BUS606: Operations and Supply Chain Management

However, Specifically in terms of safety stock, there are many safety stock calculating formulas. The following Equations are widely used. Safety stock (SS) is calculated using:

$S S=k \sigma z$

Where $k$ is a safety factor parameter based on the distribution of the total demand in a lead time and σ z is the standard deviation of total demand in a lead time ($Z$). The demand in a lead time is a random sum (lead time) of random numbers (demand). Therefore, probability theory informs us that, when lead time and demand are uncorrelated, the standard deviation of total demand in a lead time is

$\sigma_{z}=\sqrt{\mu_{t} \sigma_{d}^{2}+\mu_{d}^{2} \sigma_{t}^{2}}$

Where,

$\mu_{t}$ = mean lead time
$\mu_{d}$ = mean demand
$\sigma_{t}$ = standard deviation of time
$\sigma_{d}$ = standard deviation of demand

Different textbooks, Inventory reviews and software are also following these rules.

Yamazaki, Shida, & Kanazawa define Safety stock is defined as inventory that absorbs various differences between supply and demand. These differences result from changes in the timing of customer purchases, differences between average demand and actual demand, fluctuations in demand, machine breakdowns, employee absenteeism, material shortages, product defects, etc. Traditionally, safety stock is given by

$\mathrm{sS}=\mathrm{z} * \sigma \mathrm{t} * \sqrt{(\mathrm{T}+\mathrm{L})}$
$z$: Safety factor
$\sigma_{t}$: Standard deviation of demand per unit time
$T$ = Review period
$L$ = Lead time

Another model is proposed by Scheidler & Workman, they proposed easier method to work with safety inventory. The equation is:

$\text { Safety Stock= SS }_{\text {Demand }}+\mathrm{SS}_{\text {Supply }}+\mathrm{SS}_{\text {Strategic }}$

$\text { SS }_{\text { Demand }}=(1 \text {-Forecast Accuracy }) \times \text { Lead Time } = \text { Days of coverage needed }$

$\text { SS }_{\text { Supply }}=(1 \text {-Supplier on time delivery }) \times \text { Lead Time } = \text { Days of coverage needed }$

$\text { SS }_{\text { Strategic }} = \text { Days of coverage needed }$

Eppen & Martin, proposed another method but close to Feller (1957). The core difference is Feller (1957) proposed root over the of multiplication addition of standard deviation and mean lead time and demand. Where Eppen & Martin did not use the root instead used multiple variables to compute safety stock

We use the following notation throughout.

Let

$W$ = lead time demand, a continuous random variable,
$X$ = demand during one time period, a continuous random variable,
$Y$ = length of lead time in time periods, a discrete random variable,
$w$ = lead time demand given period lead time,
$\mu_{x}$ = mean demand during one time period,
$\mu_{y}$ = mean lead time,
$\sigma_{x}$ = demand variance during one time period,
$\sigma_{y}$ = lead time variance,
$\mu_{w} = \mu_{x} + \mu_{y}$
$\sigma_{w}^{2}=\left(\sigma_{x}^{2}\right)\left(\mu_{y}\right)+\left(\sigma_{y}^{2}\right)\left(\mu_{x}\right)^{2}$

In inventory literatures, there are many complex theories used, but in most of the cases, the users of the theories have hard time to understand and use the models. One of the main reasons is the inventory managers are busy of their daily operations. In a store or warehouse, there are multiple products. Most of the time it is not possible for them to use any complex formula to derive safety stock levels. Therefore there is need of using close to complex but simple but inclusion of different variable data used formula that will be beneficial for root level users.

In terms of Safety stock calculation, most of the time literatures use lead time and demand variation. But demand variation do no stay always same. Besides, Demand is not fixed most of the time. Therefore demand is uncertain. But we see little effort in demand forecasting efforts in the literature. Some authors considered demand errors seriously. But still comprehensive demand forecasting and error methods are not used in terms of safety stock calculations. As most of the calculations deal with certain data. Here our effort is to deal with uncertain data. In that case inclusion of demand forecasting is necessary.

Besides, Demand forecasting Economic order quantity could also be used to derive safety stock calculation. As EOQ is already recognized for assessing optimum safety stock, we could use this in our safety stock calculation. Although EOQ has some drawbacks, But still it is one of the most recognized formula to derive ultimate inventory. Besides, we will not use EOQ alone. Inclusion on other variables in an equation will reduce the error of the EOQ in our calculation.