1. Topic 6. INVENTORY MANAGEMENT

2. I. Introduction What is inventory? Types of Inventories:
stored resource used to satisfy current or future demand
Types of Inventories:
Raw Materials/Components
In-Process Goods (WIP)
Finished Goods
Supplies

3. Introduction Inventory Related Costs:
Holding Cost -- cost to carry a unit in inventory for a length of time (annual), includes interest opportunity cost, insurance, taxes, depreciation, obsolescence, deterioration, May be expressed as a percentage of unit price or as a dollar amount per unit

4. Introduction Inventory Related Costs (continued):
Order Cost -- Cost of ordering and receiving inventory, Include determining how much is needed, preparing invoices, shipping costs, inspecting goods upon receipt for quantity and quality, Generally expressed as a fixed dollar amount, regardless of order size
Inventory may also influence purchasing cost
Inventory is costly

5. Introduction Inventory Related Costs (continued):
Shortage Cost-- result when demand exceeds the inventory on hand, Include the opportunity cost of not making a sales, loss of customer goodwill, late charges, and in the case of internal customers, the cost of lost production or downtime, difficult to measure, thus may have be subjectively estimated

6. Introduction Why Hold Inventories? Meet anticipated demand
Lead time – the time period between place an order until receive the order
Average lead time demand is considered as anticipate demand
Protect against stock-out
Safety stock – more than average lead time demand inventory

7. Introduction Why Hold Inventories (continued)?
De-couple successive operations - separate production from distribution
Wine production and inventory
Smooth production process
Snowmobile production and inventory
Buy/Produce in economic lot sizes - take advantage of quantity discounts
Hedge against price increases

8. Introduction
JIT Inventory – minimum inventory needed to keep a system running, small lot sizes
Advantages
lower inventory costs
easy to identify problems and potential problems
Disadvantages
requires accurate timing and cooperation
breakdowns stop everything

9. Introduction Inventory Classification A Identify important
Annual
$ volume
of items A B C
High
Low
Few
Many
Number of Items
Inventory Classification
Identify important
Items and more inventory
control on important items
Measure of importance:
ABC analysis:
A = 70-80% of total inventory value, but only 15% of items
B = 15-25% of total inventory value, but 30% of items
C = 5% of total inventory value, but 55% of items

10. Introduction Monitor Inventory
As important as demand forecast for decision making
Universal Product Code - Bar code printed on a label that has information about the item to which it is attached
Cycle counting: taking physical counts of items and reconciling with records on a continual rotating basis, regular inventory audits, ABC approach

11. Introduction Inventory Systems
Objective: minimize annual total inventory cost and maintain satisfied service level.
service level: probability of no shortage
Total Inventory Cost is not Inventory Cost
Annual total inventory cost (TC) = annual product cost + annual inventory cost
Annual product cost = annual demand * unit price
Annual inventory cost = annual holding cost + annual setup (order) cost + annual shortage cost

12. Introduction Possible performance measures customer satisfaction
number of backorders/lost sales
number of customer complaints
inventory turnover
ratio of annual cost of goods sold to average inventory investment
days of inventory
expected number of days of sales that can be supplied from existing inventory

13. Introduction Requirements for Effective Inventory Management :
A system to keep track of the inventory on hand and on order
A classification system for inventory items
A reliable forecast of demand that includes an measure of forecast error
Reasonable estimates of inventory holding costs, ordering costs, and shortage costs
Knowledge of lead times and lead time variability

14. Introduction
1. Continuous (Perpetual) Review System: (event-triggered)
Monitor the inventory level all the time, order a fixed quantity (Q) when the inventory level drops to the reorder point (ROP)
Calculate: Q and ROP
Re-Order Point (ROP) – an inventory level when actual inventory drops to it will trigger an activity of re-order.

15. Introduction 2. Periodic Review System: (time-triggered)
Place an order every fixed period T. Each time bring the current inventory to a target level M
Calculate: T and M
3. Advantages and Disadvantages?

16. Introduction Dependent and Independent Demand:
Dependent demand: derived demand, lumpy (subassemblies and components)
cars
Independent demand: from customer side, smooth (end items and finished goods)
tires

17. II. Inventory Models On Order Quantity
Model Basics (consider as annual)
Total Cost (TC) = Product Cost + Inventory Cost
Inventory Cost = Holding Cost
+ Setup (Order) Cost + Shortage Cost
TC = Product Cost + Holding Cost

18. Inventory Models On Order Quantity
Product Cost = Annual Demand * Unit Price
Holding Cost = average inventory level * Holding Cost per unit per year
Ordering Cost = # of orders * Setup Cost per order
# of orders = annual demand / order quantity
Shortage Cost = Shortage Cost per unit
* average # of shortage per year
Best Order Quantity = a quantity that minimizes TC

19. Inventory Models On Order Quantity
EOQ Model (Economic Order Quantity), Fixed-Order-Quantity Model
Assumptions
There is one product type
Demand is known and constant
Lead time is known and constant
Receipt of inventory is instantaneous (one batch, same time)
Shortage is not allowed

20. EOQ Model (continued) Q Lead time Reorder point Place order Receive

21. EOQ Model (continued) Notation and Terminology
Q = order quantity(# of pieces per order)
Q0 = Economic Order Quantity (EOQ)
D = demand for the time period considered (units per year)
S = setup/order cost ($ per order)
H = holding cost per unit per year ($ per unit per year)
in general proportional to the price, H = I*P

22. EOQ Model (continued) Notation and Terminology (continued)
I = Interest rate (expanded) (% per year)
P = unit price ($ per unit)
IC = inventory cost = setup cost + holding cost
TC = IC + product cost
Find Out EOQ

23. EOQ Model (continued) Average Inventory Level = Holding Cost =
Number of orders per year =
Setup (Order) Cost =
Shortage Cost = 0, why?

24. EOQ Model (continued) Product Cost = IC = Total Cost (TC) =
Minimize TC Minimize IC, why?

25. EOQ Model (continued)
Observation: at the best order quantity EOQ (Q0),
holding cost = setup cost
Solve Q0, we have

26. EOQ Model (continued) The Inventory Cost Curve is U-Shaped Annual Cost
Carrying Costs
Annual
Ordering Costs QO
(EOQ)
Order Quantity (Q)

27. EOQ Model (continued) Example:
Annual demand = 10,000 unit/year, ordering cost = $50/order, unit cost (price) = $4/unit, expanded interest rate = 25%/year. EOQ? TC at EOQ?

28. EOQ Model (continued) Sensitivity of IC with related to Q
-- Example (continued)
Avg.
Inventory
Holding Cost
# of orders
per year
Order
Cost IC Q
(Q/2)
(Q/2)*H
(D/Q)
(D/Q)*S
+(D/Q)*S
500
250
$250 20
$1,000
$1,250
1000
$500 10
1500
750
$750
6.667
$333
$1,083

29. EOQ Model (continued) Conclusion: Thinking Challenge:
1. Inventory cost curve is flat around EOQ
2. Flatter when Q increases than when Q decreases from EOQ
Thinking Challenge:
If the order quantity Q = 2*EOQ, by how much IC will increase?

30. EOQ Model (continued) Sensitivity of EOQ with related to D, H, S, P, I
1. Insensitive to parameter change
2. Directions?

31. EPQ Model
EPQ (Economic Production Quantity) Model: Fixed Order Quantity Model with Incremental Replenishment
Problem description:
Assumptions
There is one product type
Demand is known and constant
Receipt of inventory is gradual and at a constant replenishment (production) rate
Shortage is not allowed

32. EPQ Model (continued) Q Production rate - usage rate Usage rate
Quantity
on hand
Usage
rate
Reorder
point
Time
Start to
produce
Finish
production
Start to
produce
Production run length

33. EPQ Model (continued) Notation and Terminology
Qp = production quantity(# of pieces/production run)
Qp0 = Best production quantity (EPQ)
p = daily production rate (units per day)
d = daily demand rate (units per day)
D = demand rate (units per year)
S = production setup (order) cost($ per setup)
H = holding cost per unit per year (again H = I*P in general)
T = production run length = Q/p

34. EPQ Model (continued) Maximum Inventory Level =
Average Inventory Level =
Annual Holding Cost =

35. EPQ Model (continued) Number of production runs per year =
Order Cost =
IC =
TC =
Minimize TC Minimize IC, why?

36. EPQ Model (continued) Observation: at EPO, holding cost = setup cost
Best Production Quantity (EPQ) formula:

37. EPQ Model (continued) Remarks: EPQ > EOQ (why?)
Example: D=2000 unit/year, S=$5/setup, H=$0.4/unit/year, p=100 unit/day, 200 working days/year. Find the best production batch size and the # of production runs/year.

38. EOQ with discount EOQ with Discount Model:
Assumptions: same as with EOQ, plus discount on all units
Terminology
Price breaks: the smallest order quantity to receive a discount price
Feasibility: the order quantity matching the claimed price is feasible, otherwise infeasible.

39. EOQ with discount (continued)
Example:
Order Price
$2.1/unit
$2.0
Great equal 700 $1.9
Idea is to compare TC curves under different prices - why TC?

40. EOQ with discount (continued)
Order Quantity
Total Cost Curve
for Price 1
Total Cost Curve
for Price 2
$ cost
Total Cost Curve
for Price 3
400
700

41. EOQ with discount (continued)
Order Quantity
Total Cost Curve
for Price 1
Total Cost Curve
for Price 2
$ cost
Total Cost Curve
for Price 3
400
700

42. EOQ with discount (continued)
Observations:
EOQ with a lower price, if feasible, is better than any order quantity with the same or higher price.
Potential best order quantity: cheapest feasible EOQ, price breaks associated with lower prices.

43. EOQ with discount (continued)
Solution Procedure:
1. Find the feasible EOQ with cheapest possible price.
2. Calculate TCs of the EOQ (from Step 1) and price breaks above EOQ.
3. Pick the order quantity with lowest TC

44. EOQ with discount (continued)
Example (continued) Annual demand = 10,000 unit/year, order cost = $5.5/order. Assuming holding costs are proportional to unit prices and annual interest rate = 20%. Find the best order quantity.

45. III. Models on Reorder Points - When to Order?
Find ROP (Re-Order Point)
ROP depends on:
Lead Time: time between placing and receiving an order
Demand Distribution: how uncertain
Desired Service Level: probability of no shortage = 1-P(s), where P(s) = probability of shortage

46. Models on Reorder Points - When to Order ? (continued)
Constant Demand Rate:
Constant daily demand rate = d, Lead time = L days
ROP = d * L = Lead time demand
Remark:
no uncertainty in demand
service level = 100%
safety stock = 0

47. Models on Reorder Points - When to Order ? (continued)
Variable Demand with Stable Average Rate
How continuous review system works?
Lead time demand: demand during the lead time
ROP Lead time demand ==>
ROP < Lead time demand ==>
ROP = Average lead time demand + Safety Stock = m + SS

48. Models on Reorder Points - When to Order ? (continued)
Remarks:
Higher the desired service level --->
More uncertain the demand --->
Two methods to determine the SS

49. Models on Reorder Points - When to Order ? (continued)
1. Determine SS and ROP based on shortage cost inf. (if available)
SS increases  Holding cost ? Shortage cost ?
Best SS minimizes total inventory cost

50. Models on Reorder Points - When to Order ? (continued)
1. Determine SS and ROP based on shortage cost inf. (continued)
-- Example: Consider a light switch carried by Litely. Litely sells 1,350 of these switches per year, and places order for 300 of these switches at a time. The carrying cost per unit per year is calculated as $5 while the stock out cost is estimated at $6 ($3 lost profit per switch and another $3 lost in goodwill, or future sales loss). Find the best SS level and ROP for Litely.

51. Models on Reorder Points - When to Order ? (continued)
Determine SS and ROP based on shortage cost inf. (continued)
1. Determine SS and ROP based on demand inf. during each lead time period:
Lead Time Demand 5 10 15 20 25 30
Probability
0.1
0.15
0.2

52. Models on Reorder Points - When to Order ? (continued)
Determine SS and ROP based on shortage cost inf. (continued)
If SS = 0, ROP = m = 15 switches

53. Models on Reorder Points - When to Order ? (continued)
Determine SS and ROP based on shortage cost inf. (continued)
# of orders per year =
For no safety stock, Litely has the following shortage table. Why?
Shortage Level
no shortage 5 10 15
Probability
0.55
0.2
0.15
0.1

54. Models on Reorder Points - When to Order ? (continued)
Determine SS and ROP based on shortage cost inf. (continued)
Determine the best SS in following table
Safety stock
Add. Holding
cost
Avg.
shortage
(per order)
Annual
shortage cost
Total cost 5 10 15

55. Models on Reorder Points - When to Order ? (continued)
2. Determine ROP and SS based on lead time demand distribution and desired service level:

56. Models on Reorder Points - When to Order ? (continued)
Case 1. Empirical Lead time demand distribution
-- Example:
Lead Time Demand
Frequency
Probability
ROP
Service Level 3 2 4 5 6 7 8

57. Models on Reorder Points - When to Order ? (continued)
Find R and SS to achieve the service level of 85% and 95%, respectively.

58. Models on Reorder Points - When to Order ? (continued)
Case 2. Lead time demand is Normally distributed with (m, )
SS = , ROP = m + SS, z = single tail normal score of desired service level.
( is the standard deviation)
Example:
Lead time demand is Normally distributed with mean = 4 and standard deviation = 3. Find ROP and SS to achieve the service level of 85% and 95%, respectively.

59. IV. Single Period Model and Marginal Analysis (Newsvendor Problem)

60. Homework (Additional problems)
Problem 1: A toy manufacturer uses approximately 36,000 silicon chips annually. The chips are used at a steady rate during the 240 days the plant operates. Annual holding cost is 50 cents per chip, and ordering cost (per order) is $25/order. Assume that each of their orders comes in one batch. Determine:
a. .the best order quantity
b. demonstrate that your order quantity is optimal by showing that annual ordering costs = annual holding costs
c. the average inventory level
d. the number of orders per year
e. the number of working days between orders (Hint: days between orders = # days in a year / # of orders per year. Why?)

61. Homework (Additional problems)
Problem 2. The Dine Corporation is both a producer and a user of brass couplings. The firm operates 200 days a year and uses the couplings at a steady rate of 50 per day. Couplings can be produced at a rate of 150 per day. Inventory holding cost is estimated at $5 per unit per year. Machine setup costs are $40 per production run. Determine:
a. the best production run size
b. demonstrate that your production run size is optimal by showing that annual set up costs = annual holding costs (Hint: find the formula of holding and setup cost for EPQ model in my lecture note.)
c. the maximum inventory level (Hint: find the formula in the derivation of EPQ)
d. the number of production runs per year
e. the cycle time and the production time within each cycle (Hint: cycle time is given by Q/d and production time is given by Q/p. Why? Think before using the formula)

62. Homework (Additional problems)
A small manufacturing firm used roughly 3,400 pounds of chemical dye each year. Currently the firm purchases 300 pounds per order and pays $3 per pound. The supplier has just announced that orders of 1,000 pounds or more will be filled at a price of $2.5 per pound. The manufacturing firm incurs a cost of $100 each time it submits an order and assigns an annual holding cost of 20% of the purchase price per pound.
a. determine the best order size that minimizes the total cost
b. if the supplier offered the discount at 2,500 pounds instead of at 1,000 pounds, what order size would minimize total cost?

63. Homework (Additional problems)
Problem 4: A product is ordered four times every year. Inventory carrying cost is $20 per unit per year, and the cost of shortage for each unit is $40. Given the following demand probabilities during the reorder period
Lead Time Demand 40 80
120
160
Probability
0.1
0.25
0.3

64. Homework (Additional problems)
Problem 4 (continued)
a) What is the average lead time demand?
b) What would be the reorder point without safety stock?
c) What would be the probabilities of the following shortage levels if the company uses the reorder point without safety stock?

65. Homework (Additional problems)
Problem 4 (continued)
d) Follow the Litely example in my lecture to find out the best safety stock level to minimize the total cost.
e) What is the reorder point to achieve the 95% service level? What is the associated safety stock? (Hint: you need to follow the example in my lecture note under Case 1)
Shortage Level 40 80
Probability