since pages 142 to 151 of the text are rather difficult to read, the following is a presentation...
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Since Pages 142 to 151 of the text are rather difficult to read,the following is a presentation of…
An Alternate to Pages 142-151 of “Supply Chain Logistics Management”
by Bowersox, Closs, Cooper
“Statistical Methods of Calculating Safety Stock Requirements
and Average Inventory”
“Statistical Methods of Calculating Safety Stock Requirements”
• Assumptions:– Daily demand is different day by day.– When the supply is replenished the number of
days it takes for the replenishment to arrive varies.
– Therefore, we have variable demand and a variable replenishment cycle.
Demand Varies
Each day we ship out a different amount
Day 1
Warehouse
Demand Varies
Each day we ship out a different amount
Day 2
Warehouse
Demand Varies
Each day we ship out a different amount
Day 3
Warehouse
Replenishment Varies
Supplier
Every time we order a replenishment of stock, delivery time is different.
Warehouse
3 Day Delivery
Replenishment Varies
Supplier
Every time we order a replenishment of stock, delivery time is different.
Warehouse
5 Day Delivery
Replenishment Varies
Supplier
Every time we order a replenishment of stock, delivery time is different.
Warehouse
6 Day Delivery
“Statistical Methods of Calculating Safety Stock Requirements”
• Assumptions:– Daily demand is different day by day.– When the supply is replenished the number of
days it takes for the replenishment to arrive varies.
– Therefore, we have variable demand and a variable replenishment cycle.
How Much Safety Stock Do We Need?
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
Mean of replenishment rate ( Standard
deviation ofdaily sales
) plusMean ofdaily salessquared (
2Standarddeviation ofreplenishmentrate
)2
The next slide shows a table listing 25 days of sales for a hypothetical company.
Day Sales in
Cases
Day Sales in
Cases
Day Sales in
Cases
1 100 10 110 19 110
2 80 11 130 20 120
3 70 12 120 21 70
4 60 13 100 22 100
5 80 14 80 23 130
6 90 15 90 24 110
7 120 16 90 25 90
8 110 17 100
9 100 18 140
From Strategic Logistics Management by Stock and Lambert
• We can see there is variability in sales from day to day.
We can see there is variability in sales from day to day.
Demand VariesEach day we ship out a different amount
Warehouse
• Now here’s a table with a hypothetical list of the required delivery times for our company’s last 16 orders.
Order Number Days required
to receive order
Order Number Days required
to receive order
Order Number Days required
to receive order
Order Number Days required
to receive order
1 7
Order Number Days required
to receive order
Order Number Days required
to receive order
1 7
2 10
Order Number Days required
to receive order
Order Number Days required
to receive order
1 7 9 8
2 10 10 9
3 10 11 9
4 13 12 10
5 12 13 10
6 11 14 11
7 8 15 11
8 9 16 12
• We can see there is variability in the time it takes to replenish our stock.
We can see there is variability in the time it takes to replenish our stock
Supplier
Every time we order a replenishment of stock, delivery time is different.
Warehouse
Different Delivery Times
How Much Safety Stock Do We Need?
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
Mean of replenishment rate ( Standard
deviation ofdaily sales
) plusMean ofdaily salessquared (
2Standarddeviation ofreplenishmentrate
)2
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
Mean of replenishment rate ( Standard
deviation ofdaily sales
) plusMean ofdaily salessquared (
2Standarddeviation ofreplenishmentrate
)2
We have seen our daily sales
Day Sales in
Cases
Day Sales in
Cases
Day Sales in
Cases
1 100 10 110 19 110
2 80 11 130 20 120
3 70 12 120 21 70
4 60 13 100 22 100
5 80 14 80 23 130
6 90 15 90 24 110
7 120 16 90 25 90
8 110 17 100
9 100 18 140
From Strategic Logistics Management by Stock and Lambert
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
Mean of replenishment rate ( Standard
deviation ofdaily sales
) plusMean ofdaily salessquared (
2Standarddeviation ofreplenishmentrate
)2
We have seen our daily sales
We will need to know the mean of daily sales and the standard deviation of daily sales
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
Mean of replenishment rate ( Standard
deviation ofdaily sales
) plusMean ofdaily salessquared (
2Standarddeviation ofreplenishmentrate
)2
First, determine the mean of daily sales
Day Sales in
Cases
Day Sales in
Cases
Day Sales in
Cases
1 100 10 110 19 110
2 80 11 130 20 120
3 70 12 120 21 70
4 60 13 100 22 100
5 80 14 80 23 130
6 90 15 90 24 110
7 120 16 90 25 90
8 110 17 100
9 100 18 140
From Strategic Logistics Management by Stock and Lambert
Mean = 100
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
Mean of replenishment rate ( Standard
deviation ofdaily sales
) plusMean ofdaily salessquared (
2Standarddeviation ofreplenishmentrate
)2
Let’s put our daily sales mean of 100 into our formula to determinesafety stock. It’s 100 squared or 10,000
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
Mean of replenishment rate ( Standard
deviation ofdaily sales
) plus (2
Standarddeviation ofreplenishmentrate
)2
Let’s put our daily sales mean of 100 into our formula to determinesafety stock. It’s 100 squared or 10,000
10,000
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
Mean of replenishment rate ( Standard
deviation ofdaily sales
) plus (2
Standarddeviation ofreplenishmentrate
)2
We have seen our replenishment rates
We will need to know the mean of the replenishment rate
and the standard deviation of the replenishment rate.
10,000
Order Number Days required
to receive order
Order Number Days required
to receive order
1 7 9 8
2 10 10 9
3 10 11 9
4 13 12 10
5 12 13 10
6 11 14 11
7 8 15 11
8 9 16 12
Order Number Days required
to receive order
Order Number Days required
to receive order
1 7 9 8
2 10 10 9
3 10 11 9
4 13 12 10
5 12 13 10
6 11 14 11
7 8 15 11
8 9 16 12
First the mean of the replenishment rate
Order Number Days required
to receive order
Order Number Days required
to receive order
1 7 9 8
2 10 10 9
3 10 11 9
4 13 12 10
5 12 13 10
6 11 14 11
7 8 15 11
8 9 16 12
First the mean of the replenishment rate
Mean = 10
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
Mean of replenishment rate ( Standard
deviation ofdaily sales
) plus (2
Standarddeviation ofreplenishmentrate
)2
Let’s put our replenishment rate mean of 10 into our formula to determinesafety stock.
10,000
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
10 ( Standard deviation ofdaily sales
) plus (2
Standarddeviation ofreplenishmentrate
)2
Let’s put our replenishment rate mean of 10 into our formula to determinesafety stock.
10,000
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
10 ( Standard deviation ofdaily sales
) plus (2
Standarddeviation ofreplenishmentrate
)2
Now we need the standard deviation of daily sales and the standard deviation of the replenishment rate.
10,000
(Observation – mean)2
N-1
Find the Standard Deviation of Daily Sales
Q
S =
(Observation – mean)2
N-1
Now Find the Standard Deviation of the Sales
Q
S =
Remember, the mean or average is 100
(Observation – 100)2
N-1
Now Find the Standard Deviation of the Sales
Q
S =
Remember, the mean or average is 100
(Observation – 100)2
N-1
Now Find the Standard Deviation of the Sales
Q
S =
Now calculate how far each day’s salesare from the mean.
Remember, the mean or average is 100
Day Sales in
Cases
Deviation from mean
Deviation
Squared
1 100
2 80
3 70
4 60
5 80
6 90
7 120
8 110
9 100
Mean = 100
From Strategic Logistics Management by Stock and Lambert
Day Sales in
Cases
Deviation from mean
Deviation
Squared
1 100
2 80
3 70
4 60
5 80
6 90
7 120
8 110
9 100
Mean = 100
From Strategic Logistics Management by Stock and Lambert
minus mean of 100 =
Day Sales in
Cases
Deviation from mean
Deviation
Squared
1 100 0
2 80
3 70
4 60
5 80
6 90
7 120
8 110
9 100
Mean = 100
From Strategic Logistics Management by Stock and Lambert
Day Sales in
Cases
Deviation from mean
Deviation
Squared
1 100 0 0 X 0 =
2 80
3 70
4 60
5 80
6 90
7 120
8 110
9 100
Mean = 100
From Strategic Logistics Management by Stock and Lambert
Day Sales in
Cases
Deviation from mean
Deviation
Squared
1 100 0 0
2 80
3 70
4 60
5 80
6 90
7 120
8 110
9 100
Mean = 100
From Strategic Logistics Management by Stock and Lambert
Day Sales in
Cases
Deviation from mean
Deviation
Squared
1 100 0 0
2 80
3 70
4 60
5 80
6 90
7 120
8 110
9 100
Mean = 100
From Strategic Logistics Management by Stock and Lambert
minus mean of 100 =
Day Sales in
Cases
Deviation from mean
Deviation
Squared
1 100 0 0
2 80 -20
3 70
4 60
5 80
6 90
7 120
8 110
9 100
Mean = 100
From Strategic Logistics Management by Stock and Lambert
Day Sales in
Cases
Deviation from mean
Deviation
Squared
1 100 0 0
2 80 -20 -20 x -20 =
3 70
4 60
5 80
6 90
7 120
8 110
9 100
Mean = 100
From Strategic Logistics Management by Stock and Lambert
Day Sales in
Cases
Deviation from mean
Deviation
Squared
1 100 0 0
2 80 -20 400
3 70
4 60
5 80
6 90
7 120
8 110
9 100
Mean = 100
From Strategic Logistics Management by Stock and Lambert
Day Sales in
Cases
Deviation from mean
Deviation
Squared
1 100 0 0
2 80 -20 400
3 70
4 60
5 80
6 90
7 120
8 110
9 100
Mean = 100
From Strategic Logistics Management by Stock and Lambert
minus mean of 100 =
Day Sales in
Cases
Deviation from mean
Deviation
Squared
1 100 0 0
2 80 -20 400
3 70 -30
4 60
5 80
6 90
7 120
8 110
9 100
Mean = 100
From Strategic Logistics Management by Stock and Lambert
Day Sales in
Cases
Deviation from mean
Deviation
Squared
1 100 0 0
2 80 -20 400
3 70 -30 -30 x -30 =
4 60
5 80
6 90
7 120
8 110
9 100
Mean = 100
From Strategic Logistics Management by Stock and Lambert
Day Sales in
Cases
Deviation from mean
Deviation
Squared
1 100 0 0
2 80 -20 400
3 70 -30 900
4 60
5 80
6 90
7 120
8 110
9 100
Mean = 100
From Strategic Logistics Management by Stock and Lambert
Day Sales in
Cases
Deviation from mean
Deviation
Squared
1 100 0 0
2 80 -20 400
3 70 -30 900
4 60
5 80
6 90
7 120
8 110
9 100
Mean = 100
From Strategic Logistics Management by Stock and Lambert
minus mean of 100 =
Day Sales in
Cases
Deviation from mean
Deviation
Squared
1 100 0 0
2 80 -20 400
3 70 -30 900
4 60 -40
5 80
6 90
7 120
8 110
9 100
Mean = 100
From Strategic Logistics Management by Stock and Lambert
Day Sales in
Cases
Deviation from mean
Deviation
Squared
1 100 0 0
2 80 -20 400
3 70 -30 900
4 60 -40 1600
5 80
6 90
7 120
8 110
9 100
Mean = 100
From Strategic Logistics Management by Stock and Lambert
Day Sales in
Cases
Deviation from mean
Deviation
Squared
1 100 0 0
2 80 -20 400
3 70 -30 900
4 60 -40 1600
5 80 -20 400
6 90 -10 100
7 120 +20 400
8 110 +10 100
9 100 0 0
Mean = 100
From Strategic Logistics Management by Stock and Lambert
Day Sales in
Cases
Deviation from mean
Deviation
Squared
10 110 +10 100
11 130 +30 900
12 120 +20 400
13 100 0 0
14 80 -20 400
15 90 -10 100
16 90 -10 100
17 100 0 0
18 140 +40 1600
Mean = 100
From Strategic Logistics Management by Stock and Lambert
Day Sales in
Cases
Deviation from mean
Deviation
Squared
19 110 10 100
20 120 20 400
21 70 -30 900
22 100 0 0
23 130 30 900
24 110 10 100
25 90 -10 100
Mean = 100
From Strategic Logistics Management by Stock and Lambert
(Observation – 100)2
N-1
Find the Standard Deviation of the Sales
Q
S =
Now add up all the squared deviations, knownas “squares” to find the “sum of squares.”
Day Sales in
Cases
Deviation from mean
Deviation
Squared
1 100 0 0
2 80 -20 400
3 70 -30 900
4 60 -40 1600
5 80 -20 400
6 90 -10 100
7 120 +20 400
8 110 +10 100
9 100 0 0
Mean = 100
From Strategic Logistics Management by Stock and Lambert
Now add up all the squares
Day Sales in
Cases
Deviation from mean
Deviation
Squared
10 110 +10 100
11 130 +30 900
12 120 +20 400
13 100 0 0
14 80 -20 400
15 90 -10 100
16 90 -10 100
17 100 0 0
18 140 +40 1600
Mean = 100
From Strategic Logistics Management by Stock and Lambert
Now add up all the squares
Day Sales in
Cases
Deviation from mean
Deviation
Squared
19 110 10 100
20 120 20 400
21 70 -30 900
22 100 0 0
23 130 30 900
24 110 10 100
25 90 -10 100Sum of squares =
10,000
Mean = 100
From Strategic Logistics Management by Stock and Lambert
Now add up all the squares
10000
N-1
Now Find the Standard Deviation of the Sales
Q
S =
Sum of squares
10000
N-1
Now Find the Standard Deviation of the Sales
Q
S =
N= number of days of sales
10000
25-1
Now Find the Standard Deviation of the Sales
Q
S =
N= number of days of sales
10000
24
Now Find the Standard Deviation of the Sales
Q
S =
Now Find the Standard Deviation of the Sales
Q
S =
416.66666
Of which the square root is…
Now Find the Standard Deviation of the Sales
Q
S =
20.4
Rounded to 20
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
( Standard deviation ofdaily sales
) plus (2
Standarddeviation ofreplenishmentrate
)2
Let’s put that daily sales standard deviation of 20 into ourformula for safety stock.
10 10,000
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
( 20 ) plus (2
Standarddeviation ofreplenishmentrate
)2
Let’s put that daily sales standard deviation of 20 into ourformula for safety stock.
10 10,000
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
( 400 ) plus (2
Standarddeviation ofreplenishmentrate
)2
Let’s put that daily sales standard deviation of 20 into ourformula for safety stock. And square it.
10 10,000
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
400 plus ( Standarddeviation ofreplenishmentrate
)2
Let’s put that daily sales standard deviation of 20 into ourformula for safety stock. And square it.
10 10,000
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
400 plus ( Standarddeviation ofreplenishmentrate
)2
And find the standard deviation for the replenishment rate
10 10,000
Order Number
Days required
to receive order
Deviation from
mean
Deviation squared
1 7
2 10
3 10
4 13
5 12
6 11
7 8
8 9
Replenishment rate mean = 10
Order Number
Days required
to receive order
Deviation from
mean
Deviation squared
1 7
2 10
3 10
4 13
5 12
6 11
7 8
8 9
Replenishment rate mean = 10
minus mean of 10 =
Order Number
Days required
to receive order
Deviation from
mean
Deviation squared
1 7 -3
2 10
3 10
4 13
5 12
6 11
7 8
8 9
Replenishment rate mean = 10
Order Number
Days required
to receive order
Deviation from
mean
Deviation squared
1 7 -3 9
2 10
3 10
4 13
5 12
6 11
7 8
8 9
Replenishment rate mean = 10
Order Number
Days required
to receive order
Deviation from
mean
Deviation squared
1 7 -3 9
2 10
3 10
4 13
5 12
6 11
7 8
8 9
Replenishment rate mean = 10
minus mean of 10 =
Order Number
Days required
to receive order
Deviation from
mean
Deviation squared
1 7 -3 9
2 10 0
3 10
4 13
5 12
6 11
7 8
8 9
Replenishment rate mean = 10
Order Number
Days required
to receive order
Deviation from
mean
Deviation squared
1 7 -3 9
2 10 0 0
3 10
4 13
5 12
6 11
7 8
8 9
Replenishment rate mean = 10
Order Number
Days required
to receive order
Deviation from
mean
Deviation squared
1 7 -3 9
2 10 0 0
3 10
4 13
5 12
6 11
7 8
8 9
Replenishment rate mean = 10
minus mean of 10 =
Order Number
Days required
to receive order
Deviation from
mean
Deviation squared
1 7 -3 9
2 10 0 0
3 10 0
4 13
5 12
6 11
7 8
8 9
Replenishment rate mean = 10
Order Number
Days required
to receive order
Deviation from
mean
Deviation squared
1 7 -3 9
2 10 0 0
3 10 0 0
4 13
5 12
6 11
7 8
8 9
Replenishment rate mean = 10
Order Number
Days required
to receive order
Deviation from
mean
Deviation squared
1 7 -3 9
2 10 0 0
3 10 0 0
4 13
5 12
6 11
7 8
8 9
Replenishment rate mean = 10
minus mean of 10 =
Order Number
Days required
to receive order
Deviation from
mean
Deviation squared
1 7 -3 9
2 10 0 0
3 10 0 0
4 13 +3
5 12
6 11
7 8
8 9
Replenishment rate mean = 10
Order Number
Days required
to receive order
Deviation from
mean
Deviation squared
1 7 -3 9
2 10 0 0
3 10 0 0
4 13 +3 9
5 12 +2 4
6 11 +1 1
7 8 -2 4
8 9 -1 1
Replenishment rate mean = 10
Order Number
Days required
to receive order
Deviation from
mean
Deviation squared
9 8 -2 4
10 9 -1 1
11 9 -1 1
12 10 0 0
13 10 0 0
14 11 +1 1
15 11 +1 1
16 12 +2 4
Replenishment rate mean = 10
Order Number
Days required
to receive order
Deviation from
mean
Deviation squared
9 8 -2 4
10 9 -1 1
11 9 -1 1
12 10 0 0
13 10 0 0
14 11 +1 1
15 11 +1 1
16 12 +2 4
Replenishment rate mean = 10
Sum of squares =40
(Observation – mean)2
N-1
Q
R =
Find the Standard Deviation of the replenishment rate
40
N-1
Q
=
Sum of squares
Find the Standard Deviation of the replenishment rate
R
N-1
Q
=
N= number orders placed
40
Find the Standard Deviation of the replenishment rate
R
16-1
Q
=
N= number orders placed
40
Find the Standard Deviation of the replenishment rate
R
15
Q
=40
Find the Standard Deviation of the replenishment rate
R
Q
=
2.66666
Of which the square root is…
Find the Standard Deviation of the replenishment rate
R
Q
=
1.634
Find the Standard Deviation of the replenishment rate
R
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
400 plus ( Standarddeviation ofreplenishmentrate
)2
Let’s put that replenishment rate standard deviation of 1.634 into our formula for safety stock.
10 10,000
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
400 plus ( 1.634 )2
Let’s put that replenishment rate standard deviation of 1.634 into our formula for safety stock.
10 10,000
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
400 plus 2.669
Let’s put that replenishment rate standard deviation of 1.634 into our formula for safety stock. And square it.
10 10,000
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
400 plus 2.669
And work our formula.
10 10,000
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
(400) + (2.669)
And work our formula.
(10) (10,000)
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
And work our formula.
30,700
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
175 cases of safety stock required.
Let’s bring it all together.
Back to our daily sales.
Day Sales in
Cases
Day Sales in
Cases
Day Sales in
Cases
1 100 10 110 19 110
2 80 11 130 20 120
3 70 12 120 21 70
4 60 13 100 22 100
5 80 14 80 23 130
6 90 15 90 24 110
7 120 16 90 25 90
8 110 17 100
9 100 18 140
From Strategic Logistics Management by Stock and Lambert
The lowest number of salesin a day was 60.
The highest number of salesin a day was 140.
Back to our replenishment rate
Order Number Days required
to receive order
Order Number Days required
to receive order
1 7 9 8
2 10 10 9
3 10 11 9
4 13 12 10
5 12 13 10
6 11 14 11
7 8 15 11
8 9 16 12
The fastest we received an order was 7 days.
The slowest we received an order was 13 days.
Therefore…
• We have daily sales variation from 60 to 140 cases.
• We have replenishment rate variability from 7 to 13 days.
• We calculated that we would need 175 cases of safety stock to provide adequate inventory for…..
• We have daily sales variation from 60 to 140 cases.
• We have replenishment rate variability from 7 to 13 days.
• We calculated that we would need 175 cases of safety stock to provide adequate inventory for…..Well, we can’t know how adequate that is, can we?
• Yes, we can know.
• Yes, we can know. By looking at service levels.
• Yes, we can know. By looking at service levels.
• And by remembering that in our formula for finding safety stock we were working with 1 standard deviation for our daily sales and our replenishment rate.
We’ve seen this before:
• Standard deviation represents an average of how far observations are away from the mean.
• Standard deviation represents an average of how far observations are away from the mean.
• There are certain characteristics of standard deviation in a normal distribution…
We’ve seen this before:
We have a mean of x
We have a mean of x
We have a standarddeviation of y
If we determine 1 standard deviation above and below the mean…
We have a mean of x
We have a standarddeviation of y
If we determine 1 standard deviation above and below the mean…
x+y = 1 standard deviation
x-y = 1 standarddeviation
We have a standarddeviation of y
We have a mean of x
If we determine 1 standard deviation above and below the mean…
x+y = 1 standard deviation
x-y = 1 standarddeviation
We have a standarddeviation of y
We have a mean of xIn a normal distributionabout 68%% of the observations will usually be within 1 standard deviation of the mean.
If we determine 1 standard deviation above and below the mean…
x+y = 1 standard deviation
x-y = 1 standarddeviation
We have a standarddeviation of y
We have a mean of xIn a normal distributionabout 68%% of the observations will usually be within 1 standard deviation of the mean.
68.26%
If we determine 1 standard deviation above and below the mean…
x+y = 1 standard deviation
x-y = 1 standarddeviation
We have a standarddeviation of y
We have a mean of x
68.26%
1 standard deviation of safety stock will give usan 84.13% service level.
If we determine 1 standard deviation above and below the mean…
x+y = 1 standard deviation
x-y = 1 standarddeviation
We have a standarddeviation of y
We have a mean of x
68.26%
1 standard deviation of safety stock will give usan 84.13% service level.Just figure 100%-68.26%
If we determine 1 standard deviation above and below the mean…
x+y = 1 standard deviation
x-y = 1 standarddeviation
We have a standarddeviation of y
We have a mean of x
68.26%
1 standard deviation of safety stock will give usan 84.13% service level.Just figure 100%-68.26%= 31.74%
If we determine 1 standard deviation above and below the mean…
x+y = 1 standard deviation
x-y = 1 standarddeviation
We have a standarddeviation of y
We have a mean of x
68.26%
1 standard deviation of safety stock will give usan 84.13% service level.Just figure 100%-68.26%= 31.74%, then divide31.74% by 2
If we determine 1 standard deviation above and below the mean…
x+y = 1 standard deviation
x-y = 1 standarddeviation
We have a standarddeviation of y
We have a mean of x
68.26%
1 standard deviation of safety stock will give usan 84.13% service level.Just figure 100%-68.26%= 31.74%, then divide31.74% by 2 = 15.87%
If we determine 1 standard deviation above and below the mean…
x+y = 1 standard deviation
x-y = 1 standarddeviation
We have a standarddeviation of y
We have a mean of x
68.26%
1 standard deviation of safety stock will give usan 84.13% service level.Just figure 100%-68.26%= 31.74%, then divide31.74% by 2 = 15.87%Add that to 68.26%
If we determine 1 standard deviation above and below the mean…
x+y = 1 standard deviation
We have a standarddeviation of y
We have a mean of x
68.26%
1 standard deviation of safety stock will give usan 84.13% service level.Just figure 100%-68.26%= 31.74%, then divide31.74% by 2 = 15.87%Add that to 68.26%15.87% + 68.26%= 84.13%
• That’s how we know 175 cases of safety stock for our hypothetical company will provide us with enough stock 84% of the time.
Safety stock =
Because when we calculated this formula, wewere using 1 standard deviation.
Mean of replenishment rate ( Standard
deviation ofdaily sales
) plusMean ofdaily salessquared (
2Standarddeviation ofreplenishmentrate
)2
Safety stock =
If we wanted higher service levels (and 84% is notvery good), we would increase the standard deviationwhen we calculated the formula.
Mean of replenishment rate ( Standard
deviation ofdaily sales
) plusMean ofdaily salessquared (
2Standarddeviation ofreplenishmentrate
)2
Service Levels
1 standard deviation of safety stock = 1-.6826 + .6826 = .8413 2
Service Levels
1 standard deviation of safety stock = 1-.6826 + .6826 = .8413 2 2 standard deviation of safety stock = 1-.9544 + .9544 = .9772 2
Service Levels
1 standard deviation of safety stock = 1-.6826 + .6826 = .8413 2 2 standard deviation of safety stock = 1-.9544 + .9544 = .9772 2 3 standard deviation of safety stock = 1-.9974 + .9974 = .9987 2
Service Levels
1 standard deviation of safety stock = 1-.6826 + .6826 = .8413 2 2 standard deviation of safety stock = 1-.9544 + .9544 = .9772 2 3 standard deviation of safety stock = 1-.9974 + .9974 = .9987 2
84% service level
Service Levels
1 standard deviation of safety stock = 1-.6826 + .6826 = .8413 2 2 standard deviation of safety stock = 1-.9544 + .9544 = .9772 2 3 standard deviation of safety stock = 1-.9974 + .9974 = .9987 2
84% service level
Almost98%servicelevel.
Service Levels
1 standard deviation of safety stock = 1-.6826 + .6826 = .8413 2 2 standard deviation of safety stock = 1-.9544 + .9544 = .9772 2 3 standard deviation of safety stock = 1-.9974 + .9974 = .9987 2
84% service level
Almost98%servicelevel.
Almost100% servicelevel
Service Levels
1 standard deviation of safety stock = 1-.6826 + .6826 = .8413 2 2 standard deviation of safety stock = 1-.9544 + .9544 = .9772 2 3 standard deviation of safety stock = 1-.9974 + .9974 = .9987 2
84% service level
Almost98%servicelevel.
Almost100% servicelevel
On the next page is a service level chart. It tells you the standarddeviation to use to achieve a specific service level.
Service Level Table
Service Level Number of standard deviations of safety stock needed.
84.1% 1
90.3% 1.3
94.5% 1.6
97.7% 2
98.9% 2.3
99.5% 2.6
99.9% 3
So how do we apply this?
So how do we apply this?
Suppose we want a
94.5% service level.
So how do we apply this?
Suppose we want a 94.5% service level.
That means that when a customer wants a product, 94.5% of the time the product will be
in stock.
Service Level Table
Service Level Number of standard deviations of safety stock needed.
84.1% 1
90.3% 1.3
94.5% 1.6
97.7% 2
98.9% 2.3
99.5% 2.6
99.9% 3
We multiply the standarddeviations of our formula
by 1.6
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
Mean of replenishment rate ( Standard
deviation ofdaily sales
) plusMean ofdaily salessquared (
2Standarddeviation ofreplenishmentrate
)2
times 1.6 times 1.6
In Our Original Formula…
• We used 1 standard deviation.
• Standard deviation of daily sales was 20
• The standard deviation of the replenishment rate was 1.634
In Our New Formula…
• We will use the standard deviation times 1.6
• Standard deviation of daily sales was 20
• The standard deviation of the replenishment rate was 1.634
• Therefore, we multiply 20 by 1.6 = 32.
• And 1.634 by 1.6 = 2.6144
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
Mean of replenishment rate ( Standard
deviation ofdaily sales
) plusMean ofdaily salessquared (
2Standarddeviation ofreplenishmentrate
)2
1 standard deviation=20
1 standard deviation=1.634
times 1.6
= 32
times 1.6
= 2.6144
Meaning our original formula will now be changed to…
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
Mean of replenishment rate ( Standard
deviation ofdaily sales
) plusMean ofdaily salessquared (
2Standarddeviation ofreplenishmentrate
)2
1 standard deviation=20
1 standard deviation=1.634
times 1.6
= 32
times 1.6
= 2.6144
Meaning our original formula will now be changed to…
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
Mean of replenishment rate ( ) plus
Mean ofdaily salessquared (
2
)2
32 2.6144
Meaning our original formula will now be changed to…
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
10 ( ) plus 10,000 (2
)2
32 2.6144
Meaning our original formula will now be changed to…
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
10 plus 10,000
= 280.333 rounded to 280
1024 6.8350XX
To Provide a 94.5% Service Level…
• We need 280 units of safety stock.
We’ve Just Seen How to Determine Safety Stock.
• But how much average inventory should we have to achieve various levels of customer service?
• We need to– Determine our service level.– Determine our Economic Ordering Quantity (EOQ)– Determine our average cycle stock.– Determine our safety stock level. – Add average cycle stock and safety stock.
To Determine Average Inventory:
• Determine our service level. Let’s say it’s 84.1%
• Determine our Economic Ordering Quantity (EOQ).
• Determine our average cycle stock.
• Determine our safety stock level.
• Add average cycle stock and safety stock.
To Determine Economic Ordering Quantity
2CoD EOQ = CiUWhere
EOQ = Economic ordering quantity.Co = ordering cost (dollars per order)Ci = Annual inventory carry costs (% product cost or value) D= Annual demand (number of units)U = Average cost or value of one unit of inventory
To Determine Economic Ordering Quantity
2CoD EOQ = CiUWhere
EOQ = Economic ordering quantity.Co = ordering cost (dollars per order)Ci = Annual inventory carry costs (% product cost or value) D= Annual demand (number of units)U = Average cost or value of one unit of inventory
We will use some data from thehypothetical organization we looked
at earlier.
Day Sales in
Cases
Day Sales in
Cases
Day Sales in
Cases
1 100 10 110 19 110
2 80 11 130 20 120
3 70 12 120 21 70
4 60 13 100 22 100
5 80 14 80 23 130
6 90 15 90 24 110
7 120 16 90 25 90
8 110 17 100
9 100 18 140
From Strategic Logistics Management by Stock and Lambert
Mean = 100
To Determine Economic Ordering Quantity
2CoD EOQ = CiUWhere
EOQ = Economic ordering quantity.Co = ordering cost (dollars per order)Ci = Annual inventory carry costs (% product cost or value) D= Annual demand (number of units)U = Average cost or value of one unit of inventory
We will use some data from thehypothetical organization we looked
at earlier.
To Determine Economic Ordering Quantity
2CoD EOQ = CiUWhere
EOQ = Economic ordering quantity.Co = ordering cost (dollars per order)Ci = Annual inventory carry costs (% product cost or value) D= Annual demand (number of units)U = Average cost or value of one unit of inventory
Our hypothetical company had mean daily sales of 100. We multiplyThat by 250 business days which gives annual demand of 25,000
To Determine Economic Ordering Quantity
2Co (25,000) EOQ = CiUWhere
EOQ = Economic ordering quantity.Co = ordering cost (dollars per order)Ci = Annual inventory carry costs (% product cost or value) D= Annual demand (number of units)U = Average cost or value of one unit of inventory
Our hypothetical company had mean daily sales of 100. We multiplyThat by 250 business days which gives annual demand of 25,000
To Determine Economic Ordering Quantity
2Co (25,000) EOQ = CiUWhere
EOQ = Economic ordering quantity.Co = ordering cost (dollars per order)Ci = Annual inventory carry costs (% product cost or value) D= 25,000U = Average cost or value of one unit of inventory
Our other values will be arbitrary for the sake of this exercise.
To Determine Economic Ordering Quantity
2(28)x (25,000) EOQ = CiUWhere
EOQ = Economic ordering quantity.Co = $28Ci = Annual inventory carry costs (% product cost or value) D= 25,000U = Average cost or value of one unit of inventory
Our other values will be arbitrary for the sake of this exercise.
To Determine Economic Ordering Quantity
2(28)x (25,000) EOQ = .32 x UWhere
EOQ = Economic ordering quantity.Co = $28Ci = 32%D= 25,000U = Average cost or value of one unit of inventory
Our other values will be arbitrary for the sake of this exercise.
To Determine Economic Ordering Quantity
2(28)x (25,000) EOQ = .32 x 4.37Where
EOQ = Economic ordering quantity.Co = $28Ci = 32%D= 25,000U = $4.37 per case
Our other values will be arbitrary for the sake of this exercise.
To Determine Economic Ordering Quantity
EOQ = 1,000 Where
EOQ = Economic ordering quantity.Co = $28Ci = 32%D= 25,000U = $4.37 per case
To Determine Average Inventory
• Determine our service level. Let’s say it’s 84.1%
• Determine our Economic Ordering Quantity (EOQ).
• Determine our average cycle stock.
• Determine our safety stock level.
• Add average cycle stock and safety stock.
To Determine Average Cycle Stock…
As we saw earlier, it is one half of order quantity
200
400
0
Days 10 20 30 40 50 60
Inventory
Orderplaced
Orderarrival
Orderplaced Average
cycleinventory
A. Order quantity of 400 units
Orderarrival
The Effect of Reorder Quantity on Average Inventory Investment with Constant
Demand and Lead Time
a6-3 a
Cycle stock is one-half the ordering quantity.
From instructor’s material: “Strategic Logistics Management” by Stock and Lambert(2001).
Cycle Stock = ½ Ordering Quantity
EOQ = 1,000 Where
EOQ = Economic ordering quantity.Co = $28Ci = 32%D= 25,000U = $4.37 per case
2= 500
To Determine Average Inventory
• Determine our service level. Let’s say it’s 84.1%
• Determine our Economic Ordering Quantity (EOQ).
• Determine our average cycle stock.
• Determine our safety stock level.
• Add average cycle stock and safety stock.
Safety stock =
Safety stock required when there is variability inboth demand and lead time.
Mean of replenishment rate ( Standard
deviation ofdaily sales
) plusMean ofdaily salessquared (
2Standarddeviation ofreplenishmentrate
)2
Using our data from our hypothetical organization, we havealready seen that for an 84.1% service level, we need 175cases of safety stock.
To Determine Average Inventory
• Determine our service level. Let’s say it’s 84.1%
• Determine our Economic Ordering Quantity (EOQ).
• Determine our average cycle stock.
• Determine our safety stock level.
• Add average cycle stock and safety stock.
To Determine Average Inventory
Average cycle stock plus safety stock for this service level (84.1%)
500 casesAverage cycle stock
+ 175Safety stock
= 675 casesAverageinventory
To Determine Average Inventory
• Determine our service level. Let’s say it’s 84.1%
• Determine our Economic Ordering Quantity (EOQ).
• Determine our average cycle stock.
• Determine our safety stock level.
• Add average cycle stock and safety stock.
End of Program.
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