# Statistical Analysis for Inventory Control

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A business must keep track of its inventory. The difficulties inherent in doing so depend on the nature of the inventory and the business: for example, some businesses store products that are relatively easy for untrustworthy employees, or even facility visitors, to steal. One inventory control problem, then, is determining whether a business is losing stock to theft, and the size of such a problem if it does exist.

## A Kalman Filter

The "Kalman filter," named after mathematical system theorist Rudolf Emil Kalman, is an important statistical technique for filtering out the "noise" in a system to fine-tune statements of the system's current status and predicted course. Kalman first developed the filter in the context of aeronautical problems—he expounded on it in a 1960 paper on research "supported in part by the U.S. Air Force Office of Scientific Research."

## Application of the Filter

But by 1984, statisticians had discovered the utility of the Kalman filter in inventory and quality-control issues. R.E. Barlow and T.Z. Irony, in a paper published in a collection of works on statistical inference edited by Malay Ghosh and Pramod Pathak (1992) note that a Kalman filter model can allow "a decision maker to decide at each time period whether the data indicate a diversion"; that is, whether inventory is being diverted from its intended use. Furthermore, if there is diversion, the Kalman filter can distinguish a "block loss" from a "trickle loss."

## Distribution Parameters

A key to using a Kalman filter for this purpose is establishing appropriate distribution parameters. In other words, there has to be an "expected holdup" (the expectation that inventory counts will almost never come out perfectly in a large warehouse or within a complicated system), and whatever leakage is to be detected has to be that in excess of the expected holdup.

The "distribution parameter" is a number indicating how much imperfection is to be allowed for. Barlow and Irony write that the parameters "should not be set arbitrarily or casually, but only after a careful assessment of process and loss uncertainties which takes into account the effect of the parameters on the resulting decision procedure."

## Future Demand Projection

Beyond the question of "diversion" of inventory, there is the planning issue of how much inventory will be necessary to meet future demand. Of necessity, statistical methods use the past to predict the future. This is itself both unreliable and unavoidable. David Frederick Ross, in "Distribution Planning and Control" (2004) lists seven errors that arise in the "data collection process." One is the misalignment of time periods and data. For example, if a bicycle retailer tried to analyze past demand for children's bikes on the basis of data aggregated by financial quarter, the whole period from January to March would be treated as a single unit. In fact, parents in much of the northern hemisphere buy a lot of bikes for their children in March, as snow melts, and relatively few in January, so that lumping would prove unhelpful.

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