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Chapter 11: The Order Up-To
Model
Medtronic’s InSync pacemaker supply chain and
objectives
 Look at problem from two persopectives …
 One distribution center (DC) in Mounds View, MN.
 About 500 sales territories throughout the country.
 Consider Susan Magnotto’s territory in Madison,
Wisconsin.
 Objective:
 Because the gross margins are high, develop a system to
minimize inventory investment while maintaining a very high
service target, e.g., a 99.9% in-stock probability or a 99.9%
fill rate.
Timing in the order up-to model
 Time is divided into periods of equal length, e.g., one hour, one month.
 During a period the following sequence of events occurs:
 A replenishment order can be submitted.
 Inventory is received.
 Random demand occurs.
 Lead times:
 An order is received after a fixed number of periods, called the lead time.
 Let l represent the length of the lead time.
Order
Receive
period 0
order
Order
Receive
period 1
order
Order
Receive
period 2
order
An example with l = 1
Time
Demand
occurs
Period 1
Demand
occurs
Period 2
Demand
occurs
Period 3
Order up-to model vs. newsvendor model
 Both models have uncertain future demand, but there are differences…
Newsvendor
Order up-to
After one period
Never
Number of
replenishments
One (maybe two or
three with some
reactive capacity)
Unlimited
Demand occurs
during
replenishment
No
Yes
Inventory
obsolescence
 Newsvendor applies to short life cycle products with uncertain demand
and the order up-to applies to long life cycle products with uncertain,
but stable, demand.
The Order Up-To Model:
Choosing a demand distribution/forecast
 Can calculate the mean m and standard deviation s from
empirical data
 Calculate from monthly numbers
 For DC: average weekly demand = 80.6 and weekly standard
deviation = 58.81
 For sales territory average daily demand = 0.29 units
Distribution Center
Sales territory
16
700
14
600
12
Units
10
400
8
6
300
4
200
2
100
Dec
Oct
Nov
Sep
Jul
Month
Aug
Jun
Apr
May
Mar
Jan
Dec
Oct
Nov
Sep
Jul
Aug
Apr
May
Mar
Jan
Jun
Month
Feb
0
0
Feb
Units
500
The Order Up-To Model:
Model design and implementation
Order up-to model definitions
 On-order inventory [a.k.a. pipeline inventory] = the number of units that
have been ordered but have not been received.
 On-hand inventory = the number of units physically in inventory ready to
serve demand.
 Backorder = the total amount of demand that has has not been
satisfied:
 All backordered demand is eventually filled, i.e., there are no lost
sales.
 Inventory level = On-hand inventory - Backorder.
 Inventory position = On-order inventory + Inventory level.
 Order up-to level, S
 the maximum inventory position we allow.
 sometimes called the base stock level.
Order up-to model implementation
 Each period’s order quantity = S – Inventory position
 Suppose S = 4.
 If a period begins with an inventory position = 1, then three units
are ordered. (4 – 1 = 3 )
 If a period begins with an inventory position = -3, then seven
units are ordered (4 – (-3) = 7)
 A period’s order quantity = the previous period’s demand:
 Suppose S = 4.
 If demand were 10 in period 1, then the inventory position at the
start of period 2 is 4 – 10 = -6, which means 10 units are ordered
in period 2.
 The order up-to model is a pull system because inventory is ordered
in response to demand.
 The order up-to model is sometimes referred to as a 1-for-1
ordering policy.
The Order Up-To Model:
Performance measures
What determines the inventory level?
 Short answer:
 Inventory level at the end of a period = S minus demand over l +1 periods.
 Explanation via an example with S = 6, l = 3, and 2 units on-hand at the
start of period 1
Period 1
Period 2 Period 3
Keep in mind:
Period 4
At the start of a period the
Inventory level + On-order equals
S.
Time
D1
D2
D3 D4
?
All inventory on-order at the
start of period 1 arrives before
the end of period 4
Nothing ordered in periods 2-4
arrives by the end of period 4
All demand is satisfied so there
Inventory level at the end are no lost sales.
of period four
= 6 - D1 – D2 – D3 – D4
Expected on-hand inventory and backorder
…
S
Period 1
…
Period 4
S – D > 0, so there is
on-hand inventory
D
D = demand
over l +1
periods
Time
S – D < 0, so there
are backorders
 This is like a Newsvendor model in which the order quantity is S and the
demand distribution is demand over l +1 periods.
 So …
 Expected on-hand inventory at the end of a period can be evaluated
like Expected left over inventory in the Newsvendor model with Q =
S.
 Expected backorder at the end of a period can be evaluated like
Expected lost sales in the Newsvendor model with Q = S.
Stockout and in-stock probabilities, on-order
inventory and fill rate
 The stockout probability is the probability at least one unit is backordered in a
period:
Stockout probabilit y  ProbDemand over l  1 periods  S 
 1  ProbDemand over l  1 periods  S 
 The in-stock probability is the probability all demand is filled in a period:
In - stock probabilit y  1 - Stockout probabilit y
 ProbDemand over l  1 periods  S 
 Expected on-order inventory = Expected demand over one period x lead time
 This comes from Little’s Law. Note that it equals the expected demand over
l periods, not l +1 periods.
 The fill rate is the fraction of demand within a period that is NOT backordered:
Fill rate  1-
Expected backorder
Expected demand in one period
Demand over l +1 periods
 DC:
 The period length is one week, the replenishment lead time is three weeks, l
=3
 Assume demand is normally distributed:
 Mean weekly demand is 80.6 (from demand data)
 Standard deviation of weekly demand is 58.81 (from demand data)
 Expected demand over l +1 weeks is (3 + 1) x 80.6 = 322.4
 Standard deviation of demand over l +1 weeks is
3  1  58.81  117.6
 Susan’s territory:
 The period length is one day, the replenishment lead time is one day, l =1
 Assume demand is Poisson distributed:
 Mean daily demand is 0.29 (from demand data)
 Expected demand over l +1 days is 2 x 0.29 = 0.58
 Recall, the Poisson is completely defined by its mean (and the standard
deviation is always the square root of the mean)
DC’s Expected backorder assuming S = 625
 Expected backorder is analogous to the Expected lost sales
in the Newsvendor model:
 Suppose S = 625 at the DC
 Normalize the order up-to level:
z
S m
s

625  322.4
 2.57
117.6
 Lookup L(z) in the Standard Normal Loss Function Table:
L(2.57)=0.0016
 Convert expected lost sales, L(z), for the standard normal into the
expected backorder with the actual normal distribution that
represents demand over l+1 periods:
Expected backorder  s  L(z)  117.6  0.0016  0.19
 Therefore, if S = 625, then on average there are 0.19 backorders at
the end of any period at the DC.
Other DC performance measures with S = 625
Fill rate  1-
Expected backorder
0.19
 1
 99.76%.
Expected demand in one period
80.6
 So 99.76% of demand is filled immediately (i.e., without being backordered)
 (Note for below: inventory level + backorder = on hand)
Expected on-hand inventory  S-Expected demand over  l  1 periods
 Expected backorder
=625 - 322.4  0.19  302.8.
 So on average there are 302.8 units on-hand at the end of a period.
Expected on-order inventory  Expected demand in one period  Lead time
= 80.6  3  241.8.
 So there are 241.8 units on-order at any given time.
Performance measures in Susan’s territory:
~~FYI but will not be covered on Exam
 Look up in the Poisson Loss Function Table expected backorders for a
Poisson distribution with a mean equal to expected demand over l+1
periods:
Mean demand = 0.29
Mean demand = 0.58
F(S)
L (S )
F(S)
L (S )
S
S
0
0.74826
0.29000
0
0.55990
0.58000
1
0.96526
0.03826
1
0.88464
0.13990
2
0.99672
0.00352
2
0.97881
0.02454
3
0.99977
0.00025
3
0.99702
0.00335
4
0.99999
0.00001
4
0.99966
0.00037
5
1.00000
0.00000
5
0.99997
0.00004
F (S ) = Prob {Demand is less than or equal to S}
L (S ) = loss function = expected backorder = expected
amount demand exceeds S
 Suppose S = 3:
 Expected backorder = 0.00335
 In-stock = 99.702%
 Fill rate = 1 – 0.00335 / 0.29 = 98.84%
 Expected on-hand = S – demand over l+1 periods + backorder = 3 –
0.58 + 0.00335 = 2.42
 Expected on-order inventory = Demand over the lead time = 0.29
Problem 11.1
A furniture store sells study desks with normally
distributed demand. The mean = 40 and the
standard deviation = 20. The lead time is 2 weeks
and inventory replenishments are ordered weekly.
a) If S = 220 and your inventory = 100 with 85 desks
on order, how many desks will you order? (35)
b) Suppose S = 220 and you are about to place an
order and your inventory level – 160 with 65 desks
on order. How many desks will you order? (order
0)
The Order Up-To Model:
Choosing an order up-to level, S, to
meet a service target
Choose S to hit a target in-stock with
normally distributed demand
 Suppose the target in-stock probability at the DC is 99.9%:
 From the Standard Normal Distribution Function Table,
F(3.08)=0.9990
 So we choose z = 3.08
 To convert z into an order up-to level:
S  m  z  s  322.4  3.08 117.6
 685
 Note that m and s are the parameters of the normal distribution
that describes demand over l + 1 periods.
Problem 11.1
A furniture store sells study desks with normally
distributed demand. The mean = 40 and the
standard deviation = 20. The lead time is 2 weeks
and inventory replenishments are ordered weekly.
c) What is the optimal order-up-to level if you want
to target a 98% in stock probability? (191.14)
Choose S to hit a target fill rate with
normally distributed demand
 Find the S that yields a 99.9% fill rate for the DC.
 Step 1: Evaluate the target lost sales
 note 1 period in the numerator and l + 1 periods in the denominator


Expected demand in one period
1  Fill rate 
L( z )  
 Standard deviation of demand over l  1 periods 
 80.6 

(1  0.999)  0.0007
 117.6 
 Step 2: Find the z that generates that target lost sales in the Standard Normal
Loss Function Table:
 L(2.81) = L(2.82) = L(2.83) = L(2.84) = 0.0007
 Choose z = 2.84 to be conservative (higher z means higher fill rate)
 Step 3: Convert z into the order up-to level:
S  322.4  2.84 117.62  656
Problem 11.1
A furniture store sells study desks with normally
distributed demand. The mean = 40 and the
standard deviation = 20. The lead time is 2 weeks
and inventory replenishments are ordered weekly.
d) What is the optimal order-up-to level if you want
to target a 98% fill rate? (S = 175.77)
The Order Up-To Model:
Appropriate service levels
Justifying a service level via cost minimization
 Let h equal the holding cost per unit per period
 e.g. if p is the retail price, the gross margin is 75%, the annual holding cost
is 35% and there are 260 days per year, then h = p x (1 -0.75) x 0.35 / 260
= 0.000337 x p
 Let b equal the penalty per unit backordered
 e.g., let the penalty equal the 75% gross margin, then b = 0.75 x p
 “Too much-too little” challenge:
 If S is too high, then there are holding costs, Co = h
 If S is too low, then there are backorders, Cu = b
 Cost minimizing order up-to level satisfies
Prob Demand over  l  1 periods  S  
Cu
b
(0.75  p)


Co  Cu h  b (0.000337  p)  (0.75  p)
 0.9996
 Optimal in-stock probability is 99.96% because
In - stock probabilit y  ProbDemand over l  1 periods  S  Critical ratio
The optimal in-stock probability is usually quite high
 Suppose the annual holding cost is 35%, the backorder penalty cost
equals the gross margin and inventory is reviewed daily.
Optimal in-stock probability
100%
98%
96%
94%
92%
90%
88%
0%
20%
40%
60%
Gross margin %
80%
100%
Problem 11.1
A furniture store sells study desks with normally
distributed demand. The mean = 40 and the
standard deviation = 20. The lead time is 2 weeks
and inventory replenishments are ordered weekly.
e) Suppose S = 120. What is the expected on-hand
inventory? (13.82)
f) Suppose S = 200. What is the fill rate? (99.69%)
g) Suppose you want to maintain a 95% in-stock
probability. What is the expected fill-rate?
(98.22%)
The Order Up-To Model:
Controlling ordering costs
Impact of the period length
 Increasing the period length leads to larger and less
frequent orders:
 The average order quantity = expected demand in a single period.
 The frequency of orders approximately equals 1/length of period.
 See “saw tooth” diagrams in the text
 Suppose there is a cost to hold inventory and a cost to
submit each order (independent of the quantity ordered)…
 … then there is a tradeoff between carrying little inventory
(short period lengths) and reducing ordering costs (long
period lengths)
Tradeoff between inventory holding costs and
ordering costs
22000
20000
18000
Total costs
16000
14000
Cost
 Costs:
 Ordering costs = $275 per order
 Holding costs = 25% per year
 Unit cost = $50
 Holding cost per unit per year = 25%
x $50 = 12.5
12000
Inventory
holding costs
10000
8000
6000
4000
 Period length of 4 weeks minimizes costs:
 This implies the average order
quantity is 4 x 100 = 400 units
 EOQ model offers an approximation for
min Q:
Q
2 K  R

h
2  275  5200
 478
12.5
Ordering costs
2000
0
0
1
2
3
4
5
6
Period length (in weeks)
7
8
9
The Order Up-To Model: Managerial insights
140
120
100
Expected inventory
 Better service requires more
inventory at an increasing rate
80
60
40
Inc re a s ing
s ta nda rd de via tion
20
0
90% 91% 92% 93% 94% 95% 96% 97% 98% 99% 100%
Fill rate
600
Expected inventory
 Shorten lead times and you will
reduce inventory
500
400
300
200
100
0
0
5
10
Lead time
15
20
3000
 Do not forget about pipeline
inventory
Inventory
2500
2000
1500
1000
500
0
0
5
10
Lead time
15
20
Order up-to model summary
 The order up-to model is appropriate for products with random demand
but many replenishment opportunities.
 Expected inventory and service are controlled via the order up-to level:
 The higher the order up-to level the greater the expected
inventory and the better the service (either in-stock probability or
fill rate).
 The key factors that determine the amount of inventory needed are…
 The length of the replenishment lead time.
 The desired service level (fill rate or in-stock probability).
 Demand uncertainty.
 When inventory obsolescence is not an issue, the optimal service level is
generally quite high.
Problem 11.1
A furniture store sells study desks with normally distributed demand.
The mean = 40 and the standard deviation = 20. The lead time is 2
weeks and inventory replenishments are ordered weekly.
a)
If S = 220 and your inventory = 100 with 85 desks on order, how
many desks will you order? (35)
b)
Suppose S = 220 and you are about to place an order and your
inventory level – 160 with 65 desks on order. How many desks will
you order? (order 0)
c)
What is the optimal order-up-to level if you want to target a 98% in
stock probability? (191.14)
d)
What is the optimal order-up-to level if you want to target a 98%
fill rate? (S = 175.77)
e)
Suppose S = 120. What is the expected on-hand inventory? (13.82)
f)
Suppose S = 200. What is the fill rate? (99.69%)
g)
Suppose you want to maintain a 95% in-stock probability. What is
the expected fill-rate? (98.22%)