1. Inventory Modeling Concepts
Module C4
Inventory Modeling Concepts

2. What is inventory? INVENTORY MODELING Items in inventory in a store
Items waiting to be shipped
Employees in a firm
Computer information in computer files
Etc.

3. COMPONENTS OF AN INVENTORY POLICY
Q = the amount to order (the order quantity)
R = when to reorder (the reorder point)

4. BASIC CONCEPT
Balance the cost of having goods in inventory to other costs such as:
Order Cost
Purchase Costs
Shortage Costs

5. HOLDING COSTS Costs of keeping goods in inventory Cost of capital Rent
Utilities
Insurance
Labor
Taxes
Shrinkage, Spoilage, Obsolescence

6. Holding Cost Rate Annual Holding Cost Per Unit
These factors, individually are hard to determine
Management (typically the CFO) assigns a holding cost rate, H, which is a percentage of the value of the item, C
Annual Holding Cost Per Unit, Ch
Ch = HC (in $/item in inv./year)

7. ORDER/SETUP COSTS
When purchasing items, this cost is known as the order cost, CO (in $/order)
These are costs associated with the ordering process that are independent of the size of the order-- invoice writing or checking, phone calls, etc.
Labor
Communication
Some transportation

8. ORDER/SETUP COSTS (Cont’d)
When these costs are associated with producing items for sale they are called set-up costs (still labeled CO-- in $/setup)
Costs associated with getting the process ready for production (regardless of the production quantity)
Readying machines
Calling in shift workers
Paperwork, communications involved

9. PROCUREMENT/PRODUCTION COSTS
These are the per unit purchase costs, C, if we are ordering the items from a supplier
These are the per unit production costs, C, if we are producing the items for sale

10. CUSTOMER SATISFACTION COSTS
Shortage/Goodwill Costs associated with being out of stock
goodwill
loss of future sales
labor/communication
Fixed administrative costs = Cb ($/occurrence)
Annualized Customer Waiting Costs =
Cs ($/item short/year)

11. BASIC INVENTORY EQUATION
(Total Annual Inventory Costs) =
(Total Annual Order/Setup-Up Costs) +
(Total Annual Holding Costs) +
(Total Annual Purchase/Production Costs) +
(Total Annual Shortage/Goodwill Costs)
This is a quantity we wish to minimize!!

12. REVIEW SYSTEMS Continuous Review -- Periodic Review --
Items are monitored continuously
When inventory reaches some critical level, R, an order is placed for additional items
Periodic Review --
Ordering is done periodically (every day, week, 2 weeks, etc.)
Inventory is checked just prior to ordering to determine an order quantity

13. TIME HORIZONS Infinite Time Horizon Single Period Models
Assumes the process has and will continue “forever”
Single Period Models
Ordering for a one-time occurence

14. EOQ-TYPE MODELS EOQ (Economic Order Quantity-type models assume:
Infinite Time Horizon
Continuous Review
Demand is relatively constant

15. THE BASIC EOQ MODEL Order the same amount, Q, each time
Reordering is instantaneous
Demand is relatively constant at D items/yr.
Infinite Time Horizon/Continuous Review
No shortages
Since reordering is instantaneous

16. AVERAGE INVENTORY
INVENTORY VS. TIME Q
Average Inventory = Q/2

17. THE EOQ COST COMPONENTS
Total Annual Order Costs:
(Cost/order)(average # orders per year) = CO(D/Q)
Total Annual Holding Costs:
(Cost Per Item in inv./yr.)(Average inv.) = Ch(Q/2)
Total Annual Purchase Costs:
(Cost Per Item)(Average # items ordered/yr.) = CD

18. THE EOQ TOTAL COST EQUATION
TC(Q) = CO(D/Q) + Ch(Q/2) + CD
This a function in one unknown (Q) that we wish to minimize

19. SOLVING FOR Q*
TC(Q) = CO(D/Q) + Ch(Q/2) + CD

20. THE REORDER POINT, r* Since reordering is instantaneous, r* = 0
MODIFICATION -- fixed lead time = L yrs.
r* = LD
But demand was only approximately constant so we may wish to carry some safety stock (SS) to lessen the likelihood of running out of stock
Then, r* = LD + SS

21. TOTAL ANNUAL COST
The optimal policy is to order Q* when supply reaches r*
TC(Q*) = COD/Q* + (Ch/2)(Q*) + CD + ChSS
<==variable cost==> fixed safety
cost stock cost
The optimal policy minimizes the total variable cost, hence the total annual cost

22. TOTAL VARIABLE COST CURVE
Ignoring fixed costs and safety stock costs:

23. EXAMPLE -- ALLEN APPLIANCE COMPANY
Juicer Sales For Past 10 weeks
Using 10-period moving average method,
D = ( …+ 130)/10 = 120/ wk = 6240/yr

24. This is an EOQ Model with:
ALLEN APPLIANCE COSTS
Juicers cost $10 each and sell for $11.85
Cost of money = 10%
Other misc. costs associated with inventory = 4%
Labor, postage, telephone charges/order = $8
Workers paid $12/hr min. to unload an order
Desires a safety stock = 13
This is an EOQ Model with:
H = = .14; Ch = .14(10) = $1.40
CO = $8 + (1/3 hr.)*($12/hr.) = $8 + $4 = $12
SS = 13

25. OPTIMAL ORDER QUANTITY FOR ALLEN

26. OPTIMAL QUANTITIES
Total Order Cost = COD/Q* = (12)(6240)/327 = $228.99
Total Holding Cost = (Ch/2)Q* = (1.40/2)(327) = $228.90
(Total Order Cost = Total Holding Cost -- except for roundoff)
# Orders Per Year = D/Q* = 6240/327 = 19.08
Time between orders (Cycle Time) = Q*/D = 327/6240 = years = 2.72 weeks
r* = SS = 13

27. TOTAL ANNUAL COST
Total Variable Cost = Total Order Cost + Total Holding Cost = $ $ = $457.89
Total Fixed Cost = CD = 10(6240) = $62,400
Total Safety Stock Cost =ChSS =(1.40)(13) = $18.20
Total Annual Cost = $ $62,400 + $18.20 = $62,876.09

28. Using the Inventory Template
Input Parameters
Note: Ch is automatically calculated
Optimal Quantities

29. WHY IS EOQ MODEL IMPORTANT?
No real-life model really is an EOQ model
Many models are variants of EOQ-type models
Many situations can be approximated by EOQ models
The EOQ model is relatively insensitive to some pretty major errors in input parameters

30. INSENSIVITY IN EOQ MODELS
We cannot affect fixed costs, only variable costs
TV(Q) = COD/Q + (Ch/2)(Q)
Now, suppose D really = 7500 (>20% error)
We did not know this and got Q* = 327
TV(327) = ((12)(7500))/327 + (1.40/2)(327) =$504.13
Q* should have been: SQRT(2(12)(7500)/1.40) = 359
TV(359) = ((12)(7500))/359 + (1.40/2)(359) =$502.00
This is only a 0.4% increase in the TVCost

31. Module C4 Review Cost Components of Inventory Models
Holding, Order/Setup, Procurement, Shortage
Objective -- Minimize Total Annual Cost
Continuous Review/Infinite Time Horizon
Basic EOQ Assumptions
Basic EOQ Formula
Quantities of Interest
Use of Template
Importance of EOQ Models