customer demand customer demand varies in both timing and quantity: time quantity individual...
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Customer Demand
• Customer demand varies in both timing and quantity:
Time
Quantity
Individual Customer Order
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
Demand
Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8 Day 9
Individual Customer Demand
Cumulative Demand for Period
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
QROP
LT LT
• Using the EOQ model, we developed a reorder point (ROP) inventory management system:
• In the EOQ model, demand is assumed to be constant
• 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:
ROP with Variable Demand
QROP
LT LT
DDLT*
*(average) Demand During Lead Time
Safety Stock
• To avoid stockouts, the reorder point should include additional inventory, safety stock, to reduce the probability of a stockout.
ROP
LT LT
Safety Stock
Q
DDLT*
*(average) Demand During Lead Time
Safety Stock
DDLTσ
DDLT
DDLTSLσZSS
Probability of aStockout
Using the standard deviation ofthe DDLT, we can set ana safety stock level based onthe probability of a stockout
Safety Stock
Cumulative Probability Z
0.50 0.00000.55 0.12570.60 0.25330.65 0.38530.70 0.52440.75 0.67450.80 0.84160.85 1.03640.90 1.28160.95 1.64490.98 2.05370.99 2.3263
0.995 2.57580.998 2.8782
Z
CumulativeProbability
For a given service level (cumulative probability),the safety stock is calculated as:
DDLTSLσZSS
Safety Stock Example
• Suppose we have the following weekly demand (consumption) data for a product:
Week Demand1 982 923 1114 885 1246 947 868 1099 9710 76
Average Demand 97.5
13.9Standard Deviationof Demand
If the lead time is one week,then we have:
If we want a 95% service level,then the safety stock should be:
So the reorder point should be:
So a ROP of 120 should be used
DDLT = 97.5
SS = (1.6449)(13.9) = 22.86
ROP = 97.5 + 22.86 = 120.36
Safety Stock using MAD
• 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
MAD Calculation
xix 97.5
10
975X
Week Demand
1 98 0.52 92 5.53 111 13.54 88 9.55 124 26.56 94 3.57 86 11.58 109 11.59 97 0.510 76 21.5
975 104
10.410
104MAD
Standard Deviation Calculation
2xix 97.5
10
975X
Week Demand
13.99
1744.51n
2XiXSD
1 98 0.252 92 30.253 111 182.254 88 90.255 124 702.256 94 12.257 86 132.258 109 132.259 97 0.25
10 76 462.25975 1744.5
Safety Stock using MAD
• The standard deviation can be estimated from the MAD using:
• 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 R
0.50 0.00000.55 0.15710.60 0.31670.65 0.48170.70 0.65550.75 0.84310.80 1.05200.85 1.29550.90 1.60190.95 2.05610.98 2.56720.99 2.90790.995 3.21980.998 3.5977
SD = 1.25 MAD
SS = (R)(MAD)
Safety Stock Example Revisited
• The following weekly demand (consumption) data for the product was: The Demand During Lead Time is:
For a 95% service level,the safety stock should be:
So the reorder point should be:
So a ROP of 119 should be used (vs. 120 calculated using the SD)
DDLT = 97.5
SS = (2.0561)(10.4) = 21.38
ROP = 97.5 + 21.38 = 118.88
Week Demand1 982 923 1114 885 1246 947 868 1099 97
10 76
Average Demand 97.5
10.4MAD
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–
2Y
2XZ σσσ
Xσ
ZσYσ
DPDPDPLT σ2σσσ
DPσDPσ
LTσDP = 1 weekLT = 2 weeks
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:
DPLT
LT2LT
2LTDP
σ2
σ
σ2σσσ1
DPσ
LTσDP = 2 weeksLT = 1 weeks
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:
LTσ
In general terms, the standard deviation of the demandfor the lead time is:
where the lead time and demand period are measured inthe same time units (typically days). The demand period islevel of aggregation used for determining demand.
Safety Stock
DPperiod demand
time leadLT σσ
So the safety stock level can be calculated as:
using the standard deviation of demand and:
using the MAD, where:
Safety Stock
Demand SL σ WZSS
MAD WRSS
Period DemandTime LeadW
Note that if the Demand Period does not equal the LeadTime, then the DDLT is calculated as:
DDLT
Demand WDDLT
Demand data for a material has been collected on a weeklybasis for 6 months. Demand appears level, with:
Mean: 270 units/weekStandard 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
Safety Stock: Example 1
• The formula for safety stock using the standard deviation is:
so for this example we have:
Demand SL σ WZSS
111111.2
40 7
10 2.3263SS
Demand data for product has been collected on a weeklybasis with the following results:
Mean: 109 units/weekMAD: 20 units/week
The lead time is 4 days. Calculate the safety stock required for a 95% customer service level.
Safety Stock: Example 2
Safety Stock: Example 2
• The formula for safety stock using the standard deviation is:
so for this example we have:
3131.08
20 74 2.0561SS
MAD WRSS
Demand Period and Lead Time in SAP
Demand period is set by the Period Indicator on the Forecasting Viewof the Material Master
The applicable periods are:M – MonthlyW – WeeklyT – Daily
Demand Period and Lead Time in SAP
In-house production is used for Lead Time for products made in-house
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.
LT LT LT
LT
Twice as manyopportunities for stockout
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:
where D is annual demand and Q is the order quantity.
Q
D
OrderAnnual SLSL
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:
• So there is only a 46.3% chance of going a year without a stockout
463.01595.095.0
800
12,000
Order Q
D
Annual SLSL
Demand Patterns
Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8 Day 9
Week 1 Week 2
Regular Demand
Day 1 Day 2 Day 3 Day 4 Day 5 Day 6 Day 7 Day 8 Day 9
Week 1 Week 2Sparse Demand
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
Time
Quantity
Expected timebetween orders
Expected orderquantity