In this series of blog posts, we have been talking about Jane, who is in the role of inventory planner at her company, and Kate, a consultant, who has been helping Jane with the concepts.

*Catch up on the first 7 parts: part 1, part 2, part 3, part 4, part 5, part 6 and part 7.*

Jane asked, “Shouldn’t safety stock have the same unit of measure as the input demand? Here it seems the units are inconsistent as we are multiplying the standard deviation by √L.”

Kate replied, “Good question. So, you remember that standard deviation (the σ) has the same unit of measure (UOM) as the input demand and the unit of measure of the variance is UOM^{2}. I see the confusion coming from √L. It so happens that the multiplying term for the lead time is √(L day/1 day) and not √L. Now √(L day/1 day) is a unit less quantity. This comes about when we convert the variance from a daily number to a number over L number of days.”

“How is that?” Jane asked.

“Remember when I told you that variances of independent quantities are additive?” Kate replied. “Now think like this. You have demand on L number of days. On each of those L days, the variance would be the same, namely, σ_{DayDemand}^{2}. One assumes that the demands on the individual days are independent, and therefore the variances can be added up. Thus, over L days, the variance is σ_{DayDemand}^{2}*(L/1). Please note we divide here by 1 as the original variance was calculated over one day. If the original variance was over n days, we would divide by n. The (L/1) is therefore a unitless number. Does that make sense to you?”

Jane thought for a moment and then said, “I suppose I follow the math. But how fair is the assumption that the demand on subsequent days is independent?”

Kate said, “Not a fair assumption in some cases. However, let us go with that assumption for now and take this discussion forward. Can you think of other reasons why there may be a variance in demand?”

Jane shook her head.

“Well, here is a hint,” said Kate. “In the calculation, so far, we have assumed a constant lead time. Is that a good assumption?”

“Not at all,” responded Jane. “We do see significant fluctuations in lead time. I am thinking of a particular product. Sometimes, it shows up in 5 days, other times it takes 8-9 days. The average lead time is about 7 days.”

“All right,” said Kate. “So, one must account for this variability in the lead time in the safety stock calculations. What is the way we will measure this variability?”

“Variance and Standard Deviation?” asked Jane.

“Right,” said Kate. “Variance and standard deviation of the lead time to be precise.”

“If σ_{LeadTime} is the standard deviation of the lead time, and μ_{Demand} is the average demand, then the expected demand over this standard deviation would be μ_{Demand}^{*} σ_{LeadTime}. However, we should look at it in terms of variances, because variances of independent quantities can be added. So, the demand variance associated with the variance of the lead time would be μ_{Demand}^{2} * σ_{LeadTime}^{2}.

Kate continued, “The total variance to be considered for the two variabilities discussed so far will be equal to σ_{DayDemand}^{2}*(L/1) + μ_{Demand}^{2} * σ_{LeadTime}^{2}. The first term is related to the variation in demand; the second term is related to the variation in the lead time.”

“What about the unit of measure of the second term?” asked Jane.

“Good point,” said Kate. “For this to be additive, they need to be of the same unit of measure. Recall, the UOM of the first term was units^{2}. In the second term, μ_{Demand}^{2} will have the UOM of (units/day)^{2}, and σ_{LeadTime}^{2} will have the UOM of day^{2}. If you multiply the two, you are left with units^{2}. So, these can be safely added. Thus, we get to the formula of the safety stock.”

And then Kate wrote on the whiteboard:

Safety Stock = z-score * √[ {σ_{DayDemand}^{2}*(L/1)} + {μ_{Demand}^{2} * σ_{LeadTime}^{2}}]

Kate then added, “Since both the quantities inside the square root have UOM of units^{2}, they will have the UOM of units when they come out of the square root. Since the z-score is unitless, we will get the safety stock in units.”

We will continue the discussion in the next blog.

*Read the next blog in the series here.*