1. Re-Order Point Problems Set 1: General

2. Notations

3. ROP Problems
a) If average demand per day is 20 units and lead time is 10 days. Assuming zero safety stock. Isafety=0, compute ROP.
b) If average demand per day is 20 units and lead time is 10 days. Assuming 70 units of safety stock. Isafety=70, compute ROP?
c) If average demand during the lead time is 200 and standard deviation of demand during lead time is 25. Compute ROP at 90% service level. Compute safety stock.
d) If average demand per day is 20 units and standard deviation of daily demand is 5, and lead time is 16 days. Compute ROP at 90% service level. Compute safety stock.
e) If demand per day is 20 units and lead time is 16 days and standard deviation of lead time is 4 days. Compute ROP at 90% service level. Compute safety stock.

4. ROP; Fixed R, Fixed L, Zero Safety Stock
a) If average demand per day is 20 units and lead time is 10 days. Assuming zero safety stock.
Isafety = 0, L=10, R=20
Compute ROP.
ROP = average demand during lead time + safety stock
ROP = LTD+ Isafety
ROP = LTD + 0
LTD = L × R
LTD = 20 × 10 = 200
ROP = 200

5. ROP; Fixed R, Fixed L, Isafety >0
b) If average demand per day is 20 units and lead time is 10 days. Assuming 70 units of safety stock.
Isafety = 70, L=10, R=20
Compute ROP.
ROP = average demand during lead time + Safety Stock
ROP = LTD + Isafety
LTD = R × L = 20 × 10 = 200
Isafety = 70
ROP = = 270

6. ROP; total demand during lead time is variable
c) If average demand during the lead time is 200 and standard deviation of demand during lead time is 25. Compute ROP at 90% service level. Compute safety stock.

7. Safety Stock and ROP
Service level
Risk of a
stockout
Probability of
no stockout
ROP
Average
demand
Quantity
Safety stock
z-scale z
Each Normal variable x is associated with a standard Normal Variable z
x is Normal (Average x , Standard Deviation x)  z is Normal (0,1)
There is a table for z which tells us
Given any probability of not exceeding z. What is the value of z.
Given any value for z. What is the probability of not exceeding z.

8. Popular z Values Risk Service level z value 0.1 0.9 1.28

9. ROP and Isafety z = (x-Average x)/(Standard Deviation of x) x = μ +z σ
x = Average x +z (Standard Deviation of x)
μ = Average x
σ = Standard Deviation of x
x = μ +z σ 
ROP = LTD +z σLTD 
ROP = LTD +Isafety

10. ROP and Isafety LTD = Lead Time Demand (Demand during lead time)
LTD = Average LTD = L×R
LTD = Standard Deviation of Lead Time Demand
Isafety = zLTD

11. ROP; total demand during lead time is variable
c) If average demand during the lead time is 200 and standard deviation of demand during lead time is 25. Compute ROP at 90% service level. Compute safety stock
LTD = 200
σLTD = 25
ROP= × 25
ROP =
ROP = 232
Isafety = 32
z = (X- μ )/σ
X= μ +z σ
ROP = LTD + Isafety
Isafety = z σLTD
ROP=LTD + zσLTD
SL = 90%
z = 1.28

12. ROP; Variable R, Fixed L
d) If average demand per day is 20 units and standard deviation of demand is 5 per day, and lead time is 16 days. Compute ROP at 90% service level. Compute safety stock.

13. ROP; Variable R, Fixed L
Previous Problem: If average demand during the lead time is 200 and standard deviation of demand during lead time is 25. Compute ROP at 90% service level. Compute safety stock.
This Problem: If average demand per day is 20 units and standard deviation of demand per day is 5, and lead time is 16 days. Compute ROP at 90% service level. Compute safety stock.
If we can transform this problem into the previous problem, then we are done, because we already know how to solve the previous problem.

14. ROP; Variable R, Fixed R
d) If average demand per day is 20 units and standard deviation of demand is 5 per day, and lead time is 16 days. Compute ROP at 90% service level. Compute ss.
What is the average demand during the lead time
What is standard deviation of demand during lead time

15. μ and σ of demand per period and fixed L
If Demand is variable and Lead time is fixed
L: Lead Time
R: Demand per period (per day, week, month)
R: Average Demand per period (day, week, month)
R: Standard deviation of demand (per period)
LTD: Average Demand During Lead Time
LTD = L × R
LTD: Standard deviation of demand during lead time
𝜎 𝐿𝑇𝐷 = 𝐿 𝜎 𝑅

16. μ and σ of demand per period and fixed L
If demand is variable and Lead time is fixed
L: Lead Time = 16 days
R: Demand per day
R: Average daily demand =20
R: Standard deviation of daily demand =5
LTD: Average Demand During Lead Time
LTD = L × R = 16 × 20 = 320
LTD: Standard deviation of demand during lead time
𝜎 𝐿𝑇𝐷 = 𝐿 𝜎 𝑅
𝜎 𝐿𝑇𝐷 = =20

17. Now It is Transformed
The Problem originally was: If average demand per day is 20 units and standard deviation of demand is 5 per day, and lead time is 16 days. Compute ROP at 90% service level. Compute safety stock.
We transformed it to: The average demand during the lead time is 320 and the standard deviation of demand during the lead time is 20. Compute ROP at 90% service level. Compute safety stock.
Which is the same as the previous problem: If average demand during the lead time is 200 and standard deviation of demand during lead time is 25. Compute ROP at 90% service level. Compute safety stock.

18. ROP; Variable R, Fixed L SL = 90%  z = 1.28 x = μ + z σ
ROP= LTD +z σLTD
LTD = 320
σLTD = 20
ROP= × 20
ROP=
ROP=
ROP = 346
Isafety = 26

19. ROP; Variable R, Fixed L
e) If demand per day is 20 units and lead time is 16 days and standard deviation of lead time is 4 days. Compute ROP at 90% service level. Compute Isafety.
What is the average demand during the lead time
What is standard deviation of demand during lead time

20. μ and σ of L and Fixed R 𝜎 𝐿𝑇𝐷 =R 𝜎 𝐿
If Lead time is variable and Demand is fixed
L: Lead Time
L: Average Lead Time
L: Standard deviation of Lead time
R: Demand per period
LTD: Average Demand during lead time
LTD = L × R
LTD: Standard deviation of demand during lead time
𝜎 𝐿𝑇𝐷 =R 𝜎 𝐿

21. μ and σ of L and Fixed R 𝜎 𝐿𝑇𝐷 =R 𝜎 𝐿 𝜎 𝐿𝑇𝐷 =20(4) =80
If Lead time is variable and Demand is fixed
L: Lead Time
L: Average Lead Time = 16 days
L: Standard deviation of Lead time = 4 days
R: Demand per period = 20 per day
LTD: Average Demand During Lead Time
LTD = 16 × 20 = 320
LTD: Standard deviation of demand during lead time
𝜎 𝐿𝑇𝐷 =R 𝜎 𝐿
𝜎 𝐿𝑇𝐷 =20(4) =80

22. μ and σ of L and Fixed R LDT = 320 LTD = 80 SL = 90% z =1.28
ROP = LTD +zLTD
ROP = (80)
ROP =
Isafety = 102.4

23. Given Safety Inventory Compute Service Level
Average lead time demand is 20,000 units. Standard deviation of lead time demand is estimated to be 5,000 units. The warehouse currently orders a 14-day supply, 28,000 units, each time the inventory level drops to 24,000 units. Suppose holding cost is $2 per unit per year.
LTD = 20,000, LTD = 5,000, ROP = 24,000, H=$2, R = 2,000 / day, Q or EOQ = 28,000.
a) Compute the service level.
ROP = LTD + Isafety  Isafety = 24,000 – 20,000 = 4,000
ROP = LTD +z LTD  Isafety = z LTD =4000
z(5,000) = 4,000  z = 4,000/5,000 = 0.8
z =0.8  P(z ≤ Z) = = 78.81%
In % of the order cycles, the warehouse will not have a stockout. Risk = 21.19%.

24. Given Safety Inventory Compute Service Level
b) Compute the cycle inventory, and average inventory.
At reorder point we order 28,000 units.
Icycle = Q/2 = 28,000/2 = 14,000
Isafety = 4,000
Average Inventory = I = Icycle + Isafety =14, ,000 =
c) Compute the total holding costs per year.
H(Average Inventory) = H(I) = 2  18,000 = 36,000/year
d) Compute the average flow time.
R = 2,000 / day, I = 18,000
RT = I 2000T = 18,000
T = 9
9 days

25. Problem 7.1
MassPC Inc. produces a 4-week supply of its PC Pal model when stock on hand drops to 500 units. It takes 1 week to produce a batch. Orders average 400 units per week, and standard deviation of forecast errors is estimated at 125 units.
ROP =
4(400)
500
L =
1 week
LTD =
400/week
𝜎 𝐿𝑇𝐷 =125
Q =
Normal table
LTD = N(400, 125).
ROP = LTD + Isafety
ROP = LTD +z LTD
500= 400+z(125)
100 = 125z
z = 100/125 = 0.8 z
Second digit
after decimal
Up to the first digit
after
decimal
Probability

26. Problem 7.1 b) Compute Isafety for 80%, 90%, 95%, 99% service levels.
Safety Stock
Service Level
Increasing service level increases the required safety inventory more than proportionately.

27. Problem 7.1
Selecting service level and safety inventory level is an important strategic decision.
A firm may choose high quality service in terms of product availability. Or
The firm may choose to be a low cost supplier by holding down inventory costs. In either case, it is positioning itself along the quality vs. cost trade-off curve.
This graph is a perfect example to show how a strategic position could be operationalized.
Cost
Quality

28. Service Level, Fill Rate
Within 100 time intervals, stockouts occur in 20.
Service Level = 80/100 = 80%.
Probability of stock-out = 20%.
Risk = # of stockout intervals/ Total # of intervals
Service Level = 1- (# of stockout intervals/ Total # of intervals)
Suppose that in each time interval in which a stockout occurred, the number of units by which we were short.
Suppose that cumulative demand during the 100 time intervals was 15,000 units and the total number of units short in the 20 intervals with stockouts was 1,500 units.
Fill rate = (15,000-1,500)/15,000 = 13,500/15,000 = 90%.
Fill Rate = Expected Sales / Expected Demand
Fill Rate = (1- Expected Stockout )/ Expected Demand