1. The document discusses different definitions of stock levels including on-hand stock, net stock, inventory position, and safety stock. 2. It also discusses replenishment control systems for items with probabilistic demand, including continuous versus periodic review and different inventory policy forms like (s,Q), (s,S), (R,S), and (R, s, S). 3. The document provides examples of how to determine safety stock levels using approaches like equal time supplies, cost minimization based on stockout costs, and meeting customer service level targets. It also provides examples of calculating safety stock for different demand distributions and cost structures.

1. INE446 Production Systems II
Lecture 3: Individual Items with Probabilistic Demand
1

2. Different Definitions of Stock Level
1. On-hand (OH) stock This is stock that is physically on the shelf; it can never be negative. This
quantity is relevant in determining whether a particular customer demand is satisfied
directly from the shelf.
2. Net stock is defined by the following equation:
𝑵𝒆𝒕 𝒔𝒕𝒐𝒄𝒌 = (𝑶𝒏 𝒉𝒂𝒏𝒅) − (𝑩𝒂𝒄𝒌𝒐𝒓𝒅𝒆𝒓𝒔)
This quantity can become negative (namely, if there are backorders). It is used in some mathematical
derivations and is also a component of the following important definition.
3. Inventory position also called the available stock is defined by the equation:
𝑰𝒏𝒗𝒆𝒏𝒕𝒐𝒓𝒚 𝒑𝒐𝒔𝒊𝒕𝒊𝒐𝒏 = (𝑶𝒏 𝒉𝒂𝒏𝒅) + (𝑶𝒏 𝒐𝒓𝒅𝒆𝒓) − (𝑩𝒂𝒄𝒌𝒐𝒓𝒅𝒆𝒓𝒔) − (𝑪𝒐𝒎𝒎𝒊𝒕𝒕𝒆𝒅)
• The on-order stock is that stock which was ordered but not yet received.
• The committed quantity is the stock that cannot be used for other purposes in the short run.
4. Safety stock (SS) or buffer is the average level of the net stock just before a replenishment
arrives as a buffer against larger-than-average demand during the effective replenishment
lead time. With backorders, it is possible that SS could be negative.
2

3. Backorders versus Lost Sales
When an item is temporarily out of stock, two possible situations may
happen:
1. Complete backordering: Any demand during stockouts is backordered
and filled as soon as a new replenishment arrives.
This situation corresponds to a captive market, common in government organizations
(particularly the military) and at the wholesale–retail link of some distribution systems (e.g.,
exclusive dealerships).
2. Complete lost sales: Any demand during stockouts is lost. The
customer seeks his order elsewhere to satisfy his or her need.
• Most common at the retail–consumer link such as demand for a loaf of bread.
• Also, when the customer accepts substitute product is clearly a lost sale for the originally
intended item but may affect the anticipated demand.
3

4. How to Control System under Probabilistic Demand?
The replenishment control system must resolve the following three
problems:
1. How often the inventory status should be determined.
2. When a replenishment order should be placed.
3. How large the replenishment order should be.
To respond to these three fundamental issues, a manager must
determine several things.
1. How important is the item.
2. Can, or should, the stock status be reviewed continuously or periodically.
3. What form should the inventory policy take.
4. What specific cost or service objectives should be set.
4

5. The Importance of the Item: A, B, and C Classification
• How critical the item under consideration is to the firm requires the A, B,
and C Classification
• Recall that
• “A” items make up roughly 20% of the total number of items but represent 80% of the
dollar sales volume.
• “B” items comprise roughly 30% of the items but represent 15% of the dollar volume.
• “C” items comprise roughly 50% of the items and represent only 5% of the dollar
volume.
• Note: Firms often include slow-moving and inexpensive items in the “A” category if
the items are critical to the business.
• The importance of the item helps direct the response to the remaining three
questions.
• This chapter is devoted to “B” items.
5

6. Continuous versus Periodic Review
• How often the inventory status should be determined specifies the
review interval 𝑅:
𝑅 = the time that elapses between two consecutive moments at which the
stock level is observed
• An extreme case is where there is continuous review; that is, the
stock status is always known, so 𝑅 → 0.
• Continuous review stock level is determined through each transaction
(shipment, receipt, demand, etc.) that triggers an immediate updating
of the status, called transactions reporting.
• Periodic review determines the stock status only every 𝑅 time units.
E.g., vending machine refills.
6

7. Continuous vs Periodic Review models
• Items may be produced on the same piece of equipment, purchased
from the same supplier, or shipped in the same transportation mode
require coordination of replenishments, thus, periodic review is
particularly appealing to get them done with the same review interval
• Periodic review also allows a reasonable prediction of the level of the
workload on the staff involved. A rhythmic, rather than random,
pattern is usually appealing to the staff.
• For slow-moving items, updates are only made when a transaction
occurs, and thus continuous review is preferred, but the seldom
transactions of those items make periodic review also viable.
• The major advantage of continuous review is that, to provide the
same level of customer service, it requires less SS (hence, lower
carrying costs).
7

8. The Form of the Inventory Policy: Four Types
• Answering When should an order be placed and what quantity should
be ordered.
• Order-Point, Order-Quantity (𝑠, 𝑄) continuous review system
• Order-Point, Order-Up-to-Level (𝑠, 𝑆) continuous review system
• Periodic-Review, Order-Up-to-Level (𝑅, 𝑆) system
• Hybrid (𝑅, 𝑠, 𝑆) periodic review system
where
• 𝑠 = reorder level
• 𝑄 = order quantity
• 𝑆 = order up to level
• 𝑅 = review period length
8

9. Order-Point, Order-Quantity (𝑠, 𝑄) continuous review system
• This is a continuous-review system (i.e., 𝑅 = 0).
• A fixed quantity 𝑄 is ordered in every replenishment
• Whenever the inventory position drops to the reorder point 𝑠 or lower.
• Note that the inventory position, and not the net stock, is used to trigger an
order.
• The (𝑠, 𝑄) system is often called a two-bin system:
• A physical form of implementation is to have two bins for storage of an item.
• As long as units remain in the first bin, demand is satisfied from it.
• The amount in the second bin corresponds to the order point 𝑠.
• When this second bin is opened, a replenishment is triggered.
• When the replenishment arrives, the second bin is refilled, and the remainder is put into the
first bin.
• It should be noted that the physical two-bin system will operate properly only when no more
than one replenishment order is outstanding at any point in time.
• Thus, to use the system, it may be necessary to adjust 𝑄 upward so that it is appreciably
larger than the average demand during a lead time.
9

10. Order-Point, Order-Up-to-Level (𝑠, 𝑆) Continuous Review System
• This is a continuous review system.
• Like the (𝑠, 𝑄) system, a replenishment is made whenever the inventory position
drops to the order point 𝑠 or lower.
• However, a variable replenishment quantity is used, ordering enough to raise the
inventory; i.e., 𝑆 − (𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑦 𝑎𝑡 𝑜𝑟𝑑𝑒𝑟 𝑖𝑛𝑠𝑡𝑎𝑛𝑐𝑒)
• The (𝑠, 𝑆) system is frequently referred to as a min-max system
• Note: If all demand transactions are unit sized (i.e., one unit at a time demand),
then the two systems are identical because the replenishment requisition will
always be made when the inventory position is exactly at 𝑠; that is, 𝑆 = 𝑠 + 𝑄.
• If transactions can be larger than unit size, the replenishment quantity in the
(𝑠, 𝑆) system becomes variable.
10

11. Periodic-Review, Order-Up-to-Level (𝑅, 𝑆) System
• This system, also known as a replenishment cycle system.
• It is frequently seen when items are ordered from the same supplier or require
resource sharing.
• The control procedure is that every 𝑅 units of time (i.e., at each review instant)
enough is ordered to raise the inventory position to the level 𝑆.
• Because of the periodic-review property, this system is much preferred to order
point systems in terms of coordinating the replenishments of related items.
For example, when ordering from overseas, it is often necessary to fill a shipping container to
keep shipping costs under control. The coordination afforded by a periodic-review system can
provide significant savings.
• In addition, the (𝑅, 𝑆) system offers a regular opportunity (every 𝑅 units of time)
to adjust the order-up-to-level 𝑆, a desirable property if the demand pattern is
changing with time.
• The main disadvantages of the (𝑅, 𝑆) system are that the replenishment
quantities vary and that the carrying costs are higher than in continuous-review
systems.
11

12. Hybrid (𝑅, 𝑠, 𝑆) periodic review system
• This is a combination of (𝑠, 𝑆) and (𝑅, 𝑆) systems.
• The idea is that every 𝑅 units of time the inventory position is
checked.
• If it is at or below the reorder point 𝑠, we order enough to raise it to 𝑆.
• If the position is above 𝑠, nothing is done until at least the next review
instant.
• The (𝑠, 𝑆) system is the special case where 𝑅 = 0.
• Alternatively, one can think of the (𝑅, 𝑠, 𝑆) system as a periodic
version of the (𝑠, 𝑆) system.
• Also, the (𝑅, 𝑆) situation can be viewed as a periodic implementation
of (𝑠, 𝑆) with 𝑠 = 𝑆 − 1.
12

13. Rules of Thumb for Selecting the Form of the Inventory Policy
13

14. Rules to determine Safety Stock (𝑆𝑆) levels
• 𝑆𝑆 Established through the Use of a Simple-Minded Approach
• 𝑆𝑆 Based on Minimizing Cost
• 𝑆𝑆 Based on Customer Service
• 𝑆𝑆 Based on Aggregate Considerations
14

15. Various of models to determine SS
• Equal time supplies
• Fixed safety factor
• Cost (𝐵1) per stockout occasion
• Fractional charge (𝐵2) per unit short
• Fractional charge (𝐵3) per unit short per unit time
• Cost (𝐵4) per customer line item backordered
• Specified probability (𝑃1) of no stockout per replenishment cycle
• Specified fraction (𝑃2) of demand to be satisfied directly from the shelf (i.e., the fill rate)
• Specified ready rate (𝑃3)
• Specified average Time Between Stockout occasions (𝑇𝐵𝑆)
• Minimization of Expected Total Stockout Occasions Per Year (ETSOPY) subject to a
specified Total Safety Stock (𝑇𝑆𝑆)
• Minimization of Expected Total Value Short Per Year (𝐸𝑇𝑉𝑆𝑃𝑌) subject to a specified 𝑇𝑆𝑆
15

16. 𝑆𝑆 Established through the Use of a Simple-Minded Approach
Examples of simple-minded approaches (no specific objectives):
• Equal Time Supplies: The 𝑆𝑆s of a broad group of (if not all) items in an
inventory population are set equal to the same time supply.
Example: reorder any item when its inventory position minus the forecasted lead
time demand drops to a 2-month supply or lower.
• Equal Safety Factors: This approach uses a common value of 𝑘 for a
broad range of items where the 𝑆𝑆 is
𝑆𝑆 = 𝑘𝜎𝐿
16

17. 𝑆𝑆 Based on Minimizing Cost
Four illustrative cases:
• Specified Fixed Cost (𝐵1) per Stockout Occasion
The only cost associated with a stockout occasion is a fixed value 𝐵1, independent of
the magnitude or duration of the stockout
• Specified Fractional Charge (𝐵2) per Unit Short
A fraction 𝐵2 of unit value is charged per unit short; that is, the cost per unit short of
an item 𝑖 is 𝐵2 × 𝑣𝑖, where 𝑣𝑖 is the unit variable cost of the item
• Specified Fractional Charge (𝐵3) per Unit Short per Unit Time
A charge 𝐵3 per dollar short (equivalently, 𝐵3 × 𝑣 per unit short) per unit time.
• Specified Charge (𝐵4) per Customer Line Item Short
A customer line item is one of perhaps many lines of a customer’s order. Therefore, if a
customer orders 10 items, but only nine are in stock, the supplier is short of one
customer line item. The assumption here is that there is a charge 𝐵4 per line item
short.
17

18. 𝑆𝑆 Based on Customer Service
• Specified probability (𝑃1) of no stockout per replenishment cycle called the
“Cycle Service Level” (𝐶𝑆𝐿)
• 𝑃1 is the fraction of cycles in which a stockout does not occur.
• A stockout is defined as an occasion when the OH stock drops to the zero level.
• Using a common 𝑃1 across a group of items, is equivalent to using a common safety
factor 𝑘.
• Specified fraction (𝑃2) of demand to be satisfied routinely from available
inventory called the “Fill Rate” (𝑃1)
• The fill rate is the fraction of customer demand that is met routinely without
backorders or lost sales.
• Specified fraction of time (𝑃3) during which the net stock is positive called
“Ready Rate”
• Specified average time (𝑇𝐵𝑆) between stockout occasions
18

19. Example with CSL Using a Discrete Distribution
• Consider a B item, with continuous review of the inventory and choose an (s,Q) policy.
• The lead time demand distribution is presented in the shown table.
• Lead time 𝐿=1 week.
• The order quantity is 20 units.
• From the demand distribution that the expected demand per week (or per lead time)
is 2.2 units.
• The firm operates for 50 weeks per year; thus, the annual demand is 50(2.2) = 110
units/year.
• Consider a cycle service level (𝐶𝑆𝐿), 𝑃1 = Prob. of no stockout per cycle = 90%.
• To attain this 𝐶𝑆𝐿, find 𝑠 the cumulative Probability 𝑃 𝐷𝐿𝑇 ≤ 𝑠 ≥ 0.9 .
• Clearly 𝑠 = 4 as the first value 𝑋 for which 𝑃 𝐷𝐿𝑇 ≤ 𝑋 ≥ 0.9 is for 𝑋 = 4.
• A reorder point 𝑠 = 3 would provide a 𝑃1 of only 80%
19

20. Example for Cost Per Unit Short 𝐵2 Using a Discrete Distribution
• Same demand data as before
• The fixed cost per order 𝐴=$18.
• The cost per unit of inventory held per year is $10.
• All unmet demand is lost rather than backordered.
• The shortage-costing method using the cost per unit short,
• Let the order quantity 𝑄 = 20 and the reorder point 𝑠 = 2.
• The firm has established that 𝐵2𝑣 = $20.
• The Safety Stock=the expected amount OH (or net stock) just before the
replenishment arrives and is computed using the shown table to be
𝑆𝑆 = = (0.1)(2) + (0.2)(1) + (0.3 + 0.2 + 0.2)(0) = 0.4
• The total annual cost is comprised of:
20
Ordering Cost = 𝐴 ×
𝐷
𝑄
= $18 ×
110
20
= $99.
Holding Cost = (𝑄/2 + 𝑆𝑆)($10 𝑝𝑒𝑟 𝑢𝑛𝑖𝑡) = (10.4)($10) = $104.
Shortage Cost = (Expected number of units short per cycle)×(Number of cycles per year)×(Cost per unit short)
where, (Expected number of units short per cycle)= = 0×0.3+1×0.2+2×0.2=0.6
Total annual cost = $99 + $104 + $66 = $269

21. 𝑆𝑆 Based on Aggregate Considerations
21

22. 22
4. When to Reorder with EOQ Ordering
• Reorder Point - When the quantity on hand of an
item drops to this amount, the item is reordered. We
call it ROP.
• Safety Stock - Stock that is held in excess of expected
demand due to variable demand rate and/or lead
time. We call it ss.
• (lead time) Service Level - Probability that demand
will not exceed supply during lead time. We call this
cycle service level, CSL.

23. 23
Optimal Safety Inventory Levels
Lead Times
time
inventory
Shortage
An inventory cycle
ROP
Q

24. 24
Safety Stock
LT Time
Expected demand
during lead time
Maximum probable demand
during lead time
ROP
Quantity
Safety stock

25. 25
Case 1: remaining inventory after Demand during Lead Time
0
ROP
Demand
During LT
LT
Inventory
DLT:
Demand
During
LT

26. 26
Case 2 : Shortage after Demand during Lead Time
ROP
0
Demand
During LT
LT
Shortage
DLT:
Demand
During
LT
𝑆ℎ𝑜𝑟𝑡𝑎𝑔𝑒 = max{ 0, (𝐷𝐿𝑇 − 𝑅𝑂𝑃)}

27. 27
Cycle Service Level
Cycle service level: percentage of cycles with shortage
ROP]
time
lead
during
[Demand
inventory]
t
[Sufficien
inventory
sufficient
has
cycle
single
a
y that
Probabilit
0.7
7
.
0
otherwise
1
shortage,
has
cycle
a
if
0
Write
10
1
0
1
0
1
1
1
0
1
1
:
cycles
10
consider
example
For

=
=
=
=
+
+
+
+
+
+
+
+
+
=
CSL
CSL
CSL

28. 28
DLT : Demand during lead time where Lead Time and Demand
are random variables
Expected value of the demand during lead time: 𝐸(෍
𝑖=1
𝐿𝑇
𝐷𝑖) = (𝐿)(𝐷)
Variance of the demand during lead time: 𝑉𝑎𝑟(෍
𝑖=1
𝐿𝑇
𝐷𝑖) = 𝐿 ⋅ 𝜎𝐷
2 + 𝐷2 ⋅ 𝜎𝐿𝑇
2
= 𝜎𝐷𝐿𝑇
2
𝐷 = Average demand per period
𝜎𝐷
2
= Variance of demand
𝐿 = Average lead time in number of periods
𝜎𝐿𝑇
2
= Variance of lead time

29. 29
Reorder Point: Assume Demand has normal distribution
ROP
Risk of a stockout = 𝜶
Service level (Probability of no stockout) = 𝟏 − 𝜶
Expected
demand Safety Stock
0 𝒛𝜶
Quantity
z-scale
𝑅𝑂𝑃 = 𝐸(𝐷𝐿𝑇) + 𝑧𝛼 𝜎𝐷𝐿𝑇
Note: 𝑠𝑠 = 𝑧𝛼 𝜎𝐷𝐿𝑇

30. 30
Finding safety stock from Cycle Service Level CSL using
EXCEL™
𝐶𝑆𝐿 = 𝑃(𝐷𝐿𝑇 ≤ 𝑅𝑂𝑃) = 𝑃(𝑁𝑜𝑟𝑚𝑎𝑙(𝐿 × 𝐷, 𝜎𝐷𝐿𝑇
2
) ≤ 𝑅𝑂𝑃) = 𝑃(𝑍 ≤
𝑅𝑂𝑃 − 𝐿 ⋅ 𝐷
𝜎𝐷𝐿𝑇
)
where, Z is the Standard normal variable
Using EXCEL ™ , one can find
𝑅𝑂𝑃−𝐿⋅𝐷
𝜎𝐷𝐿𝑇
= normsinv(CSL); where normsinv is a function in
EXCEL™
𝑆𝑆 = 𝑅𝑂𝑃 − 𝐿 × 𝐷 = 𝜎𝐷𝐿𝑇 × normsinv(CSL)
𝑅𝑂𝑃 = 𝐿 × 𝐷 + 𝑆𝑆 = 𝐿 ⋅ 𝐷 + 𝜎𝐷𝐿𝑇 × normsinv(CSL)
Note we use normal density for the demand during lead time
Note: The excel function normsinv has default values of 0 and 1 for the mean
and standard deviation. Defaults are used unless these values are specified.

31. 31
Example: Safety inventory vs. Lead time variability
D = 2,500 item-units/day; D = 500 item-units/day; CSL = 0.90
Case1: L = 7 days fixed,
Normsinv(0.9)=1.3 (=1.28), either from table or Excel, LT= 0,
DLT=sqrt(7)*500=1323 since L is fixed
SS=1323*normsinv(0.9)=1719.8
ROP=(D)(L)+ss=17500+1719.8
Case2: L = 7 days random where Lt=1,
DLT=sqrt(7*500*500+2500*2500*1)=2828 since L is random
SS=2828*normsinv(0.9)=3676
ROP=(D)(L)+ss=17500+3676

32. 32
Expected shortage per cycle
• First let us study shortage during the lead time
Expected shortage = 𝐸(0, max( 𝐷𝐿𝑇 − 𝑅𝑂𝑃))
= න
𝐷=𝑅𝑂𝑃
∞
(𝐷 − 𝑅𝑂𝑃)𝑓𝐷(𝐷)𝑑𝐷 where fD is pdf of DLT.
• Example:
𝑅𝑂𝑃 = 10, 𝐷𝐿𝑇 =
9 with prob 𝑝1 = 1/4
10 with prob 𝑝2 = 2/4
11 with prob 𝑝3 = 1/4
, what is the Expected Shortage during Lead time?
Expected shortage = ෍
i=1
3
max{0,(𝑑𝑖 − 𝑅𝑂𝑃)}𝑝𝑖 =
= max{0,(9−10)}
1
4
+ max{0,(10−10)}
2
4
+ max{0,(11−10)}
1
4
=
1
4

33. 33
Expected shortage per cycle
If we assume that DLT is normal,
3.
-
11
Table
use
)
(
))
,
0
(max(
then
Let z
E
ROP
DLT
E
L
D
ROP
z DLT
DLT


=
−

−
=
2
170
-
172
)
10
(
10
2
10
6
1
)
12
(
10
2
12
6
1
10
2
6
1
6
1
)
10
(
shortage
Expected
Shortage?
Expected
),
12
,
6
(
,
10
2
2
12
10
2
12
10
=
=








−
−








−
=
−




=
−
=
=
=
=
=
=

D
D
D
D
D
dD
D
Uniform
D
ROP
• Ex:

34. 34
Example: Finding expected shortage per cycle
Suppose that the demand during lead time has expected value 100 and
stdev 30, find the expected shortage if ROP=120.
z=(120-100)/30=0.66.
E(z)=0.153 from Table 11-3.
Expected shortage = 30*0.153=4.59

35. 35
Fill rate
• Fill rate is the percentage of demand filled from the
stock
• In a cycle
• Fill rate = 1-(Expected shortage during LT) / Q
• For normally distributed demand
Q
z
E
D
D
Q
z
E
DLT
DLT
)
(
rate)
Fill
1
(
*
year
per
short
units
of
number
Expected
)
(
1
cycle
per
Demand
cycle
per
shortage
Expected
1
rate
Fill


=
−
=
−
=
−
=

36. 36
Example: computing the fill rate
Suppose that the demand during lead time has expected value 100 and
stdev 30, find the expected shortage if ROP=120. Compute the fill rate
if order sizes are 200. Compute the annual expected shortages if
there are 12 order cycles per year.
Expected shortage per cycle=4.59 from the last example.
Fill rate = 1-4.59/20=0.7705
Annual expected shortage=12*4.59=55.08.