Roger M. Hill

The single-vendor single-buyer integrated production-inventory model with a generalised policy

Abstract This note considers a more general type of policy for the single-vendor, single-buyer integrated production-inventory model than has hitherto been presented, based on successive shipments to the buyer, within a single production batch, increasing by a fixed factor.

Another look at the single-vendor single-buyer integrated production-inventory problem

This paper considers a supply chain in which a ‘vendor’ supplies a product to a ‘buyer’. The vendor manufactures the product at a finite rate and periodically ships the output to the buyer. The buyer then consumes the product at a fixed rate. Costs are attached to manufacturing batch set up, the delivery of a shipment and stockholding at the vendor and buyer. The objective is to determine the production and shipment policy which minimizes long-run total average cost—assuming the vendor and buyer collaborate and find a way of sharing the consequent benefits. Most previous work has been based on the assumption that unit stockholding costs increase as stock moves down the supply chain, but recent research has suggested that the opposite may sometimes hold. We show how the optimal batch production and shipment policy may be derived when unit stockholding costs increase as stock moves down the chain and shipments are not necessarily equal in size.

Applying Bayesian methodology with a uniform prior to the single period inventory model

Abstract We consider the application of the Bayesian approach to parameter estimation to the single period inventory model. We assume complete prior ignorance of the values that the (single) unknown parameter of the demand distribution might take and express this by using a uniform prior over the permitted range of parameter values. Direct analytical and numerical comparisons are made for three distributions and the results show that over a wide range of parameter values, including most of those which are likely to be of practical interest, the application of Bayesian methodology produces better decisions (resulting in lower expected total cost) than the approach of using a point estimate for the parameter, with no increase in computation or complexity. This suggests that this methodology could usefully be applied to this and other decision models and also provides a strong justification for the use of the full Bayesian approach when a meaningful prior is available.

Optimal and near-optimal policies for lost sales inventory models with at most one replenishment order outstanding

In this paper we use policy-iteration to explore the behaviour of optimal control policies for lost sales inventory models with the constraint that not more than one replenishment order may be outstanding at any time. Continuous and periodic review, fixed and variable lead times, replenishment order sizes which are constrained to be an integral multiple of some fixed unit of transfer and service level constraint models are all considered. Demand is discrete and, for continuous review, assumed to derive from a compound Poisson process. It is demonstrated that, in general, neither the best (s, S) nor the best (r, Q) policy is optimal but that the best policy from within those classes will have a cost which is generally close to that of the optimal policy obtained by policy iteration. Finally, near-optimal computationally-efficient control procedures for finding (s, S) and (r, Q) policies are proposed and their performance illustrated.

The (r,Q) control of a periodic-review inventory system with continuous demand and lost sales

Abstract In this paper we consider a periodic review inventory model with lost sales during a stockout and with the constraint that at most one replenishment order may be outstanding at any time. Demands in successive review periods are independent, identically distributed variables from a continuous distribution. The fixed lead time is an integral number of review periods. We explore control policies of the (r,Q) type – that is a replenishment order of size Q is placed when the inventory position (stock in hand plus stock on order) falls to or below the re-order level r. We use asymptotic renewal theory results to estimate the `undershoot’ of the re-order level r and also to estimate the cycle stockholding cost (which turns out to take a relatively simple form). Based on these approximations we set out a policy improvement solution methodology and illustrate this with some numerical examples for which demand is normally distributed. These numerical examples suggest that a relatively simple approach, based on the economic order quantity, can provide results which are very close to optimal.

On the suboptimality of (S-1, S) lost sales inventory policies

Abstract This paper considers replenishment policies for the continuous review inventory model with Poisson demand, a fixed lead time on replenishment and lost sales during a stockout. There is a time-dependent stockholding cost and a cost per unit of sale lost but there is no (or negligible) cost associated with placing a replenishment order. The standard policy is to specify a maximum stock level, S, and place a replenishment order whenever a demand occurs and is met. This is commonly referred to as an (S−1, S) policy. It is shown that there are other replenishment policies which give lower-cost solutions. In fact it is demonstrated that an (S−1, S) policy can never be optimal if S⩾2.

Continuous-review, lost-sales inventory models with Poisson demand, a fixed lead time and no fixed order cost

Abstract This paper considers continuous-review lost-sales inventory models with no fixed order cost and a Poisson demand process. There is a holding cost per unit per unit time and a lost sales cost per unit. The objective is to minimise the long run total cost. Base stock policies are, in general, sub-optimal under lost sales. The optimal policy would have to take full account of the remaining lead times on all the orders currently outstanding and such a policy would be too complex to analyse, let alone implement. This paper considers policies which make use of the observation that, for lost sales models, base stock policies can be improved by imposing a delay between the placement of successive orders. The performance of these policies is compared with that of the corresponding base stock policy and also with the policy of ordering at fixed and regular intervals of time.

Order splitting in continuous review (Q, r) inventory models

A number of recent articles in the literature have argued the case, when lead time is variable, for splitting a replenishment order for Q between n suppliers by comparing this with the alternative of placing a single order for Q on one supplier. The split order compares favourably on the grounds that the arrival of the first component of a split order cannot be later than the arrival of an order from any one specified supplier. This note argues that an alternative comparison could be made with a policy of ordering Q/n from a single supplier (n times as often). It makes this comparison in the context of a continuous review (Q, r) inventory model but does so not by comparing aggregate costs but by fixing Q and the customer stock service level and comparing the average stock — an approach which is more appropriate to how many companies manage inventory in practice. We consider Poisson and deterministic demand processes, a general lead time distribution and both lost sales and backorder models.

Numerical analysis of a continuous-review lost-sales inventory model where two orders may be outstanding

Abstract The steady-state solutions of continuous-review lost-sales inventory models where more than one order may be outstanding do not, in general, have a simple analytic form. A numerical procedure is described for computing the steady state for the case of poisson demand and constant lead time where two orders may be outstanding. Some results are tabulated and these are compared with three alternative approximate models.

Stock replenishment policies for a stochastic exponentially-declining demand process

This paper addresses the problem of determining stock replenishment policies to meet the demand for spare parts for items of equipment which are no longer manufactured. The assumptions that the number of items still in use is decreasing and that parts fail randomly lend credence to a Poisson demand process with an underlying mean which is decreasing exponentially. We use a dynamic programming formulation in continuous time to determine that replenishment policy which minimises the mean total discounted cost of set-up/order, unit production/purchase, unsatisfied demand and stock left over at the end of the time horizon.