A comparison between the order and the volume fill rates for a base-stock inventory control system under a compound renewal demand process

The order fill rate is less commonly used than the volume fill rate (most often just denoted fill rate) as a performance measure for inventory control systems. However, in settings where the focus is on filling customer orders rather than total quantities, the order fill rate should be the preferred measure. In this paper we consider a continuous review, base-stock policy, where all replenishment orders have the same constant lead time and all unfilled demands are backordered. We develop exact mathematical expressions for the two fill-rate measures when demand follows a compound renewal process. We also elaborate on when the order fill rate can be interpreted as the (extended) ready rate. Furthermore, for the case when customer orders are generated by a negative binomial distribution, we show that it is the size of the shape parameter of this distribution that determines the relative magnitude of the two fill rates. In particular, we show that when customer orders are generated by a geometric distribution, the order fill rate and the volume fill rate are equal (though not equivalent when considering sample paths). For the case when customer inter-arrival times follow an Erlang distribution, we show how to compute the two fill rates.Backordering; continuous review; compound renewal process; inventory control; negative binomial distribution; service levels

The (r,q) policy for the lost-sales inventory system when more than one order may be outstanding

We study the continuous-review (r; q) system in which un_lled demands are treated as lost sales. The reorder point r is allowed to be equal to or larger than the order quantity q. Hence, we do not restrict our attention to the well-known case with at most one replenishment order outstanding, but our modeling streamlines exact analysis of that case. The cost structure is standard. We assume that demand is Poisson, that lead times are Erlangian and that orders do not cross in time (lead times are sequential). We determine the equilibrium distribution of the inventory on hand at the delivery instants from the solution (obtained by the Gauss-Seidel method) of the equilibrium equations of a Markov chain. To optimize r and q we develop an adapted version of the algorithm suggested by Federgruen and Zheng for the backorders model (BO). The results obtained in our numerical study show that the suggested procedure dominates standard textbook approximations. In particular, the reductions in the average cost of a simple Economic Order Quantity policy are in the range of 3-14%. Except when lead times are long and variable or when the unit cost of shortage is low, the optimal BO policy provides a fair approximation to the average cost of the best policy.inventory/production; operating characteristics; policies; probability; Markov processes; Area of review; Manufacturing; Service; Supply Chain Operations

An inventory control project in a major Danish company using compound renewal demand models

We describe the development of a framework to compute the optimal inventory policy for a large spare-parts’ distribution centre operation in the RA division of the Danfoss Group in Denmark. The RA division distributes spare parts worldwide for cooling and A/C systems. The warehouse logistics operation is highly automated. However, the procedures for estimating demands and the policies for the inventory control system that were in use at the beginning of the project did not fully match the sophisticated technological standard of the physical system. During the initial phase of the project development we focused on the fitting of suitable demand distributions for spare parts and on the estimation of demand parameters. Demand distributions were chosen from a class of compound renewal distributions. In the next phase, we designed models and algorithmic procedures for determining suitable inventory control variables based on the fitted demand distributions and a service level requirement stated in terms of an order fill rate. Finally, we validated the results of our models against the procedures that had been in use in the company. It was concluded that the new procedures were considerably more consistent with the actual demand processes and with the stated objectives for the distribution centre. We also initiated the implementation and integration of the new procedures into the company’s inventory management systemBase-stock policy; compound distribution; fill rate; inventory control; logistics; stochastic processes