1. Inventory Models (II) under SC environmentBy
Dr. Debadyuti Das

2. The Multi-Period Continuous Review [ROP or Q or (s, S)] Model
There are five determinants of reorder point quantity.
The rate of demand
The lead time
The extent of demand variability
The extent of lead time variability
The degree of stockout risk acceptable to the management (safety stock)
If demand and lead time are both constant, the reorder point is simply
ROP = d x LT

3. A View of (s, S) Policy S Inventory Level s Time Inventory Position
Lead
Time
Lead
Time
Inventory Level s
Time

4. Safety Stock Quantity Maximum probable demand Expected demandLT
Time
Expected demand
during lead time
Maximum probable demand
ROP
Quantity
Safety stock
Safety stock reduces risk of
stockout during lead time

5. Reorder Point The ROP based on a normal
Risk of
a stockout
Service level
Probability of
no stockout
Expected
demand
Safety
stock z
Quantity
z-scale
The ROP based on a normal
Distribution of lead time demand

6. Notations D = average demand per unit time
d = standard deviation of daily demand
dLT = standard deviation of demand during lead time
LT = replenishment lead time in days
LT = standard deviation of lead time
h = holding cost of one unit for one day
K = fixed cost
SL = Service level (for example, 95%). This implies that the probability of stocking out is 5%
SS = Safety inventory

7. Notations ROP = Reorder point
The reorder point (s) has two components:
To account for average demand during lead time: LT D
To account for deviations from average (we call this safety stock) z  STD   d
Since there is a fixed cost, we order more than up to the reorder point: Q=(2 K D)/h
The total order-up-to level is: S=Q+s

8. Three possible situations
Case 1: Only demand is variable
Case 2: Only lead time is variable
Case 3: Both demand and lead time are variable

9. Case1: Example
The distributor has historically observed weekly demand of: D = d = 32.1
Replenishment lead time is 2 weeks, and desired service level SL = 97%
Average demand during lead time is:  2 = 89.2
Safety Stock is:  32.1  2 = 85.3
Reorder point is thus 175, or about 3.9 weeks of supply at warehouse and in the pipeline

10. Case1: Example (Contd.) Q=679 Order-up-to level (S) thus equals:
Reorder Point + Q = = 854
Cycle inventory = Q/2 = 340
Average Inventory = cycle inventory + SS = = 425
Average Flow Time = Avg inventory / throughput = 425 /44.6 = 9.53

11. Case 2: Example Only lead time is variable D = 600,
Average lead time (LT) = 6
S.D. of LT (LT) = 2
Desired service level = 90 %
Demand during average LT = (D X LT)= (600 X 6) = 3600
Safety stock = z D (LT) = 1.28 X 600 X 2 = 1536
ROP = 5136

12. Case 3: Example Both demand and lead time are variable
Average demand (D) = 25,
Average lead time (LT) = 10
d = 3, LT = 2
Desired service level = 95 %
ROP = D X LT + Z (LT d2 + D2 LT2)
= 25 X 10(3)2 + (25)2 (2)2
=334

13. Impact of demand and supply uncertainty on safety stock
Average demand
S.D. of demand
Average lead time
S.D. of
L.T.
Safety stock
Safety stock in days
Remarks
100 30 15 5
1026
10.3
Base case
232
2.3
No supply uncertainty
1000 10
No demand uncertainty
1006
Reduce demand uncertainty
2.5
526
5.3
Reduce supply uncertainty
7.5
1003
Reduction in lead time

14. Periodic Review Policy (P Model or Fixed order Model or Base stock policy)
Orders are placed at fixed time intervals
Order quantity for next interval?
Suppliers might encourage fixed intervals
May require only periodic checks of inventory levels
Risk of stockout

15. Fixed order Model
At each review, inventory position is raised to the base-stock level.
The base-stock level includes two components:
Average demand during r+L days (the time until the next order arrives): (r+L)*D
Safety stock during that time: z* d * r+L

16. Base-Stock Policy Inventory Level Time r L Inventory Position
Base-stock Level
Inventory
Position r L

17. Issues involved in FOI Model
Items from the same supplier may yield savings in:
Ordering
Packing
Shipping costs
May be practical when inventories cannot be closely monitored
However,
Requires a larger safety stock
Increases carrying cost
Costs of periodic reviews

18. Risk pooling (Centralization vs Decentralization)
Ordering cost: Rs. 257 per order
Purchase price: Rs. 30
Inventory carrying cost per unit per year: 20% of Rs. 30 i.e. Rs 6
Annual demand: 30000
Q*: 1600
Cycle stock: Q*/2: 800
Service level: 97.7 %
d: 100; d : 30; L: 15
Transportation cost in centralization case: Rs 1/unit
Transportation cost in decentralization case: Rs 1.10/unit

19. Risk Pooling (Centralization vs Decentralization)
For the same service level, which system will require more inventory? Why?
For the same total inventory level, which system will have better service? Why?
What are the factors that affect these answers?

20. Risk pooling (Centralization vs Decentralization)
Analysis of decentralization vs centralization
Decentralized system
Centralized system
Remarks
No. of stock points 16 1
Cycle stock
800
3200
Increases cycle stock by 4 times
Safety stock
232
928
Increases by 4 times
Total inventory
( ) x 16 = 16,512
= 4128
Total inventory carrying cost
16512 x 6 = 99,072
4128 x 6 = 24,768
Incremental transportation cost
300 x 100 x 16 x 0.1 = 48,000