1. Operations Management Dr. Ron Tibben-Lembke
Inventory Management
Operations Management
Dr. Ron Tibben-Lembke

2. Purposes of Inventory Meet anticipated demand
Demand variability
Supply variability
Decouple production & distribution
permits constant production quantities
Take advantage of quantity discounts
Hedge against price increases
Protect against shortages

3. %
2007

5. Source: CSCMP, Bureau of Economic Analysis

6. Two Questions Two main Inventory Questions: How much to buy?
When is it time to buy?
Also:
Which products to buy?
From whom?

7. Types of Inventory Raw Materials Subcomponents Work in progress (WIP)
Finished products
Defectives
Returns

8. Inventory Costs
What costs do we experience because we carry inventory?

9. Inventory Costs Costs associated with inventory: Cost of the products
Cost of ordering
Cost of hanging onto it
Cost of having too much / disposal
Cost of not having enough (shortage)

10. Shrinkage Costs How much is stolen? Where does the missing stuff go?
2% for discount, dept. stores, hardware, convenience, sporting goods
3% for toys & hobbies
1.5% for all else
Where does the missing stuff go?
Employees: 44.5%
Shoplifters: 32.7%
Administrative / paperwork error: 17.5%
Vendor fraud: 5.1%

11. Inventory Holding Costs
Category % of Value Housing (building) cost 4% Material handling 3% Labor cost 3% Opportunity/investment 9% Pilferage/scrap/obsolescence 2% Total Holding Cost 21%

12. Inventory Models Fixed order quantity models Fixed order period models
How much always same, when changes
Economic order quantity
Production order quantity
Quantity discount
Fixed order period models
How much changes, when always same

13. Economic Order Quantity
Assumptions
Demand rate is known and constant
No order lead time
Shortages are not allowed
Costs:
S - setup cost per order
H - holding cost per unit time

14. EOQ Inventory Level Q* Decrease Due to Optimal Constant Demand Order
Quantity
Decrease Due to
Constant Demand
Time

15. EOQ Inventory Level Instantaneous Q* Receipt of Optimal Optimal
Order
Quantity
Instantaneous
Receipt of Optimal
Order Quantity
Time

16. EOQ
Inventory
Level Q*
Optimal
Order
Quantity
Time

17. EOQ w Lead Time Inventory Level Q* Optimal Order Quantity Time

18. EOQ
Inventory
Level Q*
Reorder
Point
(ROP)
Time
Lead Time

19. EOQ Inventory Level Q* Average Inventory Q/2 Reorder Point (ROP) Time
Lead Time

20. Total Costs Average Inventory = Q/2 Annual Holding costs = H * Q/2
# Orders per year = D / Q
Annual Ordering Costs = S * D/Q
Cost of Goods = D * C
Annual Total Costs = Holding + Ordering + CoG

21. How Much to Order?
Annual Cost
Holding Cost
= H * Q/2
Order Quantity

22. How Much to Order? Annual Cost Ordering Cost = S * D/Q Holding Cost
= H * Q/2
Order Quantity

23. How Much to Order? Total Cost = Holding + Ordering Annual Cost
Order Quantity

24. How Much to Order? Total Cost = Holding + Ordering Annual Cost
Optimal Q
Order Quantity

25. Optimal Quantity
Total Costs =

26. Optimal Quantity
Total Costs =
Take derivative with respect to Q =

27. Optimal Quantity Total Costs = Take derivative with respect to Q =
Set equal
to zero

28. Optimal Quantity Total Costs = Take derivative with respect to Q =
Set equal
to zero
Solve for Q:

29. Optimal Quantity Total Costs = Take derivative with respect to Q =
Set equal
to zero
Solve for Q:

30. Optimal Quantity Total Costs = Take derivative with respect to Q =
Set equal
to zero
Solve for Q:

31. Adding Lead Time Use same order size Order before inventory depleted
R = * L where:
= average demand rate (per day)
L = lead time (in days)
both in same time period (wks, months, etc.) d d

32. A Question:
If the EOQ is based on so many horrible assumptions that are never really true, why is it the most commonly used ordering policy?
Profit function is very shallow
Even if conditions don’t hold perfectly, profits are close to optimal
Estimated parameters will not throw you off very far

33. Quantity Discounts
How does this all change if price changes depending on order size?
Holding cost as function of cost:
H = I * C
Explicitly consider price:

34. Discount Example
D = 10,000 S = $20 I = 20% Price Quantity EOQ c = 5.00 Q < Q >=

35. Discount Pricing X 633 X 666 X 716 Total Cost Price 1 Price 2 Price 3
,000
Order Size

36. Discount Pricing X 633 X 666 X 716 Total Cost Price 1 Price 2 Price 3
,000
Order Size

37. Discount Example
Order 666 at a time: Hold 666/2 * 4.50 * 0.2= $ Order 10,000/666 * 20 = $ Mat’l 10,000*4.50 = $45, , Order 1,000 at a time: Hold 1,000/2 * 3.90 * 0.2= $ Order 10,000/1,000 * 20 = $ Mat’l 10,000*3.90 = $39, ,590.00

38. Discount Model 1. Compute EOQ for next cheapest price
2. Is EOQ feasible? (is EOQ in range?)
If EOQ is too small, use lowest possible Q to get price.
3. Compute total cost for this quantity
Repeat until EOQ is feasible or too big.
Select quantity/price with lowest total cost.

39. Inventory Management -- Random Demand

40. Random Demand Don’t know how many we will sell
Sales will differ by period
Average always remains the same
Standard deviation remains constant

41. Impact of Random Demand
How would our policies change?
How would our order quantity change?
How would our reorder point change?

42. Mac’s Decision How many papers to buy? Average = 90, st dev = 10
Cost = 0.20, Sales Price = 0.50
Salvage = 0.00
Cost of overestimating Demand, CO
CO = = 0.20
Cost of Underestimating Demand, CU
CU = = 0.30

43. Optimal Policy
G(x) = Probability demand <= x Optimal quantity: Mac: G(x) = 0.3 / ( ) = 0.6 From standard normal table, z = =Normsinv(0.6) = Q* = avg + zs = *10 = = 93

44. Optimal Policy If units are discrete, when in doubt, round up
If u units are on hand, order Q - u units
Model is called “newsboy problem,” newspaper purchasing decision
By time realize sales are good, no time to order more
By time realize sales are bad, too late, you’re stuck
Similar to the problem of # of Earth Day shirts to make, lbs. of Valentine’s candy to buy, green beer, Christmas trees, toys for Christmas, etc., etc.

45. Random Demand – Fixed Order Quantity
If we want to satisfy all of the demand 95% of the time, how many standard deviations above the mean should the inventory level be?

46. Probabilistic Models Safety stock = x m From statistics, Safety stock
Therefore, z =
Safety stock
& Safety stock = zsL sL
From normal table z.95 = 1.65
Safety stock = zsL= 1.65*10 = 16.5
R = m + Safety Stock
= = ≈ 367

47. Random Example What should our reorder point be?
demand over the lead time is 50 units,
with standard deviation of 20
want to satisfy all demand 90% of the time
(i.e., 90% chance we do not run out)
To satisfy 90% of the demand, z = 1.28
Safety stock = zσL= 1.28 * 20 = 25.6
R = = 75.6

48. St Dev Over Lead Time
What if we only know the average daily demand, and the standard deviation of daily demand?
Lead time = 4 days,
daily demand = 10,
standard deviation = 5,
What should our reorder point be, if z = 3?

49. St Dev Over LT
If the average each day is 10, and the lead time is 4 days, then the average demand over the lead time must be 40.
What is the standard deviation of demand over the lead time?
Std. Dev. ≠ 5 * 4

50. St Dev Over Lead Time Standard deviation of demand =

51. Service Level Criteria
Type I: specify probability that you do not run out during the lead time
Probability that 100% of customers go home happy
Type II: proportion of demands met from stock
Percentage that go home happy, on average
Fill Rate: easier to observe, is commonly used
G(z)= expected value of shortage, given z. Not frequently listed in tables

52. Two Types of Service
Cycle Demand Stock-Outs Sum 1,450 55
Type I:
8 of 10 periods
80% service
Type II:
1,395 / 1,450 =
96%

53. Fixed-Time period models

54. Fixed-Time Period Model
Every T periods, we look at inventory on hand and place an order
Lead time still is L.
Order quantity will be different, depending on demand

55. Fixed-Time Period Model: When to Order?
Inventory Level
Target maximum
Time
Period

56. Fixed-Time Period Model: : When to Order?
Inventory Level
Target maximum
Time
Period
Period

57. Fixed-Time Period Model: When to Order?
Inventory Level
Target maximum
Time
Period
Period

58. Fixed-Time Period Model: When to Order?
Inventory Level
Target maximum
Period
Time

59. Fixed-Time Period Model: When to Order?
Inventory Level
Target maximum
Period
Time

60. Fixed-Time Period Model: When to Order?
Inventory Level
Target maximum
Period
Time

61. Fixed Order Period Standard deviation of demand over T+L =
T = Review period length (in days)
σ = std dev per day
Order quantity (12.11) =

62. Inventory Recordkeeping
Two ways to order inventory:
Keep track of how many delivered, sold
Go out and count it every so often
If keeping records, still need to double-check
Annual physical inventory, or
Cycle Counting

63. Cycle Counting
Physically counting a sample of total inventory on a regular basis
Used often with ABC classification
A items counted most often (e.g., daily)
Advantages
Eliminates annual shut-down for physical inventory count
Improves inventory accuracy
Allows causes of errors to be identified

64. Fixed-Period Model Answers how much to order
Orders placed at fixed intervals
Inventory brought up to target amount
Amount ordered varies
No continuous inventory count
Possibility of stockout between intervals
Useful when vendors visit routinely
Example: P&G rep. calls every 2 weeks

65. ABC Analysis Divides on-hand inventory into 3 classes
A class, B class, C class
Basis is usually annual $ volume
$ volume = Annual demand x Unit cost
Policies based on ABC analysis
Develop class A suppliers more
Give tighter physical control of A items
Forecast A items more carefully

66. Classifying Items as ABC
% Annual $ Volume
Items %$Vol %Items
A 80 15
B 15 30
C 5 55 A B C
% of Inventory Items

67. ABC Classification Solution

68. ABC Classification Solution