s comma upper S right-parenthesis"/> form. After each transaction, the inventory position is compared with a control number, , the order point. If the inventory position is less than (or in some cases at or than ), a replenishment order is released. The replenishment order can be for a standard order quantity or, alternatively, enough may be ordered to raise the inventory position to the value , the order‐up‐to level, or OUT level. (This is also known as the ‘replenishment level’.) If all demand transactions are unit sized, the two systems are identical because the replenishment requisition will always be made when the inventory position is exactly at (so that ). If the demand sizes vary, then the replenishment quantity in the policy also varies. In Figure 2.3, we show graphically the operation of the policy and its equivalence to , assuming unit sized transactions. We further assume, for ease of presentation, that no stockouts occur (i.e. backordered demand does not need to be accounted for).
Figure 2.3 Continuous review
and
policies for unit sized transactions.
When the inventory position drops to (exactly) the order point (), then an order for a fixed quantity is placed or, equivalently, enough is ordered to raise the inventory position up to the OUT level . Note that, at the time of ordering, the inventory position always equals the stock on hand because there are no pending receipts from the supplier(s). Once the order is placed, the inventory position increases to (stock on hand) + (stock on order) (=) and decreases thereafter in exactly the same way as the stock on hand. The quantity ordered is received after the lead time (), at which point in time the inventory position equals again the stock on hand.
The problem is to find the optimal values of the control parameters and . (See Technical Note 2.5 for discussion of equivalence with optimisation.) Optimality here refers to a solution that has been explicitly developed based on minimising the total inventory cost (Ordering cost + Holding cost + Backorder cost). In theory, optimisation of the two control parameters, and , should occur in parallel recognising that cost interactions exist between them. Alternatively, the parallel optimisation may be for and (rather than and ) (see, for example, Wagner 1975). In practice, though, in our experience, a separate (independent) optimisation of these parameters is the norm because of its relative simplicity. Further arguments in support of this approach relate to its near‐optimal behaviour (Porteus 1985).
2.5.2 Periodic Review Systems
In periodic review systems, the inventory decision rules most usually take the form of a or policy, the former being the one to be discussed in more detail in this book. Under the regime of both policies, after every periods (constant inventory review interval) enough is ordered to raise the inventory position up to the ‘order‐up‐to level’, . The difference between the two policies is that the policy requires the inventory position to be less than or equal to (or in certain cases strictly less than ) before an order is placed. Therefore, the policy always results in higher ordering costs because even a unit‐sized transaction during the review interval will trigger a replenishment requisition, whilst the policy will place an order only if the cumulative demand, over the review interval, exceeds some minimum level. The can be viewed as the periodic implementation of the system because the reduces to for .
