Optimal order-limit policies for an (r, Q) inventory system

The reorder-point-reorder-quantity policies referred to as (r,Q) policies are widely used in industry and extensively studiedin the inventory literature. It should be noted, however, thatshortages often occur in practice even when the optimal reorderpoint and reorder quantity are achieved. In this paper, we presenta mathematical model to control a shortage period by an emergencyorder. The problem is the determination of the best timing todeliver items after a shortage occurs. By applying the conceptof repair-limit replacement policy in the context of maintenancetheory, the optimal order time limits based on three kinds ofcost criteria are derived in the framework of a simple (r, Q)inventory system. Finally, some examples of the inventory controlmodel with a stochastic lead time are given to explain the optimalorder-limit policies.     

Optimal ordering policies for continuous review perishable inventory models

This paper extends the notions of perishable inventory models to the realm of continuous review inventory systems. The traditional perishable inventory costs of ordering, holding, shortage or penalty, disposal and revenue are incorporated into the continuous review framework. The type of policy that is optimal with respect to long run average expected cost is presented for both the backlogging and lost-sales models. In addition, for the lost-sales model the cost function is presented and analyzed.     

Determining the optimal constrained multi-item (Q, r) inventory policy by maximising risk-adjusted profit

^** Email: john.betts{at}infotech.monash.edu.au^*** Email: robertj{at}unimelb.edu.au The predominant approach to determining replenishment batchsizes for capital constrained multi-item inventories is to assumethat at some point in time the replenishment of all items willcoincide, and that batch sizes are small enough that the constraintis not violated when this event occurs. However, when an inventoryconsists of a large number of independently replenished components,the probability that all replenishments coincide is very small.The standard approach thus results in unnecessarily conservativebatch sizes that under-utilise the available resource, resultingin lower profit than would be the case if a small risk of violatingthe constraint was tolerated. In this paper, a new approachto determining constrained batch sizes is presented where, fora certain average investment, the probability of exceeding abinding, or fixed, constraint on capital is determined. Thisprobability is used to define an adjustment factor to be appliedto expressions for company profit so that an optimal trade-offbetween maximising profit and reducing risk of failure is obtainedsimply by optimising this adjusted profit. By optimising profitadjusted for the risk of exceeding the constraint, the new modelyields batch sizes that are larger, and result in greater profitabilitythan those recommended under traditional models.     

Cooperation and game-theoretic cost allocation in stochastic inventory models with continuous review

We study cooperation strategies for companies that continuously review their inventories and face Poisson demand. Our main goal is to analyze stable cost allocations of the joint costs. These are such that any group of companies has lower costs than the individual companies. If such allocations exist they provide an incentive for the companies to cooperate.     

Sensitivity of continuous review stochastic (s,S) inventory systems to ordering delays

This paper examines continuous review stochastic (s, S) inventory systems with ordering delays. That is, systems where the difference between the time the order should be placed and the time the order is actually placed is non-trivial. The traditional inventory ordering, holding and penalty costs are included and the average cost for an (s, S) policy is developed and examined. Computational results are presented for two cases. In the first case, the manager is aware of the delay and uses the policy that minimizes all costs. We present the increase in cost due to having a delay. In the second case the manager is unaware of the delay and uses the (integer) square root formula. We present the increase in cost due to using the square root formula when it is inappropriate and in fact our computational results indicate that there may very well be a large increase in cost due to being unaware of the ordering delay.     

Order and chaos in Hofstadter's Q(n) sequence

A number of observations are made on Hofstadter's integer sequence defined by Q(n) = Q(n − Q(n − 1)) + Q(n − Q(n − 2)), for n > 2, and Q(1) = Q(2) = 1. On short scales, the sequence looks chaotic. It turns out, however, that the Q(n) can be grouped into a sequence of generations. The k‐th generation has 2^k members that have “parents” mostly in generation k − 1 and a few from generation k − 2. In this sense, the sequence becomes Fibonacci type on a logarithmic scale. The variance of S(n) = Q(n) − n/2, averaged over generations, is ≅2^αk, with exponent α = 0.88(1). The probability distribution p*(x) of x = R(n) = S(n)/n^α, n ≫ 1, is well defined and strongly non‐Gaussian, with tails well described by the error function erfc. The probability distribution of x_m = R(n) − R(n − m) is given by p_m(x_m) = λ_m p*(x_m/λ_m), with λ_m → √2 for large m. © 1999 John Wiley & Sons, Inc.     

Reorder quantities for (<Emphasis Type="Italic">Q,R</Emphasis>) inventory models

The reorder level R and the reorder quantity Q are the parameters to be decided in a continuous review inventory policy. Their optimal values can be approached through iterative methods, but these are tedious and inconvenient for control routines. A frequent practice is to set Q as the economic reorder quantity and compute R accordingly. Yet this practice may introduce a substantial cost penalty. The paper re-arranges conventional theoretical expressions in order to facilitate the use of numerical approximations to help find a quasi-optimal solution. This approach is then applied to Gamma distributed demands showing that computations are straightforward.     

Optimality of (s, S) policies for jump inventory models

This paper is concerned with the optimality of (s, S) policies for a single-item inventory control problem which minimizes the total expected cost over an infinite planning horizon and where the demand is driven by a piecewise deterministic process. Our approach is based on the theory of quasi-variational inequality.     

Evolutionary Pareto optimizers for continuous review stochastic inventory systems

Multi-objective inventory control has been studied for a long time. The trade-off analysis of cycle stock investment and workload, so called the exchange curve concept, possibly dates back to several decades ago. A classical way to such trade-off analysis is to utilize the Lagrangian relaxation technique or interactive method to search for the optimum in a sequence of single objective optimization problems. However, the field of optimization has been changed over the last few decades since the concept of evolutionary computation was introduced. In this paper, a continuous review stochastic inventory system with three objectives about cost and shortage is resolved by evolutionary computation in order to plan for the control policies under backordering and lost sales. Two evolutionary optimizers, multi-objective electromagnetism-like optimization (MOEMO) and multi-objective particle swarm optimization (MOPSO), are employed to well and fast approximate the non-dominated policies in term of lot size and safety stock. Trade-offs are observed in a non-dominated set that no one excels the others in all objectives. Computational results show that the evolutionary Pareto optimizers could generate trade-off solutions potentially ignored by the well-known simultaneous method. Comparisons between the results of backordering and lost sales indicate that decision makers will make more deliberate choices about lot sizing and safety stocking when unsatisfied demand is completely lost.     

Continuous review inventory models for perishable items ordered in batches

This paper is an in-depth treatment of an inventory control problem with perishable items. We focus on two prototypes of perishability for items that have a common shelflife and that arrive in batches with zero lead time: (i) sudden deaths due to disasters (e.g., spoilage because of extreme weather conditions or a malfunction of the storage place) and (ii) outdating due to expirations (e.g., medicine or food items that have an expiry date). By using known mathematical tools we generalize the stochastic analysis of continuous review (s, S) policies to our problems. This is achieved by integrating with each inventory cycle stopping times that are independent of the inventory level. We introduce special cases of compound Poisson demand processes with negative jumps and consider demands (jumps) that are exponentially distributed or of a unit (i.e., Poisson) demand. For these special cases we derive a closed form expression of the total cost, including that of perishable items, given any order up to level. Since the stochastic analysis leads to tractable expressions only under specific assumptions, as an added benefit we use a fluid approximation of the inventory level to develop efficient heuristics that can be used in general settings. Numerical results comparing the solution of the heuristics with exact or simulated optimal solutions show that the approximation is accurate.     

On sufficient conditions for cost rate function unimodality in periodic review (s, S) inventory models

The cost rate function that arises in the stationary analysis of a class of periodic review regenerative inventory systems is known to be unimodal if the renewal density of the underlying demand sequence is decreasing. We prove that the same result holds, under zero leadtimes, if the renewal density is concave increasing.     

Exact evaluation of (R,Q)-policies for two-level inventory systems with Poisson demand

In this paper we show how to exactly evaluate holding and shortage costs for a two-level inventory system with one warehouse and N different retailers. Lead-times (transportation times) are constant, and the retailers face different Poisson demand processes. All facilities apply continuous review (R, Q)-policies. We express the policy costs as a weighted mean of costs for one-for-one ordering policies.     

基于(Q,r)补货策略的寄售库存模式研究

本文运用数学建模和数值分析的方法研究了基于(Q,r)补货策略的寄售库存模式中合同参数的优化问题。在该模式下,供应商确定Q、r值,并维持(z,Z)的库存水平,零售商对低于下限z或超过上限Z的库存收取罚金b。研究结果表明,零售商可以通过设置合理的罚金b来激励供应商选择合理的r和Q,使得最低的期望库存水平高于z,并且使成本达到最优。     

A study concerning splitting and jointly continuous topologies on C(Y,Z)

Let Y and Z be two fixed topological spaces and C(Y, Z) the set of all continuous maps from Y into Z. We construct and study topologies on C(Y, Z) that w^{e call F}n(τn)-family-open topologies. Furthermore, we find necessary and su?cient conditions such that these topologies to be splitting and jointly continuous. Finally, we present questions concerning a further study on this area.     

A multi-product continuous review inventory system in stochastic environment with budget constraint

Common characteristics of inventory systems include uncertain demand and restrictions such as budgetary and storage space constraints. Several authors have examined budget constrained multi-item stochastic inventory systems controlled by continuous review policies without considering marginal review shortage costs. Existing models assume that purchasing costs are paid at the time an order is placed, which is not always the case since in some systems purchasing costs are paid when order arrive. In the latter case the maximum investment in inventory is random since the inventory level when an order arrives is a random variable. Hence payment of purchasing costs on delivery yields a stochastic budget constraint for inventory. In this paper with mixture of back orders and lost sales, we assume that mean and variance of lead time demand are known but their probability distributions are unknown. After that, we apply the minimax distribution free procedure to find the minimum expected value of the random objective function with budget constraint. The random budget constraint is transformed to crisp budget constraint by chance-constraint technique. Finally, the model is illustrated by a numerical example.     

The impact of data collection on fill rate performance in the (R,s, Q) inventory model

In this paper we consider the determination of the reorder point s in an (R, s, Q) inventory model subject to a fill rate service level constraint. We assume that the underlying demand process is a compound renewal process. We then derive an approximation method to compute the reorder level such that a target service level is achieved. Restrictions on the input parameters are given, within which this method is applicable. Moreover, we will investigate the effects on the fill rate performance in case the underlying demand process is indeed a compound renewal process, while the demand process is modelled as a discrete-time demand process. That is, the time axis is divided in time units (for example, days) and demands per time unit are independent and identically distributed random variables. It will be shown that smooth and erratic behaviour of the inter-arrival times have different impacts on the performance of the fill rate when demand is modelled as a discrete-time process and in case the underlying demand process is a compound renewal process.