Part II: When to Order? Inventory
Management Under Uncertainty
 Demand
or Lead Time or both uncertain
 Even “good” managers are likely to run out once
in a while (a firm must start by choosing a service
level/fill rate)
 When can you run out?
– Only during the Lead Time if you monitor the
system.
 Solution: build a standard ROP system based on
the probability distribution on demand during the
lead time (DDLT), which is a r.v. (collecting
statistics on lead times is a good starting point!)
1
The Typical ROP System
Average Demand
ROP
set as demand that accumulates during
lead time
ROP = ReOrder Point
Lead Time
2
The Self-Correcting EffectA
Benign Demand Rate after ROP
Hypothetical Demand
Average Demand
ROP
Lead Time
Lead Time
3
What if Demand is “brisk” after
hitting the ROP?
Hypothetical Demand
Average Demand
ROP = EDDLT + SS
ROP >
EDDLT
Safety
Stock
Lead Time
4
When to Order
 The
basic EOQ models address how much
to order: Q
 Now, we address when to order.
 Re-Order point (ROP) occurs when the
inventory level drops to a predetermined
amount, which includes expected demand
during lead time (EDDLT) and a safety
stock (SS):
ROP = EDDLT + SS.
When to Order
 SS
is additional inventory carried to reduce
the risk of a stockout during the lead time
interval (think of it as slush fund that we dip into when
demand after ROP (DDLT) is more brisk than average)
 ROP depends
–
–
–
–
on:
DDLT,
Demand rate (forecast based).
EDDLT &
Length of the lead time.
Demand and lead time variability. Std. Dev.
Degree of stockout risk acceptable to
management (fill rate, order cycle Service Level) 6
The Order Cycle Service Level,(SL)
 The
percent of the demand during the lead time
(% of DDLT) the firm wishes to satisfy. This
is a probability.
 This is not the same as the annual service
level, since that averages over all time periods
and will be a larger number than SL.
 SL should not be 100% for most firms.
(90%? 95%? 98%?)
 SL rises with the Safety Stock to a point.
7
Quantity
Safety Stock
Maximum probable demand during
lead time (in excess of EDDLT)
defines SS
Expected demand
during lead time
(EDDLT)
ROP
Safety stock (SS)
LT
Time
8
Variability in DDLT and SS
 Variability
in demand during lead time (DDLT)
means that stockouts can occur.
– Variations in demand rates can result in a temporary
surge in demand, which can drain inventory more
quickly than expected.
– Variations in delivery times can lengthen the time a
given supply must cover.
 We
will emphasize Normal (continuous)
distributions to model variable DDLT, but discrete
distributions are common as well.
 SS buffers against stockout during lead time.
9
Service Level and Stockout Risk
 Target
service level (SL) determines how
much SS should be held.
– Remember, holding stock costs money.
 SL =
probability that demand will not
exceed supply during lead time (i.e. there
is no stockout then).
 Service level + stockout risk = 100%.
10
Computing SS from SL for Normal
DDLT
 Example
10.5 on p. 374 of Gaither &
Frazier.
 DDLT is normally distributed a mean of
693. and a standard deviation of 139.:
– EDDLT = 693.
– s.d. (std dev) of DDLT =  = 139..
– As computational aid, we need to relate this to
Z = standard Normal with mean=0, s.d. = 1
» Z = (DDLT - EDDLT) / 
11
Reorder Point (ROP)
Service level
Risk of
a stockout
Probability of
no stockout
Expected
demand
0
ROP
Quantity
Safety
stock
z
z-scale
12
Area under standard Normal pdf from -  to +z
Z = standard Normal with mean=0, s.d. = 1
Z = (X -  ) / 
See G&F Appendix A
See Stevenson, second from last page
P(Z <z)
Standard
Normal(0,1)
0
z
z-scale
0
P(Z  z)
.5
. 67
.75
.84
.80
1.28
.90
1.645
2.0
.95
.98
2.33
.99
3.5
.999813
z
Computing SS from SL for Normal DDLT
to provide SL = 95%.
 ROP =
EDDLT + SS
= EDDLT + z ().

z is the number of standard deviations SS is set above
EDDLT, which is the mean of DDLT.
z is read from Appendix B Table B2. Of Stevenson OR- Appendix A (p. 768) of Gaither & Frazier:
– Locate .95 (area to the left of ROP) inside the table (or as close
as you can get), and read off the z value from the margins: z =
1.64.
Example: ROP = 693 + 1.64(139) = 921
SS = ROP - EDDLT = 921 - 693. = 1.64(139) = 228
 If we double the s.d. to about 278, SS would double!
 Lead time variability reduction can same a lot of
inventory and $ (perhaps more than lead time itself!)
14
Summary View
Q+SS =
Target
Holding Cost = C[ Q/2 + SS]
(1) Order trigger by crossing ROP
(2) Order quantity up to (SS + Q)
Not full due to brisk
Demand after trigger
ROP = EDDLT + SS
ROP >
EDDLT
Safety
Stock
Lead Time
15
Part III: Single-Period Model: Newsvendor
 Used
to order perishables or other items with
limited useful lives.
– Fruits and vegetables, Seafood, Cut flowers.
– Blood (certain blood products in a blood bank)
– Newspapers, magazines, …
 Unsold
or unused goods are not typically carried
over from one period to the next; rather they are
salvaged or disposed of.
 Model can be used to allocate time-perishable
service capacity.
 Two costs: shortage (short) and excess (long).
16
Single-Period Model
 Shortage
or stockout cost may be a charge for
loss of customer goodwill, or the opportunity cost
of lost sales (or customer!):
Cs = Revenue per unit - Cost per unit.
 Excess (Long) cost applies to the items left over
at end of the period, which need salvaging
Ce = Original cost per unit - Salvage value per
unit.
(insert smoke, mirrors, and the magic of
17
Leibnitz’s Rule here…)
The Single-Period Model: Newsvendor
 How
do I know what service level is the best one, based
upon my costs?
 Answer: Assuming my goal is to maximize profit (at
least for the purposes of this analysis!) I should satisfy
SL fraction of demand during the next period (DDLT)
 If Cs is shortage cost/unit, and Ce is excess cost/unit,
then
Cs
SL 
Cs  Ce
18
Single-Period Model for Normally
Distributed Demand


Computing the optimal stocking level differs slightly
depending on whether demand is continuous (e.g.
normal) or discrete. We begin with continuous case.
Suppose demand for apple cider at a downtown street
stand varies continuously according to a normal
distribution with a mean of 200 liters per week and a
standard deviation of 100 liters per week:
– Revenue per unit = $ 1 per liter
– Cost per unit = $ 0.40 per liter
– Salvage value = $ 0.20 per liter.
19
Single-Period Model for Normally
Distributed Demand
 Cs
= 60 cents per liter
 Ce = 20 cents per liter.
 SL = Cs/(Cs + Ce) = 60/(60 + 20) = 0.75


To maximize profit, we should stock enough product to
satisfy 75% of the demand (on average!), while we
intentionally plan NOT to serve 25% of the demand.
The folks in marketing could get worried! If this is a
business where stockouts lose long-term customers, then
we must increase Cs to reflect the actual cost of lost
customer due to stockout.
20
Single-Period Model for Continuous
Demand

demand is Normal(200 liters per week, variance =
10,000 liters2/wk) … so  = 100 liters per week
 Continuous
example continued:
– 75% of the area under the normal curve
must be to the left of the stocking level.
– Appendix shows a z of 0.67 corresponds to a
“left area” of 0.749
– Optimal stocking level = mean + z () = 200
+ (0.67)(100) = 267. liters.
21
Single-Period & Discrete Demand: Lively
Lobsters




Lively Lobsters (L.L.) receives a supply of fresh, live lobsters from
Maine every day. Lively earns a profit of $7.50 for every lobster sold,
but a day-old lobster is worth only $8.50. Each lobster costs L.L.
$14.50.
(a) what is the unit cost of a L.L. stockout?
Cs = 7.50 = lost profit
(b) unit cost of having a left-over lobster?
Ce = 14.50 - 8.50 = cost – salvage value = 6.
(c) What should the L.L. service level be?
SL = Cs/(Cs + Ce) = 7.5 / (7.5 + 6) = .56 (larger Cs leads to SL >

.50)
Demand follows a discrete (relative frequency) distribution as given on next
page.
22
Lively Lobsters: SL = Cs/(Cs + Ce) =.56
Probability that demand
Cumulative
Demand
Relative
Relative
follows a
discrete
Frequency Frequency
(relative
(pmf)
(cdf)
Demand
frequency)
19
0.05
0.05
distribution:
20
0.05
0.10
21
0.08
0.18
Result: order 25
22
0.08
0.26
Lobsters,
23
0.13
0.39
because that is
the smallest
24
0.14
0.53
amount that
25
0.10
0.63
will serve at
26
0.12
0.75
least 56% of
27
0.10
you do
the demand on
a given night.
28
0.10
you do
29
0.05
1.00
* pmf = prob. mass function
will be less than or equal to x
P(D < 19 )
P(D < 20 )
P(D < 21 )
P(D < 22 )
P(D < 23 )
P(D < 24 )
P(D < 25 )
P(D < 26 )
P(D < 27 )
P(D < 28 )
P(D < 29 )
23