Supply Chain
Safety Stock
1. Safety Stock Definition
Safety stock is the extra inventory kept to avoid stockouts. The safety stock is based on the standard deviation of demand during the lead time.
2. Safety Stock Formula
The basic safety stock formula accounts for variability in demand and lead time:
\[SS = Z \cdot \sqrt{L \cdot \sigma_d^2 + \bar d^2 \cdot \sigma_L^2}\]Where:
- $SS$ = Safety stock.
- $Z$ = z-score corresponding to the desired service level (e.g., $Z = 1.65$ for 95% service level).
- $\sigma_d$ = Standard deviation of daily demand.
- $\bar d$ = Average daily demand.
- $L$ = Lead time (number of days or a random variable).
- $\sigma_L^2$ = Variance of lead time (if lead time is random).
3. Derivation of Safety Stock Formula
Step 1: When Lead Time is Constant
When lead time $L$ is fixed,
the total demand $D_{LT}$ is the sum of daily demand over $L$ days:
\[D_{LT} = D_1 + D_2 + ... + D_L\]where $D_1, D_2, …, D_L$ are i.i.d. (independent and identically distributed) demand values.
the total demand variance during lead time is:
\[\text{Var}(D_{LT}) = L \cdot \sigma_d^2\]This is because the linear combination property of the variance for independent variables. variance
The safety stock for a constant lead time becomes:
\[SS = Z \cdot \sigma_d \cdot \sqrt{L}\]Step 2: When Lead Time is Random
When lead time $L$ is random (i.e., it has variability), we need to account for the uncertainty in lead time. The law of total variance is used to compute the total variance of demand during lead time. law of total variance
The total variance of demand during lead time is:
\[\text{Var}(D_{LT}) = \mathbb{E}[\text{Var}(D_{LT} \mid L)] + \text{Var}(\mathbb{E}[D_{LT} \mid L])\]Where:
- $\mathbb{E}[\text{Var}(D_{LT} \mid L)] = L \cdot \sigma_d^2$ (the variance of demand over a fixed lead time).
- $\text{Var}(\mathbb{E}[D_{LT} \mid L]) = \bar d^2 \cdot \sigma_L^2$ (the additional variance due to variability in lead time).
So, the total variance of demand during lead time is:
\[\text{Var}(D_{LT}) = L \cdot \sigma_d^2 + \bar d^2 \cdot \sigma_L^2\]The standard deviation of demand during lead time is:
\[\sigma_{LT} = \sqrt{L \cdot \sigma_d^2 + \bar d^2 \cdot \sigma_L^2}\]Thus, the final safety stock formula when lead time is random is:
\[SS = Z \cdot \sqrt{L \cdot \sigma_d^2 + \bar d^2 \cdot \sigma_L^2}\]4. Intuition Behind the Formula
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First Part ($L \cdot \sigma_d^2$): This represents the variance in demand over a fixed lead time. It assumes that lead time is constant and calculates how demand varies during that period.
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Second Part ($\bar d^2 \cdot \sigma_L^2$): This represents the variance due to uncertainty in lead time. It reflects the additional variability that comes from the fact that lead time $L$ is not fixed, and it influences the total demand during that uncertain period.