1. Lecture 24 Order Quantities (Continued)
Books
Introduction to Materials Management, Sixth Edition, J. R. Tony Arnold, P.E., CFPIM, CIRM, Fleming College, Emeritus, Stephen N. Chapman, Ph.D., CFPIM, North Carolina State University, Lloyd M. Clive, P.E., CFPIM, Fleming College
Operations Management for Competitive Advantage, 11th Edition, by Chase, Jacobs, and Aquilano, 2005, N.Y.: McGraw-Hill/Irwin.
Operations Management, 11/E, Jay Heizer, Texas Lutheran University, Barry Render, Graduate School of Business, Rollins College, Prentice Hall

2. Objectives Probabilistic Models and Safety Stock Probabilistic Demand
Other probabilistic models
Fixed period system
EOQ consequences
Period order quantity model

3. Probabilistic Models and Safety Stock
Used when demand is not constant or certain
Use safety stock to achieve a desired service level and avoid stockouts
ROP = d x L + ss
Annual stockout costs = the sum of the units short x the probability x the stockout cost/unit x the number of orders per year

4. Safety Stock Example Number of Units Probability 30 .2 40 ROP  50 .3
ROP = 50 units Stockout cost = $40 per frame
Orders per year = 6 Carrying cost = $5 per frame per year
Number of Units
Probability 30 .2 40
ROP  50 .3 60 70 .1
1.0

5. Additional Holding Cost
Safety Stock Example
ROP = 50 units Stockout cost = $40 per frame
Orders per year = 6 Carrying cost = $5 per frame per year
Safety Stock
Additional Holding Cost
Stockout Cost
Total Cost 20
(20)($5) = $100 $0
$100 10
(10)($5) = $ 50
(10)(.1)($40)(6) = $240
$290
$ 0
(10)(.2)($40)(6) + (20)(.1)($40)(6) = $960
$960
A safety stock of 20 frames gives the lowest total cost
ROP = = 70 frames

6. Probabilistic Demand Inventory level Time Safety stock 16.5 units
ROP 
Place order
Inventory level
Time
Minimum demand during lead time
Maximum demand during lead time
Mean demand during lead time
ROP = safety stock of 16.5 = 366.5
Receive order
Lead time
Normal distribution probability of demand during lead time
Expected demand during lead time (350 kits)

7. Probabilistic Demand Probability of no stockout 95% of the time
Mean demand 350
Risk of a stockout (5% of area of normal curve)
ROP = ? kits
Quantity
Safety stock
Number of standard deviations z

8. Probabilistic Demand
Use prescribed service levels to set safety stock when the cost of stockouts cannot be determined
ROP = demand during lead time + ZsdLT
where Z = number of standard deviations
sdLT = standard deviation of demand during lead time

9. Probabilistic Example
Average demand = m = 350 kits
Standard deviation of demand during lead time = sdLT = 10 kits
5% stockout policy (service level = 95%)
Using Appendix I, for an area under the curve of 95%, the Z = 1.65
Safety stock = ZsdLT = 1.65(10) = 16.5 kits
Reorder point = expected demand during lead time + safety stock
= 350 kits kits of safety stock
= or 367 kits

10. Other Probabilistic Models
When data on demand during lead time is not available, there are other models available
When demand is variable and lead time is constant
When lead time is variable and demand is constant
When both demand and lead time are variable

11. Other Probabilistic Models
Demand is variable and lead time is constant
ROP = (average daily demand x lead time in days) + ZsdLT
where sd = standard deviation of demand per day
sdLT = sd lead time

12. Probabilistic Example
Average daily demand (normally distributed) = 15
Standard deviation = 5
Lead time is constant at 2 days
90% service level desired
Z for 90% = 1.28
From Appendix I
ROP = (15 units x 2 days) + Zsdlt
= (5)( 2)
= = ≈ 39
Safety stock is about 9 iPods

13. Other Probabilistic Models
Lead time is variable and demand is constant
ROP = (daily demand x average lead time in days)
= Z x (daily demand) x sLT
where sLT = standard deviation of lead time in days

14. Probabilistic Example
Z for 98% = 2.055
From Appendix I
Daily demand (constant) = 10
Average lead time = 6 days
Standard deviation of lead time = sLT = 3
98% service level desired
ROP = (10 units x 6 days) (10 units)(3)
= =
Reorder point is about 122 cameras

15. Other Probabilistic Models
Both demand and lead time are variable
ROP = (average daily demand x average lead time) + ZsdLT
where sd = standard deviation of demand per day
sLT = standard deviation of lead time in days
sdLT = (average lead time x sd2) + (average daily demand)2 x sLT2

16. Probabilistic Example
Average daily demand (normally distributed) = 150
Standard deviation = sd = 16
Average lead time 5 days (normally distributed)
Standard deviation = sLT = 1 day
95% service level desired
Z for 95% = 1.65
From Appendix I
ROP = (150 packs x 5 days) sdLT
= (150 x 5) (5 days x 162) + (1502 x 12)
= (154) = 1,004 packs

17. Fixed-Period (P) Systems
Orders placed at the end of a fixed period
Inventory counted only at end of period
Order brings inventory up to target level
Only relevant costs are ordering and holding
Lead times are known and constant
Items are independent from one another

18. Fixed-Period (P) Systems
Target quantity (T)
On-hand inventory
Time Q1 Q2 Q3 Q4 P P

19. Fixed-Period (P) Example
3 jackets are back ordered No jackets are in stock
It is time to place an order Target value = 50
Order amount (Q) = Target (T) - On-hand inventory - Earlier orders not yet received + Back orders
Q = = 53 jackets

20. Fixed-Period Systems Inventory is only counted at each review period
May be scheduled at convenient times
Appropriate in routine situations
May result in stockouts between periods
May require increased safety stock

21. EOQ Assumptions Demand is relatively constant and is known.
The item is produced or purchased in lots or batches and not continuously.
Order prep costs & inventory-carrying costs are constant and known.
Replacement occurs all at once.

22. EOQ Consequences Average inventory = EOQ lot size / 2
# of orders per year
= Annual demand / lot size

23. Basic EOQ Model Basic EOQ: Demand is constant over time
Inventory drops at a uniform rate over time
When the inventory reaches 0, the new order is placed and received, and the inventory level again jumps to Q units
The optimal order quantity will occur at a point where the total setup cost is equal to the total holding cost
Basic EOQ:

24. Basic EOQ Model (cont.) Benefit of EOQ model: Reorder Points:
It is a robust model, meaning that it gives satisfactory answers even with substantial variation in the parameters.
Reorder Points:
Lead Time - the time between the placement and receipt of an order.
The when-to-order decision is expressed in terms of a reorder point, the inventory level at which an order should be placed.

25. Inventory Level Over Time (Basic EOQ Model)
Maximum Inventory Level
Average

26. Production Order Quantity Model
Production Order Quantity Model is useful when:
Inventory continuously flows or builds up over a period of time after an order has been placed or when units are produced and sold simultaneously.
Takes into account the daily production (or inventory flow) rate and the daily demand rate.
All other EOQ assumptions are valid.
Production Order
Quantity

27. Inventory Level Over Time (Production Model)
Production Portion
of Cycle
Maximum Inventory Level
Demand
Portion

28. Period Order Quantities
Calculate or determine EOQ
Determine avg. weekly usage
Divide EOQ by avg. weekly usage to determine period
Order the amount needed during the next period to satisfy demand during that period

29. Practice Question 1. Sarah’s Silk Screening
 Sarah’s Silk Screening sells souvenir shirts. Sarah is trying to decide how many to produce for the upcoming naming of the College of Management. The University will allow her to sell the shirts only on one day, the day that the school naming is announced. Sarah will sell the T-shirts for $20 each. When the event day is over, she will be allowed to sell the remaining stock to the Bookstore for $4 each. It costs Sarah $8 to make the specialty shirt. She estimates mean demand to be 545, with a standard deviation is How many shirts should she make?

30. Practice Question 2. The Great Southern Automotive Co.
 The Great Southern Automotive Co. buys steering wheels from a supplier. One particular steering wheel has a known and constant demand rate of 2,000 units per year. The fixed cost of ordering is $100 and the inventory holding cost is $2 per unit per year. It takes 2 weeks for an order to arrive. Compute
The optimal EOQ
The reorder point
The average inventory level
The time between successive orders
The total annual cost
If demand was variable with a standard deviation of 4 units per week, and the firm aims for 98% customer satisfaction, what would the reorder point be?

31. End of Lecture 24