Management Inventory 2
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Zhibin Yang
Assistant Professor, Decision
Sciences
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1
DSC 335 Roadmap
Operations Strategy
Decision Making Tools
Process strategy/analysis
Capacity analysis/planning
Process Management
Littlefield Game 1
Waiting line models
Case: Kristens Cookie
Lean systems
Quality management
Inventory management
Case: Blanchard
Supply Chain Mgmt.
Supply chain dynamics
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Littlefield Game 2
2
Beer game
Inventory Management (2)
Continuous review system
Periodic review system
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3
Outline of Inventory Management (2)
q
Continuous review systems
q
Periodic review systems
q
ABC analysis
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4
Two types of Inventory Control Systems
q
Continuous review system (Q-system)
When to order: when inventory declines to ROP
q Event-trigger restocking
q Also known as: Reorder Point (ROP) system
q How much: a fixed quantity is ordered every time
q EOQ is a continuous review system with certain demand
q
q
Periodic review system (P-system)
When to order: an order is placed after a fixed period of time
q Time-triggered restocking
q How much: An order of variable amount
q
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5
Continuous Review, Certain Demand EOQ
Inventory
Q
Slope
= D (units/yr)
= d (units/day)
Reorder
Point
(ROP)
Receive
order
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Place
order
Time
Receive
order
Reorder Point: ROP = dL
Lead time:
L (days)
6
When Demand is Uncertain
q
Average demand rate: d
q
Is ROP = d L good enough?
Stockout may occur, only
during delivery lead time!!!
Inventory
dL
Stockout!
slope d
slope d
0
Time
Receive order
Place order
L
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Receive order
Place order
L
7
Adding Cushion Safety Stock (SS)
ROP = dL + SS
Inventory
Increase
ROP by SS
ROP
dL
SS
Time
0
L
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L
8
Safety Stock (SS)
q
q
q
q
When demand has unpredictable variability, stockout occur
when actual demand during lead time exceeds ROP
Safety stock is held to cushion against uncertainties
ROP = average demand during lead time + safety stock
What determines safety stock?
q Service level= Probability of no stockout during lead time
= 1 (Probability of stockout during lead time)
q Variability of the demand (rate)
q Lead time
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What Determines Safety Stock?
Probability distribution of demand
during lead time (L days)
Distribution of
demand of higher
variability
Probability of no
stockout in lead time
(= service level)
0
dL
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SS
ROP
10
Probability of
stockout during
lead time
Demand during
lead time
What Determines Safety Stock? (contd)
Inventory
L
ROP
L
dL
Service Level
Distribution of
demand during
lead time
SS
Increased demand
variability
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11
Compute ROP for a Given Service Level
If demand has a normal distribution
Probability distribution of
demand during lead time
SS = z dLT
ROP = d L + z dLT
Service level
0
dL
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SS
ROP
12
Demand during
lead time
Standard Deviation of Lead-Time Demand
dLT
One-day demand
Lead time L = 2
Two-day demand
d d
_
d=10
dLT
20
__
dLT = 32
d = 3
Daily demand variability d
Lead time L days
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dLT
13
dLT = d L
Demand
z Value for a Service Level
using Normal Distribution Table
The area under standard
normal distribution
function represents the
service level.
Find the corresponding
z value for the given
service level.
Example: a service level
of 0.95 has a z value of
1.6 + (0.04+0.05)/2 =
1.645
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Service
level
z
0.0
0.1
0.2
0.3
0.4
.00
0.5000
0.5398
0.5793
0.6179
0.6554
.01
0.5040
0.5438
0.5832
0.6217
0.6591
.02
0.5080
0.5478
0.5871
0.6255
0.6628
.03
0.5120
0.5517
0.5910
0.6293
0.6664
.04
0.5160
0.5557
0.5948
0.6331
0.6700
.05
0.5199
0.5596
0.5987
0.6368
0.6736
.06
0.5239
0.5636
0.6026
0.6406
0.6772
.07
0.5279
0.5675
0.6064
0.6443
0.6808
.08
0.5319
0.5714
0.6103
0.6480
0.6844
.09
0.5359
0.5753
0.6141
0.6517
0.6879
0.5
0.6
0.7
0.8
0.9
0.6915
0.7257
0.7580
0.7881
0.8159
0.6950
0.7291
0.7611
0.7910
0.8186
0.6985
0.7324
0.7642
0.7939
0.8212
0.7019
0.7357
0.7673
0.7967
0.8238
0.7054
0.7389
0.7704
0.7995
0.8264
0.7088
0.7422
0.7734
0.8023
0.8289
0.7123
0.7454
0.7764
0.8051
0.8315
0.7157
0.7486
0.7794
0.8078
0.8340
0.7190
0.7517
0.7823
0.8106
0.8365
0.7224
0.7549
0.7852
0.8133
0.8389
1. 0
1.1
1.2
1.3
1.4
0.8413
0.8643
0.8849
0.9032
0.9192
0.8438
0.8665
0.8869
0.9049
0.9207
0.8461
0.8686
0.8888
0.9066
0.9222
0.8485
0.8708
0.8907
0.9082
0.9236
0.8508
0.8729
0.8925
0.9099
0.9251
0.8531
0.8749
0.8944
0.9115
0.9265
0.8554
0.8770
0.8962
0.9131
0.9279
0.8577
0.8790
0.8980
0.9147
0.9292
0.8599
0.8810
0.8997
0.9162
0.9306
0.8621
0.8830
0.9015
0.9177
0.9319
1.5
1.6
1.7
1.8
1.9
0.9332
0.9452
0.9554
0.9641
0.9713
0.9345
0.9463
0.9564
0.9649
0.9719
0.9357
0.9474
0.9573
0.9656
0.9726
0.9370
0.9484
0.9582
0.9664
0.9732
0.9382
0.9495
0.9591
0.9671
0.9738
0.9394
0.9505
0.9599
0.9678
0.9744
0.9406
0.9515
0.9608
0.9686
0.9750
0.9418
0.9525
0.9616
0.9693
0.9756
0.9429
0.9535
0.9625
0.9699
0.9761
0.9441
0.9545
0.9633
0.9706
0.9767
2.0
2.1
2.2
2.3
2.4
0.9772
0.9821
0.9861
0.9893
0.9918
0.9778
0.9826
0.9864
0.9896
0.9920
0.9783
0.9830
0.9868
0.9898
0.9922
0.9788
0.9834
0.9871
0.9901
0.9925
0.9793
0.9838
0.9875
0.9904
0.9927
0.9798
0.9842
0.9878
0.9906
0.9929
0.9803
0.9846
0.9881
0.9909
0.9931
0.9808
0.9850
0.9884
0.9911
0.9932
0.9812
0.9854
0.9887
0.9913
0.9934
0.9817
0.9857
0.9890
0.9916
0.9936
2.5
2.6
2.7
2.8
2.9
0.9938
0.9953
0.9965
0.9974
0.9981
0.9940
0.9955
0.9966
0.9975
0.9982
0.9941
0.9956
0.9967
0.9976
0.9982
0.9943
0.9957
0.9968
0.9977
0.9983
0.9945
0.9959
0.9969
0.9977
0.9984
0.9946
0.9960
0.9970
0.9978
0.9984
0.9948
0.9961
0.9971
0.9979
0.9985
0.9949
0.9962
0.9972
0.9979
0.9985
0.9951
0.9963
0.9973
0.9980
0.9986
0.9952
0.9964
0.9974
0.9981
0.9986
3.0
3.1
3.2
3.3
3.4
0.9987
0.9990
0.9993
0.9995
0.9997
0.9987
0.9987
0.9991
0.9991
0.9993
0.9994
0.9995
0.9995
0.9997 14 0.9997
0.9988
0.9991
0.9994
0.9996
0.9997
0.9988
0.9992
0.9994
0.9996
0.9997
0.9989
0.9992
0.9994
0.9996
0.9997
0.9989
0.9992
0.9994
0.9996
0.9997
0.9989
0.9992
0.9995
0.9996
0.9997
0.9990
0.9993
0.9995
0.9996
0.9997
0.9990
0.9993
0.9995
0.9997
0.9998
Sensitivity Analysis: Safety Stock
q
Safety stock increases (decreases) with an increase in:
demand (per day) variability or forecast error
q service level
q delivery lead time
q
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Differences: EOQ versus Continuous Review
System (Q-system)
EOQ
q
q
q
q
Q-system
Demand: deterministic and
constant
q
Objective: to minimize annual
total costs (holding + ordering)
Order size: EOQ
q
When to order: when inventory
level drops below ROP
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q
16
q
Demand: random (with
constant mean demand)
Objective: to achieve
customer service level
Order size: EOQ calculated
using mean demand
When to order: same as
EOQ
Example: Inventory of Bird-feeders
q
q
A museum of natural history opened a gift shop which operates
52 weeks per year. Managing inventories has become a problem.
Top-selling SKU is a bird feeder.
q The average sales (d) are 18 units per week with a standard
deviation of 5 units.
q The supplier charges $60 per unit (c).
q Ordering cost (S) is $45.
q Annual holding cost is 25% (i) of a feeders value.
q The lead time is constant at two (2) weeks.
q Management wants a 90% service level
Determine the order quantity (Q), safety stock (SS) and reorder
point (ROP)
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Solution
q
D= 18*52/year, S=$45/order, H=i*c=25%*60=$15/year
Set Q=EOQ, if a service level requirement is imposed.
q
Q = EOQ = 75 units
q
d = 5,d = 18 units/week, and L = 2 weeks
q
dLT = d (L)1/2 = 5 (2)1/2 = 7.07.
q
90% service level: z = 1.28
q
Safety stock (SS) = z dLT = 1.28(7.07) = 9 units
q
Reorder point (ROP) =d L + SS = 2(18) + 9 = 45 units
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18
Exercise
q
q
Weisss paint store uses a reorder point system to control the stock
level of its white latex paint product in the gallon size. Demand
is observed to be normally distributed with a monthly mean of 28
gallons and a standard deviation of 8. Re-supply lead time is
2.5 months. Costs associated with replenishment are $15 per
order and carrying cost is 30 percent per year. The store
manager targets a 98 percent in-stock probability during the
lead time. A gallon of paint costs the store $6.
When should the managers plan for replenishment of this item
and what should the reorder quantity be?
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Solution
q
Monthly demand, d=28, with d = 8; Annual demand, D=2812
q
Ordering (setup) cost, S=15/order; Holding cost, H = ic = 30%(6)
q
L=2.5 months.
q
Service level = 98%, z = 2.05
Determine the replenishment quantity
2 DS
2(28 12)(15)
Q = EOQ =
=
= 74.9, or 75 gallons
ic
0.30(6)
Next, the find reorder point quantity
ROP = d ( L) + z d L = 28(2.5) + 2.05(8) 2.5 = 70 + 26 = 96
The average inventory =
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EOQ
+ z d L = 37.5 + 25.9 = 63.4 units
2
20
Extra Exercise
Wood County Hospital consumes 500 boxes of bandages per week. The
price of bandages is $70 per box. The cost of processing an order is $60,
and the cost of holding one box for a year is 15% of the value of the
material. Assume 52 weeks per year.
a) The hospital orders bandages in lot sizes of 900 boxes. What extra
costs does the hospital incur, which it could save by applying the
EOQ concept?
b) Demand is normally distributed with a standard deviation of weekly
demand being 100 boxes. The lead-time is 1/2 week. What safety
stock is necessary if the hospital uses continuous review inventory
system, and a 97% service level? What should the reorder point be?
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Solution
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22
Summary of Continuous Review Model
q
q
q
Optimal Order Quantity: Q, use EOQ
Reorder Point = Avg lead time demand + Safety stock
ROP =d L + SS
Safety Stock is determined based on
q Service
q
level; Variability of demand; lead time
For normal demand
qz
can be compute for a given service level
q d = demand variability
q L = lead time
q SS = z dLT = z d L
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Two types of Inventory Control Systems
q
Continuous review system (Q-system)
When to order: when inventory declines to ROP
q Event-trigger restocking
q Also known as: Reorder Point (ROP) system
q How much: a fixed quantity is ordered every time
q EOQ is a continuous review system with uncertain demand
q
q
Periodic review system (P-system)
When to order: an order is placed after a fixed period of time
q Time-triggered restocking
q How much: An order placed for a variable amount
q
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Periodic Review Systems
q
Key differences compared to continuous review model:
When to order: every T days (review cycle/period)
q How much to order: variable, raising inventory up to a target
position
q
q
Objective: to minimize inventory cost with service level
guarantee
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Periodic Review Model
Target Inventory Position
Inventory
Level
Order
Arrival
Order
Arrival
Order
Arrival
Order
Arrival
Safety
Stock
Place
order
Place
order
Place
order
Place
order
Stockout
L
L
T
L
T
T
Exposure period (or protection interval): time between
placing order n and receiving order n+1
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26
Time (t)
Exposure Period (Protection Interval)
q
q
q
q
At time t1, you review inventory and find I1 units on hand. You
decide to order q1, raising inventory position to I1+q1
Once youve placed an order at t1, only one orders will be
received until between t1 to t1+L+T.
Therefore, I1+q1 must be sufficient to cover demand through a
period of length L+T (Exposure Period or Protection Interval).
I1+q1 is referred to as Target Inventory Position.
q2
q1
I2
I1
L
t1
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q3
T
L
t2
Exposure Period 27
I3
T
t3
L
Time (t)
T
Target Inventory Position to Achieve a
Service Level
q
q
Notation: L lead time; T review period; L+T : standard
deviation of demand during protection interval (exposure period);
Periodic review models:
q Target inventory position = (L+T)d +SS, where SS = z L+T
q z is computed from Normal Distribution Table
q
q
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L +T = d L + T
Order quantity, Q = (L+T) d + z L+T (on-hand
inventory)
28
Continuous Review vs. Periodic Review
q Continuous review: SS = z dLT
q
q
Periodic review: SS = z L+T
Why?
What if you run out of inventory?!
q Continuous review Order more immediately
q Periodic review have to wait till next order time
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29
Example
q
q
Bird feeder example. Recall that demand for the bird feeder is
normally distributed with a mean of 18 units per week and a
standard deviation in weekly demand of 5 units. The lead time
is 2 weeks, and the business operates 52 weeks per year. The
continuous review (Q) system for this problem called for an EOQ
of 75 units and a safety stock of 9 units for a cycle-service level
of 90 percent.
For the same service level requirement (90%), what is the target
inventory position for the periodic review ( P) system? (assume
we set review period T to be the time between orders in the Q
system.)
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30
Solution
q
Average annual demand
D = (18 units/week)(52 weeks/year) = 936 units
q
Time between reviews (weeks)
T=
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EOQ
(52) =
D
75
(52) = 4.2 or 4 weeks
936
31
(contd)
q
We now find the standard deviation of demand over the
protection interval (L+T) = 6:
L +T = d T + L = 5 6 = 12.25 units
q
Before calculating Target Inventory Position, we also need a z
value. For a 90 percent cycle-service level z = 1.28.
q
The safety stock, SS = z L+T =1.28(12.25) = 15.68 or 16 units
q
Target Inventory Position
= Average demand during the protection interval + Safety stock
= d(L+T) + SS
= (18 units/week)(6 weeks) + 16 units = 124 units
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32
Exercise
q
q
Weisss paint store keeps an inventory of white latex paint
product in the gallon size. The manager reviews the stock level
every 4 months. Demand is observed to be normally distributed
with a monthly mean of 28 gallons and a standard deviation of 8.
Re-supply lead time is 2.5 months. The store manager targets a 98
percent in-stock probability during the lead time.
What are the target inventory position?
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Solution
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34
Summary
Continuous Review
Periodic Review
Service level
Prob. of no stockout during
the lead time (L)
Prob. of no stockout during
the exposure period (L+T)
When to order
Inventory hits ROP
ROP = d L + SS
Every T days
How much to order
Fixed quantity: Order EOQ
q = Target IP On-hand
Target IP = (L+T)d + SS
Safety Stock
(normal demand)
SS = z dLT
SS = z L+T
Average Cycle
Inventory
EOQ / 2
d T / 2
Average Inventory
EOQ / 2 + SS
(+d L) *
d T / 2 + SS
* Depending on the ownership of the pipeline inventory.
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35
(+d L) *
Aggregate Control of Inventories
q
q
q
Manages the entire inventory or broad groups of
products collectively
Simplifies administration when thousands of items are
involved
Two tools:
1.
2.
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Turnover ratio as a performance measure
ABC classification of items
36
Performance measure: Turnover Ratio
Annual sales
Turnover ratio =
Average value of inventory
Example: A fruit grower stocks its dried fruit products in 12
warehouses around the country. What is the turnover ratio for
the distribution system?
Warehouse
no.
1
2
3
4
5
6
Annual
warehouse
throughput, $
21,136,032
16,174,988
78,559,012
17,102,486
88,228,672
40,884,400
Average
inventory
level, $
2,217,790
2,196,364
9,510,027
2,085,246
11,443,489
5,293,539
Warehouse
no.
7
8
9
10
11
12
Totals
$425,295,236
TO ratio =
= 9.7
$43,701,344
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37
Annual
warehouse
throughput, $
43,105,917
47,136,632
24,745,328
57,789,509
16,483,970
26,368,290
425,295,236
Average
inventory level,
$
6,542,079
5,722,640
2,641,138
6,403,076
1,991,016
2,719,330
43,701,344
ABC Classification
q
Large number of stock-keeping units (SKU)
Demand volume and value of items vary
q
q
Classify SKUs into 3 classes, based on their aggregate
values
The breakdown between classes is arbitrary
Use a Pareto chart
q
q
Number of parts
(percentage of total)
Value of all parts
(percentage of total)
A
5% to 15%
75%
B
30%
15%
C
50% to 60%
10%
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38
ABC Classification Using a Pareto Chart
100
Percentage of dollar value
90
Class A
80
70
60
50
40
30
Class B
Class C
20
10
0
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10 20 30 40 50 60 70 80 90 100
Percentage of SKUs
39
ABC Classification
q
Step 1: Classify items into A, B, C according to $ value
q
q
Dollar value=dollar cost of one unit * annual demand
Step 2: Determine level of inventory for items in each
classification
q
Class A: requires frequent review to reduce average lot size and high
turnover
q
Class B: intermediate level of control, less frequent review plus
adequate safety stock
Class C: loose control
q
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40
ABC Classification Example
Unit
cost
$510
320
350
60
80
30
20
30
20
10
Unit
sales
60
50
40
90
60
130
180
100
120
170
Part
no.
9
8
2
1
4
3
6
5
10
7
Part
Value
Value Value Cumulative Part Cumulative
6.0%
$30,600 35.8%
35.8% 10.0%
20.0
16,000 18.7
11.0
54.5
30.0
14,000 16.4
15.0
70.9
6.3
5,400
24.0
77.2
40.0
50.0
5.6
4,800
30.0
82.8
4.6
3,900
60.0
43.0
87.4
4.2
3,600
70.0
61.0
91.6
3.5
3,000
80.0
71.0
95.1
2.8
2,400
90.0
83.0
97.9
2.1
1,700
100.0
100.0 100.0
$ 85,400
Set different service
levels for each class
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Class
A
B
C
Items
9,8,2
1, 4, 3
6, 5, 10, 7
% Value % Parts
15
70.9
16.5
28
12.6
57
41
41
A
B
C