* 12.0 Managing Economies of scale in a supply chain- safety stocks/13.0 Determining optimal level of product availability
12.1 Safety Stock calculation

– Calculate ROP to achieve 95% service level
Let us consider 9 weeks of fluctuating demand:

Week | Weekly demand | 1 | 20 | 2 | 10 | 3 | 0 | 4 | 110 | 5 | 20 | 6 | 10 | 7 | 20 | 8 | 20 | 9 | 60 | Total | 270 |

We can calculate the mean as ∑ x / n = 270/9 = 30 units a week
We can also calculate the variance (S2) and then determine the standard deviation (S).

Week | Deviation from Mean(x - )))) | Squared Deviation( x- )) 2 ) | 1 | 20-30=-10 | (-10)2 = 100 | 2 | 10-30=-20 | (-20)2 = 400 | 3 | 0-30=-30 | (-30)2 = 900 | 4 | 110-30=80 | (80)2 = 6400 | 5 | 20-30=-10 | (-10)2 = 100 | 6 | 10-30=-20 | (-20)2 = 400 | 7 | 20-30=-10 | (-10)2 = 100 | 8 | 20-30=-10 | (-10)2 = 100 | 9 | 60-30=30 | (30)2 = 900 | | | ∑(x - )2)= 9400 |

S2 (Variance) = 9400/9-1 using formula ∑(x - )2 / n-1 = 1175

Therefore S (Standard Deviation) = SQRT 1175 = 34.3 (or 34 units)

* 12.1.1 The interpretation of standard deviation of 34
This means that the demand for the 9 weeks differ from the mean ( 30 units ) by an average of 34.
If a desired service level of say 95% is required, we can use normal distribution tables to determine the Z-score ( which is 1.645 for 95%), then applying :-
ROL = Mean + Z-score X Standard Deviation( where Z-score is the desired service level indicator)

= 30 + 1.645 X 34
= 86 units (safety stock is 56 units. If 30 units is the mean and lead time is 1 week).

However, for varying levels of service, the Z-score differs as follows:

90% Z-score = 1.28, therefore ROL = 30 + 1.28 X34 = 73 units
80% Z-score = 0.84, therefore ROL = 30 + 0.84 X 34 = 59 units
70% Z-score = 0.49, therefore ROL = 30 +0.49 X 34 = 47 units