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Inventory Models

Developing inventory models provides an analyst with a better understanding of how an inventory system operates based on the underlying structure and parameterization of the model. In its simplest form, a model is a representation of system inputs and outputs, as well as descriptions of the internal operations of the process. Inventory modeling offers several advantages such as creating an ability to compress the time scale of the system’s operational performance and conduct what-if analyses offline or separately from the system being modeled. But there are also some disadvantages. Models are not 100% accurate, and complex models are time- consuming to create and interpret. For this reason, the modeling approach should be standardized. Modeling also requires setting parameters and decision rules that mimic how the process works. Parameters are quantitatively linked to a system’s model in a probabilistic sense, in that random inputs are transformed, based on their underlying probability distribution at an operational level, into an output or event. In addition, a model’s initial starting and ending states as well as simulation duration must be set. Statistical tests are used to verify the accuracy of the model.

The perpetual inventory model (PIM) is used to monitor inventory status day to day in a perpetual manner. A common PIM model is shown in Figure 13.7. At the beginning of an order cycle T, an economic order quantity Qj is ordered from a supplier. This quantity Q, is expected to be linearly depleted during the order cycle T. During the order cycle, the depletion rate may be higher or less than the original forecast for the item. If demand is higher than expected, the reorder point quantity will be reached earlier in the order cycle, causing the PIM system to release


Perpetual inventory model (PIM).

an order earlier than planned in the original forecast. The reorder point quantity is calculated by multiplying the reorder point lead time by the average expected daily demand d and comparing the calculated quantity against the current on-hand inventory quantity with the safety-stock quantity netted out of the calculation. If the current inventory is equal to or less than a reorder point quantity, then another economic order quantity Q, is placed by the MRPII system. Using this model, it can be seen that an optimum inventory quantity may vary during an order cycle, but its expected value is Vi Qj + the safety-stock quantity, assuming a linear depletion rate proportional to forecasted demand. A qualitative depiction of a safety-stock quantity is shown in Figure 13.7, and an algorithm is shown in Figure 13.8 with supporting definitions.

Figure 13.9 is a generalized depiction of how an optimum inventory quantity for an item is impacted by its lead time and demand variation. When building the PIM inventory model, inventory quantities, their associated standard costs, and other information such as lead time, demand variation, location, supplier, and product type can be aggregated across an entire inventory population. Once aggregated, sensitivity analyses can be conducted using the PIM inventory model to identify where projects should be deployed to reduce lead time and demand variation and therefore inventory investment. The model also shows that, for any lead time and demand variation combination, there is an optimum inventory quantity for an item. Using this model, the actual on-hand inventory quantity of an item can be compared to its calculated optimum quantity to determine whether there is an excess or a shortage of inventory relative to the unit service-level target.

Figure 13.8 shows the PIM inventory model’s algorithm. This algorithm will show where additions and reductions in inventory investment are possible. Steps 1 and 2 calculate an optimum inventory quantity. Additional modeling assumptions, such as safety-stock calculations, are also shown in Figure 13.8. As an example, it shows how the total standard deviation (ct,) is calculated using lead time and demand. The formula shows that safety stock will always be required unless the variation of lead time and demand are both zero. Figure 13.8 also defines the terms used in the PIM model. In step 3, excess inventory is calculated by subtracting an optimum from the average on-hand inventory quantity. If the resultant number is positive, there is too much inventory for an item and its location, and its quantity should be reduced from its current level while still meeting the service target. However, there may be several complicating factors that


Perpetual inventory model (PIM) algorithm.

prevent an organization from achieving a calculated optimum inventory quantity. These may include large lot sizes representing multiples of lead time, various operational issues, and obsolete inventory that prevents the immediate reduction of an item’s inventory quantity. The purpose of this analysis is to identify items with too much or too little inventory. A particularly useful attribute of this analysis is that optimum inventory investment can be aggregated upward, item by item, through the population to identify significant amounts of excess or obsolete inventory. This provides


Graphical view of the perpetual inventory model (PIM).

a financial justification to deploy improvement projects to reduce the causes of high inventory investment.

Development of a useful analytical model requires that an organization build into its assumptions relevant characteristics of a system. As an example, every item in an inventory population must be described, at a minimum, by criteria such as t lead time, demand variation, and relevant demographics such as product family, facility, customer, supplier, and similar descriptive factors that are important to an efficient aggregation of business benefits.

Inventory is impacted by product demand patterns. Irregular demand patterns will require more inventory than predictable patterns that can be forecasted more accurately. Because the unit standard deviation of demand of a product (or item) is an important input in a safety-stock calculation, it is important to ensure it is estimated correctly. There are several ways to filter out the impact of irregular demand components. These range from statistically identifying outliers to truncating a time series using only observations from the most recent portion of its historical demand pattern. This assumes there is less variation in latter portions of a time series.

Lead time is another important input in an inventory model. Process issues will increase lead time. Examples include late deliveries, poor quality,

FIGURE 13.10

Impact of lead time and lot size on the model. Top: The model uses lead time to calculate inventory investment. The lead time must reflect actual on-time delivery performance to ensure adequate safety stock. Bottom: Minimum buys and large lot sizes force inventory levels higher.

and large lot sizes. Figure 13.10 shows the system lead time is 10 days versus an actual lead time of 15 days. It is important that a correct lead-time estimate is used to calculate an item’s optimum inventory quantity. Minimum lot sizes also have a significant impact on inventory investment. As an example, if the actual system lead time is 10 days but the minimum lot size is 20 days, then the average inventory in this situation will be Vi x 20 days versus Vi x 10 days, or 10 days versus 5 days. The reorder quantity for the item will be calculated using the actual lead time of 10 days. It is evident that a larger lot size requires an adjustment to properly calculate an optimum inventory quantity. Other adjustments may also be necessary to make an inventory model useful for analysis and improvement purposes.

A PIM inventory model can be used in its current form, with minor adjustments, when modeling finished goods inventories. It is may also be relatively easy to apply to WIP inventories if material flows are not complicated. Regardless of the model, lead times must be calculated based on a systems critical path and its bottleneck. In assemble-to-order and make- to-order (sometimes called engineer-to-order) production systems, the process network should be mapped and its lead times carefully calculated to estimate the impact of a system’s bottleneck on the order-to-cash lead time of the process. If a bottleneck is not managed well, WIP inventory will build up within a process or one or more operations within process may be starved for WIP inventory.

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