2014/5/12
Michael C. Bergerac
Former Chief Executive
Revlon, Inc.
Inventory Strategy
•• Forecasting
Forecasting
• Inventorydecisions
Inventory decisions
•• Purchasing
Purchasing and
and supply
supply
scheduling
schedulingdecisions
decisions
•• Storage
Storage fundamentals
fundamentals
• Storage decisions
Customer
service
goals
service
goals
• The
product
• The product
Logistics service
service
•• Logistics
• Ord. proc. & info. sys.
sys.
Transport Strategy
• Transport fundamentals
•• Transport
Transport decisions
decisions
Location Strategy
•Location
Location decisions
Strategy
• The
network
planning process
Location
decisions
• The network planning process
9-1
G
ZING N
I
IN
GI LL
NN O
N A
LA RG TR
N
P O O C
PLANNING
“Every management mistake ends up in inventory.”
ORGANIZING
Inventory Policy
Decisions
CONTROLLING
Inventory Decisions in
Strategy
9-2
1.What are Inventories?
• Finished product held for sale
• Goods in warehouses
• Work in process
• Goods in transit
• Staff hired to meet service needs
• Any owned or financially controlled
Section 1. Appraisal of
Inventories?
raw material, work in process, and/or
finished good or service held in
anticipation of a sale but not yet sold
9-3
9-4
2. Where are Inventories?
Production
Inbound
transportation
Outbound
transportation
Finished goods
warehousing
Customers
Production
materials
Inventories
in-process
-Allows purchasing to take place under most favorable price
terms
Shipping
Finishedgoods
-Provides immediacy in product availability
• Encourage production, purchase, and transportation
economies
-Allows for long production runs
-Takes advantage of price-quantity discounts
-Allows for transport economies from larger shipment sizes
• Act as a hedge against price changes
Receiving
Material
sources
3. Reasons for Inventories
• Improve customer service
• Protect against uncertainties in demand and lead times
-Provides a measure of safety to keep operations
running when demand levels and lead times cannot be known
for sure
Inventory
locations
• Act as a hedge against contingencies
-Buffers against such events as strikes, fires, and
disruptions in supply
9-4
9-6
1
2014/5/12
5. Types of Inventories
4. Reasons Against Inventories
• Pipeline
• They consume capital resources that might be put to
better use elsewhere in the firm
• They too often mask quality problems that would more
immediately be solved without their presence
• They divert management’s attention away from careful
planning and control of the supply and distribution
channels by promoting an insular attitude about
channel management
-Inventories in transit
• Speculative
-Goods purchased in anticipation of price increases
• Regular/Cyclical/Seasonal
-Inventories held to meet normal operating needs
• Safety
-Extra stocks held in anticipation of demand and
lead time uncertainties
• Obsolete/Dead Stock
9-7
-Inventories that are of little or no value due to being
out of date, spoiled, damaged, etc.
9-8
1. Nature of Demand
• Perpetual demand
-Continues well into the foreseeable future
Section 2. Classifying
inventory management
problem
• Seasonal demand
-Varies with regular peaks and valleys throughout
the year
• Lumpy demand
-Highly variable (3  Mean)
• Regular demand
-Not highly variable (3 < Mean)
Accurately forecasting
demand is singly the
most important factor
in good inventory
management
• Terminating demand
-Demand goes to 0 in foreseeable future
• Derived demand
-Demand is determined from the demand of another
item of which it is a part
9-10
9-9
Nature of Demand
2. Inventory Management
Philosophies
• Pull
-Draws inventory into the stocking location
-Each stocking location is considered independent
-Maximizes local control of inventories
• Push
-Allocates production to stocking locations based on
demand
-overall
Encourages economies of scale in production
• Just-in-time
-Attempts to synchronize stock flows so as tojust
meet demand as it occurs
-Minimizes the need for inventory
9-12
9-11
2
2014/5/12
Pull vs. Push Inventory Philosophies
Inventory Management
Philosophies (Cont’d)
PUSH - Allocate supply to each
warehouse based on the forecast
for each warehouse
PULL - Replenish inventory with
order sizes based on specific needs
of each warehouse
• Supply-Driven
Demand
forecast
-Supply quantities and timing are unknown
-All supply must be accepted and processed
-Inventories are controlled through demand
Q1
• Aggregate Control
Warehouse#1
A1
-Classification of items:
›Groups items according to their sales level
A2
Q2
Plant
Warehouse#2
A3
based on the 80-20 principle
Demand
forecast
Q3
›Allows different control policies for 3 or more
A = Allocation quantity to each warehouse
Q = Requested replenishment quantity
by each warehouse
broad product groups
9-13

Warehouse#3
Demand
forecast
9-11
Inventory management involves balancing product
availability (customer service) with the costs of providing
a given level of product availability.
Section 3. Inventory
Objectives
16
9-15
1.Product Availability
A primary objective of inventory management is to ensure that
product is available at the time and in the quantities desired.
 The probability of fulfillments capability from current stock
(item fill rate) is referred to as the service level for a single
item.

Service level  1- expected number of units out of stock demand
Total annual demand


Service level is expressed as a value between 0 and 1.
Customers frequently request more than one items at a time.
17
• For Example: Suppose that five items are requested on
an order where each item has a fill rate of 0.95. Filling the
entire order without any item being out of stock would be:
0.955≈0.77
• Weighted average fill rate (WAFR) is found by multiplying
the frequency with which each combination of items
appears on the order by the probability of filling the order
completely.
• Example:
18
3
2014/5/12
Relevant Costs (Cont’d)
2. Costs Relevant to Inventory
Management
• Carrying costs
• Procurement costs
-Cost for holding the inventory over time
-The primary cost is the cost of money tied up in
-Cost of preparing the order
-Cost of order transmission
-Cost of production setup if appropriate
-Cost of materials handling or processing at the
inventory, but also includes obsolescence,
insurance, personal property taxes, and storage
costs
-Typically, costs range from the cost of short term
receiving dock
capital to about 40%/year. The average is about
25%/year of the item value in inventory.
-Price of the goods
9-19
9-20
3. Inventory Management Objectives
Relevant Costs (Cont’d)
• Out-of-stock costs
-Lost sales cost
›Profit immediately foregone
›Future profits foregone through loss of goodwill
-Backorder cost
›Costs of extra order handling
›Additional transportation and handling costs
›Possibly additional setup costs
Good inventory management is a careful balancing act
between stock availability and the cost of holding
inventory.
Customer Service,
i.e., StockAvailability
Inventory Holdingcosts
• Service objectives
-Setting stocking levels so that there is onlya
specified probability of running out of stock
• Cost objectives
-Balancing conflicting costs to find the most
economical replenishment quantities and timing
9-21
9-22
Inventory’s Conflicting Cost Patterns
Section 4. Push Inventory
Control
Minimum cost
reorder quantity
Cost
Total cost
Procurement cost
Stockout cost
Replenishment quantity
9-16
9-24
4
2014/5/12
Push Inventory Control


Push method is appropriate where




A framework of push approaches involves the following
steps:
Determine through forecasting or other means the
requirements for the period between now and the next
expected production run or vendor purchase.
 Find the current on-hand quantities at each stocking point.
 Establish the stock availability level at each stocking
point.
 Calculate total requirements from the forecast plus
additional quantities needed to cover uncertainty in the
demand forecast.
 Apportion the excess over total net requirements to the
stocking points on the basis of the average demand rate,
that is, the forecast demand.
 Sum the net requirements and prorate the excess quantities
to find the amount to be allocated to each stocking point.

Production or purchase quantities exceed the short-term
requirements of the inventory into which the quantities
are to be shipped.
Production or purchase quantities cannot be stored at
the production site for lack of space or other reasons.
Production or purchase is the dominant force in
determining the replenishment quantities in the channel.
The following questions need to be addressed:
How much inventory should be maintained at each
stocking points?
How much of a purchase order or production run should
be allocated to each stocking point?
 How should the excess supply over requirements be
appointed among the stocking points?


25
26
Chapter 9

Example: When the tuna boats are sent to the fishing
grounds, a packer of tuna products must process all
the tuna caught since storage is limited and, for
competitive reasons, the company does not want to
sell the excess of this valued product to other
packers. Therefore, this packer processes all fish
brought in by fleet and then allocates the production
to its three field warehouses on a monthly basis.
There is only enough storage at the plant for one
month’s demand. The current production run is
125,000lb
.

Compute the total requirement for each warehouse:
Total requirements = Forecast + (z × Forecast error)

Net requirements are found as the difference between total
requirements and the quantities on hand in the warehouse.
Net requirements = 125,000 – 110,635,
which is the excess production that needs to be prorated to
the ware houses.
27

28
Prorating（ 分 派 ） the excess production in
proportion to the average demand rate for each
warehouse.
Section 5. Basic Pull
Inventory Control
The proportion of the excess allocated（分配） to
warehouse 1 should be (10,000 / 13,000) ×14,365 =
1,105
29
9-30
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2014/5/12
1. Single Order Purchasing
Glossary of Terms
D  average annual demand, units
d  average perioddemand, units
S  procurement cost per order, $/order
I  carrying costs as a percent of product value, % per year
C  product value, $ per unit
Make a one-time purchase of an item. How much to order?
Procedure: Balance incremental profit against incremental loss.
sd  standard deviation of demand (d),units
k  out - of - stock cost, $ per unit
p  purchase price
Profit = Price per unit  Cost per unit
Loss = Cost per unit  Salvage value per unit
s '  standard deviation of compound demand distribution
E(z)  partial expectation or unit normalloss integral
P  probability of being in - stock duringlead time (Q - system)
If CPn is probability of n units being sold, then
or duringlead time plus order cycle time (P - system
Q  order quantity
ROP  reorder point quantity, units
T  order interval, e.g., days
MAX  target inventory level, units
z  normal deviate or number of standard deviations from mean
Daily stocking of
newspapers in
vending
machines is a
good example
CPn x Loss = (1  CPn) x Profit
or
CPn = Profit/(Profit + Loss)
on compound demand distribution
r  safety stock, or z x s' ,units
TC  total relevant cost, $
SL  service level as a percent of total annual demand
LT ,sLT  average and standard deviation of lead time
Now, increase order quantity until CPn just matches cumulative
probability of selling additional units.
CPn  probability of n units being sold
9-18
9-31
Single Order Purchasing (Cont’d)
Example A clothing item is purchased for a seasonal sale. It
costs $35, but it has a sale price of $50. After the season is
over, it is marked down by 50% to clear the merchandise.
The estimated quantities to be sold are:
Number of
items, n
10
15
20
25
30
35
Probability of
selling exactly n
items
0.15
0.20
0.30
0.20
0.10
0.05
1.00
Cumulative
probability
0.15
0.35
0.65
0.85
0.95
1.00
Single Order Purchasing (Cont’d)
Solution
Profit = $50 35 = $15
Loss = $35  (0.5)(50) = $10
CPn = 15/(15 + 10) = 0.60
CPn is between 15 and 20 items, round up and order 20
items.
9-19
9-34
Reorder Point Method Under Certainty
for a Single Item
2. Simple Two-Bin Pull Method
Given:
d = 50 units/week
I = 10%/year
S = $10/order
C = $5/unit
LT = 3 weeks
Quantity on-hand
plus on-order
Note: No uncertainty
in demand or lead
time—manage
regular (cycle) stock
only
Q
Reorder
point, R
Develop a simple control system by finding the
replenishment quantity (Q) and the reorder point
(ROP).
0
The relevant total cost is:
TC  ordering cost  carrying costs
 DS ICQ
Q
2
Order
Placed
9-21
Lead
time
Order
Received
Lead
Time
time
Order
Received
Order
Placed
9-22
6
2014/5/12
Two-Bin Method (Cont’d)
Noninstantaneous resupply
At times, production or supply continues while demand
is depleting inventories. This requires a slight
modification of the EOQ formula. That is,
Using differential calculus, the optimal value for Q will
be:
p
Qp*  2DS
IC p d
Q*  2DS/IC  2(50x52)(10)/(0.10x5) 322 units
Famous EOQ
formula
The reorder point is:
Justadd
this term
where
ROP = d(LT) = 3(50) = 150 units
p = output or supply rate
d = demand rate
Rule When the inventory level drops to 150 units
(ROP) then reorder 322 units (Q*).
and p > d. ROP remains unchanged.
9-23
9-36
Advanced Pull Inventory Control
When demand and lead time cannot be known for sure,
we must plan for the situation where not enough stock
may be on hand to fill customer requests.
 In addition to the regular stock that is maintained for
meeting average demand and average lead time, an
incremental quantity is added to inventory.
 The amount of this safety stock sets the level of stock
availability provided to customers by controlling the
probability of a stock-out occurring.
 Two basic inventory control methods:
 The reorder point method
 The period review method

Section 6. Advanced Pull
Inventory Control
40
9-39
Quantity on hand
1. Reorder Point Control for a Single Item
Week 1
Q
Receive
order
0
P
Stockout
LT
LT
Time
9-25
Week 2
+
DDLT
ROP
DDLT
P
Q
Place
order
Reorder Point Control (Cont’d)
Finding the reorder point requires an understanding of
the demand-during-lead-time distribution
Week 3
=
+
sd=10
sd=10
sd=10
d =100
d =100
d =100
Weekly demand is normally distributed with a
mean of d = 100 and a standard deviation of
sd = 10
Lead time is 3 weeks
S’=17.3
z
X = 300 ROP
X  d  LT  100(3)  300
s'  s LT
 10 3  17.3
d
9-27
7
2014/5/12
Reorder Point Control (Cont’d)

Demand during lead time (DDLT) has a mean of
Given:
d = 50 units/week
sd = 10 units/week
I = 10%/year
S = $10/order
X'  LT  d


Demand during lead time (DDLT) has a standard
deviation
s'd  L T  sd
2
C = $5/unit
LT = 3 weeks
P = 99% during lead time
ROP  d  LT  z  s'd
Find Q* and ROP
From the EOQ formula
Q*  2(50x52)(10) 322 units
0.10(5)
Good method for products:
1. Of high value
2. That are purchased from
one vendor or plant
3. Having few economies of
scale in production,
purchasing, or
transportation
9-24
43
2. Reorder Point Control -Total relevant
cost
Reorder Point Control (Cont’d)
The total relevant cost equation is now extended to
include the costs of safety stock as well as out-of-stock.
Hence,
Now,
X '  d(LT ) 50(3)150 units
s'  sd LT 10 3 17.32 units
TCOrdercostcarrying cost,regularcost
carrying cost,safetycoststockoutcost
 DS ICQ ICzs' k Ds'E (z)
Q
2
Q
Hence,
ROP  X '  zs '
 1502.33(17.32)  190 units
where 2.33 is the normal deviate at a probability of
0.01 taken from a normal distribution table.
9-29
9-45
3. Pull Methods ---Reorder point control with
demand and lead time uncertainties
The combined effect of these two uncertainties is particularly
hard to estimate accurately. It is the standard deviation of the
demand-during-lead-time distribution that is the problem,
especially if the level of demand and the length of the lead time
are related to each other. Ideally, we would simply observe the
actual demand occurring over each lead time period. If the
demand and lead time are independent of each other and each
are represented by separate distributions, we may estimate the
standard deviation (s′) from
The total relevant cost equation is now extended to
include the costs of safety stock as well as out-of-stock.
The out-of-stock cost (k) is $2/unit. The price term is
dropped. Hence,
TC  D S IC Q ICr  k D s ' E (z)
Q
2
Q
 2,600(10) (0.1)(5) 322 (0.1)(5)(40)
322
2
 2 2,600 (17.32)(0.0034)
322
$182.20
s'  LT(s 2d)d 2(s 2LT)
where E(z) = 0.0034 from a unit normal loss table at a
z value of 2.33
9-29
Caution: Can result in very
high safety stock levels when
lead-time variability is high
After computing s’, calculation of the ordering policy would
9-37
be identical to that presented previously.
8
2014/5/12
Periodic Control for a Single Item
4.Pull Methods---Periodic review control
with demand uncertainty
Quantity on hand
C = $5/unit
LT = 3 weeks
P = 0.99
k = $2/unit
Q2
Q1
~
The inventory is reviewed at the time interval (T) to
determine the quantity on hand. The replenishment
quantity (Q) to be ordered is the difference between a
target level called MAX and the quantity on hand. We
need to find MAX and T*.
Good methodfor
Given:
d = 50 units/week
sd = 10 units/week
I = 10%/year
S = $10/order
M
products:
1. Of low value
2. That are purchased
from the same vendor
3. Having economies of
scale in production,
purchasing, and
transportation
9-38
Periodic Review (Cont’d)
q
Stock
level
reviewed
Order
received
0
LT
Time
LT
T
T
T = review interval
q = quantity on hand
Qi = order quantity
M = maximum level
M - q = replenishmentquantity
LT = lead time
9-39
Periodic Review (Cont’d)
DD(T * + LT)
Estimate Q* from the EOQ formula as if under demand
certainty conditions. Recall that this is Q* = 322 units.
Now,
P
T* = Q*/d = 322/50 = 6.4 weeks
Construct the demand-during-lead-time-plus-ordercycle-time distribution.
s′
T is order
review time
s' s Td * LT
Z(s′)
X
= d(T* + LT)
MAX
9-52
9-51
Periodic Review (Cont’d)
Periodic Review (Cont’d)
where
The total relevant cost for this design is:
X d(T * LT)50(6.43) 470
s'  s T * LT 10 6.43 30.66
TC = DS/Q + ICQ/2 + ICr + ks’(D/Q)E(z)
d
= 2600(10)/322 + (.10)(5)(322/2)
+ (.10)(5)(71) + 2 (30.66)(2600/322)(.0034)
Find MAX
MAX = d(T* + LT) + z(s’)
= 50(6.4 + 3) + 2.33(30.66)
= 470 + 71.44 = 541 units
= $198
Note Compare this cost with that of the reorder point
method to see that periodic review control carries a
slight premium in cost due to more safety stock.
Rule Review the inventory every 6.4 weeks and place
an order for the difference between the MAX level of 541
units and the quantity on hand + quantity on order –
backorders.
9-42
9-54
9
2014/5/12
Pull Methods (Cont’d)
Pull Methods (Cont’d)
In/
Date Customer Sales hand
10/26
10/26
10/30
10/30
11/2
11/9
11/29
12/1
12/13
12/14
12/15
1/8
1/8
1/8
1/9
1/17
1/23
1/24
1/26
1/26
1/29
1/29
2/2
Size
8½x14
Bal Fwd
100M
Progression
Ogleby
Mid Ross
Unt Sply
Berea Lit
Dol Fed
Card Fed
Belmont
Shkr Sav
BFK
100M
Card Fed
Pt of View
Am Safety
Foster
Gib Prtg
Bel-Gar
Copies
Slvr Lake
100M
Sagamore
M/Wgt
12.72
20000
25000
15000
50000
25000
10000
20000
15000
5000
500
30000
10000
5000
15000
5000
5000
20000
5000
20000
Basis
20
On
80500
180500
160500
135500
120500
70500
45500
35500
15500
500
500*
0
100000
70000
60000
55000
40000
35000
30000
10000
5000
105000
85000
Grain
L
In/
Date Customer Sales hand
On
In/
Date Customer Sales Hand
2/2
2/5
2/6
2/6
2/6
2/6
2/8
2/14
2/15
2/16
2/21
2/26
2/27
2/28
2/28
3/1
3/2
3/8
3/8
3/12
3/12
3/12
3/20
35000
30000
15000
0*
0*
0*
0*
100000
150000
145000
130000
125000
75000
72500
47500
12500
2500
0
0*
150000
110000
60000
45000
3/30
3/30
3/30
3/30
4/2
4/2
4/9
4/12
5/7
5/8
5/8
5/11
5/14
5/15
5/16
5/16
5/16
5/16
5/16
5/22
Coding
Color
White
Copies
Bel-Gar
Bel-Gar
Superior
Unt Sply
Berea Prtg
Sagamore
100M
50M
Bel-Gar
Bel-Gar
Inkspot
Lcl 25UAW
Ptrs Dvl
Shkr Sav
Copies
Untd Tor
Sagamore
Sagamore
150M
Untd Tor
Preston
Midland
50000
5000
15000
25000
15000
15000
5000
5000
15000
5000
50000
2500
25000
35000
10000
2500
12500
40000
50000
15000
Finish
RmSeal
Grade
Advantage Bond
Sup Meats
Copies
Ptrs Dvl
Belmont
Berea Prtg
Berea Prtg
REM
Mid Ross
Ohio Ost
Inkspots
Prts Dvl
100M
BVR
Guswold
ESB
Superior
J Stephen
Am Aster
Am Aster
Sagamore
21200
M. Base Cost
2.64
Location
F 14
25000
50
5000
10000
4950
15050
500
5000
5000
5000
2500
5000
10000
15000
50000
5000
15000
10000
15000
Date
4/2
Min
Max
Ctn. Skid Cont.
5M
On
Supply chain example
20000
19950
14950
4950
0
0*
0*
0*
0*
0*
0*
100000
95000
85000
70000
20000
15000
0
0*
0*
Suppose that inventory is to be maintained on a
distributor’s shelf for an item whose demand is
forecasted to be d = 100 units per day and sd = 10 units
per day. A reorder point is the method of inventory
control. The supply channel is shown in the diagram.
Determine the average inventory to be held at the
distributor where we have:
I = 10%/year
S = $10/order
125M
250M
Att.
9-44
*
No stock or insufficient stock to meet demand
9-56
Supply Chain Example (Cont’d)
Supply Chain Example (Cont’d)
Supplier
Solution The reorder point inventory theory applies.
However, determining the statistics of the demandduring-lead-time distribution requires taking the leadtime for the entire channel into account.
Processing time
X p  1, s 2 p 0.1
Transport time
Inbound transport
C = $5/unit
P = 0.99 during lead time
Recall,
X i  4 , s 2i  1.0
s'  LT(s2 d)d 2 (s 2 LT
)
where
2  s2 s 2 s 2
sLT
p
o
i
Outbound transport
 0.1 1.0  0.25
 1.35days
Transport time
Pool point
Distributor
X o  2 , s 2o 0.25
9-57
9-58
Pull Methods (Cont’d)
Supply Chain Example (Cont’d)
Joint ordering
Average lead time
LT  Xp  Xi  Xo 1427 days
Now
s'  7x102 1002 x1.35  14,200 119.16 days
and
Q*  2(100)(10)  63 units
0.1(5)
*
Q
AIL  z(s' ) 63  2.33(199.16)309 units
2
2
Perpetual inventory control for most firms is the problem of
managing items jointly rather than singly. This occurs since
more than one item is typically purchased from the same
vendor. The approach to joint ordering is to find a common
order review interval (T) and then to set separate target levels
(MAX) based on specific item costs and service levels.
A common review time may be specified, or it may be
computed based on appropriate economics.
T* 
2(O 
S)
i
(IC D )
ii
where
O = common procurement cost, $/order
Note: Q* = T*xd
9-59
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10
2014/5/12
Joint Ordering Example
Joint Ordering Example (Cont’d)
Given
Find common review time
Item
A
Average daily demand ( d)
30
Demand std. dev. ( sd)
8
Average lead time (LT)
14
25
Annual carrying cost (I)
30
Procurement cost (S)
with common cost (O)
In-stock probability (P)
80
Product value (C)
170
25
Out-of-stock cost (k)
365
Selling days per year
2[80(3020)]
4.35 days
[0.25/365][170(30)200(75)]
T* 
B
75 units
10 units
14 days
25 %
20 $/order
80 $/order
92 %
200 $/unit
45 $/unit
365 days
Find target quantity (MAX) for item A
s'  s
A
d
T * LT 8 4.3514 34.3 units
A
A
then z@80%=0.84
MAXA  X z(s'A)30(4.3514)0.84(34.3)579 units
9-61
9-62
Joint Ordering Example (Cont’d)
Pull Methods (Cont’d)
which has an average inventory of
Avg. Inventoryi T *(di /2) zi (s'i)
Avg. InventoryA  4.35(30/2)0.84(34.3)94.1units
The Min-Max variant
Find target quantity (MAX) for item B
s'  s
B
d
This is basically a reorder point system, but the order
quantity is incremented by the amount of the difference
between the reorder point quantity and the quantity on
hand + quantity on order  backorders. This takes into
account that demand does not decrement inventory
levels evenly. Therefore, inventory levels may fall
below the reorder point at the time that it is reached.
T * LT 10 4.3514  42.8 units
B
B
then for z@90%=1.41
MAXB 75(4.3514)1.41(42.8)1437 units
which has an average inventory of
Avg. inventoryB  4.35(75/2)1.41(42.8) 223 units
9-63
9-64
Pull Methods (Cont’d)
Min-Max Inventory Control
Add increment ROP q to order size
Quantity on hand
M
The T, R, M variant
Q1
~
Q2
This is a combination of the min-max and the periodic
review systems. The stock levels are reviewed
periodically, but control the release of the replenishment
order by whether the reorder point is reached. This
method is useful where demand is low, such that small
quantities might be released under a periodic review
method.
Q*
ROP
q
LT
LT
Time
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2014/5/12
Pull Methods (Cont’d)
Inventory level
Pull Methods (Cont’d)
Inventory not
below R, so don’t
place an order
T,R,M variant
Stock to demand (a periodic review method)
Q1
This is an important periodic review method, not so much
because of its accuracy but because of its popularity in
practice. The method is synchronized with the period of
the forecast. The target quantity (MAX) is developed as
follows.
An example
Q2
R
q
LT
• Set the period of the forecast, say 4weeks
• Add time for lead time, say 1 week
• Add an increment of time for safety stock, say 1 week
Time
LT
T
T
T = review time
R = reorder point M – Q = replenishment quantity
9-68
9-56
Stock to Demand (Cont’d)
Pull Methods (Cont’d)
Therefore, MAX is 6/4 times the monthly forecast. The
replenishment quantity is determined as follows.
At the time (T) of the monthly stock-level review, make a
forecast and determine the MAX level.
Multiple item, multiple-location control
The theory that has been discussed previously is
useful when designing inventory control systems for
the practical problem of controlling many items at
many locations. Consider how a specialty chemical
company designed such a practical system. TASO is
the time to accumulate a stock order (truckload) for all
items in warehouse.
Units
MAX = Forecast x 6/4
12,500
Plus: Backorders
0
Less: Quantity on hand* -5,342
Less: Quantity on order
-4,000
Order quantity (Q)
3,158
*Quantity on hand = actual quantity on hand +
quantity on order – backorders
9-69
Q3
Q2
~
Q1
Stock
order
Order
received
0
~
LT
TASO
TASO
M = maximum level
TASO = time to accumulate stock order
LT
TASO
Time
Qi = order quantity
LT = lead time
9-60
Multi-Echelon Inventories
Control the entire channel inventory levels, not just a
single echelon.
How much stock here when
retailers also carry stock?
Warehouse
echelon
R1
d1 , sd 1
Warehouse
lead-time, LTw
S
Supplier
R2
W
d 2, ds2
Warehouse
R3
End customer demand
Quantity on hand
M
Multiple-Item, Multiple-Location Control
9-70
d 3 , sd 3
Retailer
9-72
12
2014/5/12
Multi-Echelon Inventories (Cont’d)
Multi-Echelon Inventories (Cont’d)
Example
An item has the following cost characteristics. Item
values are CR=$10/unit and CW=$5/unit. Carrying cost
is I = 20%/year. Ordering costs are SR=$40/order and
SW=$75/order. Lead times are LTR=0.25 month and
LTW=0.5 months. In-stock probability for retailers and
warehouses is 90%. Monthly demand statistics are:
Monthly
avg.,units
Std.dev.,
units
Retailer 1
202.5
16.8
Retailer 2
100.5
15.6
Retailer 3
302.5
18.0
Combined
605.5
32.4
Solution
Based on reorder point inventory control, the retailers’
inventory statistics are
Retailer 1
Retailer 2
Retailer 3
Reorder qty, Q
312
220
381
Reorder point, ROP
61
35
87
Avg. inv., AIL
167
120
202
The warehouse echelon order quantity is
QW 2DWSW  2(605.5x12)(75) 1,043.98, or 1,044units
ICW
0.20(5)
ROPW  dW xLTW  zsW LTW  605.5(.5)1.28(32.4) .5
332 units
9-73
Multi-Echelon Inventories (Cont’d)
The warehouse echelon inventory is
AILW QW  zs WLT W
2
1,043.98 1.28(32.4) 0.5
2
551.32. or 551units
Product items can be grouped according to 80-20
curve, each with different stocking policies
100
90
Total sales (%)
80
The average warehouse inventory is the warehouse
echelon inventory less the retailers’ inventory, or 551
– 167 –120 – 202 = 62 units.
70
60
50
40
30
Rule When the total warehouse inventory (sum of
retailers’ inventory, inventory at the warehouse and on
order, and retailers’ orders less any inventory committed
to customers drops below 332 units, order 1,044 units.
A items
B items
10
0
0
20
9-75
Areas under
Standardized
Normal
Distribution
.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
.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
.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
.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
.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
.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
.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
.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
.08
.09
0.5319 0.5359
0.5714 0.5753
0.6103 0.6141
0.6480 0.6517
0.6844 0.6879
0.7190 0.7224
0.7517 0.7549
0.7823 0.7852
0.8106 0.8133
0.8365 0.8389
0.8599 0.8621
0.8810 0.8830
0.8997 0.9015
0.9162 0.9177
1.4
1.5
1.6
1.7
1.8
0.9192
0.9332
0.9452
0.9554
0.9641
0.9207
0.9345
0.9463
0.9564
0.9649
0.9222
0.9357
0.9474
0.9573
0.9656
0.9236
0.9370
0.9484
0.9582
0.9664
0.9251
0.9382
0.9495
0.9591
0.9671
0.9265
0.9394
0.9505
0.9599
0.9678
0.9279
0.9406
0.9515
0.9608
0.9686
0.9292
0.9418
0.9525
0.9616
0.9693
0.9306
0.9429
0.9535
0.9625
0.9699
0.9319
0.9441
0.9545
0.9633
0.9706
1.9
2.0
2.1
2.2
2.3
0.9713
0.9772
0.9821
0.9861
0.9893
0.9719
0.9778
0.9826
0.9864
0.9896
0.9726
0.9783
0.9830
0.9868
0.9898
0.9732
0.9788
0.9834
0.9871
0.9901
0.9738
0.9793
0.9838
0.9875
0.9904
0.9744
0.9798
0.9842
0.9878
0.9906
0.9750
0.9803
0.9846
0.9881
0.9909
0.9756
0.9808
0.9850
0.9884
0.9911
0.9761
0.9812
0.9854
0.9887
0.9913
0.9767
0.9817
0.9857
0.9890
0.9916
2.4
2.5
2.6
2.7
2.8
2.9
3.0
3.1
3.2
3.3
3.4
0.9918
0.9938
0.9953
0.9965
0.9974
0.9981
0.9987
0.9990
0.9993
0.9995
0.9997
0.9920
0.9940
0.9955
0.9966
0.9975
0.9982
0.9987
0.9991
0.9993
0.9995
0.9997
0.9922
0.9941
0.9956
0.9967
0.9976
0.9982
0.9987
0.9991
0.9994
0.9995
0.9997
0.9925
0.9943
0.9957
0.9968
0.9977
0.9983
0.9988
0.9991
0.9994
0.9996
0.9997
0.9927
0.9945
0.9959
0.9969
0.9977
0.9984
0.9988
0.9992
0.9994
0.9996
0.9997
0.9929
0.9946
0.9960
0.9970
0.9978
0.9984
0.9989
0.9992
0.9994
0.9996
0.9997
0.9931
0.9948
0.9961
0.9971
0.9979
0.9985
0.9989
0.9992
0.9994
0.9996
0.9997
0.9932
0.9949
0.9962
0.9972
0.9979
0.9985
0.9989
0.9992
0.9995
0.9996
0.9997
0.9934
0.9951
0.9963
0.9973
0.9980
0.9986
0.9990
0.9993
0.9995
0.9996
0.9997
0.9936
0.9952
0.9964
0.9974
0.9981
0.9986
0.9990
0.9993
0.9995
0.9997
0.9998
40
60
Total items(%)
80
100
9-72
Unit Normal
Loss
Integrals
A table entry is the proportion of the area
under the curve from a z of 0 to a
positive value of z. To find the area from
a z of 0 to a negative z, subtract the
tabled value from 1.
z
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
C items
20
Aggregate Inventory Control
9-74
9-104
9-105
13