1. Individual Items with
Probabilistic Demand

2. TERMINOLOGY
• On-hand stock: Physically present on shelf. Cannot
be -ve number.
• Net stock: (on-hand) - (Backorders)
• Inventory position: (on-hand) +
(on order) -
(Backorders) - (Committed)
• Safety stock: Average level of net stock just before a
replenishment arrives

3. Backorder vs. Lost sales
• Complete backordering: Government organizations,
wholesale-retail link of few distribution systems
(exclusive dealership)
• Complete lost sales: Retail-consumer link mostly for
FMCGs
• Combination of the two
• Numerical value of SS depends on the degree to
which backorders or lost sales occur

4. Three Key Issues
• How
often
the
inventory
status
should
determined?
• When a replenishment order should be placed?
• How large the replenishment order should be?
be

5. 4 Questions for Inventory
Policies
• How important is the item?
• Can, or should, the stock
status be reviewed
continuously or periodically?
• What form should the inventory policy take?
• What specific cost or service objectives should be
set?

6. IMPORTANCE OF THE ITEM
• A items comprise roughly 20% of the total number
of items, but represent 80% of the dollar sales
volume
• B items comprise roughly 30% of the items and 15%
of the dollar volume
• C items comprise of balance 50% of items and only
5% of the total dollar volume

7. CONTINUOUS vs. PERIODIC
•How
often
should
the
inventory
status
be
determined? - Periodic Review
•Transactions Reporting - Continuous review
•COMPARISON
•Items may be produced on the same piece
of
equipment, purchased from same supplier or shipped
in the same transportation mode - Periodic Review

8. CONTINUOUS Vs. PERIODIC
• COMPARISON
• Periodic review
allows a reasonable prediction of
the level of the workload on the staff. While the
replenishment can be done at any moment in
continuous review making it less predictable
• Continuous
review is expensive. Especially in fast
moving goods (FMCG). So reviewing errors and
costs are high

9. CONTINUOUS Vs. PERIODIC
• COMPARISON
• For slow moving
goods, periodic review is still
better as it can trace any pilferage or spoilage during
the review period but if it continuous review it
doesn’t detect unless a transaction happens
• Continuous
review, to provide the same level of
customer service, requires less safety stock than
periodic review

10. Four Types of Control Systems
•Order Point, Order Quantity (s, Q) System
•Order Point, Order-up-to-level (s, S) System
•Periodic-Review, Order-up-to-level (R, S) System
•(R, s, S) System

11. (s, Q) System
•Continuous review (R = 0)
•Inventory position and not
Net stock is used to
trigger an order.
•Two-bin
system: Amount
in
the
second
bin
corresponds to order point.
•(s, Q) system is easy to understand even for the stock
clerk

12. (s, Q) System
•One disadvantage is that it cannot accommodate a large
order. For ex., if Q=10 units, D=15 units occurring when
Inventory Level
the position is 1 unit just above s. Order in multiples.
s+Q
s
A
L
B
Time

13. (s, S) System
•Order
quantity is variable enough to raise the
position to order-up-to level ‘S’
•For unit-sized demand, (s,S) is same as (s,Q) system.
Inventory Level
S
s
A
L
B
Time

14. (s, S) System
•Also called as min-max system as inventory position,
except for momentary drop is always between s & S.
•Optimal value of (s, S) is difficult to find and is most
of the times arbitrarily fixed.

15. •Used
(R, S) System
by those companies not using computer
controls
•Control
procedure is that every R units of time,
enough is ordered to raise the inventory position to
S
•Carrying costs are high
•Provides opportunity to adjust the position to S if
demand pattern is changing with time

16. S
0
R
L
L

17. (R, s, S) System
•Combination of (R, S) and (s, S) systems.
•Every R units of time, check the inventory position, if
it is below s, then order to raise the level to S. If the
level is above s, nothing is done until the next review.
•(s, S) is special case where R=0
•(R, s, S) is a periodic version of (s, S) system

18. (R, s, S) System
•(R, s, S)
provides the lower total cost comprising
replenishment, holding and shortage costs than any
other method but is difficult to get all 2 parameters

19. Specific Cost & Service
Objectives
• Two types of risks to be balanced:
• If demand is large, stockout may occur
• If demand is lower, then large inventory is carried

20. Specific Cost & Service Objectives
• Four methods of modeling these are:
• Safety stocks established through
use of simple
approach
• Safety stocks based on minimizing cost
• Safety stocks based on customer service levels
• Safety stocks based on aggregate consideration

21. SINGLE PERIOD MODEL
• Christmas trees Cost (C) = $4 per tree and SPrice (S) = $9,
Profit = $10000, so demand was around 2000 units
• Xbar = 2000 and σ = 100 trees
• Let salvage value be V = $1
• Marginal Profit (MP) on a tree = S - C = 9 - 4 = $5
• Marginal Loss (ML) on a tree = C - V = 4 - 1 = $3
• To order Q rather than Q - 1, the expected profit on marginal
tree must be ≥ expected loss

22. SINGLE PERIOD MODEL
• Let ‘p’ be the probability of selling the marginal tree
• Profit criterion is expected profit ≥ expected loss,
• i.e. p(MP) ≥ (1 - p)ML
• p(MP) ≥ ML - p(ML)
• p(MP+ML) ≥ ML
• Minimum acceptable probability of selling the Qth tree is given
by p ≥ ML / (MP + ML) or SOR ≥ ML / (MP + ML)
• Q = xbar + z
SOR
σ

23. MULTI PERIOD MODEL
• Service Level and Safety Stocks: In reality, sometimes it is difficult
to calculate ML (Stockout costs which has intangibles)
• When
stockout costs are not available, surrogate is the
customer service level
• Service level can be classified in two ways: i) Order Service
Level (OSL) and ii) Unit Service Level (USL)

24. ORDER SERVICE LEVEL (OSL)
• Proportion of cycles that customer demand was satisfied. Order
Service Level represents the probability of not having a stockout
during the placement of order.
• Suppose OSL = 0.90 and number of orders/cycles = D/Q = 20,
then customer demand will be satisfied in OSL * No. Of orders
= 0.90 * 20 = 18 orders.
• It also means that during 2 orders, stockout can be expected.
(0.10 * 20 or 20 - 18 = 2)
• OSOR doesn’t tell how many units were short or not filled
during any cycle

25. UNIT SERVICE LEVEL (USL)
• USL indicates the percentage of units of demand filled during
any period of time, whereas the USOR specifies the quantities
of units unfilled or short during that period.
• Suppose USL = 0.95 and D = 5000 units for an item. Then 0.95
* 5000 = 4750 units of customer demand will be filled on an
average during the year.
• It also means that 250 units of stockout can be expected. i.e. If
there are 10 orders then approx 25 units per order will be
short of.

26. PERCENT ORDER SERVICE LEVEL
⨱
L
and MADL are obtained from forecasting systems,
then σL must be calculated i.e. σL = 1.25MAD.
• Suppose the following details are available,
L
⨱
• Suppose the
= 200, MADL = 32, L
= 5 days, D = 10000, S = 1500 and h = 30, then using EOQ we
get, Sqrt(2DS / h) = Sqrt ((2*10000*1500)/30) = 1000, so D/Q =
10 orders or cycles
• Suppose the manager is willing to accept a 0.375 probability of
stockout on any cycle. Then OSL = 1 - 0.375 = 0.625
+ Safety Stock = L + Z0.375 * σL
⨱
L
⨱
• And the ROP =

27. PERCENT UNIT SERVICE LEVEL
• Percent Unit Service Level tells us what percentage of units
demanded can be supplied from stock.
• It is usually called as fill rate.
⨱
L
L
x
0
s
•
• x - R = σ (z - k)
k
z

28. PERCENT UNIT SERVICE LEVEL
• USOR = E(X - R) / Q
• USOR = σ g(k) / Q
• g(k) = (Q * USOR) / σ
• Specify USL, then find g(k), then k, then Safety Stock, then R
• Suppose USL = 0.99, then USOR = 0.01, then g(k) = Q*0.01/σ ,
L
L
L
then go to the table, pick ‘k’ value corresponding to calculated
g(k), then calculate ROP = xL + kσL

29. BACKORDER COST
• When we are provided with stockout costs and not the service
levels, then how to calculate safety stock
• Increasing the safety stock by one more unit has the same effect
as adding one more unit to reorder level ‘s’
• Raising
the reorder point by 1 unit will cost (holding cost)
(Q*h) / D = (1000*30) / 10000 = $3 per cycle
• If we do not add one more unit and suffer the stockout, then
backorder penalty is given by $b / unit with SOR probability.

30. • Per
BACKORDER COST
cycle marginal cost of adding 1 unit to R = per cycle
marginal cost of not adding 1 unit to R
• Qh / D = OSOR*(b)
• So, b = Qh/ (D * OSOR) i.e. Backorder cost ‘b’
• At 99% USL, 14 units were safety stock, so, z σ = 14 where σ =
L
L
40.
• z * 40 = 14 leading to z = 0.35 leading to OSOR = 0.363 or
36.3%
• b = (30 * 1000) / (0.363 * 10000) = $8.26