The continuous-review (r,Q) inventory model with lost sales and stochastic lead-time is considered. It is assumed that Qr, so that at most a single order may be outstanding at any time. This model is much treated in the literature, beginning with Hadley and Whitin (1963) and there are many heuristics, but no simple exact algorithm. We derive the model for both continuous and discrete (Poisson) demand and stochastic lead-time, and devise a simple iterative algorithm that is certain to locate the global minimum if the lead-time demand distribution function is log-concave. It iterates between a square-root formula for Q and a simple equation for r. The algorithm is applicable to discrete demand when amended by a final comparison between two adjacent policies. We finally show that the model is applicable without the constraint Qr, but then lead-time demand is a function of r and Q, the investigation of which is left to future research.

The probability distribution of customer waiting times due to stock-outs and backlogging is studied, including mean and variance, when demand follows a compound Poisson process and the inventory is governed by an (nQ,R) inventory policy under continuous review. Existing results are reviewed and new exact contributions are presented, particularly covering the case when R<–1, a case that quite often may be optimal for central warehouses, but is missing in the literature. Waiting times are investigated for two cases: For an average unit of demand, assuming partial deliveries, and for a customer demanding

d units, assuming full deliveries only. Here d may be interpreted as the constant order size from a specific customer/retailer. The results are easily generalized to the waiting time of an average customer. Besides exact formulae, various approximations and simple heuristics are suggested for calculation of means and variances. The results should be valuable for approximately optimal inventory control in distribution

The discounted continuous-review (R,Q) inventory model with continuous and stochastic demand is investigated.  New optimality conditions are derived, clarifying the difference to the average cost case, also graphically. Supported by depreciation theory, applied to the value of a setup, the results suggest an insightful and very precise approximation – The Shrewd Accountant’s Heuristic – based on a new average-cost model. It deepens and extends the work of Hadley (1964). Three examples are worked out in detail and the model is generalized to Poisson demand and to stochastic lead-times.

Customer waiting time is a critical variable in heuristic modelling of multi-echelon inventory systems. It is not sufficiently investigated in the literature, which is large.

The paper reviews and comments the literature and extends it by investigating customer waiting times in (nQ,R) inventory systems with continuous review and compound Poisson demand and with R+1<0 allowed. Two basic cases are considered: the waiting time for an average unit of demand assuming partial deliveries, and the waiting time for a customer demanding d units when only full deliveries are allowed.

The probability distributions are characterized and exact and approximate formulae are derived for means and variances. There are some computational results of various heuristics for the mean and variance of waiting time in each of the cases.

Assume that in periods with stochastic demand remain until the next replenishment arrives at a central warehouse. How should the available inventory be allocated among N retailers? This paper presents a new policy and a new lower bound for the expected cost of this problem. The lower bound becomes tight as N -> infinity. The infinite horizon problem then decomposes into N independent m-period problems with optimal retailer ship-up-to levels that decrease over the in periods, and the warehouse is optimally replenished by an order-up-to level that renders zero (local) warehouse safety stock at the end of each replenishment cycle. Based on the lower bound solution, we suggest a heuristic for finite N. In a numerical study it outperforms the heuristic by Jackson [Jackson, P. L. 1988. Stock allocation in a two-echelon distribution system or what to do until your ship comes in. Management Sci. 34(7) 880-895], and the new lower bound improves on Clark and Scarf's [Clark, A. J., H. Scarf. 1960. Optimal policies for a multi-echelon inventory problem. Management Sci. 6(4) 475-490] bound when N is not too small. Moreover, the warehouse zero-safety-stock heuristic is comparable to Clark and Scarf's warehouse policy for lead times that are not too long. The suggested approach is quite general and may be applied to other logistical problems. In the present application it retains some of the risk-pooling benefits of holding central warehouse stock.

The discounted continuous-review (R,Q) inventory model with continuous and stochastic demand is investigated, generalizing and refining Hadley's (1964) work. A new optimality condition is derived, clarifying the difference to the average cost case. Based on depreciation theory, applied to the value of a set-up,a very easy and extremly precise approximation is suggested, based on the average cost model- The Shrewd Accountant's Heuristic. A few examples are worked out in detail.

Formulas for (r,Q) inventory policies always rely on approxi-mations. We demonstrate that this classical problem in fact is pseudo-convex, so that it can be exactly optimized by a (new) square-root formula, and an equation for r. So, no approxima-tions are required. The result assumes a quasi-convex inventory cost function - typically assured for log-concave demand distributions. A new more precise theory on how to use (r,Q) policy parameters to approximate (s,S) policies is also presented.

A stationary, serial inventory system with fixed order quantities is considered. The detailed assumptions coincide with those of Chen (2000) except that holding costs are linear at each stage and that inability to satisfy customer demand at the final stage causes shortage costs of dimensions [$/unit/year], [$/unit] and [$/year]. It is shown that for the serial equivalents of four of the single-stage cost-rate models in Rosling (2002), it is optimal to control the serial system by (nQ,r) policies, provided the lead-time demand distribution functions are log-concave. For compound Poisson demand processes, log-concavity requires that the customer demand density be non-increasing. The work generalizes the results by Chen (2000) and Axsäter (2003).