A Measure on the Percent Change in Safety
Stock needs
with a Change in the Forecast Error
Nick
T. Thomopoulos, Ph.D.
Illinois
Institute of Technology
This paper shows how the need for safety
stock increases when the forecast error increases – and conversely -- how the need for
safety stock decreases when the forecast error decreases. A way to measure the change in safety stock
with forecast error changes is presented.
The paper also shows how much safety stock is needed relative to the
total inventory – here called the total stock.
The results of this paper brings forth the
obvious need to use a sound statistical forecasting system to run the inventory
operation. In essence, the smaller the
forecast error, the less need for safety stock, the less inventory and the
higher inventory profit margin..
Two of the key measures on the performance of the inventory system
are the amount of stock on-hand and the service level. The service level SL is typically measured as
the ratio of (demand filled) over (total demand). The on-hand inventory is the
available inventory – usually measured in dollars. The on-hand inventory is conveniently grouped
into two partitions: cycle stock and safety stock. The cycle stock is the portion of stock
carried to meet the average flow of demands as planned by the forecasts over
the future time periods. The safety
stock is the stock carried to meet the uncertainty associated with the
forecasts of the demands. The
uncertainty in demands is a measure of the forecast error. The typical measure of the forecast errors is
the standard deviation of the one month ahead forecast error and is denoted
here as s.
Safety stock and service level
References 1, 2, 3 show how to compute the safety stock to yield
the service level goal as desired by the management. On an individual part, the data used to
determine the safety stock is listed below:
SL = desired
service level
F = average
monthly forecast
L = lead time
to procure the part from the supplier
Q = the size
of the order quantity
s = the standard deviation of the one-month
ahead forecast error
For the analysis of this paper, references 2, 3 show
how the order size and forecast error are converted in units of the average
monthly forecast as shown below:
M = Q/F =
months-in-buy
cov = s/F =
coefficient of variation
This way, the data is independent from the forecast size and is
defined in relative terms. The data to
determine the safety stock is now reduced to the following:
SL = desired service level
L = lead time
in months
M = the order
size in months supply
cov = the coefficient of variation
Reference 3 gives a series of tables that list the months of
safety stock for a wide variation of situations using the data above. Three tables of this paper list some of the
results from the reference. The tables
are arranged as follows:
Table 1. L = 0.25 (one
week) and M = .50 (one-half month)
Table 2. L = 1.00 (one
month) and M = 1.00 (one month)
Table 3. L = 2.00 (two
months) and M = 1.00 (one month)
Each of the tables give safety stock
results from five values of the service level: (SL = 0.900, 0.925, 0.950, 0.975
and 0.990) and eight values of the coefficient of variation (cov = 0.1, 0.2,
0.3, 0.4, 0.5, 0.6, 0.8 and 1.0). Note
the tables are split into two sections: upper and lower sections.
The upper section lists the months of safety stock needed to yield
the service level goals in association with the lead time, the month-in-buy and
the coefficient of variation.
The lower section lists the percent increase in safety stock that
is needed to yield the service level goal (with L and M fixed) and with a 10
percent increase in the coefficient of variation (cov). In essence it is a measure of how much more
safety stock (in percent) is needed when the forecast error has an increase of
10%. It also is a measure on how much
less safety stock (in percent) is needed when the forecast error has a decrease
of 10%.
For clarity, it is helpful to describe how the above percentages
are obtained. As an example consider
Table 2 where L = 1.00 and M = 1.00.
Suppose the service level of interest is SL = 0.95 and the cov entries
are 0.30 and 0.50. The months of safety
stock for this situation is ss = 0.18 and 0.45 months supply,
respectively. Below
lists the results in tabular form.
SL L M cov ss
.95 1.00 1.00 0.30 0.18
.95 1.00 1.00 0.40 0.31
When the cov changes from 0.30 to 0.40, the percent increase in
cov is: (0.40-0.30)/0.30 = 0.33 (33%).
In a corresponding way, the percent increase in the safety stock is: (0.31-0.18)/0.18
= 0.72 (72%). So 0.72/0.33 = 2.18 is the percent increase
in safety stock to accommodate a one percent increase in the cov. Using 10 percent as the base increase in cov,
the corresponding increase in safety stock becomes 21.8%. The associated table entry lists 21.67% which
differs due to rounding.
Although the discussion above describes how safety stock increases
as the cov increases, the analogy holds as cov decreases. So when the cov decreases by 10%, Tables 1-3
list the percent decrease in safety stock that is expected.
Cycle stock, safety stock and total stock
Table 4 is a related table that lists the months of safety stock
relative to the cycle stock and the total stock. Note the average cycle stock is half the
months-in-buy and thereby cs = M/2 is used here. Note also the total stock is the sum of cycle
stock and safety stock (ts = cs + ss).
Table 4 lists cs, ss and ts when the service level is SL = 0.95 and when
L, M and cov range the same as in Tables 1-3.
Table 4 shows that the safety stock can represents
a large portion of the total stock. For
example, the safety stock is
near 50 percent of the total stock when the coefficient of
variation (cov) is .50. The safety stock is over 50
percent of the total stock when cov is greater than 0.50.
Summary
As stated at the beginning, this paper shows how much less safety
stock is needed when the forecast error decreases by 10 percent. With this
knowledge, it behooves the inventory management to insist on using a good forecast
system – one built with sound statistical techniques. Note that in many
inventory operations, safety stock represents a large portion of the total
inventory – 50 percent is common – and thereby a reduction in the safety stock
will significantly reduce the total inventory – and this leads to lower inventory costs
and higher profits.
References
1. Brown, R.G., Smoothing, Forecasting and
Prediction of Discrete Time Series,
2. Thomopoulos, N.T., Applied
Forecasting Methods,
3. Thomopoulos, N.T., Strategic
Inventory Management and Planning,
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Table 1. L
= .25 and M = .50
Months of safety stock vs. lead time, months-in-buy,
coefficient of variation and service level;
and the corresponding percent increase in safety stock
when cov increases 10%
-----------------------------------------------------------------
----months----
------------------SL---------------------
L M
cov 0.900 0.925
0.950 0.975 0.990
------------------
-------------------ss---------------------
0.25 0.50
0.10 0.00 0.00
0.00 0.02 0.04
0.25 0.50
0.20 0.00 0.01
0.03 0.08 0.13
0.25 0.50
0.30 0.02 0.05
0.09 0.15 0.22
0.25 0.50
0.40 0.07 0.11
0.16 0.23 0.32
0.25 0.50
0.50 0.12 0.17
0.22 0.31 0.42
0.25 0.50
0.60 0.18 0.23
0.30 0.40 0.52
0.25 0.50
0.80 0.31 0.37
0.46 0.59 0.74
0.25 0.50
1.00 0.45 0.53
0.63 0.78 0.97
% increase in ss at 10% increase in
cov
-------------------------------------------
0.25 0.50
0.20 - - -
30.00 22.50
0.25 0.50
0.30 - -
40.00 17.50 13.85
0.25 0.50
0.40 75.00 36.00
23.33 16.00 13.64
0.25 0.50
0.50 28.57 21.82
15.00 13.91 12.50
0.25 0.50
0.60 25.00 17.65
18.18 14.52 11.90
0.25 0.50
0.80 21.67 18.26
16.00 14.25 12.69
0.25 0.50
1.00 18.06 17.30
14.78 12.88 12.43
-----------------------------------------------------------------
-----------------------------------------------------------------
Table 2. L
= 1 and M = 1
Months of safety stock vs. lead time, months-in-buy,
coefficient of variation and service level;
and the corresponding percent increase in safety stock
when cov increases 10%
-----------------------------------------------------------------
----months----
------------------SL---------------------
L M cov
0.900 0.925 0.950
0.975 0.990
------------------
-------------------ss---------------------
1.00 1.00
0.10 0.00 0.00
0.00 0.03 0.09
1.00 1.00
0.20 0.00 0.01
0.07 0.16 0.25
1.00 1.00
0.30 0.04 0.10
0.18 0.30 0.43
1.00 1.00
0.40 0.14 0.21
0.31 0.46 0.63
1.00 1.00
0.50 0.24 0.33
0.45 0.63 0.83
1.00 1.00
0.60 0.36 0.47
0.60 0.80 1.04
1.00 1.00
0.80 0.62 0.75
0.92 1.18 1.48
1.00 1.00
1.00 0.90 1.05
1.25 1.57 1.94
%
increase in ss at 10% increase in cov
--------------------------------------------
1.00 1.00
0.20 - - -
43.33 17.78
1.00 1.00
0.30 - -
31.43 17.50 14.40
1.00 1.00
0.40 75.00 33.00
21.67 16.00 13.95
1.00 1.00
0.50 28.57 22.86
18.06 14.78 12.70
1.00 1.00
0.60 25.00 21.21
16.67 13.49 12.65
1.00 1.00
0.80 21.67 17.87
16.00 14.25 12.69
1.00 1.00
1.00 18.06 16.00
14.35 13.22 12.43
-----------------------------------------------------------------
-----------------------------------------------------------------
Table 3. L
= 2 and M =1
Months of safety stock vs. lead time, months-in-buy,
coefficient of variation and service level; and the
corresponding percent increase in safety stock
when cov increases 10%
----------------------------------------------------------------
----months---- ------------------SL--------------------
L M
cov 0.900 0.925
0.950 0.975 0.990
------------------
-------------------ss---------------------
2.00 1.00
0.10 0.00 0.00
0.00 0.08 0.15
2.00 1.00
0.20 0.03 0.09
0.16 0.27 0.40
2.00 1.00
0.30 0.16 0.24
0.34 0.50 0.68
2.00 1.00
0.40 0.32 0.42
0.55 0.74 0.96
2.00 1.00
0.50 0.50 0.61
0.77 1.00 1.27
2.00 1.00
0.60 0.69 0.82
1.00 1.27 1.59
2.00 1.00
0.80 1.09 1.26
1.48 1.83 2.24
2.00 1.00
1.00 1.53 1.73
2.00 2.42 2.92
%
increase in ss at 10% increase in cov
-------------------------------------------
2.00 1.00
0.20 - - -
23.75 16.67
2.00 1.00
0.30 - -
22.50 17.04 14.00
2.00 1.00
0.40 30.00 22.50
18.53 14.40 12.35
2.00 1.00
0.50 22.50 18.10
16.00 14.05 12.92
2.00 1.00
0.60 19.00 17.21
14.94 13.50 12.60
2.00 1.00
0.80 17.39 16.10
14.40 13.23 12.26
2.00 1.00
1.00 16.15 14.92
14.05 12.90 12.14
-----------------------------------------------------------------
-------------------------------------------------------------------------------------------------------------------------
Table 4.
Months of cycle stock (cs), safety stock (ss) and total stock (ts) versus
lead time months (L), months-in-buy (M), coefficient of variation
(cov) and service level (SL) = 0.95.
-------------------------------------------------------------------------------------------------------------------------
L = .25 M = .50 L = 1.00 M = 1.00 L= 2.00 M = 1.00
SL cov cs ss ts cs ss ts
cs ss
ts
-----------------
-----------------------
-----------------------
-----------------------
0.95 0.10 0.25
0.00 0.25 0.50 0.00
0.50 1.00 0.01
1.01
0.95 0.20 0.25
0.03 0.28 0.50 0.07
0.57 1.00 0.16
1.16
0.95 0.30 0.25
0.09 0.34 0.50 0.18
0.68 1.00 0.34
1.34
0.95 0.40 0.25
0.16 0.41 0.50 0.31
0.81 1.00 0.55 1.55
0.95 0.50 0.25
0.22 0.47 0.50 0.45
0.95 1.00 0.77
1.77
0.95 0.60 0.25
0.30 0.55 0.50 0.60
1.10 1.00 1.00
2.00
0.95 0.80 0.25
0.46 0.71 0.50
0.92 1.42 1.00 1.48
2.48
0.95 1.00 0.25
0.63 0.88 0.50 1.25
1.75 1.00 2.00
3.00
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