Determining Safety Stock with Uncertain Demand

In inventory demand, most of the cases data are interdependent. To deal with the mutually depended data, we work with linier or multi-linier regression. In our case linier regression will be used. If more variables are there multi-linier regression will be useful.

The regression formula is following:

$b=\frac{\sum Y X-n \bar{Y} \bar{X}}{\sum X^{2}-n X^{2}}$

$a=\bar{D}-b X$

$D(F)=a+b X$

1st step: Demand forecast using linier regression

In our case, Lead time and Demand are two interdependent factors. In our test case, we will consider following data for our study. It is hypothetical data.

In this case, lead time and demand are mutually dependent variables. We have data of 6 period. Lead time for 1st month is 8 days and demand was 1200. Same goes on with subsequent months. We will calculate the regression forecast of each period. In this way forecast of 7th period will derive, assuming the lead time is known, demand is unknown. In this way we will find the forecast of 7th period ans will regard as Forecasted Demand of period 7.

2nd step: Find out forecast error

After bringing the forecasted data, now it is time to work with forecast error. Generally, Error is the differences between the actual and forecast values, in equation, it is defined:

$e_{t}=y_{t}+\hat{y}_{t}$

However, We are working with popular method of forecasting Mean Absolute error (MAE), Sum of Squired Error (SSE), Mean squired Error (MSE), Squire root of MSE(RMSE).

Forecasts and error measures for a simple regression model

*Calculation A: Demand forecasting & Error calculation. Step 1 & 2.

3rd step: Deriving EOQ

EOQ:

The Economic Order Quantity (EOQ) formula is one of the most pervasive formulas in the business and engineering literature. It is the level of inventory that minimizes total inventory holding costs and ordering costs. Engineers study the EOQ formula in engineering economy classes and in industrial engineering classes. Business students study the EOQ formula in both finance and operations management classes. Sometimes the EOQ formula is studied for practical reasons as part of specific applications in inventory systems.

EOQ decision rules

These rules indicate what quantities could be ordered, so as to balance optimum the cost associated with them. Balancing the fixed costs per lot against the carrying costs is the basis for arriving at the economic order quantity. The development of the formula used the following assumptions:

• Demand is continuous and constant (Le. rate of depletion of inventory is constant);
• The process continues infinitely; and
• There are no quantity constraints (on order quantity or storage capacity). Initially, the development also assumes the following (later these assumptions would be relaxed to develop variants of the basic rule):
• Replenishment is instantaneous;
• No shortages are allowed; and
• Costs are time and quantity invariant.

The development uses the following is the notation:

$Q$ = order quantity

$h$ = inventory holding cost

$K$ = order cost in dollars per order;

$D$ = demand rate pieces per unit time (year); and

The rule is

$\mathrm{EOQ}=\frac{\sqrt{2 \mathrm{KD}}}{h}$

For our case

Demand=Forecasted Demand of period 7= received from Calculation above.

Cost of Order= we assume in this case 1000

Cost of the product: we assume=1100

Carrying cost= we assume =20% of Purchase price.

$\mathrm{EOQ}=\frac{\sqrt{2 \mathrm{KD}}}{h}=\mathrm{EOQ}=\frac{\sqrt{2 \times 1100 \times 1000}}{220}=100$

*Calculation B: EOQ calculation

4th step: Set review period

In this step, we have to set a review period. The review period is supposed to be set by the organization. As it depends on the policy, practice, supplier and capacity of organization.

In our case we are assuming, review period= 10 Days

5th step: Get mean absolute error

In this step, we will get mean absolute error to find the errors among actual demand and forecasting of different period. These will help to reduce the errors likely to happen in our forecasted 7th period.

$M A P E=\frac{\sum_{i=1}^{n}\left|y_{i}-x_{i}\right|}{\mathrm{n}}$

6th step: Derive mean lead time

Lead time is another important factors in demand forecasting. In this step we are trying to get the central tendency or mean of lead time so far occurred during supply of a particular product. Lead time means the time took to receive the product after ordering in a specific time.

Mean lead time, $\mu_{t}=\sum X / n$

7th step: Derive standard deviation of Lead time

We use standard deviation of lead time to from lead time history so far and calculate the deviation of lead time happed so far. This also reduces the errors in lead time we are considering in our equation. In our case, 7th period lead time 12 days assumption error is minimized the standard deviation of lead time calculation.

Standard deviation of lead time, $\sigma_{L} \mathrm{~T}=\frac{\sum_{i=1}^{N}\left(x_{i}-\bar{x}\right)^{2}}{N-1}$

8th Step: Calculating all data altogether

The proposed model is mention below. The rational of this model is discussed later part of this writing.

$\text { Safety Stock }=\frac{E O Q^{*}}{\text { Re view Period }}+\text { mean absolute error of demand } \times \text { mean lead time } \times \text { Standard deviation of lead time }$

* EOQ after regression demand forecasted and error calculated demand

We can write this model, in short,

$\text { Safety } \quad \text { Stock }=\frac{E O Q^{*}}{R}+M A E \times \mu t \times \sigma L$

In our case, the calculation is:

$=\frac{100}{10}+140 \times 9.71 \times 1.70==2476.05$

*Calculation C: Safety stock calculation