, and warehousing of inventory. Unlike typical economic lot sizing models, the warehousing cost structure examined here is not the simple unit rate type, but rather a more realistic step function of the warehouse space to be acquired. In the cases when only one type of stock‐keeping unit (SKU) is warehoused, or when multiple SKUs are warehoused, but, with separable inventory costs, closed form solutions are obtained for the optimal warehouse size. For the case of multi‐SKUs with joint inventory replenishment cost, a heuristic with a provable performance bound of 94% is provided. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 299–312, 2001     

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     

A two-echelon inventory model with purchases,dispositions, shipments,returns and transshipments

This paper presents a one-period two-echelon inventory model with one warehouse in the first echelon and n warehouses in the second echelon. At the beginning of the period the stock levels at all facilities are adjusted by purchasing or disposing of items at the first echelon, returning or shipping items between the echelons and transshipping items within the second echelon. During the period, demands (which may be negative) are placed on all warehouses in the second echelon and an attempt is made to satisfy shortages either by an expedited shipment from the first echelon to the second echelon or an expedited transshipment within the second echelon. The decision problem is to choose an initial stock level at the first echelon (by a purchase or a disposition) and an initial allocation so as to minimize the initial stock movement costs during the period plus inventory carrying costs and system shortage costs at the end of the period. It is shown that the objective function takes on one of four forms, depending on the relative magnitudes of the various shipping costs. All four forms of the objective function are derived and proven to be convex. Several applications of this general model are considered. We also consider multi-period extensions of the general model and an important special case is solved explicitly.     

Algorithms for minimizing total cost,bottleneck time and bottleneck shipment in transportation problems

We consider the transportation problem of determining nonnegative shipments from a set of m warehouses with given availabilities to a set of n markets with given requirements. Three objectives are defined for each solution: (i) total cost, TC, (ii) bottleneck time, BT (i.e., maximum transportation time for a positive shipment), and (iii) bottleneck shipment, SB (i.e., total shipment over routes with bottleneck time). An algorithm is given for determining all efficient (pareto-optimal or nondominated) (TC, BT) solution pairs. The special case of this algorithm when all the unit cost coefficients are zero is shown to be the same as the algorithms for minimizing BT. provided by Szwarc and Hammer. This algorithm for minimizing BT is shown to be computationally superior. Transportation or assignment problems with m=n=100 average about a second on the UNIVAC 1108 computer (FORTRAN V)) to the threshold algorithm for minimizing BT. The algorithm is then extended to provide not only all the efficient (TC, BT) solution pairs but also, for each such BT, all the efficient (TC, SB) solution pairs. The algorithms are based on the cost operator theory of parametric programming for the transportation problem developed by the authors.     

Exact analysis of (R,s, S) inventory control systems with lost sales and zero lead time

We study an (R, s, S) inventory control policy with stochastic demand, lost sales, zero lead‐time and a target service level to be satisfied. The system is modeled as a discrete time Markov chain for which we present a novel approach to derive exact closed‐form solutions for the limiting distribution of the on‐hand inventory level at the end of a review period, given the reorder level (s) and order‐up‐to level (S). We then establish a relationship between the limiting distributions for adjacent values of the reorder point that is used in an efficient recursive algorithm to determine the optimal parameter values of the (R, s, S) replenishment policy. The algorithm is easy to implement and entails less effort than solving the steady‐state equations for the corresponding Markov model. Point‐of‐use hospital inventory systems share the essential characteristics of the inventory system we model, and a case study using real data from such a system shows that with our approach, optimal policies with significant savings in inventory management effort are easily obtained for a large family of items.     

A computationally efficient approach for determining inventory levels in a capacitated multiechelon production‐distribution system

The system under study is a single item, two‐echelon production‐inventory system consisting of a capacitated production facility, a central warehouse, and M regional distribution centers that satisfy stochastic demand. Our objective is to determine a system base‐stock level which minimizes the long run average system cost per period. Central to the approach are (1) an inventory allocation model and associated convex cost function designed to allocate a given amount of system inventory across locations, and (2) a characterization of the amount of available system inventory using the inventory shortfall random variable. An exact model must consider the possibility that inventories may be imbalanced in a given period. By assuming inventory imbalances cannot occur, we develop an approximation model from which we obtain a lower bound on the per period expected cost. Through an extensive simulation study, we analyze the quality of our approximation, which on average performed within 0.50% of the lower bound. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 377–398, 2000     

Computing demand properties at the wholesale warehouse level

Computational formulas are given for the mean, variance, and autocorrelation function of the demand process at an upper-echelon facility (warehouse). The demand process at the warehouse is induced by the aggregated inventory replenishment processes of N independently operated lower-echelon facilities (stores) in parallel. Each store, we assume, employs an (s,S) inventory replenishment policy with complete backlogging to satisfy its own random, independently and identically distributed demand. The formulas result from an analysis of the stochastic replenishment process at a single store. Examples of the properties of the demand process at the upper-echelon facility are presented for several lower-echelon environments.     

On the first come–first served rule in multi‐echelon inventory control

A two‐echelon distribution inventory system with a central warehouse and a number of retailers is considered. The retailers face stochastic demand and replenish from the warehouse, which, in turn, replenishes from an outside supplier. The system is reviewed continuously and demands that cannot be met directly are backordered. Standard holding and backorder costs are considered. In the literature on multi‐echelon inventory control it is standard to assume that backorders at the warehouse are served according to a first come–first served policy (FCFS). This allocation rule simplifies the analysis but is normally not optimal. It is shown that the FCFS rule can, in the worst case, lead to an asymptotically unbounded relative cost increase as the number of retailers approaches infinity. We also provide a new heuristic that will always give a reduction of the expected costs. A numerical study indicates that the average cost reduction when using the heuristic is about two percent. The suggested heuristic is also compared with two existing heuristics. © 2007 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Coherent systems based on sequential order statistics

The sequential order statistics (SOS) are a good way to model the lifetimes of the components in a system when the failure of a component at time t affects the performance of the working components at this age t. In this article, we study properties of the lifetimes of the coherent systems obtained using SOS. Specifically, we obtain a mixture representation based on the signature of the system. This representation is used to obtain stochastic comparisons. To get these comparisons, we obtain some ordering properties for the SOS, which in this context represent the lifetimes of k‐out‐of‐n systems. In particular, we show that they are not necessarily hazard rate ordered. © 2011 Wiley Periodicals, Inc. Naval Research Logistics, 2011     

Transshipment through crossdocks with inventory and time windows

A major challenge in making supply meet demand is to coordinate transshipments across the supply chain to reduce costs and increase service levels in the face of demand fluctuations, short lead times, warehouse limitations, and transportation and inventory costs. In particular, transshipment through crossdocks, where just‐in‐time objectives prevail, requires precise scheduling between suppliers, crossdocks, and customers. In this work, we study the transshipment problem with supplier and customer time windows where flow is constrained by transportation schedules and warehouse capacities. Transportation is provided by fixed or flexible schedules and lot‐sizing is dealt with through multiple shipments. We develop polynomial‐time algorithms or, otherwise, provide the complexity of the problems studied. © 2005 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

A decomposable transshipment algorithm for a multiperiod transportation problem

A dynamic version of the transportation (Hitchcock) problem occurs when there are demands at each of n sinks for T periods which can be fulfilled by shipments from m sources. A requirement in period t₂ can be satisfied by a shipment in the same period (a linear shipping cost is incurred) or by a shipment in period t₁ < t₂ (in addition to the linear shipping cost a linear inventory cost is incurred for every period in which the commodity is stored). A well known method for solving this problem is to transform it into an equivalent single period transportation problem with mT sources and nT sinks. Our approach treats the model as a transshipment problem consisting of T, m source — n sink transportation problems linked together by inventory variables. Storage requirements are proportional to T² for the single period equivalent transportation algorithm, proportional to T, for our algorithm without decomposition, and independent of T for our algorithm with decomposition. This storage saving feature enables much larger problems to be solved than were previously possible. Futhermore, we can easily incorporate upper bounds on inventories. This is not possible in the single period transportation equivalent.     

联合补充博弈的费用分摊

就一个仓库、多个零售商,对联合订货费用函数的模型进行分析,给出了一个求解最佳订货周期的多项式时间的算法,且算法的时间复杂性为O（nlogn）。利用文献[8]中的技巧,给出了该库存博弈的核。