BUS606: Operations and Supply Chain Management

Inventory control models in the research literature and in textbooks typically assume that the demand distribution and all its parameters are known. However, in practice such information is not available, and future demands have to be forecasted based on historical observations. No forecasting procedure produces perfect forecasts. For unbiased forecasting procedures, not only methodical forecasting but also continuous forecast errors calculation of demand is necessary.

There are many forecast method found in the literature. Surprisingly, some inventory textbooks do not mention forecasting at all and also they do consider to discuss the use of forecasts and errors consideration in the decision models. Some inventory textbooks, such as Silver, Pyke, and Peterson and Axsäter, do indicate that parameters have to be estimated in practical applications, and that the resulting forecast error should be taken into account. Some books considers per-period demand forecast errors (e.g. Estimated via Mean Squared Error, MSE, or Mean Absolute Deviation, MAD).

The lead time forecast errors can correlated and has been pointed out by some researchers. Silver et al. noted that the relationship between the forecast error during the lead time and that during the forecast interval. Beutel and Minner discuss a framework of least squares demand forecasting and inventory decision making.

A common approach to inventory decision making under un-known demand parameters is the Bayesian approach, which was initiated by Dvoretzky, Kiefer, and Wolfowitz, Scarf, Karlin, and Iglehart. Applications of Bayesian estimation in dynamic programming efforts are given by e.g. Azoury and Lariviere and Porteus. It has been found by them that specific, optimal order quantities are numerically tough to derive. This led to the development of heuristics, that essentially reduce multi-period problems to single-period problems, and typically assume a negligible lead time. Numerical Bayesian studies mainly involve single-parameter demand distributions, or, if the distribution has more than one parameter, impose that one or more parameters are known. Besides, Rajashree Kamath and Pakkala use a lognormal distribution with a known variance. This restriction is theoretically convenient, because it ensures that the posterior demand distribution is again (log) normal, but it is practically incorrect, as the variance has to be estimated as well.

A different approach is tried by Hayes, who introduces the method of minimizing the Expected Total Operating Cost, which follows from the sampling distribution of the parameter estimators. Janssen, Strijbosch, and Brekelmans seek the optimal 'upward biases' in the traditional stock level calculation via simulation. Similarly, Strijbosch and Moors consider a batch ordering (r, Q) policy under normally distributed demand with unknown parameters and zero lead time.

Lippman and McCardle examined the relationship between equilibrium inventory levels and the allocation rule. Along with them Netessine and Rudi established the uniqueness of the equilibrium for the n-product case, and extended the work of Parlar . Avsar and Gursoy treated the same condition on an infinite horizon. Ahn and Olsen worked on multiple periods with consideration of a subscription sale. Nettesine, Rudi, and Wang presented four scenarios of customers' backlogging and transfer behaviour in a dynamic environment.

Khmelnitsky and Gerchak analysed a continuous review policy by using a deterministic inventory model in which demand is a function of the instantaneous inventory level, where shortages are possible and production capacity is limited. Baker and Urban investigated the continuous, deterministic case of an inventory system in which the demand rate for an item is a function of that inventory level. Kelle and Silver identified a reorder point for purchasing new containers in which products are sold. Yuan and Cheung developed for this model a continuous review (s, S) policy with returns based on the sum of the on-hand stock and the number of items in the field.

Decroix and Neuts proposed an optimal order-up-to policy with regular and emergency replenishments for the case in which the regular order lead time is one period and the emergency lead time is zero. Whittemore and Saunders derived the optimal policy for the case in which the regular and replenishment lead times are multiples of the periodic review period. Rosenshine and Obee investigated a standing order inventory systemin which a constant quantity is received every period, and a fixed-size emergency order can be placed when the inventory falls below the reordering point.

Blumenfeld, Hall, and Jordan analysed the trade-off between expediting cost and safety stock cost. Emergency orders are considered to be sufficiently large to avoid stock outs and to be received with a zero lead time. Chiang and Gutierrez analysed inventory models in which the lead time is shorter than the periodic review period with a periodic review system. Teunter and Vlachos allowed more than one emergency order per review period. Alfredsson and Verrijdt showed the effectiveness of emergency flexibility, including transshipments from other retailers. Chartniyom et al. proposed a solution approach to avoiding stock outs that incorporates two options, lateral transshipment and emergency order.

Weiss proposed an optimal reorder point and order-up-to level with continuous review and a zero lead time. Kalpakam and Arivarignan investigated a model in which there is an exponential lifetime and a zero reorder point. Nahmias, Perry, and Stadje considered the condition in which the demand rate is not affected by the productlifetime.Avinadav and Arponen assumed that the demand rate increases in proportion to lifetime. Avinadav, Herbon, and Spiegel modelled demand that decreases in proportion to price and polynomially with lifetime.

There are other streams available in literatures. But in terms of safety stock, most of them considered it as inclusive effort while working with inventory as a whole. Moreover, an optimal policy has been proposed for considering the production system, the order sizes, the planning horizon etc. Considering all these, the authors some time focus on safety stock more or less. But still no one skips the specification of safety stock one way or another. However, they have common effort, making the models complex and realistic.