LOW-TURN SPARE PARTS INVENTORY MANAGEMENT

As pointed out in another article in this same magazine (August/2002 edition), spare parts inventory management constitutes a separate chapter of inventory management. This is because spare parts have substantial differences in terms of acquisition costs (generally higher), response times (generally longer), inventory turnover (generally lower) and demand distribution ( definitely not adhering to the normal distribution) when compared, for example, to non-durable consumer goods and their raw materials.

Specifically, the impossibility of assuming the distribution of demand adhering to the normal curve, makes the answer to the following question quite complex: what should be the reorder point or the safety stock of a given spare part so that the probability of a shortage is as small as you want? Several books on statistics, material management and logistics present, in their appendices, tables with the probabilities of the accumulated normal distribution, which would make the answer to this question relatively “simpler”. The problem is that it is not known, in advance, the magnitude of the error in planning inventories resulting from the assumption of normal distribution when demand definitely does not have this profile.

A way normally used to deal with such a situation is to consider that the demand profile adheres to the Poisson distribution. The properties of this distribution make it particularly interesting for understanding how different levels of safety stocks would affect the probability of product shortages, especially in low-turnover environments, that is, annual consumption between 1 and 300 units per year. For example:

• the Poisson distribution is discrete, that is, it is possible to calculate the probability of occurrence of a certain level of consumption based on its historical average. In other words, it would be possible to answer questions like: “given that the historical consumption of a certain spare part is 50 units per year, what is the probability that consumption will be exactly 4 pieces in the next month?”
• the Poisson distribution assumes independence between events, that is, the level of consumption in a month is not affected by consumption in the previous month and will not affect consumption in the following months either.
• in the Poisson distribution the variance is equal to the average consumption in a given period.

This article aims to present the Poisson distribution, illustrating not only how it can be used in the management of low-turnover spare parts inventories, but also how to implement it in the Excel Spreadsheet and analyze the results obtained in order to expand the elements for decision making.

EXAMPLES OF DECISION-MAKING SUPPORT

The following formula shows how to calculate the probability (px(t)), for a given period of time (t), of the consumption of spare parts being equal to x units, given that the historical average consumption, for the same time horizon , it's from ? units.

Where:x = consumption of spare parts per time interval; t = considered time interval; l = historical rate of consumption of spare parts per unit of time; px(t) = probability of having x requests for spare parts during the time interval t.

Table 1 presents an example considering a spare part with historical consumption of 2 units per year. For each level of consumption likely to occur in the next twelve months, individual probabilities and cumulative probabilities are calculated.

Table 1. Evaluation of spare parts consumption for the next 12 months, considering an average historical consumption of 2 units per year and adherence to the Poisson distribution

This table allows reaching the following conclusions for the case where the historical average consumption is 2 units per year:

• The probability of not having spare parts requests (x = ZERO) in the next 12 months is 13,50%. Consequently, there is an 86,50% probability of having at least one request for spare parts in the next 12 months. On the other hand, the probability of having nine or more requests for spare parts in the next twelve months is zero.
• Keeping four spare parts in stock guarantees a 94,70% probability of not running out of stock, while 5 spare parts guarantees a 98,30% probability.
• In a way, the accumulated probability allows evaluating the level of service, in terms of the probability of not having a shortage of parts in stock, for a given quantity of items in stock.
• The sum of the individual probabilities equals 100%. Furthermore, the probability of having at least one request is equal to the difference between 100% and the probability of having zero requests per time interval.

The following set of graphs shows the calculated probabilities for different levels of consumption based on different values ​​for historical average consumption (? – lambda). It is interesting to note that, as historical average consumption increases, the demand profile becomes more symmetrical in relation to the average. The opposite reasoning is also valid: lower values ​​of historical average consumption would lead to a distribution with a more asymmetrical profile. This result is interesting for representing situations where a “peak” in consumption would cause more inconvenience than a “trough”, as in the case of some low-turnover spare parts.

Graphics. Graphic representation of the probabilities of occurrence for different levels of historical consumption

The Microsoft Excel spreadsheet allows you to easily implement calculations relating to individual probabilities and cumulative probabilities. The POISSON function immediately calculates, for a given spare parts consumption (x) and a given historical consumption rate (?), these two probabilities. For this purpose, the following formulas must be used.

= POISSON (x, ? , false) – individual probability for x
= POISSON (x, ? , true) – cumulative probability up to and including x

For these formulas to be used correctly, it must be observed that both requests for spare parts (x) and the historical consumption rate (?) are referenced to the same time interval.

CONSIDERING RESPONSE TIME

As in the management of inventories of consumer goods and their raw materials, the stock level of spare parts that guarantees a certain probability of not having shortages must, in reality, be determined for the time interval that the company is most vulnerable, that is, the resupply cycle. The resupply cycle normally comprises the time interval that goes from placing the order with the supplier until it is received. It is during this period that unexpected increases in the consumption of spare parts can lead to stock-outs.

A basic step would be to estimate the expected average consumption in the resupply cycle and its variability, that is, the order point. among professionals and academics in the area:

Where:D = demand per unit of time; TR = resupply cycle response time, in time units k = factor of safety sD = standard deviation of demand per unit of time

What few articles and books comment on is that this is a result applicable to any distribution of demand or consumption (Normal, Poisson, etc.). In this way, the reorder point can be “rewritten” in terms of the historical average consumption of spare parts per unit of time (D = l) and the variance of this consumption, which in the case of the Poisson distribution is equal to the historical average consumption (s_D = Olive oil).

As previously mentioned, the reorder point (PP) must be determined based on the desired probability of not having a stockout during the response time. The Excel spreadsheet POISSON function can be extremely helpful in determining the reorder point. Through an iterative procedure, it can be evaluated whether the cumulative probability of consumption of spare parts in the response time (?*TR) is smaller or greater than the probability of not having a lack of product in stock determined by the point of order.

For example, let's assume that for a given spare part the desired probability of not having a stock shortage is 90%, the response time is 2 months (2/12th of the year) and the historical average consumption of 4 parts per year. Testing the values ​​1, 2 and 3 as possible ordering points in Excel's POISSON function, we have:

POISSON(PP,?*TR,true)
POISSON(1,4*2/12,true)=85,6% - Reject as it is less than 90% probability
POISSON(2,4*2/12,true)=97,0%- Accept as it is greater than 90% probability
POISSON(3,4*2/12,true)=99,5% - Accept as it is also greater than 90%

In this case, 2 is chosen as the ordering point, as it was the first value to ensure a 90% probability of not having a product shortage in the response time of the resupply cycle. The order point equal to 2 is equivalent to a factor of safety (k) equal to 1,625. This result is obtained from the last equation. It should be noted that if this safety factor were used in a table with values ​​for the cumulative Normal distribution, the probability obtained would be 94,8%. This illustrates that considering the demand adhering to the Normal distribution in these circumstances could bring wrong information about the level of service that is actually being provided.

CONCLUSION

This article complements the discussion started in the August/2002 issue on spare parts inventory management. The first issue identified a procedure for dealing with very low turnover parts (consumption of less than one unit per year). This number presents the aspects that must be considered in low-turnover parts (consumption between one and 300 units per year), mainly in relation to the assumption about consumption distribution (Normal vs. Poisson). Depending on the magnitude of consumption and the level of service you want to provide, considering the Normal distribution can lead to wrong decisions regarding order points and stock levels. In this sense, analyzing the issue from the perspective of the Poisson distribution opens the way for inventory reductions, which can be considerable depending on the company and the sector of the economy.

BIBLIOGRAPHY

(1) More information about the Poisson distribution and other distributions can be obtained from educational and training websites. For example, some interesting websites are:

http://www.math.csusb.edu/faculty/stanton/probstat/poisson.html

http://info.bio.cmu.edu/Courses/03438/PBC97Poisson/PoissonPage.html

http://www.itl.nist.gov/div898/handbook/eda/section3/eda366i.htm

http://hyperphysics.phy-astr.gsu.edu/hbase/math/poifcn.html#c2

http://engineering.uow.edu.au/Courses/Stats/File41.html

(2) For those interested in discussing the impact of different probability distributions on inventory management, there are several articles in this regard. In essence, all these articles seek to answer the following question: “what is the probability of not having a product shortage in stock considering a given distribution of demand in the response time?” In some cases it is possible to determine an equation that provides an analytical solution to such a question, in others, more complex procedures are required.

For an example with Poisson distribution: VINCENT, P, 1983, “Practical Methods for Accurate Fill Rates”, INFOR, Vol.21, No.2, May, pp.109-120.

For an example with Bernoulli distribution: JANSEN, F, HEUTS, R., KOK, T., 1998, “On the (R,s,Q) Inventory Model when Demand is Modeled as a Compound Bernoulli Process”, European Journal of Operational Research, 104, pp. 423-436.

For examples with Gamma distribution: DAS, C., 1976, “Approximate Solution to the (Q,r) Inventory Model for Gamma Lead Time Demand”, Management Science, Vol.22, Issue 9, pp. 1043-1047 and NAMIT, K., CHEN, J., 1999, ”Solutions to the (Q,r) Inventory Model for Gamma Lead Time Demand”, International Journal of Physical Distribution & Logistics Management, Vol.29, No.2 , pp. 138-151.

For examples where the impact of considering the Normal distribution is evaluated in situations where demand is not adherent to this distribution: MENTZER, JT, KRISHNAN, R., 1985, “The Effect of the Assumption of Normality on Inventory Control/Customer Service” , Journal of Business Logistics, Vol. 6, No.1, pp.101-120. and LAU, H., 1989, “Toward an Inventory Control System Under Non-Normal Demand and Lead-Time Uncertainty”, Journal of Business Logistics, Vol.10, No.1, pp. 88-103.