### Slide 1

```Customer Demand
• Customer demand varies in both timing and quantity:
Individual Customer Order
Quantity
Time
Customer Demand
• If demand for a product comes from many, independent
customers, then we don’t need to be concerned about
individual customer orders, but rather cumulative
demand over a period of time.
Customer Demand
Cumulative Demand for Period
Individual Customer Demand
Day 1
Day 2
Day 3
Day 4
Day 5
Demand
Day 6
Day 7
Day 8
Day 9
Customer Demand
• In statistics, when there is reason to suspect the
presence of a large number of small effects acting
additively and independently, it is reasonable to assume
that the observations will be normally distributed.
• Therefore, if demand for a product comes from many,
independent customers, we can assume that the
variability in cumulative demand over a period of time
can be described by the normal distribution.
EOQ and Reorder Point Systems
• Using the EOQ model, we developed a reorder point
(ROP) inventory management system:
Q
ROP
LT
LT
• In the EOQ model, demand is assumed to be constant
ROP with Variable Demand
• When demand is not constant, the reorder point
calculation should consider demand variability. If the
reorder point is only based on average demand,
stockouts will occur:
Q
Q
Q
LT
ROP
DDLT*
LT
Safety Stock
• To avoid stockouts, the reorder point should include
additional inventory, safety stock, to reduce the
probability of a stockout.
Q
ROP
DDLT*
Safety Stock
LT
LT
Safety Stock
Using the standard deviation of
the DDLT, we can set an
a safety stock level based on
the probability of a stockout
σDDLT
Probability of a
Stockout
SS ZSLσDDLT
DDLT
Safety Stock
Cumulative
Probability
Z
For a given service level (cumulative probability),
the safety stock is calculated as:
SS ZSLσDDLT
Cumulative
Probability
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
0.98
0.99
0.995
0.998
Z
0.0000
0.1257
0.2533
0.3853
0.5244
0.6745
0.8416
1.0364
1.2816
1.6449
2.0537
2.3263
2.5758
2.8782
Safety Stock Example
• Suppose we have the following
weekly demand (consumption)
data for a product:
Week
1
2
3
4
5
6
7
8
9
10
If the lead time is one week,
then we have:
Demand
98
92
111
88
124
94
86
109
97
76
DDLT = 97.5
If we want a 95% service level,
then the safety stock should be:
SS = (1.6449)(13.9) = 22.86
Average Demand
97.5
Standard Deviation
of Demand
13.9
So the reorder point should be:
ROP = 97.5 + 22.86 = 120.36
So a ROP of 120 should be used
• Many times, Safety Stock levels are calculated using the
Mean Absolute Deviation as a measure of variability
rather than the Standard Deviation. There are two
reasons for this:
– Historical: Before calculators, the calculation of a
standard deviation was not a trivial task, while the
calculation of the Mean Absolute Deviation is fairly
simple to perform by hand
– Robustness: The Mean Absolute Deviation measure
is not as easily affected by outlier points as it is using
the absolute value of the deviation rather than the
squared deviation
Week
1
2
3
4
5
6
7
8
9
10
Demand
98
92
111
88
124
94
86
109
97
76
975
xi  x
0.5
5.5
13.5
9.5
26.5
3.5
11.5
11.5
0.5
21.5
104
X
975
 97.5
10
104
 10.4
10
Standard Deviation Calculation
Week
1
2
3
4
5
6
7
8
9
10
Demand
98
92
111
88
124
94
86
109
97
76
975
xi  x 2
0.25
30.25
182.25
90.25
702.25
12.25
132.25
132.25
0.25
462.25
1744.5
X
975
 97.5
10
SD 
2


X

X
 i
n 1
1744.5

 13.9
9
• The standard deviation can be
• As a result, we can define a
safety factor R which can be
used to determine the safety
stock based on the MAD and
the desired service level:
Cumulative
Probability
0.50
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
0.98
0.99
0.995
0.998
R
0.0000
0.1571
0.3167
0.4817
0.6555
0.8431
1.0520
1.2955
1.6019
2.0561
2.5672
2.9079
3.2198
3.5977
Safety Stock Example Revisited
• The following weekly demand
(consumption) data for the
The Demand During Lead Time is:
product was:
Week
1
2
3
4
5
6
7
8
9
10
Demand
98
92
111
88
124
94
86
109
97
76
DDLT = 97.5
For a 95% service level,
the safety stock should be:
SS = (2.0561)(10.4) = 21.38
So the reorder point should be:
ROP = 97.5 + 21.38 = 118.88
Average Demand
97.5
10.4
So a ROP of 119 should be used
(vs. 120 calculated using the SD)
Demand Period vs. Lead Time Period
• In the previous example, the demand period (the period
of time used to accumulate customer demand) was one
week, which was the same as the lead time.
• Suppose the lead time was two weeks. Then the
variability of the demand for a two week period would be
greater than the MAD calculated from demand data
aggregated weekly.
• We have assumed that customer demand is
independent, i.e. that the demand for the product comes
from a number of unrelated customers. In that case,
then we can use a theorem from statistics to determine
the appropriate variability of demand during lead time
when the demand period is different from the lead time
period
Demand Period vs. Lead Time Period
• Suppose we have two
independent, normally
distributed random variables:
– X: mean X, standard
deviation X
– Y: mean Y, standard
deviation Y
• Then the sum of these
variables, Z = X + Y has mean:
– Z = X + Y
and standard deviation
– σZ  σ2X  σ2Y
σZ
σY
σX
Demand Period vs. Lead Time Period
• Suppose that the demand period is 1 week (customer
demand is measured on a weekly basis) and the lead
time is two weeks. Then the standard deviation for the
lead time can be calculated as:
σ DP
σ DP
DP = 1 week
LT = 2 weeks
σ LT
σLT

σDP  σDP

2 σDP
Demand Period vs. Lead Time Period
• Suppose that the demand period is 2 weeks (customer
demand is accumulated in 2 week intervals) and the lead
time is one week. Then the standard deviation for the
lead time can be calculated as:
σ DP
σ LT
σDP
σLT
2  σ2
σLT
LT
1

σDP
2

σ LT

2 σLT
DP = 2 weeks
LT = 1 weeks
Safety Stock
In general terms, the standard deviation of the demand
σLT

demand period
σDP
where the lead time and demand period are measured in
the same time units (typically days). The demand period is
level of aggregation used for determining demand.
Safety Stock
So the safety stock level can be calculated as:
SS ZSL W σDemand
using the standard deviation of demand and:
Time
W
Demand Period
DDLT
Note that if the Demand Period does not equal the Lead
Time, then the DDLT is calculated as:
DDLT  W Demand
















Safety Stock: Example 1
Demand data for a material has been collected on a weekly
basis for 6 months. Demand appears level, with:
Mean: 270 units/week
Standard deviation: 40 units/week
The lead time is 10 days. Calculate the safety stock
required for a 99% customer service level.
Safety Stock: Example 1
• The formula for safety stock using the standard deviation
is:
SS ZSL W σDemand
so for this example we have:
SS  2.3263 10 40
7
111.2 111
Safety Stock: Example 2
Demand data for product has been collected on a weekly
basis with the following results:
Mean: 109 units/week
The lead time is 4 days. Calculate the safety stock
required for a 95% customer service level.
Safety Stock: Example 2
• The formula for safety stock using the standard deviation
is:
so for this example we have:
SS  2.0561 4 20
7
 31.08  31
Demand Period and Lead Time in SAP
Demand period is set by the Period Indicator on the Forecasting View
of the Material Master
The applicable periods are:
M – Monthly
W – Weekly
T – Daily
Demand Period and Lead Time in SAP
In-house production is used for
Plnd delivery time + GR processing time +
Purchasing proc. time is used for
Lead Time for externally procured materials
Exposure to Stockout
• Stockouts usually occur when stock gets low—for
example, during the lead time period before a new order
arrives:
Periods of maximum exposure to stockout
LT
LT
LT
Exposure to Stockout
• The more frequently we order, the more chances there
are of stocking out.
Twice as many
opportunities for
stockout
LT
LT
LT
LT
LT
LT
LT
LT
LT
LT
Exposure to Stockout
• To fully evaluate the customer service level, we should
calculate the customer service level on an annual basis:
SLAnnual  SLOrder






D






 Q 

where D is annual demand and Q is the order quantity.
Exposure to Stockout
• For example, if we used a service level of 95% in
calculating the safety stock, the annual demand D is
12,000 units and the order quantity Q is 800 units, then
we have:


SLAnnual  SL Order


 12,000


 800


 0.95




D






 Q









 0.95




15




 0.463
• So there is only a 46.3% chance of going a year without
a stockout
Demand Patterns
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Day 7
Day 8
Day 9
Week 2
Week 1
Regular Demand
Day 1
Day 2
Day 3
Day 4
Day 5
Day 6
Week 1
Day 7
Day 8
Week 2
Sparse Demand
Day 9
Demand Patterns
• In developing the safety stock calculations, it was
assumed that demand was generated from a “large”
number of independent sources, and
• The individual demands are aggregated over a time
period sufficiently long so that there are a number of
individual demands contributing to each period demand.
• If these conditions are not met, then the safety stock
values may not perform as expected.
Demand Patterns
• If demand is sparse, then a more detailed approach to
inventory planning that considers the expected time
between orders as well as the expected order quantity
Quantity
Expected
order
quantity
Expected time
between orders
Time
```