a simple heuristic for joint inventory and pricing models with lead time and backorders we study a joint inventory and pricing problem in a single stage system with a positive lead time we consider both additive and multiplicative demand forms this problem is in general intractable due to its computational complexity we develop a simple heuristic that resolves this issue the heuristic involves a myopic pricing policy that generates each period s price as a function of the initial inventory level and a base stock policy for inventory replenishment in each period the firm monitors its so called price deflated inventory position and places an order to reach a target base stock level the price deflated inventory position weights the on hand and pipeline inventory according to a factor that reflects the sensitivity of price to the net inventory level to assess the effectiveness of our heuristic we construct an upper bound to the exact system the upper bound is based on an information relaxation approach and involves a penalty function derived from the proposed heuristic a numerical s

serial supply chains with economies of scale bounds and approximations we consider two models of stochastic serial inventory systems with economies of scale for which the forms of optimal policies are known in the first model each stage has a fixed order quantity while in the second model there is a fixed order cost for external supplies for each model we show that the optimal policy parameters can be bounded and approximated by a series of independent single stage optimal policy parameters we further construct closed form bounds and approximations for the single stage solutions and apply them to the serial systems these results provide simple and effective solutions that will help to facilitate implementations in practice they also allow us to see the connections between the serial and single stage systems and sharpen our intuition on optimal policy parameters and system behavior

evaluating echelon stock r nq policies in serial production inventory systems with stochastic demand this paper studies echelon stock r nq policies in serial production inventory systems with stochastic demand we provide a recursive procedure to compute the steady state echelon inventory levels of the systems which can be used to evaluate the long run average holding and backorder costs as well as other performance measures the procedure is based upon an observation of a relationship between the inventory status of adjacent stages in a serial system we also device exact formulas for replenishment frequencies and setup costs our results apply to both continuous review systems with compound poisson demand and periodic review systems with independent identically distributed demands a preliminary numerical study was conducted to explore the cost effectiveness of echelon stock r nq policies for two stage systems with simple poisson demand we compared among the minimum costs of echelon stock r nq policies a lower bound on the minimum achievable costs and the minimum costs of installation stock r nq policies finally we present a modification of an existing approximate evaluation procedure

optimal policies and approximations for a serial multiechelon inventory system with time correlated demand since clark and scarf s pioneering work most advances in multiechelon inventory systems have been based on demand processes that are time independent this paper revisits the serial multiechelon inventory system of clark and scarf and develops three key results first we provide a simple lower bound approximation to the optimal echelon inventory levels and an upper bound to the total system cost for the basic model of clark and scarf second we show that the structure of the optimal stocking policy of clark and scarf holds under time correlated demand processes using a martingale model of forecast evolution third we extend the approximation to the time correlated demand process and study in particular for an autoregressive demand model the impact of lead times and autocorrelation on the performance of the serial inventory system

exact evaluation of batch ordering inventory policies in two echelon supply chains with periodic review this paper studies a two echelon supply chain with stochastic and discrete consumer demand batch order quantities periodic inventory review and deterministic transportation times reorder point policies manage inventories at every location average inventory backorders and fill rates are evaluated exactly for each location safety stock is evaluated exactly at the lower echelon and a good approximation is detailed for the upper echelon numerical data are presented to demonstrate the model s utility it is found that system costs generally increase substantially if the upper echelon is restricted to carry no inventory of if the upper echelon is required to provide a high fill rate in many cases it is optimal to set the upper echelon s reorder point to yield near zero safety stock yet in some cases this simple heuristic can significantly increase supply chain operating costs finally policies selected under the assumption of continuous inventory review can perform poorly if implemented in an environment with periodic review

optimal policy for a two stage assembly system under random demand this paper considers an inventory system in which an end product is assembled from two components each of which is ordered from an external supplier only the end product has final demand which is assumed to be random using the functional equation approach of dynamic programming we characterize the forms of both the optimal order policy for the components and the optimal assembly policy of the end product for a multiperiod problem of arbitrary but finite length we believe that this work represents the only exact analysis of an mrp type assembly system when demand is stochastic

inventory cost rate functions with nonlinear shortage costs this article considers five cost rate models for inventory control each summarizing the expected holding and shortage costs per period as a function of the inventory position all models have linear holding costs and shortage cost coefficients of dimension unit period unit and period the latter two coeficients may be the shadow costs of a fill rate and a ready rate service constraint respectively one of the cost rate models is a new suggestion intended to facilitate modeling of periodic review inventory systems if and only if conditions on the demand process are presented for which the cost rate is quasi convex in the inventory position the typical sufficient condition requires that the cumulative demand distribution be logconcave a condition that is met by most demand distributions commonly used in the inventory literature the results simplify optimization and extend the known optimality of s s and nq r policies to cost structures common in applications and to the presence of typical service constraints as a prerequisite for the study a series of new monotonicity results are derived for compound renewal processes

a quantity discount approach to supply chain coordination abstract quantity discounts provide a practical foundation for inventory coordination in supply chains however typical supply chain participants may encounter difficulties in implementing the coordination policy simply because 1 specified lot size adjustments may deviate from the economic lot sizes and 2 the buying firm may face amplified overstocking risks related to increased order quantities the main objective of this study is to develop a quantity discount model that resolves the practical challenges associated with implementing quantity discount policies for supply chain coordination between a supplier and a buyer the proposed buyer s risk adjustment b ra model allows the supplier to offer discounts that capitalize on the original economic lot sizes and share the buyer s risk of temporary overstocking under uncertain demand the analytical results suggest that the proposed b ra discount approach is a feasible alternative for supply chain coordination under uncertain demand conditions

effective inventory control policies with a minimum order quantity and batch ordering abstract in this paper we consider a single item periodic review stochastic inventory system with both minimum order quantity moq and batch ordering requirements in each time period the firm can order either none or at least as much as the moq at the same time if an order is placed the order quantity is required to be an integral multiple of a given specific batch size we first adopt a heuristic policy which is specified by two parameters s t applying a discrete time markov chain approach we compute the system cost and optimize this s t policy under the long run average cost criterion we also consider a simpler one parameter policy the so called s policy which is a special case of the s t policy in an intensive numerical study we find that 1 both policies perform well in comparison with other policies and 2 the s policy also performs well and is compatible with the s t policy only in a few cases where demand variation is small the latter outperforms the former significantly we also evaluate the effects of some important parameters on system performance

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

technical note optimal control of an assembly system with multiple stages and multiple demand classes we consider an assembly system with multiple stages multiple items and multiple customer classes the system consists of m production facilities each producing a different item items are produced in variable batch sizes one batch at a time with exponentially distributed batch production times demand from each class takes place continuously over time according to a compound poisson process at each decision epoch we must determine whether or not to produce an item and should demand from a particular class arise whether or not to satisfy it from existing inventory if any is available we formulate the problem as a markov decision process and use it to characterize the structure of the optimal policy in contrast to systems with exogenous and deterministic production lead times we show that the optimal production policy for each item is a state dependent base stock policy with the base stock level nonincreasing in the inventory level of items that are downstream and nondecreasing in the inventory level of all other items for inventory allocation we show that the optimal policy is a multilevel state dependent rationing policy with the rationing level for each demand class nonincreasing in the inventory level of all nonend items we also show how the optimal control problem can be reformulated in terms of echelon inventory and how the essential features of the optimal policy can be reinterpreted in terms of echelon inventory

integrating resource and production decisions in a simple multi stage assembly system we consider a simple multi stage assembly system with stationary stochastic demand for a single final product that is made to stock the assembly tasks are performed by human resources this workforce consists of two parts regular workforce with a long term contract and temporary workforce that are contracted based on the actual demand contracting resources early is always cheaper than contracting later when we separate resource and production decisions we can first determine order up to levels and production decisions in a standard way and afterwards find the best mix of regular and temporary workforce next to this decoupled approach we present an integrated approach where these decisions influence each other in a set of examples we show that the integrated approach can be very beneficial

demand evolution in stochastic inventory systems riskiness increase abstract this paper characterizes demand evolution in the steady state under common inventory policies in a single location stochastic inventory system our results show that downstream demand nonstrictly second degree stochastically dominates upstream orders and the bullwhip effect occurs we provide the link between demand evolution and utility theory for stochastic inventory systems our model is general in that ultimate customer demand follows an arbitrary stationary distribution and the facility may arbitrarily select policy parameters

modified echelon r q policies with guaranteed performance bounds for stochastic serial inventory systems we consider the classic continuous review nstage serial inventory system with a homogeneous poisson demand arrival process at the most downstream stage stage 1 any shipment to each stage regardless of its size incurs a positive fixed setup cost and takes a positive constant lead time the optimal policy for this system under the long run average cost criterion is unknown finding a good worst case performance guarantee remains an open problem we tackle this problem by introducing a class of modified echelon r q policies that do not require qi 1 qi to be a positive integer stage i 1 ships to stage i based on its observation of the echelon inventory position at stage i if it is at or below ri and stage i 1 has positive on hand inventory then a shipment is sent to stage i to raise its echelon inventory position to ri qias close as possible we construct a heuristic policy within this class of policies which has the following features first it has provably primitive dependent performance bounds in a two stage system the performance of the heuristic policy is guaranteed to be within 1 k1 k2 times the optimal cost where k1 is the downstream fixed cost and k2 is the upstream fixed cost we also provide an alternative performance bound which depends on efficiently computable optimal r q solutions to n single stage systems but tends to be tighter second the heuristic is simple it is efficiently computable and it performs well numerically it is even likely to outperform the optimal integer ratio echelon r q policies when k1 is dominated by k2 third the heuristic is asymptotically optimal when we take some dominant relationships between the setup or holding cost primitives at an upstream stage and its immediate downstream stage to the extreme for example when h2 h1 0 where h1 is the downstream holding cost parameter and h2 is the upstream holding cost parameter

optimal policies for multi echelon inventory problems with batch ordering in many production distribution systems materials flow from one stage to another in fixed lot sizes for example a retailer orders a full truckload from a manufacturer to qualify for a quantity discount a factory has a material handling system that moves full containers of parts from one production stage to the next in this paper we derive optimal policies for multi stage serial and assembly systems where materials flow in fixed batches the optimal policies have a simple structure and their parameters can be easily determined this research extends the multi echelon inventory theory in several ways it generalizes the clark scarf model by allowing batch transfers of inventories rosling 1989 shows that assembly systems can be interpreted as serial systems under the assumption that there are no setup costs we show that the series interpretation still holds when materials flow in fixed batches which satisfy a certain regularity condition finally veinott 1965 identifies an optimal policy for a single location inventory system with batch ordering this paper generalizes his result to multi echelon settings

analysis of batch ordering inventory models with setup cost and capacity constraint abstract stochastic periodic review batch ordering inventory problems appear in many industrial settings however few literature deals with the optimal ordering polices for such problems no mention to the inclusion of the fixed ordering cost and the production capacity in this paper we consider a single item periodic review batch ordering inventory system with the consideration of the setup cost and the capacity constraint for each order over a finite planning horizon by proposing several new convex notions we show that a batch based s s policy is optimal for the unlimited ordering capacity case while for the limited ordering capacity case a modified r q policy is optimal for the setting with zero ordering setup cost and a batch based x y band policy for the setting with positive ordering setup cost moreover we analytically study the sensitivity of the policy parameters with respect to the capacity and batch order size and derive the bounds on the optimal policy parameters we further extend our analysis to the infinite horizon setting and show that the structure of the optimal policy remains similar finally the numerical experiments provide some insights into the impact of model parameters on the benefit of reducing the batch size and increasing the ordering capacity and indicate that ignoring batch requirement may lead to a significant cost increment

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

coordination mechanisms in decentralized serial inventory systems with batch ordering this paper studies a periodic review serial supply chain in which materials are ordered and shipped according to r nq policies three information scenarios are considered depending on the level of information available echelon local and quasilocal in the echelon scenario each stage can access the inventory and cost information within its echelon comprising the stage itself and all downstream stages in the local scenario each stage only accesses its own local information finally in the quasilocal scenario each stage knows its local information plus the actual customer demands we propose coordination schemes that regulate the stages to achieve the supply chain s optimal cost under each information setting all these coordination schemes fit comfortably within an emerging practice called supply chain finance which includes the organization and technology needed to implement them

an investigation of ricardian equivalence in a common trends model abstract to empirically discriminate between ricardian and non ricardian behaviour we have to use a method that distinguishes between expected and unexpected as well as permanent and transitory changes in taxes and government consumption one method that provides these distinctions between different types of changes is the common trends framework used in this paper which is a var model with cointegrating constraints in the empirical study of us data there is some support for the ricardian hypothesis but there are also some deviations from its predictions however the differences do not lend support to a keynesian view of budget deficits but rather to the view of expansionary fiscal contractions this can be motivated theoretically by uncertainty about future taxes in combination with prudent households

coordinating inventory control and pricing strategies under batch ordering in this paper we investigate joint pricing and inventory control problems in a finite horizon single product periodic review setting with certain uncertain supply capacities the demands in different periods are random variables whose distributions depend on the posted price exhibiting the additive form the order quantity in each period is required to be of integral multiples of a given specific batch size denoted by q inventory replenishment incurs a linear ordering cost referred to as the cost rate function the sum of holding and backorder costs can either be convex or quasi convex the objective is to determine a joint ordering and pricing decision that can maximize the total expected profit over the planning horizon we first consider the case in which the cost rate function is convex and show that the modifiedr q list price policy is optimal for the system with certain and limited capacities a special case of which is the r q list price policy when capacities become unlimited as supply capacities become random the optimal policy follows a new structure wherein the optimal order up to level and posted price must be coordinated to make the optimal safety stock level follow the r q policy we further consider the case of a quasi convex cost rate function which may arise when a service level constraint is used as a surrogate for the shortage cost we demonstrate that the r q list price policy is optimal for the system without supply capacity constraints in addition extensions to several other models are discussed the enabling technique is based on the notion of q jump convexity and its variants

cointegration and exchange rate dynamics abstract we utilize systems methods of estimation which impose the unit root structure to estimate both the long run equilibrium relationship the cointegrating vector and the short run dynamics of an open economy macroeconomic model in which the exchange rate and the domestic price level are endogenously determined for the united states the model is estimated over the flexible exchange rate period and over two sub periods chosen by structural change tests on the exchange rate both rational and theories consistent expectations are utilized the models with theories consistent expectations outperform those with rational expectations

technical note on optimal policies for inventory systems with batch ordering we study a periodically reviewed multiechelon inventory system in series such that order quantities at every stage have to be multiples of a given stage specific batch size the batch sizes are nested in the sense that the batch size for every stage is an integer multiple of the batch size for its downstream stage the problem is that of determining the policy that minimizes the expected discounted sum of costs over a finite horizon the result is that an echelon r nq policy is optimal when demands are independent across periods or more generally markov modulated we also comment on algorithmic implications of our result and on extensions