HOW TO DETERMINE SERVICE LEVEL FOR INVENTORY ITEMS

INTRODUCTION
 
CHOOSING YOUR SERVICE LEVEL TO OPTIMIZE INVENTORY

Sales cast determines the optimal reorder point as a function of expected demand, lead time and service level. Frequently, new users are confronted for the first time with the notion of Service Level. The following article gives a short introduction to the topic and guidance on how to set appropriate service levels. 
Definition: Service Level expresses the probability of being able to service incoming orders (or demand) within a reference period without delay from stock on hand.
The implicit assumption within this statement: It is not economic to always be able to service an order from stock on hand. Deciding on the right service level for a certain product is essentially balancing inventory costs vs. the cost of a stock out. Service level is therefore an important variable for calculating the appropriate safety stock; the higher the desired service level, the more safety stock needs to be held.


Unfortunately, the cost functions describing the problem are extremely very business specific. While inventory costs can often be determined rather easily, the cost of stock outs are much more complicated to determine. A customer that does not find the product in store might choose an alternative that is in store, postpone the purchase to a later date or buy at the competition. In grocery retail for example, out-of-shelf situations of certain must have products are known to drive customers out of the store, taking their business to a competitor.

As this example illustrates, the associated cost functions are not only business, but product specific. When considering that most manufacturers and retailers are dealing with hundreds to hundreds of thousands of products, it becomes obvious that an overly scientific approach is neither advisable nor feasible.

The good news is that in practice it mostly proves fully sufficient to work with a simple framework that can be fine-tuned over time.

HOW TO GET STARTED

Service levels are considered by many retailers as part of their core IP, and tightly guarded. Nevertheless, some ballpark figures should provide a good starting point: A typical service level in retail is 95%, with very high priority items reaching 98% or even 99%. We have seen a number of customers successfully choosing a very pragmatic approach when setting service level at a uniform 95% starting point, to subsequently improving and adjusting these to their needs.

It is important to understand the relationship between service level and safety stock. Graph 1 illustrates the relationship. Dividing by 2 the distance to 100% multiplies the safety stock by 2. For example, if an increase in service level from 95% to 97.5% will double the necessary safety stock. Service Levels approaching 100% get extremely expensive very fast, and a service level of 100% is the mathematical equivalent to infinite safety stock.

Graph 1: Relationship safety stock vs. service level

Choosing categories

It is in our experience fully sufficient to differentiate between 3-5 service level categories that cover the product portfolio from must have items to the lowest priority items. As an example, we chose a three-value system:

• High: 98%
• Medium: 95%
• Low: 90%

Categorizing products

Product rankings allow a structured and sensible way to allocate products to the categories we defined previously. Rankings that are often used solely or in combination include turnover, profitability, number of orders, COGS.

Example product ranking by turnover

• Top 80% of turnover: High service level
• Next 15% of turnover: Medium service level
• Next 5% of turnover: Low service level

Example product ranking by gross margin contribution

• Top 80% of gross margin: High service level
• Next 15% of gross margin: Medium service level
• Next 5% of gross margin: Low service level

Once the categories have been defined and service levels have been assigned, Sales cast will determine the reorder point (including safety stock levels) as a function of these values. We often see that a lot of potential for inventory reduction is not only leveraged by the accuracy of our forecast, but also by the more sophisticated method and frequent update of the reorder point provided by Lokad.

Who still feels rather insecure regarding the correct service level should remember that it is not important, and also rather unrealistic, to have the perfectly fine-tuned service levels right out of the gates. What is important is that the new attention to this notion, in combination with Sales cast forecasts and reorder point analysis, will improve the status quo with a high certainty.

OPTIMAL SERVICE LEVEL FORMULA FOR INVENTORY OPTIMIZATION

Service level (inventory) represents the expected probability of not hitting a stock-out. This percentage is required to compute the safety stock. Intuitively, the service level represents a trade-off between the cost of inventory and the cost of stock-outs (which incur missed sales, lost opportunities and client frustration among others). In this article, we detail how to optimize the service level value. Then, the analysis is refined for the special case of perishable food.

Model and formula

The classical supply chain literature is somewhat fuzzy concerning the numerical values that should be adopted for service level. Below, we propose to compute an optimal service level by modeling the respective cost of inventory and stock-outs.

Let's introduce the following variables:

• p be the service level, i.e. the probably of not having a stock-out.
• H be the carrying cost per unit for the duration of the lead time (1).
• M be the marginal unit cost of stock-out (2).

(1) The time scope considered here is the lead-time. Hence, instead of considering the more usual annual carrying cost Hy, we are considering H=d365Hy assuming that d is the lead time expressed in days.

(2) The stock-out cost includes a minima the gross margin, i.e. instant profit that would have been generated if no stock-out had been encountered. However, the loss of gross margin is not the only cost: for example, customer frustration and loss of customer loyalty should also be taken into account. As a rule of thumb, we have observed that many food retailers consider M to be equal to 3 times the gross margin.

The optimal service level is given by (the reasoning is detailed below):

p=Φ⎛⎝⎜2ln(12π−−√MH)−−−−−−−−−−−−−⎷⎞⎠⎟

Where Φ is the cumulative distribution function associated to the normal distribution. This value can be computed easily in Excel, Φ is the NORMSDIST function. Also, for sake of numerical computation: 2π−−√≈2.50

Cost function

In order to model the cost function, let's introduce two more variables:

• Q the amortized inventory quantity (3), a function that depends on p.
• O the average overflowing demand when a stock-out it hit.

(3) We are adopting here an Amortized Analysis viewpoint. The inventory level is varying all the time, but our goal here, in order to make the analysis practical, is to obtain a service level value that is decoupled from the demand forecast itself. Hence, we will assume Q to be equal to the reorder point (check our guide about safety stock for the detail).

For a given service level, the total cost C(p) that combines both inventory holding cost and stock-out costs can be written:

C(p)=Q(p)H+(1−p)MO

Where Q(p)H is the inventory cost and MO the stock-out cost, only happening with a probability 1−p. Using the formula introduced in our safety stock tutorial, and since Q(q) is equal to the reorder point, we have Q(p)=Z+σΦ−1(p) where Z is the lead demand, σ expected forecast error and Φ(p)−1 the inverse of the cumulative distribution function associated to the standard normal distribution (zero mean and variance of one).

Analysis of average missed sales

The analysis of O, the average missed sales, is subtle. Considering that distribution of the demand is the normal distribution N and that q is the available inventory, O(q) is the conditional mean of the demand x when x>q (minus the available inventory q), that is:

O(q)=∫∞qxN(x)dx∫∞qN(x)dx−q=1q+o(1q)

This result can be interpreted as: if the demand is rigorously following the normal distribution, then the average quantity of missed sales in the (conditional) event of a stock-out quickly converges to zero as the inventory level grows.

Yet, in our experience, the forecast error does not have a convergence as good as the one the normal distribution would predict. Hence, below, we will assume that O=σ that is to say that the average amount of missed sales is equal to the average forecast error. Obviously, this is more a rule of thumb than an in-depth analysis; but we have found that in practice, this approximation gives sensible results.

MINIMIZATION OF THE COST FUNCTION

By applying the replacements defined here above to the expression of C(p), we obtain:

C(p)=(Z+σΦ−1(p))H+(1−p)Mσ

Then, the C(p) expression can be differentiated in p with:

∂C∂p=σH∂Φ−1(p)−σM

Since we are looking for the minimum value, we try to solve ∂C∂p=0 which gives:

σH∂Φ−1(p)−σM=0

Which can be simplified in σ, with:

H∂Φ−1(p)−M=0

And finally:

∂Φ−1(p)=MH

Then, we will use the relationship between ∂Φ−1 and Φ−1 :

∂Φ−1(p)=2π−−√e12Φ−1(p)2

For a visual proof of the equality you can compare (1) and (2).

This relationship let us obtain:

2π−−√e12Φ−1(p)2=MH

12Φ−1(p)2=ln(12π−−√MH)

We take the positive root solution of the equation (the negative root corresponds to a local maximum of the function) with:

Φ−1(p)=2ln(12π−−√MH)−−−−−−−−−−−−−⎷

And finally by applying Φ to both sides, we obtain the optimal service level with:

p=Φ⎛⎝⎜2ln(12π−−√MH)−−−−−−−−−−−−−⎷⎞⎠⎟

Discussion of the formula

The first interesting aspect of the formula is that the optimal service level only depends on H (inventory cost) and M (stock-out cost). However, there is an implicit dependency on the lead-time as H has been defined as the carrying cost for the duration of the lead time.

Second, a greater inventory cost lowers the optimal service level; and similarly a greater stock-out cost increases the service level. This behavior is rather intuitive because the service level is a trade-off between more inventory and more stock-outs.

Then, the formula is not valid for all values of M and H. We need 2π−−√MH>1 or the logarithm will produce a negative value which is not tractable considering the outer square root. This gives M>2π−−√H, which can be approximated as M>2.5H. If this condition is not verified, then it means that the initial cost function C(p) has not minimum, or rather that the minimum is −∞ for p=0. From a practical angle, M<2.5H could be interpreted as a pathological situation where the most profitable stock level is zero stock (i.e. 100% stock-outs).

Practical example

Let's consider a 1L milk pack at 1.50shs selling price sold with 10% margin (i.e. 0.15shs of gross margin). Let's assume that the lead time is 4 days. The annual carrying cost is 1.50shs (the value is high because milk is a highly perishable product). We assume that the stock-out cost is 3 time the gross margin, that is to say 0.45shs. This gives M=0.45 and H=43651.5≈0.0055.

Based on those values and on the formula for optimal service level obtained here above, we obtain p≈98.5% which is a typical value for must-have fresh products stored in warehouses feeding grocery store networks.

CONCLUSION:

Several definitions of service levels are used in the literature as well as in practice. These may differ not only with respect to their scope and to the number of considered products but also with respect to the time interval they are related to. These performance measures are the Key Performance Indicators (KPI) of an inventory node which must be regularly monitored. If the controlling of the performance of an inventory node is neglected, the decision maker will not be able to optimize the processes within a supply chain.

REFERENCE:

Relationships between Service Level Measures for Inventory Systems John E. Boylan and F. R. Johnston The Journal of the Operational Research Society
Vol. 45, No. 7 (Jul., 1994), pp. 838-844 Published by: Palgrave Macmillan Journals