1. CHAPTER 4 INVENTORY MANAGEMENT

2. LEARNING OBJECTIVES Define inventory and functions of inventory
Conduct an ABC analysis.
Explain and apply the EOQ and POQ model to solve typical problems.
Compute a ROP and safety stock

3. Inventory A stock or store of goods
A stock of item kept to meet demand
Inventory Management
How much
When
Classified
Accuracy

4. Objective of Inventory Control
To achieve satisfactory levels of customer service while keeping inventory costs within reasonable bounds
To keep enough inventory to meet customer demand and also be cost effective.
Level of customer service
Costs of ordering and carrying inventory

5. Functions of Inventory
To meet anticipated demand
To decouple operations
To protect against stock-outs
To take advantage of order cycles
To help hedge against price increases
To take advantage of quantity discounts

6. Types of Inventory

7. Types of Inventory Raw material Purchased but not processed
Work-in-process
Undergone some change but not completed
A function of cycle time for a product
Maintenance/repair/operating (MRO)
Necessary to keep machinery and processes productive

8. Types of Inventory Finished-goods inventories
(manufacturing firms) or merchandise (retail stores)
Goods in transit
Completed product awaiting shipment

9. The Material Flow Cycle
Cycle time
95% 5%
Input Wait for Wait to Move Wait in queue Setup Run Output
inspection be moved time for operator time time

10. Effective Inventory Management
A system to keep track of inventory
A reliable forecast of demand
Knowledge of lead times
Reasonable estimates of
Holding costs
Ordering costs
Shortage costs
A classification system

11. Inventory Counting Systems
Periodic System
Physical count of items made at periodic intervals
Perpetual Inventory System
keeps track of removals from inventory continuously, thus monitoring current levels of each item

12. Inventory Counting Systems
Two-Bin System
Two containers of inventory; reorder when the first is empty
Universal Bar Code
printed on a label that has information about the item to which it is attached

13. Key Inventory Terms
Lead time: time interval between ordering and receiving the order
Item cost: Cost per item plus any other direct costs associated with getting the item to the plant
Holding (carrying) costs: cost to carry an item in inventory for a length of time, usually a year
Ordering costs: costs of ordering and receiving inventory
Shortage costs: costs when demand exceeds supply

14. Independent Versus Dependent Demand
Independent Demand A
Dependent Demand
B(4)
C(2)
D(2)
E(1)
D(3)
F(2)
Independent demand is uncertain. Dependent demand is certain.

15. ABC Classification System
Classifying inventory according to some measure of importance and allocating control efforts accordingly.
A - very important
B - moderately important
C - least important
High A
Annual
$ value
of items B C
Low
Low
High
Percentage of Items

16. ABC Analysis A Items 80 – 70 – 60 – Percent of annual dollar usage
80 –
70 –
60 –
50 –
40 –
30 –
20 –
10 –
0 –
| | | | | | | | | |
Percent of inventory items
A Items
B Items
C Items

17. ABC Analysis Example

18. Inventory Models for Independent Demand
Need to determine when and how much to order
Basic economic order quantity
Production order quantity
Quantity discount model

19. Basic EOQ Model Demand is known, constant, and independent
Lead time is known and constant
Receipt of inventory is instantaneous and complete
Quantity discounts are not possible
Only variable costs are setup and holding
Stockouts can be completely avoided

20. Cycle-Inventory Levels
Receive order
On-hand inventory (units)
Time
Inventory depletion (demand rate) Q
Average
cycle
inventory Q — 2
1 cycle

21. Total Annual Cycle-Inventory Costs
Objective is to minimize total costs
Annual cost
Order quantity
Total cost = (H) (S) D Q 2
Curve for total cost of holding and setup
Setup (or order) cost curve
Minimum total cost
Optimal order quantity (Q*)
Holding cost curve
Holding cost = (H) Q 2
Ordering cost = (S) D Q
Table 11.5

22. Minimum Total Cost Q 2 H D S =
The total cost curve reaches its minimum where the carrying and ordering costs are equal. Q 2 H D S =

23. Example
Bird feeder sales are 18 units per week, and the supplier charges $60 per unit. The cost of placing an order (S) with the supplier is $45.
Annual holding cost (H) is 25% of a feeder’s value, based on operations 52 weeks per year.
Management chose a 390-unit lot size (Q) so that new orders could be placed less frequently.
What is the annual cycle-inventory cost (C) of the current policy of using a 390-unit lot size?

24. Costing out a Lot Sizing Policy
Museum of Natural History Gift Shop:
What is the annual cycle-inventory cost (C) of the current policy of using a 390-unit lot size?
D = (18 /week)(52 weeks) = 936 units H = 0.25 ($60/unit) = $15
C = (H) (S) = (15) (45) Q 2 D
936
390
C = $ $108 = $3033

25. Lot Sizing at the Museum of Natural History Gift Shop
D = 936 units; H = $15; S = $45; Q = 390 units; C = $3033
Q = 468 units; C = ?
Would a lot size of 468 be better?
C = (H) (S) = (15) (45) Q 2 D
936
468
C = $ $90 = $3600
Q = 468 is a more expensive option.
The best lot size (EOQ) is the lowest point on the total annual cost curve!

26. Lot Sizing at the Museum of Natural History Gift Shop— — —
0 —
| | | | | | | |
Lot Size (Q)
Annual cost (dollars)
Current
cost Q
Total cost
Holding cost
Lowest
cost
Best Q (EOQ)
Ordering cost

27. Computing the EOQ Bird Feeders: C = $1,124.10 EOQ = 2DS H
D = annual demand S = ordering or setup costs per lot H = holding costs per unit
D = 936 units H = $15 S = $45
EOQ =
2(936)45 15
= or 75 units
C = (H) (S) Q 2 D
C = (15) (45) 75 2
936
C = $1,124.10

28. Computing EOQ using the Excel Solver

29. Expected number of orders Demand Order quantity D Q*
Expected time between orders
Number of working days per year N
EOQ D
TBOEOQ =

30. Understanding the Effect of Changes
A Change in the Demand Rate (D): When demand rises, the lot size also rises, but more slowly than actual demand.
A Change in the Setup Costs (S): Increasing S increases the EOQ and, consequently, the average cycle inventory.
A Change in the Holding Costs (H): EOQ declines when H increases.

31. EOQ: Robust Model ?
Errors in Estimating D, H, and S: Total cost is fairly insensitive to errors, even when the estimates are wrong by a large margin. The reasons are that errors tend to cancel each other out and that the square root reduces the effect of the error.

32. When to Reorder with EOQ Ordering
Reorder Point - When the quantity on hand of an item drops to this amount, the item is reordered
Safety Stock - Stock that is held in excess of expected demand due to variable demand rate and/or lead time.
Service Level - Probability that demand will not exceed supply during lead time.

33. Determinants of the Reorder Point
The rate of demand
The lead time
Demand and/or lead time variability
Stockout risk (safety stock)

34. Safety Stock Quantity Maximum probable demand Expected demandLT
Time
Expected demand
during lead time
Maximum probable demand
ROP
Quantity
Safety stock
Safety stock reduces risk of
stockout during lead time

35. Number of working days in a year Lead time for a new order in days
Reorder Point
Inventory level (units)
Time (days) Q*
d = D
Number of working days in a year
ROP =
Lead time for a new order in days
Demand per day
ROP (units)
Slope = units/day = d
= d x L
Lead time = L

36. Production Order Quantity Model
Used when inventory builds up over a period of time after an order is placed
Used when units are produced and sold simultaneously
Inventory level
Time
Part of inventory cycle during which production (and usage) is taking place
Demand part of cycle with no production
Maximum inventory t

37. Production Order Quantity Model

38. Production Order Quantity Model
Q = Number of pieces per order p = Daily production rate
H = Holding cost per unit per year d = Daily demand/usage rate
t = Length of the production run in days
= –
Maximum inventory level
Total produced during the production run
Total used during the production run
= pt – dt

39. Production Order Quantity Model
= –
Maximum inventory level
Total produced during the production run
Total used during the production run
= pt – dt
However, Q = total produced = pt ; thus t = Q/p
Maximum inventory level
= p – d = Q 1 – Q p d
Holding cost = (H) = – H d p Q 2
Maximum inventory level

40. Production Order Quantity Model
Setup cost = (D/Q)S
Holding cost = HQ[1 - (d/p)] 1 2
(D/Q)S = HQ[1 - (d/p)] 1 2
Q2 =
2DS
H[1 - (d/p)]
Q* =
2DS
H[1 - (d/p)] p
EPQ.xls

41. Quantity Discount Models
Reduced prices are often available when larger quantities are purchased
Trade-off is between reduced product cost and increased holding cost
Total cost = Setup cost + Holding cost + Product cost
TC = S H + PD D Q 2

42. Total Cost With PD Cost Adding Purchasing cost doesn’t change EOQ
EOQ/POQ
TC with PD
TC without PD PD
Quantity
Adding Purchasing cost doesn’t change EOQ

43. Quantity Discount Models
A typical quantity discount schedule
Discount Number
Discount Quantity
Discount (%)
Discount Price (P) 1
0 to 999
no discount
$5.00 2
1,000 to 1,999 4
$4.80 3
2,000 and over 5
$4.75
Table 12.2

44. Quantity Discount Models
Steps in analyzing a quantity discount
For each discount, calculate Q*
If Q* for a discount doesn’t qualify, choose the smallest possible order size to get the discount
Compute the total cost for each Q* or adjusted value from Step 2
Select the Q* that gives the lowest total cost

45. Quantity Discount Models
Total cost $
Order quantity
1,000
2,000
Total cost curve for discount 2
Total cost curve for discount 1
Total cost curve for discount 3
Q* for discount 2 is below the allowable range at point a and must be adjusted upward to 1,000 units at point b a b
1st price break
2nd price break
Figure 12.7

46. Quantity Discount Models
2DS IP
Calculate Q* for every discount
Q1* = = 700 cars/order
2(5,000)(49)
(.2)(5.00)
Q2* = = 714 cars/order
2(5,000)(49)
(.2)(4.80)
Q3* = = 718 cars/order
2(5,000)(49)
(.2)(4.75)

47. Quantity Discount Models
2DS IP
Calculate Q* for every discount
Q1* = = 700 cars/order
2(5,000)(49)
(.2)(5.00)
Q2* = = 714 cars/order
2(5,000)(49)
(.2)(4.80)
1,000 — adjusted
Q3* = = 718 cars/order
2(5,000)(49)
(.2)(4.75)
2,000 — adjusted

48. Quantity Discount Models
Discount Number
Unit Price
Order Quantity
Annual Product Cost
Annual Ordering Cost
Annual Holding Cost
Total 1
$5.00
700
$25,000
$350
$25,700 2
$4.80
1,000
$24,000
$245
$480
$24,725 3
$4.75
2,000
$23.750
$122.50
$950
$24,822.50
Table 12.3
Choose the price and quantity that gives the lowest total cost
Buy 1,000 units at $4.80 per unit

49. Quantity Discount Example: Collin’s Sport store is considering going to a different hat supplier. The present supplier charges $10 each and requires minimum quantities of 490 hats. The annual demand is 12,000 hats, the ordering cost is $20, and the inventory carrying cost is 20% of the hat cost, a new supplier is offering hats at $9 in lots of Who should he buy from?
Since the EOQ of 516 is not feasible, calculate the total cost (C) for each price to make the decision
4000 hats at $9 each saves $19,320 annually. Space?

50. Probabilistic Models and Safety Stock
Used when demand is not constant or certain
Use safety stock to achieve a desired service level and avoid stockouts
ROP = d x L + ss
Annual stockout costs = the sum of the units short x the probability x the stockout cost/unit x the number of orders per year

51. Safety Stock Example Number of Units Probability 30 .2 40 ROP  50 .3
ROP = 50 units Stockout cost = $40 per frame
Orders per year = 6 Carrying cost = $5 per frame per year
Number of Units
Probability 30 .2 40
ROP  50 .3 60 70 .1
1.0

52. A safety stock of 20 frames gives the lowest total cost
Safety Stock Example
ROP = 50 units Stockout cost = $40 per frame
Orders per year = 6 Carrying cost = $5 per frame per year
Safety Stock
Additional Holding Cost
Stockout Cost
Total Cost 20
(20)($5) = $100 $0
$100 10
(10)($5) = $ 50
(10)(.1)($40)(6) = $240
$290
$ 0
(10)(.2)($40)(6) + (20)(.1)($40)(6) = $960
$960
A safety stock of 20 frames gives the lowest total cost
ROP = = 70 frames

53. Probabilistic Demand Inventory level Time Figure 12.8 Safety stock
16.5 units
ROP 
Place order
Inventory level
Time
Minimum demand during lead time
Maximum demand during lead time
Mean demand during lead time
ROP = safety stock of 16.5 = 366.5
Receive order
Lead time
Normal distribution probability of demand during lead time
Expected demand during lead time (350 kits)
Figure 12.8

54. Probabilistic Demand Probability of no stockout 95% of the time
Mean demand 350
Risk of a stockout (5% of area of normal curve)
ROP = ? kits
Quantity
Safety stock
Number of standard deviations z

55. Probabilistic Demand
Use prescribed service levels to set safety stock when the cost of stockouts cannot be determined
ROP = demand during lead time + ZsdLT
where Z = number of standard deviations
sdLT = standard deviation of demand during lead time

56. Probabilistic Example
Average demand = m = 350 kits
Standard deviation of demand during lead time = sdLT = 10 kits
5% stockout policy (service level = 95%)
Using Appendix I, for an area under the curve of 95%, the Z = 1.65
Safety stock = ZsdLT = 1.65(10) = 16.5 kits
Reorder point = expected demand during lead time + safety stock
= 350 kits kits of safety stock
= or 367 kits

57. Other Probabilistic Models
When data on demand during lead time is not available, there are other models available
When demand is variable and lead time is constant
When lead time is variable and demand is constant
When both demand and lead time are variable

58. Other Probabilistic Models
Demand is variable and lead time is constant
ROP = (average daily demand x lead time in days) + ZsdLT
where sd = standard deviation of demand per day
sdLT = sd lead time
Variance = daily variance x no. of days of lead time
Standard D (sum of daily variance during lead time)

59. Probabilistic Example
Average daily demand (normally distributed) = 15
Standard deviation = 5
Lead time is constant at 2 days
90% service level desired
Z for 90% = 1.28
From Appendix I
ROP = (15 units x 2 days) + Zsdlt
= (5)( 2)
= = ≈ 39
Safety stock is about 9 iPods

60. Other Probabilistic Models
Lead time is variable and demand is constant
ROP = (daily demand x average lead time in days) + Z x (daily demand) x sLT
where sLT = standard deviation of lead time in days

61. Probabilistic Example
Z for 98% = 2.055
From Appendix I
Daily demand (constant) = 10
Average lead time = 6 days
Standard deviation of lead time = sLT = 3
98% service level desired
ROP = (10 units x 6 days) (10 units)(3)
= =
Reorder point is about 122 cameras

62. Other Probabilistic Models
Both demand and lead time are variable
ROP = (average daily demand x average lead time) + ZsdLT
where sd = standard deviation of demand per day
sLT = standard deviation of lead time in days
sdLT = (average lead time x sd2) + (average daily demand)2 x sLT2

63. Probabilistic Example
Average daily demand (normally distributed) = 150
Standard deviation = sd = 16
Average lead time 5 days (normally distributed)
Standard deviation = sLT = 1 day
95% service level desired
Z for 95% = 1.65
From Appendix I
ROP = (150 packs x 5 days) sdLT
= (150 x 5) (5 days x 162) + (1502 x 12)
= (154) = 1,004 packs

64. Total Q System Costs Total cost = Annual Holding Cost +
Annual setup/ordering Cost +
Annual safety stock holding cost

65. Fixed-Period (P) Systems
Orders placed at the end of a fixed period
Inventory counted only at end of period
Order brings inventory up to target level
Only relevant costs are ordering and holding
Lead times are known and constant
Items are independent from one another

66. Fixed-Period (P) Systems
Target quantity (T)
On-hand inventory
Time Q1 Q2 Q3 Q4 P P
Figure 12.9

67. Fixed-Period (P) Example
3 jackets are back ordered No jackets are in stock
It is time to place an order Target value = 50
Order amount (Q) = Target (T) - On-hand inventory - Earlier orders not yet received (SR)+ Back orders
Q = = 53 jackets

68. Periodic Review Systems: Calculations for TI
Targeted Inventory level:
TI = d(p + LT) + SS
d = average period demand
p = order interval (days, wks)
LT = lead time (days, wks)
SS = zσd
Replenishment Quantity (Q)=TI-OH

69. Periodic Review Systems Example
The KVS Pharmacy stocks a popular brand of over-the-counter flu and cold medicine. The average demand for the medicine is 6 packages per day, with a standard deviation of 1.2 packages. A vendor for the pharmaceutical company checks KVS’s stock every 60 days. During one visit the store had 8 packages in stock. The lead time to receive an order is 5 days. Determine the order size for this order period that will enable KVS to maintain a 95% service level.
Q = d(p + LT) + zσd OH
= 6(60 + 5) (1.2) =

70. Total P System Costs Same three cost element as Q system
Order quantity, Q will be the average consumption of inventory during the p periods between order; Q =dP

71. Q System Example
Suppose that the average demand for bird feeders is 18 units per week with a standard deviation of 5 units. The lead time is constant at 2 weeks. Determine the safety stock and reorder point for a 90% cycle-service level. What is the total cost of the Q system?
sdLT = sd LT = = 7.1
Safety stock = zsdLT = 1.28(7.1) = 9.1 or 9 units
Reorder point = dL + safety stock = 2(18) + 9 = 45 units
C = ($15) ($45) + 9($15) 75 2
936
C = $ $ $135 = $

72. P System Example d(P+LT) = sd P + LT = 5 6 = 12 units
Bird feeder demand is normally distributed with a mean of 18 units per week and a standard deviation in weekly demand of 5 units, operating 52 weeks a year. Lead time (L) is 2 weeks and EOQ is 75 units with a safety stock of 9 units and a cycle-service level of 90%. Annual demand (D) is 936 units. What is the equivalent P system and total cost?
P = (52) = (52) = 4.2 or 4 weeks
EOQ D 75
936
Time between reviews =
Standard deviation of demand over the protection period
d(P+LT) = sd P + LT = = 12 units
T = Average demand during the protection interval + Safety stock
= d (P + LT) + zsd(P + LT)
= (18 units/week)(6 weeks) (12 units) = 123 units

73. P System Example continued
© 2007 Pearson Education
P System Example continued
d = 18 units L = 2 weeks Cycle/service level = 90% EOQ = 75 units
D = (18 units/week)(52 weeks) = 936 units Safety Stock during P = Holding Costs = $15/unit Ordering Costs = $45
The time between reviews (P) = 4 weeks
Average demand during P + Safety stock = T = 123 units
The total P-system cost for the bird feeders is:
C = ($15) ($45) + 15($15)
4(18) 2
936
C = $540 + $585 + $225 = $1350
The P system requires 15 units in safety stock, while the Q system only needs 9 units. If cost were the only criterion, the Q system would be the choice.

74. Fixed-Period (P) Systems
Inventory is only counted at each review period
May be scheduled at convenient times
May require only periodic checks of inventory levels
May result in stockouts between periods
May require increased safety stock

75. Comparison of Q and P Systems
Convenient to administer
Orders for multiple items from the same supplier may be combined
Inventory Position (IP) only required at review
Systems in which inventory records are always current are called Perpetual Inventory Systems
Q Systems
Review frequencies can be tailored to each item
Possible quantity discounts
Lower, less-expensive safety stocks

76. Single Period Inventory Model
The SPI model is designed for products that share the following characteristics:
Sold at their regular price only during a single-time period
Demand is highly variable but follows a known probability distribution
Salvage value is less than its original cost so money is lost when these products are sold for their salvage value
Objective is to balance the gross profit of the sale of a unit with the cost incurred when a unit is sold after its primary selling period

77. SPI Model Example: Tee shirts are purchase in multiples of 10 for a charity event for $8 each. When sold during the event the selling price is $20. After the event their salvage value is just $2. From past events the organizers know the probability of selling different quantities of tee shirts within a range from 80 to 120
Payoff Table
Prob. Of Occurrence
Customer Demand
# of Shirts Ordered Profit
80 $960 $960 $960 $960 $960 $960
90 $900 $1080 $1080 $1080 $1080 $1040
Buy $840 $1020 $1200 $1200 $1200 $1083
110 $780 $ 960 $ $1320 $1320 $1068
120 $720 $ 900 $1080 $1260 $1440 $1026
Sample calculations:
Payoff (Buy 110)= sell 100($20-$8) –(( ) x ($8-$2))= $1140
Expected Profit (Buy 100)= ($840 X .20)+($1020 x .25)+($1200 x .30) +
($1200 x .15)+($1200 x .10) = $1083