3. The Effect of
Demand Uncertainty
• Recall that EOQ Model does not consider demand uncertainty
• What is the effect of demand uncertainty?
• Most companies treat the world as if it were predictable:
• Production and inventory planning are based on forecasts of demand made
far in advance of the selling season
• Companies are aware of demand uncertainty when they create a forecast, but
they design their planning process as if the forecast truly represents reality
4. The Effect of Demand Uncertainty
•But, does forecast truly represent reality ?
•See next slide about demand forecast
5. Demand Forecast
•The three principles of all forecasting
techniques:
•Forecasting is always wrong
•The longer the forecast horizon, the
worse is the forecast
•Aggregate forecasts are more accurate
6. Why is it more and more important to
consider demand uncertainty?
•Recent technological advances have increased the
level of demand uncertainty:
•Short product life cycles
•Increasing product variety
7. Demand Variability: Example 1
Product Demand
150
75
225
100
150
50
125
61 48 53
104
45
0
50
100
150
200
250
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
Feb
Mar
Month
Demand
(000's)
8. Demand Variability: Example 1
Histogram for Value of Orders Placed in a Week
0
5
10
15
20
25
$25,000
$50,000
$75,000
$100,000
$125,000
$150,000
$175,000
$200,000
Value of Orders Placed in a Week
Frequency
9. Reminder:
The Normal Distribution
0 10 20 30 40 50 60
Average = 30
Standard Deviation = 5
Standard Deviation = 10
Notations:
Average = mean = m
and
Standard Deviation = std dev = s
10. Demand Uncertainty and Bell Curve
Returns on Safety
Stock Investment
mean mean
+1 std dev
mean
+2 std dev
mean
-1 std dev
mean
-2 std dev
34%
34%
14% 14%
2%
2%
13. Risk Pooling: Demand variability is reduced if one aggregates
demand across locations (or time periods)
3,500 3,900 4,300
3,100
2,700 3,500 3,900 4,300
3,100
2,700
7,000 7,566 8,132
6,434
5,868
34%
14%
34%
14%
2% 2%
34%
14%
2%
Demand during 2 days
Demand during
1 day
Demand during
1 day
14. Risk Pooling: Demand variability is reduced if one aggregates
demand across locations (or time periods)
Lm
34%
14%
2%
Demand during L days
…
Lm+ sL Lm+ 2sL
Demand during
1 day with
mean=m
and
Standard Deviation = s
34% 34% 34% 34%
14% 14% 14% 14%
15. What is important when we develop a multi-period
inventory model with demand uncertainty?
how to handle demand during lead time?
Anything else? Let's understand inventory
further and recall some important
inventory characteristics
16. Understanding Inventory
• The inventory policy is affected by:
• Demand Characteristics
• Lead Time
• Number of Products
• Objectives
• Service level
• Minimize costs
• Cost Structure
18. EOQ: Calculating Total Cost in a Cycle Time T (i.e.,
total holding costs and ordering costs during the
cycle time T)
• Purchase Cost during the cycle time T: $C per unit of product * Demand
during a period of T units of time = CQ = CDT
• Holding Cost during the cycle time T: (Average Inventory level) * (Holding
Cost Rate h) *(Period of T units of time)
= (Q/2)*h*T
• Fixed ordering cost (Setup Cost) during the cycle time T :
Fixed Order Cost K
• Goal: Find the Order Quantity Q that
Minimizes These Costs:
• Note: Holding Cost Rate h is the cost for holding one unit of product for one
unit of time, e.g., h= $7 per unit of product weekly = $1 per unit of product
daily
19. Understanding Inventory
• The inventory policy is affected by:
• Demand Characteristics
• Lead Time
• Number of Products
• Objectives
• Service level
• Minimize costs
• Cost Structure
20. The Multi-Period Inventory Model
• We consider a single product inventory model (In Week 4 we will
introduce models which can handle multiple products)
• We assume that the demand is random and follows a normal distribution
• Ordering cost has two components: (1) Fixed ordering cost K per order + (2) a
variable ordering cost C which is proportional to the amount ordered.
• Inventory holding cost is charged per item per unit time use h to denote
• We assume that if an order arrives and there is no inventory, the order is lost
• The distributor has a required service level. This is expressed as the likelihood
that the distributor will not stock out during lead time.
• Intuitively, how will this affect our policy?
21. A distributor or a Distribution Center (DC) holds
inventory to:
• Satisfy demand during lead time
• Protect against demand uncertainty
• Balance fixed costs and holding costs
22. A distributor or a Distribution Center (DC) holds
inventory to:
• Satisfy demand during lead time
in EOQ setting, we order when inventory level drops to DL.
• Protect against demand uncertainty
in EOQ setting, no demand uncertainty. So this part is ignored.
• Balance fixed costs and holding costs
in EOQ setting, we order (2KD)/h so that we can balance our
fixed ordering costs and holding costs, which allows us to minimize the
total ordering and holding cost.
23. What will change when we consider demand
uncertainty on top of an EOQ setting?
The DC holds inventory to:
• Satisfy demand during lead time
in EOQ setting, we order when inventory level drops to DL.
With demand uncertainty, it makes more sense to say “Satisfy EXPECTED demand during lead
time”.
• Protect against demand uncertainty
in EOQ setting, no demand uncertainty. So this part is ignored.
With demand uncertainty, we need to add some safety stock in addition to the “EXPECTED
demand” to Protect against demand uncertainty. How to model this if demand is normally distributed?
• Balance fixed costs and holding costs
in EOQ setting, we order (2KD)/h so that we can balance our
fixed costs and holding costs, which allows us to minimize the total ordering and holding cost.
With demand uncertainty, this is still valid. That is, everytime we place an order, we still want to
balance our fixed ordering costs and holding costs, which allows us to minimize the total
ordering/holding cost
24. The Multi-Period Inventory Model: Continuous
Review Policy v.s. Periodic Review Policy
• Continuous Review: also called (Q, R) Policy
• Q = order quantity
• R = reorder point
• How to determine Q and R?
• Periodic Review:
• Short review period (e.g. daily): (s, S) Policy
• Set s = R
• Set S = R + Q
• Long review period (e.g. weekly, monthly, etc.):
• Always order after an inventory position review (This implies we have to pay fixed cost after each review!)
• In this case, what is the role of fixed costs here? Do we still care about reorder point? (No!)
• We use a Base-Stock Policy: in each review period, review inventory position and order enough to raise the
inventory position to the base-stock level
• How to determine an effective base-stock level?
25. Continuous Review: also called (Q, R) Policy
Time
Inventory
Status
R+Q
R
0
Lead
Time
Lead
Time
Inventory Position (dotted line)
Q
Inventory Level (solid line)
26. Continuous Review policy – Notation
(assuming demand is normally distributed)
• AVG = average daily demand
• STD = standard deviation of daily demand
• L = replenishment lead time in days
• h = holding cost of one unit for one day
• SL = service level (for example, 95%). This implies that the probability of stocking
out is 100%-SL (for example, 5%)
• Also, the Inventory Position at any time is the actual inventory on hand plus
items already ordered, but not yet delivered minus items that are backordered.
• Actual inventory on hand is called "inventory level".
27. Continuous Review Policy - Analysis
• The reorder point R has two components:
• To account for average demand during lead time:
LAVG
• To account for deviations from average (we call this safety stock)
z STD L
where z is chosen from statistical tables to ensure that the probability of stockouts during
leadtime is 100%-SL.
• The reorder point R = LAVG + z STD L
28. Continuous Review policy - Example
• The distributor has historically observed weekly demand of:
AVG = 44.6 STD = 32.1
Replenishment lead time is 2 weeks, and desired service level SL = 97%
• Average demand during lead time is:
44.6 2 = 89.2
• Safety Stock is:
1.88 32.1 2 = 85.3
• Reorder point is thus 175, or about 3.9 weeks of supply at warehouse and in the
pipeline
29. Continuous Review Policy – Now that we know the Re-order Point,
how much should we order each time?
That is, Q = ?
• What are the factors we should consider?
• Fixed costs = 0 (Model One)
• Fixed costs > 0 (Model Two)
30. Continuous Review policy- Model One: Fixed
Costs = 0
• It costs you NOTHING to place an order, how much should
you order every time?
• Big Q vs. small Q High inventory holding cost vs. low inventory holding
cost
• Q should be as small as possible Just maintain the inventory level
at around reorder point R
31. Continuous Review policy - Model Two: Fixed
Costs > 0
• Intuitively, how will this affect our policy?
• In addition to previous costs, a fixed cost K is paid every time an order is placed.
• The reorder point will be the same as the previous model, in order to meet the
service requirement:
R = LAVG + z STD L
• What about order quantity Q?
• Extend the EOQ model Q=(2 K AVG)/h
32. Continuous Review policy - Model Two:
The Order-Up-To Level
• We have used the extended EOQ to balance the fixed costs and the holding costs:
Q=(2 K AVG)/h
• If there was no variability in demand, we would order Q when inventory level was
at L AVG. Why?
• However, there is demand variability, so we need safety stock
z STD * L
• The total order-up-to level is:
S = Q+R = Q+ L AVG + z STD * L
33. Continuous Review Policy - Model Two: Example
• Consider the previous example, but with the following additional info:
• fixed cost of $4500 when an order is placed
• $250 product cost
• holding cost 18% of product
• Weekly holding cost:
h = (.18 250) / 52 = 0.87
• Order quantity
Q=(2 4500 44.6 / 0.87 = 679
• Order-up-to level:
S = R + Q = 175 + 679 = 854
34. Evaluating Inventory holding cost Evaluating Inventory Level
(Continuous Review Policy)
L
STD
z
(1) Inventory level before receiving an order (lowest point) =
(2) Inventory level after receiving an order (highest point) =
[(1)+(2)]/2 = Average Inventory =
L
STD
z
Q
+
L
STD
z
Q
+
2
Inventory level as a function of time in a (Q,R) policy
34
35. The Multi-Period Inventory Model: Continuous
Review Policy v.s. Periodic Review Policy
• Continuous Review: also called (Q, R) Policy
• Q = order quantity
• R = reorder point
• How to determine Q and R?
• Periodic Review:
• Short review period (e.g. daily): (s, S) Policy
• Set s = R
• Set S = R + Q
• Long review period (e.g. weekly, monthly, etc.):
• Always order after an inventory position review (This implies we have to pay fixed cost after each review!)
• In this case, what is the role of fixed costs here? Do we still care about reorder point? (No!)
• We use a Base-Stock Policy: in each review period, review inventory position and order enough to raise the
inventory position to the base-stock level
• How to determine an effective base-stock level?
36. (Q, R) Policy (continuous review policy) vs.
(s,S) Policy (short periodic review policy)
• (Q, R) Policy is also called continuous review policy because we
assume that we can review our inventory level continuously (i.e., all
the time) and can place an order anytime we want.
• In a periodic review environment, we do not know the inventory level
all the time, thus when the inventory reaches “R” – the reorder point
under the continuous review policy, we may not know and can NOT
order “Q” in every order as suggested by the continuous review
policy. Therefore, we need a more general policy.
• For Short Review Period, we can use (s, S) Policy: Whenever the
inventory position drops below a certain level, s, we order to raise the
inventory position to level S.
• s: reorder point
• S: order-up-to level
37. Periodic Review Policy: Short Review Period –
use (s, S) Policy – How to set s and S?
• When the review period is short, the situation is very
close to continuous review environment, thus we can
use the continuous review policy to approximate the
reorder point “s” and order-up-to level “S”.
• Periodic Review: Short review period (e.g. daily): (s, S)
Policy
• Set s = R
• Set S = R + Q
38. The Multi-Period Inventory Model: Continuous
Review Policy v.s. Periodic Review Policy
• Continuous Review: also called (Q, R) Policy
• Q = order quantity
• R = reorder point
• How to determine Q and R?
• Periodic Review:
• Short review period (e.g. daily): (s, S) Policy
• Set s = R
• Set S = R + Q
• Long review period (e.g. weekly, monthly, etc.):
• Always order after an inventory position review (This implies we have to pay fixed cost after each review!)
• In this case, what is the role of fixed costs here? Do we still care about reorder point? (No!)
• We use a Base-Stock Policy: in each review period, review inventory position and order enough to raise the
inventory position to the base-stock level
• How to determine an effective base-stock level?
39. Periodic Review (Long Review Period):
Use Base-Stock Policy (Re-order Point is not Needed)
,
L L L
r1 r2 r3
Units
Time
Base Stock
Inventory Level
Expected UB
Expected LB
r r
r r
40. Periodic Review Policy
The Base-Stock level (inventory position) should cover
____ days?
Base-Stock level =
41. Periodic Review Policy – Long Review Period
How to determine a good base-stock level?
Base-Stock level = (r+L)AVG + z STD r+L
In the above formula, r is the review period.
For example, if we review every month, then r = 1 month.
42. Periodic Review Policy
Base-Stock level = (r+L)AVG + z STD r+L
= (r+L)AVG + z STD r+L - L AVG
= rAVG + z STD r+L
= (r+L)AVG + z STD r+L - (r+L) AVG
= z STD r+L
Expected Inventory level after receiving an order
Expected Inventory level before receiving an order
Average Inventory
= 0.5 [(r AVG + z STD r+L)+(z STD r+L)]
= 0.5 r AVG + zSTDr+L
43. What if Lead Time is Random?
• We need information about the lead time:
AVGL is average lead time
STDL is standard deviation of lead time
• We calculate the safety stock and reorder point:
s = AVG AVGL
+ zAVGL STD2 + AVG2 STDL2
Order-up-to level S = Q + s
