1. Order Quantities when Demand is Approximately Level
Chapter 5
Inventory Management
Dr. Ron Tibben-Lembke

2. 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)

3. 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%

4. Inventory Holding Costs
Category % of Value
Housing (building) cost 6%
Material handling 3%
Labor cost 3%
Opportunity/investment 11%
Pilferage/scrap/obsolescence 3%
Total Holding Cost 26%

5. 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

6. Classifying Items as ABC
% Annual $ Usage A B C
% of Inventory Items

7. ABC Classification Solution
Stock #
Vol.
Cost
$ Vol. %
ABC
206
26,000
$ 36
$936,000
105
200
600
120,000
019
2,000 55
110,000
144
20,000 4
80,000
207
7,000 10
70,000
Total
1,316,000

8. ABC Classification Solution

9. Economic Order Quantity
Assumptions
Demand rate is known and constant
No order lead time
Shortages are not allowed
Costs:
A - setup cost per order
v - unit cost
r - holding cost per unit time

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

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

12. EOQ
Inventory
Level Q*
Optimal
Order
Quantity
Time
Lead Time

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

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

15. Total Costs Average Inventory = Q/2 Annual Holding costs = rv * Q/2
# Orders per year = D / Q
Annual Ordering Costs = A * D/Q
Annual Total Costs = Holding + Ordering

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

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

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

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

20. Optimal Quantity
Total Costs =

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

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

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

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

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

26. Sensitivity
Suppose we do not order optimal EOQ, but order Q instead, and Q is p percent larger
Q = (1+p) * EOQ
Percentage Cost Penalty given by:
EOQ = 100, Q = 150, so p = 0.5
50*(0.25/1.5) = 8.33 a 8.33% cost increase

27. Figure 5.3 Sensitivity

28. 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?

29. Benefits of EOQ 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

30. Tabular Aid 5.1 For A = $3.20 and r = 0.24%
Calculate Dv =total $ usage (or sales)
Find where Dv fits in the table
Use that number of months of supply
D = 200, v = $16, Dv=$3,200
From table, buy 1 month’s worth
Q = D/12 = 200/12 = 16.7 = 17

31. How do you get a table?
Decide which T values you want to consider: 1 month, etc.
Use same v and r values for whole table
For each neighboring set of T’s, put them into

32. How do you get a table? For example, A = $3.20, r = 0.24
To find the breakpoint between 0.25 and 0.5
Dv = 288 * 3.2 / (0.25 * 0.5 * 0.24)
= / 0.03 = 30,720
So if Dv is less than this, use 0.25, more than that, use 0.5
Find 0.5 and 0.75 breakpoint:
Dv = 288 * 3.2/(0.5 * 0.75 * 0.24) = 10,2240

33. Why care about a table? Some simple calculations to get set up
No thinking to figure out lot sizes
Every product with the same ordering cost and holding cost rate can use it
Real benefit - simplified ordering
Every product ordered every 1 or 2 weeks, or every 1, 2, 3, 4, 6, 12 months
Order multiple products on same schedule:
Get volume discounts from suppliers
Save on shipping costs
Savings outweigh small increase from non-EOQ orders

34. Uncoordinated Orders
Time

35. Simultaneous Orders Time
Same T = number months supply allows firm to order at
same time, saving freight and ordering expenses
Adjusted some T’s, changed order times

36. Offset Orders Same T = number months supply allows firm to control
maximum inventory level by coordinating replenishments
With different T, no consistency

37. Quantity Discounts
How does this all change if price changes depending on order size?
Explicitly consider price:

38. Discount Example D = 10,000 A = $20 r = 20% Price Quantity EOQ
v = Q <
3.90 Q >=

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

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

41. Discount Example Order 666 at a time: Hold 666/2 * 4.50 * 0.2= $299.70
Mat’l 10,000*4.50 = $45, ,599.70
Order 1,000 at a time:
Hold 1,000/2 * 3.90 * 0.2= $390.00
Order 10,000/1,000 * 20 = $200.00
Mat’l 10,000*3.90 = $39, ,590.00

42. Discount Model 1. Compute EOQ for each price
2. Is EOQ ‘realizeable’? (is Q in range?)
If EOQ is too large, use lowest possible value. If too small, ignore.
3. Compute total cost for this quantity
4. Select quantity/price with lowest total cost.

43. Adding Lead Time Use same order size Order before inventory depleted
R = DL where:
D = annual demand rate
L = lead time in years