1. Inventory Management

2. Agenda Independent Demand Inventory Dependent vs. independent demand
Basic Economic Order Quantity (EOQ) model. Also known as Economic Lot Size Model
Models with Demand and Supply Uncertainty
Fixed ordering costs: the base-stock model (s,S)
No fixed ordering costs: the base-stock model (S)
Risk pooling

3. Why do companies hold inventory? Why might they avoid doing so?
To meet anticipated customer demand
To account for differences in production timing (smoothing)
To protect against uncertainty (demand surge, price increase, lead time slippage)
To maintain independence of operations (buffering)
To take advantage of economic purchase order size
WHY NOT?
Requires additional space
Opportunity cost of capital
Spoilage / obsolescence

4. Independent vs. Dependent Demand
Independent Demand (Demand not related to other items or the final end-product)
Ford Taurus
Dependent Demand
(Derived demand items for component parts,
subassemblies,
raw materials, etc.)
Body Assy.
Wheel Assy. (4)
E(1)
Wheel (1)
Tire (1) 6

5. Two Decisions in Inventory Management
When is it time to reorder?
If it is time to reorder, how much?

6. Economic Order Quantity Model: Where it all started….
Time Between Orders
(Cycle Time) T = Q/D
Demand Rate, D Q
On-hand Inventory
Average Cycle
Inventory, Q/2
Q/2
Time

7. Basic EOQ Assumptions Constant Demand Rate Instantaneous replenishment
Orders received in full after lead-time.
Constant Unit Price (no discounts)

8. Economic Order Quantity Cost Model: Constant Demand, No Shortages
TC = total annual inventory cost
D = annual demand (units / year)
Q = order quantity (units)
K = cost of placing an order or setup cost ($)
h = annual inventory carrying cost ($ / unit /year)
Total Annual
Inventory
Cost =
Annual
Ordering
Cost
Annual
Holding
Cost +
TC = (D / Q) K (Q / 2) h

9. Trade-off in EOQ Model: Inventory Level vs. Number of Orders
Many orders,
low inventory
level
On-hand Inventory Q
Time Q
Few orders,
high inventory
level
On-hand Inventory
Time

10. Cost Relationships for Basic EOQ (Constant Demand, No Shortages)
Total
Cost
Ordering
Cost
TC – Annual Cost
Carrying
Cost Q*
EOQ balances carrying
costs and ordering costs in this model.
Order Quantity (how much)

11. EOQ Results (How Much to Order) (Constant Demand, No Shortages)
2 D K h
Economic Order Quantity
Number of Orders per year
Length of order cycle
Total cost
= D / Q*
T = Q* / D
= TC = (D / Q*) K + (Q* / 2) h

12. EOQ Example (How Much) EOQ:
D = 1,000 units per year
K = $20 per order
h = $8.33 per unit per month
BE CAREFUL!
= $100 per unit per year
EOQ:
Number of orders per year = 1000/20 = 50 orders
Length of order cycle = T = 365 days/50 orders = 7.3 days
Total cost = 20(1000/20)+100*20/2 = $2,000

13. EOQ Example (cont.) (D = 1,000, K = $20, h = $100)
Question: What if the company can only order in multiples of 12? (That is, order either 0 or 12 or 24 or 36, etc…)?
Answer: Q* = 20. Closest matches (above and below) are 12 and 24. Need to compute TC for both, and decide:
TC(Q) = (D / Q) K + (Q / 2) h
Q = 12  TC(12) = 20(1000/12)+100*12/2 = $2,266.67
Q = 24  TC(24) = 20(1000/24)+100*24/2 = $2,033.33
So, the company should order in lots of Q = 24

14. Robustness of EOQ model
Annual Cost
Very Flat Curve - Good!!
Total
Cost
DTC
Q*-DQ
Q*+DQ Q*
Order Quantity
Would have to mis-specify Q* by quite a bit
before total annual inventory costs would change significantly.
Example

15. Example: EOQ Robustness
Suppose that in the last problem, you have mis-specified the order costs by 100% and the holding costs by 50%. That is,
K used in the computation = $40/order (actual cost = $20 / order)
h used in computation = $150 / unit / year (actual = $ 100)
Then, using these wrong costs, you would have gotten
Your actual TC (computed substituting Q’ into TC, using correct costs of K = $20, and h = $100:
Only 1% above minimum TC!

16. Variations of EOQ Some assumptions so far Some variations of EOQ
Instantaneous replenishment (zero lead time)
Certain and constant demand rate
Constant price
Some variations of EOQ
Positive lead times and uncertain lead times
Uncertain demand
EOQ with quantity discounts

17. EOQ with Positive Lead Time
On-hand Inventory
Time Q
Demand Rate, D
Time Between Orders
(Cycle Time) T = Q/D
Average Cycle
Inventory, Q/2
Reorder
Point, s
Place
Order
Receive
order
Lead
Time, L
Q/2

18. Determining When to Reorder
Quantity to order (how much…) determined by EOQ
Reorder point (when…)determined by finding the inventory level that is adequate to protect the company from running out during delivery lead time
With constant demand and constant lead time, (EOQ assumptions), the reorder point is exactly the amount that will be sold during the lead time.
Example: Daily demand (d) = 1,000/365 = 2.74 /day
Delivery lead time (L) of 2 days
s = d*L = (2.74) (2) = 5.5  6 units

19. Effects of Demand / Lead Time Variability on Reorder Point (When)
Variable demand
Expected demand at
average demand rate d
QUESTION: How much inventory is
needed during lead time L? s
Safety Stock level
KEY POINT: s is larger when there is uncertainty about demand or L
Place
order
Receive
order L

20. Calculation of Appropriate Safety Stock Level
Safety stock: stock carried to provide a level of protection against stockouts due to uncertainty of demand during lead time
Stockout Criterion: Find s such that the probability of stockout (during the lead time) is 
Demand during lead time is a random variable
Estimate distribution from historical data (build histogram of demand + frequencies)
Normal is frequently used if distribution is unknown

21. Computing s …
Assumption: Demand during lead-time is normally distributed
1-
Probability {Demand during lead-time < s} =
Probability distribution of demand during lead time
Service Level
1-  s

22. Computing s: Taking Advantage of the Normal Distribution
1- s
Probability distribution of demand during lead time:
Mean = ; Std Dev =   
1- z
1 - 
.90
.95
.98
.99
.999 z
1.28
1.64
2.05
2.33
3.09
From normal table or, in Excel, use: =normsinv (0.90)

23. Issue…
The parameters  and  refer to mean and standard deviation of demand during lead time
Normally, companies have statistics on demand and lead time per unit of time (say, days, weeks, months)
AVG = average demand per unit of time
STD = standard deviation of demand per unit of time
AVGL = average lead time
STDL = standard deviation of lead time
Just be consistent: if demand is given on a certain time unit, say, days, then use lead time in the same time unit (in this case, days)
How to we compute  and  from AVG, STD, AVGL, and STDL?

24. More specifically…. If lead time is constant, If demand is constant,
Standard deviation of demand during lead time ()
Safety factor (std normal table)
Mean demand during lead time ()
Safety stock SS
If lead time is constant,
If demand is constant,
Note: This is a very good approximation even when demand is not normally distributed.

25. The (s,S) Policy: When There Are Fixed Ordering Costs
placed
arrives
Average demand
during lead time
Safety Stock s S
s should be set to cover the lead time demand and together with a safety stock that insures the stock out probability is  (When)
S depends on the fixed order cost – EOQ (How much)

26. The (s,S) Policy: Fixed Ordering Costs
Need to define inventory position (IP)
IP = On-Hand + On-Order– Backorder
Order when: inventory position (IP) drops below s
Order how much: bring IP to S (“big S”)
Compute Q using the EOQ formula, using mean demand D = AVG (be careful about units…):
Set S = s + Q

27. Example: (s,S) Model
Consider inventory management for a certain SKU at Home Depot. Supply lead time is variable (since it depends on order consolidation with other stores) and has a mean of 5 days and standard deviation of 2 days.
Daily demand for the item is variable with a mean of 30 units and a coefficient of variation of 0.20.
Assume a 95% service level.
There are fixed ordering costs that are estimated at $50. Assume that holding costs are 15% of the product cost ($80) per year. Also, assume that the store is open 360 days a year. Propose an inventory policy for this SKU.

28. Solution Variable definitions and preliminary calculations:
Assume 95% service level  z = 1.64
Compute s
Compute Q

29. Example: (s,S) Model
Consider inventory management for a certain SKU at WalMart. Supply lead time is variable and has a mean of 1 week and standard deviation of 2 weeks.
Weekly demand for the item is variable with a mean of 125 units and a standard deviation of 50.
Assume a 90% service level.
There are fixed ordering costs that are estimated at $30. Assume that holding costs are 20% of the product cost ($40) per year. Also, assume that the store is open 52 weeks a year. Propose an inventory policy for this SKU.

30. Solution Variable definitions and preliminary calculations:
Assume 90% service level  z = 1.28
Compute s
Compute Q

31. The Base-Stock Policy s: No Fixed Ordering Costs
Inventory policy: keep IP constant at s units (s is called the base stock level).
When: IP drops below s
So, s is also the reorder point for this model
How much: order to bring IP back to s
Example: suppose inventory level on-hand is 10, s = 20, and there are 2 units already in order. Then, IP = = 12 units. The firm should order 20 – 12 = 8 units.

32. Example: Base-Stock Model s
Consider inventory management for a certain SKU at King Soopers Supply lead time is variable and has a mean of 2 days and standard deviation of 4 days. Daily demand for the item is variable with a mean of 24 units and a coefficient of variation of Propose an inventory policy for this SKU. Assume a 98% service level.
Assume 98% service level  z = 2.05

33. Summary of Inventory Models
Use EOQ
How much: Q (EOQ formula)
When: d*L (reorder point)
yes
Use (s, S) policy
How much: necessary to bring IP back to S, where S = s + Q (Q is from EOQ formula)
When: IP drops below s (base-stock policy formula)
Is demand rate and lead time constant?
yes
Are there fixed ordering costs? no no
Use base stock (s) policy
When: IP drops below s
How much: necessary to bring IP back to s

34. Risk Pooling Means and variances are additive
Stock is based on std. Deviations
Square root law: stock for combined demands is less than the combined stocks

35. HP Example: Benefits of a Universal Product
Because of a different power supplies, HP had two laser printers, one for Europe and one for N. America. A universal product (with a universal power supply) has been proposed, but costs $30 extra. Is it worthwhile? Below is monthly demand for HP for the two markets (in thousands). Assume a one-month constant lead-time (STDL = 0) for both markets.
N. America
N(200,60)
Europe
N(150,50)
Consider z = 2 (~ 98% of service level)

36. HP Example (cont.): Benefits of a Universal Product
Demand seen by HP (NA and Europe)