1. Time Series & Stationarity Theory
A time series is a sequence of observations $Y_1, Y_2, \dots, Y_t$ ordered by time.
Stationarity Conditions
- Constant Mean: $\mathbb{E}[Y_t] = \mu \quad \forall t$
- Constant Variance: $\text{Var}(Y_t) = \sigma^2 < \infty \quad \forall t$
- Autocovariance: $\text{Cov}(Y_t, Y_{t-k}) = \gamma(k) \quad \forall t, k$
Differencing Operator ($\Delta$)
$$\Delta Y_t = Y_t - Y_{t-1}$$
2. $\text{ARIMA}(p,d,q)$ Model Equations
Autoregressive Component ($\text{AR}(p)$)
$$\Delta^d Y_t = c + \sum_{i=1}^p \phi_i \Delta^d Y_{t-i} + \varepsilon_t$$Moving Average Component ($\text{MA}(q)$)
$$\Delta^d Y_t = c + \varepsilon_t + \sum_{j=1}^q \theta_j \varepsilon_{t-j}$$Combined $\text{ARIMA}(1,1,1)$ Equation
$$\Delta Y_t = c + \phi_1 \Delta Y_{t-1} + \varepsilon_t + \theta_1 \varepsilon_{t-1}$$
3. Inventory Optimization Formulas
$$\mu = \frac{1}{n} \sum_{i=1}^n Y_i$$
$$\sigma = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (Y_i - \mu)^2}$$
Safety Stock ($SS$) Formula
$$SS = Z \times \sigma \times \sqrt{L}$$
Reorder Point ($ROP$) Formula
$$ROP = D_L + SS = (D_{daily} \times L) + (Z \times \sigma \times \sqrt{L})$$