# Good distribution assumptions for customer demand in a supply chain

Why do we normally assume normal distribution/Poisson distribution for customer demand in a supply chain? Are there other good distribution assumptions?

• I am not sure about this, but one reason could be that, as supply chains become more complex, the actual demand depends on the interaction of many random variables of unknown distribution. If we assume the final demand has an additive relation to them, and that their moments are bounded (which should be the case in real situations), by Lyapunov's theorem the resulting random variable tends to be normal. Of course, there is rarely such a thing as a normal distribution in the real-world: only truncated normals! :-) – Alberto Santini Jun 20 '19 at 20:54

I agree with @QianZhang's answer (nice theoretical properties, easy to implement), and I would add that there is some theoretical justification too. If demands come from customer arrivals, then Poisson is a reasonable demand distribution since customer arrivals are well modeled by Poisson (often). And if the mean is large enough, then normal is a good approximation for Poisson.

Even if demands don't come from a Poisson arrival process, normal is often a good approximation for whatever process they do come from. Although, as suggested by the question Supply Chain Public Data Repository, supply chain data is hard to come by, especially at the transactional level (e.g. data on individual demands so that we can check the distribution, rather than aggregate measures like mean), so it might be hard to confirm my claim empirically.

In my understanding, using normal/Poisson distribution for customer demand is mainly for two reasons.

• These distributions have nice properties for theoretical analysis in supply chain models
• These distributions are easier to implement or already been implemented for computational concerns

Question: Why do we normally assume normal distribution/Poisson distribution for customer demand in a supply chain?

Based on my experience in the industry, I have seen that generally, business users use simple thumb rules-based methods for safety stock or inventory models. The next level of sophistication for these users is usage of Normal or Poisson distribution. Usage of these distributions may not be most optimal but these distribution-based models are far more superior as compared to thumb rule-based methods. Now the next level of sophistication is the implementation of actual distribution. When you want to implement the actual distribution then first you need to conduct certain tests to identify the actual fit. The next level of complication arises when you want to combine Actual Demand Distribution with Actual Lead time distribution for computation of safety stocks in the supply chain. Unfortunately not much info is available on how to combine the actual distribution of Demand with the actual distribution of lead time to compute optimal safety stock. In my experience, for most of the time Forecast Error or Demand distribution is not normally or Poisson distributed.

Question: Are there other good distribution assumptions?

Answer: There are some approximations available. These are based on research work mentioned in book Inventory and Production Management in Supply Chains by Edward, Silver & Spike

• Std Deviation to mean demand $$\le 0.5$$ and Lead time demand $$>10$$: normal distribution.

• Std Deviation to mean demand $$> 0.5$$ and Lead time demand $$>10$$: gamma distribution

• Lead time demand $$<10$$, then use Poisson, Binomial, and Negative Binomial distributions

Theoretically it is quite naturally convincing to assume that demand time points are independent. In other words knowing that an item was bought in t1 does not give one any good clue in understanding next sales time t2. It is like process renews/regenerate itself after each event and hence time between events are independent of each other. This particular phenomenon can be modeled elegantly by exponential distribution which itself have this property.

Also when time between events are exponentially distributed then number of event occurrences is Poisson distributed. So making Poisson distribution assumption makes sense when you have to generate data (Don't have actual data). For more details you can search Poisson process or some elementary literature in queuing theory.

Normal distribution could be assumed when demand hovers around certain average like newspapers sales etc.

However when supplied with data it makes better sense to check actual distribution than just assume some distribution and model. I worked on slow moving commodity sales project and sales data was not following any Poisson distribution.