Re-Order Point Problems
Set 1: General
Notations
Lead Time Demand : LTD
Average Lead Time Demand : LTD
Std. Dev. Lead Time Demand :  LTD
Safety Stock : Isafety
Reorder Point : ROP
Demand per period : R
Average Demand : R
Std. Dev. of Demand :  R
Lead Time : L
Average Lead Time : L
Std. Dev. of Lead Time :  L
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
2
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.
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
3
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
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
4
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 = 200 + 70 = 270
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
5
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.
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
6
Safety Stock and ROP
Risk of a
stockout
Service level
Probability of
no stockout
ROP
Average
demand
Quantity
Safety stock
0
z
z-scale
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
a) Given any probability of not exceeding z. What is the value of z.
b) Given any value for z. What is the probability of not exceeding z.
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
7
Popular z Values
Risk Service level
0.1
0.9
0.05
0.95
0.01
0.99
Flow Variability; Safety Inventory
z value
1.28
1.65
2.33
z
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
0.00
0.5000
0.5398
0.5793
0.6179
0.6554
0.6915
0.7257
0.7580
0.7881
0.8159
0.8413
0.8643
0.8849
0.9032
0.9192
0.9332
0.9452
0.9554
0.9641
0.9713
0.9772
0.9821
0.9861
0.9893
0.9918
0.9938
0.9953
0.9965
0.9974
0.9981
0.01
0.5040
0.5438
0.5832
0.6217
0.6591
0.6950
0.7291
0.7611
0.7910
0.8186
0.8438
0.8665
0.8869
0.9049
0.9207
0.9345
0.9463
0.9564
0.9649
0.9719
0.9778
0.9826
0.9864
0.9896
0.9920
0.9940
0.9955
0.9966
0.9975
0.9982
0.02
0.5080
0.5478
0.5871
0.6255
0.6628
0.6985
0.7324
0.7642
0.7939
0.8212
0.8461
0.8686
0.8888
0.9066
0.9222
0.9357
0.9474
0.9573
0.9656
0.9726
0.9783
0.9830
0.9868
0.9898
0.9922
0.9941
0.9956
0.9967
0.9976
0.9982
0.03
0.5120
0.5517
0.5910
0.6293
0.6664
0.7019
0.7357
0.7673
0.7967
0.8238
0.8485
0.8708
0.8907
0.9082
0.9236
0.9370
0.9484
0.9582
0.9664
0.9732
0.9788
0.9834
0.9871
0.9901
0.9925
0.9943
0.9957
0.9968
0.9977
0.9983
Ardavan Asef-Vaziri
0.04
0.5160
0.5557
0.5948
0.6331
0.6700
0.7054
0.7389
0.7704
0.7995
0.8264
0.8508
0.8729
0.8925
0.9099
0.9251
0.9382
0.9495
0.9591
0.9671
0.9738
0.9793
0.9838
0.9875
0.9904
0.9927
0.9945
0.9959
0.9969
0.9977
0.9984
0.05
0.5199
0.5596
0.5987
0.6368
0.6736
0.7088
0.7422
0.7734
0.8023
0.8289
0.8531
0.8749
0.8944
0.9115
0.9265
0.9394
0.9505
0.9599
0.9678
0.9744
0.9798
0.9842
0.9878
0.9906
0.9929
0.9946
0.9960
0.9970
0.9978
0.9984
Sep-2012
0.06
0.5239
0.5636
0.6026
0.6406
0.6772
0.7123
0.7454
0.7764
0.8051
0.8315
0.8554
0.8770
0.8962
0.9131
0.9279
0.9406
0.9515
0.9608
0.9686
0.9750
0.9803
0.9846
0.9881
0.9909
0.9931
0.9948
0.9961
0.9971
0.9979
0.9985
0.07
0.5279
0.5675
0.6064
0.6443
0.6808
0.7157
0.7486
0.7794
0.8078
0.8340
0.8577
0.8790
0.8980
0.9147
0.9292
0.9418
0.9525
0.9616
0.9693
0.9756
0.9808
0.9850
0.9884
0.9911
0.9932
0.9949
0.9962
0.9972
0.9979
0.9985
0.08
0.5319
0.5714
0.6103
0.6480
0.6844
0.7190
0.7517
0.7823
0.8106
0.8365
0.8599
0.8810
0.8997
0.9162
0.9306
0.9429
0.9535
0.9625
0.9699
0.9761
0.9812
0.9854
0.9887
0.9913
0.9934
0.9951
0.9963
0.9973
0.9980
0.9986
0.09
0.5359
0.5753
0.6141
0.6517
0.6879
0.7224
0.7549
0.7852
0.8133
0.8389
0.8621
0.8830
0.9015
0.9177
0.9319
0.9441
0.9545
0.9633
0.9706
0.9767
0.9817
0.9857
0.9890
0.9916
0.9936
0.9952
0.9964
0.9974
0.9981
0.9986
8
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
ROP = LTD +z σLTD
σ = Standard Deviation of x

ROP = LTD +Isafety
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
9
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
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
10
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
z = (X- μ )/σ
LTD = 200
X= μ +z σ
σLTD = 25
ROP = LTD + Isafety
ROP= 200+ 1.28 × 25
Isafety = z σLTD
ROP = 200 + 32
ROP=LTD + zσLTD
SL = 90%
ROP = 232
Isafety = 32
z = 1.28
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
11
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.
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
12
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.
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
13
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
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
14
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
𝜎𝐿𝑇𝐷 = 𝐿𝜎𝑅
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
15
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
𝜎𝐿𝑇𝐷 = 𝐿𝜎𝑅
Flow Variability; Safety Inventory
𝜎𝐿𝑇𝐷 = 16 5 = 20
Ardavan Asef-Vaziri
Sep-2012
16
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.
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
17
ROP; Variable R, Fixed L
SL = 90%  z = 1.28
x=μ+zσ
ROP= LTD +z σLTD
LTD = 320
σLTD = 20
ROP= 320+ 1.28 × 20
ROP= 320 + 25.6
ROP= 320 + 26
ROP = 346
Isafety = 26
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
18
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
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
19
μ and σ of L and Fixed R
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
𝜎𝐿𝑇𝐷 =R𝜎𝐿
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
20
μ and σ of L and Fixed R
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
𝜎𝐿𝑇𝐷 =R𝜎𝐿
𝜎𝐿𝑇𝐷 =20(4) =80
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
21
μ and σ of L and Fixed R
LDT = 320
LTD = 80
SL = 90%
z =1.28
ROP = LTD +zLTD
ROP = 320 +1.28(80)
ROP = 320 + 102.4
Isafety = 102.4
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
22
Comparing the two problems
I.
Average demand per day is 10 units and standard
deviation of demand per day is 3 units. The lead
time is 5 days
II. Demand per day is 10 units. The average lead time is
3 days and standard deviation of lead time is 1 days.
1
2
3
Flow Variability; Safety Inventory
4
5
1
2
Ardavan Asef-Vaziri
3
Sep-2012
4
5
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) = 0.7881 = 78.81%
In 78.81 % of the order cycles, the warehouse will not have a
stockout. Risk = 21.19%.
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
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 + 4,000 = 18000.
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
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
25
Problem 7.1
MassPC Inc. produces a 4-week supply of its PC Pal model when
stock on hand drops to 525 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 = 550 L = 1 week LTD = 400/week 𝜎𝐿𝑇𝐷 =125 Q = 4(400)
Normal table
LTD = N(400, 125).
ROP = LTD + Isafety
ROP = LTD +z LTD
525= 400+z(125)
125 = 125z
z = 125/125 = 1
z
SL  P( LTD  ROP)  0.84134
Flow Variability; Safety Inventory
Second digit
after decimal
Up to
the first
digit
after
decimal
Ardavan Asef-Vaziri
Sep-2012
Probability
26
Problem 7.1
b) Compute Isafety for 80%, 90%, 95%, 99% service levels.
% change
Z
0.8
0.9
0.95
0.99
100
113
119
124
0.84
1.28
1.64
2.33
σLTD Is =zσLTD % change ROP = LTD +Is
125
125
125
125
105
160
206
291
100
152
195
276
505
560
606
691
Safety Stock
SL
Service Level
Increasing service level increases the required safety inventory
more than proportionately.
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
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
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
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
Flow Variability; Safety Inventory
Ardavan Asef-Vaziri
Sep-2012
29