# Understanding Hadley and Whitin's Reorder Point Formula for Logistics

• jacophile
In summary, the conversation discussed the variance of the reorder point process in an inventory management system and how it is affected by the demand during lead time and the lead time itself. It was shown that the re-order point can be calculated using the mean of the demand and the inverse CDF of the desired service level. The lead time was also discussed as a random variable, independent of the demand. This led to the formula for the distribution of the sales demand during the replenishment lead time. Finally, the conversation concluded with the derivation of the expected value and variance of the sales demand during lead time, which is a commonly used result in logistics.
jacophile
Hi, I am trying (in vain) to understand a common result in logistics due to Hadley and Whitin.

It is about the variance of the reorder point process in an inventory management system.
It is assumed that the demand X during lead time has a normal distribution with mean E(X) and variance Var(X). Then, the re-order point is set equal to:

R = E(X) + Z StdDev(X)

Where
X is a RV representing the demand for one lead time
R is the re-order point i.e. the inventory level at which a replenishment order should be placed
Z is the inverse CDF of the desired service level
The service level is the probability that there will be sufficient stock to meet demand

This makes sense to me because it transposes to

$$SL = F(\frac {R - x} {StdDev(X)})$$

so R is the mean of this process and Z.StdDev(X) is the safety stock

The lead time is the time it takes for a replenishment order to arrive and is also a RV independant of the demand

So,

X = d x LT

Where
d is the demand per unit time process (a RV) and
LT is a RV representing the lead time and
d an LT are independant

So my understanding is that

$$E(X) = E(d)E(LT)$$
$$Var(X) = Var(d)(E(LT))^2 + Var(LT)(E(d))^2 + Var(d) Var(LT)$$

The result that is normally given is (due to Hadley and Whitin 1963)

$$R = E(d)E(LT) + Z \sqrt {E(LT)Var(d) + E(d)^2Var(LT)}$$

But I can not see why.

can anyone help me out?

Hmm, I guess my post was too confusing. Any way I think I worked it out..

Just to re-state the problem (hopefully more clearly),

What is the distribution of the sales demand for a stock item during the replenishment lead time (DLT), given that the daily demand D and the lead time N are independent random variables and the sequence of daily demand D, during the lead time period are iid.

The replenishment lead time N is the number of days between placing an order for more stock and actualy taking delivery of it. The general idea is to have enough stock buffer to cover the variance in D and N.

I was trying to imagine how to multiply D and N but I realized that the demand during lead time (DLT) is actually a sum of a random number of random variables:

$$DLT\;=\;\sum_{i=1}^{N}D_i$$

Then I discovered the characteristic function and moment generating functions (I had no idea about these things so it was a massive eye-opener...) by which I understand I can proceed as follows:

$$Define \;R \equiv e^{DLT \; t}$$

$$E(R \mid N=n)\;=\;E(e^{DLT \; t} \mid N=n) \; = \; E(e^{(D_1+D_2+...+D_n) \;t}) \; = \; (M_D(t))^n \; \;$$

where $$M_D(t)$$ is the moment generating function for D

Now let the condition be a rv so that the expectation above also becomes a rv defined as

$$E(R \mid N) \;= \; (M_D(t))^N$$

so the mean of this rv is

$$E(E(R \mid N)) \; = \; E(R) \; = \; E((M_D(t))^N)$$

Also,

$$E(R) \; = \; E(e^{DLT \; t}) \; = \; M_{DLT}(t)$$

by the definition of the mgf.

$$M_{DLT}(t)\; = \; E((M_D(t))^N).$$

I had no trouble convincing myself that the zero'th, first and second derivatives of the mgf are the zero'th, first and second moments about zero, so now I can go ahead and differentiate

$$M_{DLT}'(t)\; = \; E(N M_D^{N-1}(t) M'_D(t))$$

$$M_{DLT}''(t)\; = \; E(N(N-1) M_D^{N-2}(t) M'_D(t) M'_D(t)+N M_D^{N-1}(t)M''_D(t))$$

It is then straight forward to show that

$$E(DLT) \;=\; M_{DLT}'(0)\; = \; E(D)E(N)$$

and that

$$Var(DLT) \;=\; E(N^2)Var(D)+E(N)Var(D)$$

which is exactly the result I was confused about.

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## 1. What is Hadley and Whitin's Reorder Point Formula for Logistics?

Hadley and Whitin's Reorder Point Formula for Logistics is a mathematical formula used in supply chain management to determine the optimal time to reorder inventory. It takes into account factors such as demand, lead time, and safety stock to calculate the point at which new inventory should be ordered to prevent stockouts and maintain a consistent level of inventory.

## 2. How does the formula work?

The formula takes into account the average demand during lead time, the standard deviation of demand during lead time, and the desired service level to calculate the reorder point. The formula is: Reorder Point = Average Demand during Lead Time + (Safety Factor x Standard Deviation of Demand during Lead Time), where the safety factor is determined by the desired service level.

## 3. What is the purpose of using Hadley and Whitin's Reorder Point Formula?

The purpose of using this formula is to optimize inventory levels and prevent stockouts. By calculating the reorder point, companies can ensure that they have enough inventory on hand to meet customer demand while also minimizing excess inventory and associated costs.

## 4. What are the limitations of Hadley and Whitin's Reorder Point Formula?

While the formula is a useful tool for supply chain management, it does have some limitations. It assumes that demand and lead time are constant, which may not always be the case in real-world scenarios. It also does not take into account external factors such as supplier reliability and unexpected events that may affect inventory levels.

## 5. How can Hadley and Whitin's Reorder Point Formula be applied in real-world situations?

The formula can be applied in various industries and businesses that deal with inventory management. It can be used to determine the optimal reorder point for both raw materials and finished products. This can help companies maintain a balance between inventory levels and customer demand, leading to improved efficiency and cost savings.

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