Inventory Management process

1. The following table lists the components required to
assembly an end item and quantities on hand.
Item End B C D E F G H
Amou
0 10 10 25 12 30 5 0
nt on
hand
B (2) C D (3)
E (2) F (3) G (2) E (2)
H (4) E (2)
End Item
If 20 Units of the end items are to be assembled, how many
additional units of E are required.

2. B: 20 x 2 = 40 – 10
= 30
C: 20 x 1 = 20 – 10
= 10
D: 20 x 3 = 60 –
25 = 35
E: 30 x 2 = 60 – 12
= 48
E: 10 x 2 = 20 E: 35 x 2 = 70
Total: 48 + 20 + 70 = 138

3. Inventory Management

4. Inventory Management
• Inventories are important to all types of
organizations
– They have to be counted, paid for, used in
operations, used to satisfy customers, and
managed
– Too much inventory reduces profitability
– Too little inventory damages customer
confidence
– Inventory trade-offs

5. Functions of Inventory
Inventories are resources maintained in various
forms:
- Raw Materials
- Purchased and Manufactured parts
- Sub assemblies
- Finished Products
Important: Since Inventories represent a sizable investment
in a logistic system, we must be aware of the functions they
perform

6. Important categories of Stocks
Inventory
To satisfy the
expected customer
demand
(Anticipation
Inventory)
To protect
against price
increases and
to take
advantage of
Quantity
Discounts
To avoid
stock outs
(Safety Stock
or Buffer
Stock)
To provide buffer
between successive
operations
(Decoupling
Inventory or Work-in-process
Inventory)
To satisfy periods of
seasonal high
demand (Seasonal
Inventory)
To minimize
the total cost
by ordering
the Economic
Order
Quantity
(Cycle Stock)
To act as a buffer
between various
elements of the Supply-
Chain (Suppliers-
Producers-Distributors-
Wholesalers-Retailers-
Customers) (Pipeline or
Transit Stock)

7. Inventory Related Costs
Procurement Costs
Cost of Goods
Ordering Cost
Cost / Order generally fixed
Not dependant on order quantity
- Administrative cost
- Handling
- Transportation
- Inspection of arrivals

8. Inventory Holding Costs
Costs associated with existence of Inventories
(Supply exceeds demand)
Cost/unit/unit time
iC
(i = inventory carrying cost rate)
- Storage and handing
- Interest or tied up capital
- Property taxes
- Insurance
- Spoilage
- Obsolescence
- pilferage

9. Shortage Costs
Costs associated with stockouts
(Demand exceeds Supply)
(Cost/unit) (Cost/unit /unit time)
- Loss of customer goodwill
- Loss of sales

10. Types of Inventory Management Systems
Inventory Management
Systems
Independent Demand Inventory
Management Systems
Dependent Demand
Inventory Management
Systems
Material
Requirements
Planning (MRP)
Systems
Just-In-Time
(JIT)
Systems
Hybrid MRP-JIT
Systems
For Retailers For Manufacturers
ABC
Classification
of Items
EOQ Model for
Manufacturers
Category
A Items
Category
B Items
Category
C Items
Periodic
Review
System
Basic Economic
Order Quantity
(EOQ) Model
EOQ Model
with Quantity
Discounts
EOQ Model with
Differential
Discounting
EOQ Model
with Safety
Stock
EOQ Model
with Intentional
Shortages

11. ABC Analysis
• Stock-keeping units (SKU)
• Identify the classes so management
can control inventory levels
• A Pareto chart

12. ABC Analysis
10 20 30 40 50 60 70 80 90 100
Percentage of SKUs
100 —
Percentage of dollar value
90 —
80 —
70 —
60 —
50 —
40 —
30 —
20 —
10 —
0 —
Class C
Class A
Class B

13. ABC Analysis
• Class A
– 5 – 15 % of units (approx.)
– 70 – 80 % of value (approx.)
• Class B
– 30 % of units (approx.)
– 15 % of value (approx.)
• Class C
– 50 – 60 % of units (approx.)
– 5 – 10 % of value (approx.)

14. Solved Problem 1
Booker’s Book Bindery divides SKUs into three classes,
according to their dollar usage. Calculate the usage
values of the following SKUs and determine which is
most likely to be classified as class A.
SKU Number Description
Quantity Used
per Year
Unit Value
($)
1 Boxes 500 3.00
2 Cardboard
(square feet)
18,000 0.02
3 Cover stock 10,000 0.75
4 Glue (gallons) 75 40.00
5 Inside covers 20,000 0.05
6 Reinforcing tape
(meters)
3,000 0.15
7 Signatures 150,000 0.45

15. Solved Problem 1
SOLUTION
The annual dollar usage for each item is determined by multiplying the
annual usage quantity by the value per unit. As shown in Figure 12.11,
the SKUs are then sorted by annual dollar usage, in declining order.
Finally, A–B and B–C class lines are drawn roughly, according to the
guidelines presented in the text. Here, class A includes only one SKU
(signatures), which represents only 1/7, or 14 percent, of the SKUs but
accounts for 83 percent of annual dollar usage. Class B includes the
next two SKUs, which taken together represent 28 percent of the SKUs
and account for 13 percent of annual dollar usage. The final four SKUs,
class C, represent over half the number of SKUs but only 4 percent of
total annual dollar usage.

16. Solved Problem 1
SKU
Number
Description
Quantity
Used per
Year
Unit Value
($)
Annual Dollar
Usage ($)
1 Boxes 500  3.00 = 1,500
2 Cardboard
(square feet)
18,000  0.02 = 360
3 Cover stock 10,000  0.75 = 7,500
4 Glue (gallons) 75  40.00 = 3,000
5 Inside covers 20,000  0.05 = 1,000
6 Reinforcing tape
(meters)
3,000  0.15 = 450
7 Signatures 150,000  0.45 = 67,500
Total 81,310

17. Solved Problem 1

18. Class C
Class B
Percentage of SKUs
Percentage of Dollar Value
100 –
90 –
80 –
70 –
60 –
50 –
40 –
30 –
20 –
10 –
Class
0 –
A
10 20 30 40 50 60 70 80 90 100

19. Unsolved Problem 1
Lockwood Ind. Is considering the use of ABC analysis to focus on the most
critical SKUs in its inventory. For a random sample of 8 SKUs, following table
shows the annual dollar usage. Rank the SKUs and assign them to A,B or C
class
SKU Dollar Value ($) Annual Usage
1 .01 1200
2 .03 120,000
3 .45 100
4 1.00 44,000
5 4.50 900
6 .90 350
7 .30 70,000
8 1.50 200

20. Cumulative
%
Cumulative
%
SKU
#
Descripti
on
Qty
Used/Year
Value Dollar
Usage
Pct of
Total
of Dollar
Value
of SKUs Class
4 44,000 $1.00 $44,000 60.0% 60.0% 12.5% A
7 70,000 $0.30 $21,000 28.6% 88.7% 25.0% A
5 900 $4.50 $4,050 5.5% 94.2% 37.5% B
2 120,000 $0.03 $3,600 4.9% 99.1% 50.0% B
6 350 $0.90 $315 0.4% 99.5% 62.5% C
8 200 $1.50 $300 0.4% 99.9% 75.0% C
3 100 $0.45 $45 0.1% 100.0% 87.5% C
1 1,200 $0.01 $12 0.0% 100.0% 100.0% C
Total $73,322

21. SKUs

22. OBJECTIVE OF ABC ANALYSIS:
Rationalization of Ordering Policies
EQUAL TREATMENT TO ALL
Item no. Annual Consumption
Value (Rs.)
Number of
Orders
Value per order Average inventory
1
2
3
60,000
4,000
1,000
4
4
4
15,000
1,000
250
7500
500
125
Total Inventory: Rs. 8125
PREFERENTIAL TREATMENT on basis of ABC analysis
1
2
3
60,000
4,000
1,000
8
3
1
7,500
1,333
1,000
3750
667
500
Total Inventory
Rs 4917
The optimum no of orders can be arrived at by using models of inventory
control eg. EOQ
time
inventory
Annual cost
Q*
Total cost
carriage
ordering

23. Economic Order Quantity
The lot size, Q, that minimizes total annual inventory
holding and ordering costs
 Five assumptions
1. Demand rate is constant and known with certainty
2. No constraints are placed on the size of each lot
3. The only two relevant costs are the inventory holding
cost and the fixed cost per lot for ordering or setup
4. Decisions for one item can be made independently of
decisions for other items
5. The lead time is constant and known with certainty

24. Calculating EOQ
Inventory
depletion (demand
rate)
Receiv
e order
1 cycle
On-hand inventory
(units)
Time
Q
Average
cycle
inventory
Q
2

25. Calculating EOQ
• Annual holding cost
Annual holding cost = (Average cycle inventory)
 (Unit holding cost)
 Annual ordering cost
Annual ordering cost = (Number of orders/Year)
 (Ordering costs)
 Total annual cycle-inventory cost
Total costs = Annual holding cost
+ Annual ordering cost

26. Annual cost (dollars)
Calculating EOQ
Lot Size (Q)
Holding cost
Ordering cost
Total cost
Graphs of Annual Holding, Ordering, and Total Costs

27. Calculating EOQ
 Total annual cycle-inventory and ordering cost
where
Q
2
D
Q
C = (H) + (S)
C = total annual cycle-inventory cost
Q = lot size
H = holding cost per unit per year
D = annual demand
S = ordering or setup costs per lot

28. The Cost of a Lot-Sizing Policy
EXAMPLE 12.1
• A museum of natural history opened a gift shop which operates
52 weeks per year.
• Managing inventories has become a problem.
• Top-selling SKU is a bird feeder.
• Sales are 18 units per week, the supplier charges $60 per unit.
• Ordering cost is $45.
• Annual holding cost is 25 percent of a feeder’s value.
• Management chose a 390-unit lot size.
• What is the annual cycle-inventory cost of the current policy of
using a 390-unit lot size?
• Would a lot size of 468 be better?

29. The Cost of a Lot-Sizing Policy
SOLUTION
We begin by computing the annual demand and holding cost as
D =
H =
(18 units/week)(52 weeks/year) = 936 units
0.25($60/unit) = $15
The total annual cycle-inventory cost for the current policy is
Q
2
D
Q
C = (H) + (S)
390
2
936
390
= ($15) + ($45)
= $2,925 + $108 = $3,033
The total annual cycle-inventory cost for the alternative lot
size is
468
2
936
468
C = ($15) + ($45) = $3,510 + $90 = $3,600

30. The Cost of a Lot-Sizing Policy
3000 –
2000 –
1000 –
Total
cost = (H) + (S)
Q
2
Holding cost = (H)
0 – | | | | | | | |
50 100 150 200 250 300 350 400
Lot Size (Q)
Annual cost (dollars)
Current
Q
Current
cost
Lowest
cost
Best Q
(EOQ)
Q
2
D
Q
D
Q
Ordering cost = (S)

31. Calculating EOQ
• The EOQ formula:
EOQ =
2DS
H
 Time between orders
EOQ
D
TBOEOQ = (12 months/year)

32. Finding the EOQ, Total Cost, TBO
EXAMPLE 12.2
For the bird feeders in Example 12.1, calculate the EOQ and its total
annual cycle-inventory cost. How frequently will orders be placed if
the EOQ is used?
SOLUTION
Using the formulas for EOQ and annual cost, we get
2DS
H
EOQ = =
= 74.94 or 75 units
2(936)(45)
15

33. Finding the EOQ, Total Cost, TBO
Figure shows that the total annual cost is much less than the $3,033 cost
of the current policy of placing 390-unit orders.

34. Finding the EOQ, Total Cost, TBO
When the EOQ is used, the TBO can be expressed in various
ways for the same time period.
TBOEOQ =
EOQ
D
EOQ
D
75
936
= = 0.080 year
TBOEOQ = (12 months/year)
EOQ
D
TBOEOQ = (52 weeks/year)
EOQ
D
TBOEOQ = (365 days/year)
75
936
= (12) = 0.96 month
75
936
= (52) = 4.17 weeks
75
936
= (365) = 29.25 days

35. Application 12.1
Suppose that you are reviewing the inventory policies on an $80
item stocked at a hardware store. The current policy is to replenish
inventory by ordering in lots of 360 units. Additional information is:
D = 60 units per week, or 3,120 units per year
S = $30 per order
H = 25% of selling price, or $20 per unit per year
What is the EOQ?
2DS
H
EOQ = =
= 97 units
2(3,120)(30)
20
SOLUTION

36. Application 12.1
What is the total annual cost of the current policy (Q = 360), and
how does it compare with the cost with using the EOQ?
Current Policy EOQ Policy
Q = 360 units Q = 97 units
C = (360/2)(20) + (3,120/360)(30)
C = 3,600 + 260
C = $3,860
C = (97/2)(20) + (3,120/97)(30)
C = 970 + 965
C = $1,935

37. Application 12.1
What is the time between orders (TBO) for the current policy and
the EOQ policy, expressed in weeks?
TBO360 =
TBOEOQ =
(52 weeks per year) = 6 weeks
360
3,120
(52 weeks per year) = 1.6 weeks
97
3,120
SOLUTION

38. Problem 2
Nelson’s Hardware Store stocks a 19.2 volt cordless drill that is a popular
seller. Annual demand is 5,000 units, the ordering cost is $15, and the
inventory holding cost is $4/unit/year.
a. What is the economic order quantity?
b. What is the total annual cost for this inventory item?

39. Solved Problem 2
SOLUTION
a. The order quantity is
2DS
H
EOQ = =
2(5,000)($15)
$4
= 37,500 = 193.65 or 194 drills
b. The total annual cost is
Q
2
D
Q
C = (H) + (S) =
($4) + ($15) = $774.60
194
2
5,000
194

40. Inventory Control Systems
 Nature of demand
 Independent demand
 Dependent demand

41. Inventory Control Systems
• Continuous review (Q) system
– Reorder point system (ROP) and fixed
order quantity system
– For independent demand items
– Tracks inventory position (IP)
– Includes scheduled receipts (SR), on-hand
inventory (OH), and back orders (BO)
Inventory position = On-hand inventory + Scheduled receipts
– Backorders
IP = OH + SR – BO

42. Selecting the Reorder Point
Time
On-hand inventory
Order
received
L L
Order
received
TBO TBO
Order
received
TBO
L
Order
placed
Order
placed
Order
placed
IP IP IP
R
Q Q Q
OH OH OH
Order
received
Q System When Demand and Lead Time Are Constant
and Certain

43. Application
The on-hand inventory is only 10 units, and the reorder point R is 100.
There are no backorders and one open order for 200 units. Should a new
order be placed?
IP = OH + SR – BO = 10 + 200 – 0 = 210
R = 100
SOLUTION
Decision: Place no new order

44. Placing a New Order
EXAMPLE Demand for chicken soup at a supermarket is always 25
cases a day and the lead time is always 4 days. The shelves were
just restocked with chicken soup, leaving an on-hand inventory of
only 10 cases. No backorders currently exist, but there is one open
order in the pipeline for 200 cases. What is the inventory position?
Should a new order be placed?
SOLUTION
R = Total demand during lead time = (25)(4) = 100 cases
IP = OH + SR – BO
= 10 + 200 – 0 = 210 cases

45. Continuous Review Systems
Selecting the reorder point with variable demand and
constant lead time
Reorder point = Average demand during lead time
+ Safety stock
= dL + safety stock
where
d = average demand per week (or day or months)
L = constant lead time in weeks (or days or months)

46. Reorder Point
1. Choose an appropriate service-level policy
– Select service level or cycle service level
– Protection interval
2. Determine the demand during lead time
probability distribution
3. Determine the safety stock and reorder
point levels

47. Demand During Lead Time
• Specify mean and standard deviation
• Standard deviation of demand during lead time
σdLT = σd
2L = σd L
Safety stock and reorder point
Safety stock = zσdLT
where
z = number of standard deviations needed to achieve the
cycle-service level
σdLT = stand deviation of demand during lead time
Reorder point = R = dL + safety stock

48. Demand During Lead Time
Average
demand
during
lead time
Cycle-service level = 85%
Probability of stockout
(1.0 – 0.85 = 0.15)
zσdLT
R
Finding Safety Stock with a Normal Probability Distribution for an 85 Percent Cycle-
Service Level

49. Reorder Point for Variable Demand
EXAMPLE
Let us return to the bird feeder in Example.
The EOQ is 75 units. Suppose that the
average demand is 18 units per week with
a standard deviation of 5 units. The lead
time is constant at two weeks. Determine
the safety stock and reorder point if
management wants a 90 percent cycle-service
level.

50. Reorder Point for Variable Demand
SOLUTION
In this case, σd = 5, d = 18 units, and L = 2 weeks, so
σdLT = σd L = 5 2 = 7.07. Consult the body of the table in the
Normal Distribution appendix for 0.9000, which corresponds to
a 90 percent cycle-service level. The closest number is 0.8997,
which corresponds to 1.2 in the row heading and 0.08 in the
column heading. Adding these values gives a z value of 1.28.
With this information, we calculate the safety stock and reorder
point as follows:
Safety stock = zσdLT = 1.28(7.07) = 9.05 or 9 units
Reorder point = dL + Safety stock = 2(18) + 9 = 45 units

51. Application 12.3
Suppose that the demand during lead time is normally distributed
with an average of 85 and σ= 40. Find the safety stock, and
dLT reorder point R, for a 95 percent cycle-service level.
SOLUTION
Safety stock = zσ=
1.645(40) = 65.8 or 66 units
dLT R = Average demand during lead time + Safety stock
R = 85 + 66 = 151 units
Find the safety stock, and reorder point R, for an 85 percent
cycle-service level.
Safety stock = zσdLT = 1.04(40) = 41.6 or 42 units
R = Average demand during lead time + Safety stock
R = 85 + 42 = 127 units

52. Reorder Point for Variable Demand
and Lead Time
• Often the case that both are variable
• The equations are more complicated
Safety stock = zσdLT
R = (Average weekly demand  Average lead time)
where
+ Safety stock
= dL + Safety stock
d = Average weekly (or daily or monthly) demand
L = Average lead time
σ= Standard deviation of weekly (or daily or monthly) demand
d σ= Standard deviation of the lead time
LT σ= Lσ2 + d2σ2
dLT d
LT

53. Reorder Point
EXAMPLE 12.5
The Office Supply Shop estimates that the average demand for a
popular ball-point pen is 12,000 pens per week with a standard
deviation of 3,000 pens. The current inventory policy calls for
replenishment orders of 156,000 pens. The average lead time from
the distributor is 5 weeks, with a standard deviation of 2 weeks. If
management wants a 95 percent cycle-service level, what should
the reorder point be?

54. Reorder Point
SOLUTION
We have d = 12,000 pens, σd = 3,000 pens, L = 5 weeks,
and σLT = 2 weeks
σdLT = Lσd
2 + d2σLT
2 =
(5)(3,000)2 + (12,000)2(2)2
= 24,919.87 pens
From the Normal Distribution appendix for 0.9500, the
appropriate z value = 1.65. We calculate the safety stock and
reorder point as follows:
Safety stock = zσdLT =
(1.65)(24,919.87)
= 41,117.79 or 41,118 pens
Reorder point = dL + Safety stock =
(12,000)(5) + 41.118
= 101,118 pens

55. Application 12.4
Grey Wolf lodge is a popular 500-room hotel in the North
Woods. Managers need to keep close tabs on all of the
room service items, including a special pint-scented bar
soap. The daily demand for the soap is 275 bars, with a
standard deviation of 30 bars. Ordering cost is $10 and
the inventory holding cost is $0.30/bar/year. The lead
time from the supplier is 5 days, with a standard
deviation of 1 day. The lodge is open 365 days a year.
What should the reorder point be for the bar of soap if
management wants to have a 99 percent cycle-service?

56. Application 12.4
SOLUTION
d = 275 bars
L = 5 days
σ= 30 bars
d σ= 1 day
LT σ = Lσ2 + d2σ2 =
283.06 bars dLT d
LT
From the Normal Distribution appendix for 0.9900, z = 2.33.
We calculate the safety stock and reorder point as follows;
Safety stock = zσdLT =
Reorder point + safety stock = dL + safety stock
(2.33)(283.06) = 659.53 or 660 bars
= (275)(5) + 660 = 2,035 bars

57. Continuous Review Systems
• Two-Bin system
– Visual system
– An empty first bin signals the need to place an order
Calculating total systems costs
Total cost = Annual cycle inventory holding cost
+ Annual ordering cost
+ Annual safety stock holding cost
Q
2
D
Q
C = (H) + (S) + (H) (Safety stock)

58. Application 12.5
The Discount Appliance Store uses a continuous review system
(Q system). One of the company’s items has the following
characteristics:
Demand = 10 units/wk (assume 52 weeks per
year)
Ordering and setup cost (S) = $45/order
Holding cost (H) = $12/unit/year
Lead time (L) = 3 weeks (constant)
Standard deviation in weekly demand = 8 units
Cycle-service level = 70%

59. Application 12.5
SOLUTION
What is the EOQ for this item?
D = 10/wk  52 wks/yr = 520 units
2DS
H
EOQ = =
= 62 units
2(520)(45)
12
What is the desired safety stock?
σdLT = σd L = 8 3 = 14 units
Safety stock = zσdLT = 0.525(14) = 8 units

60. Application 12.5
What is the desired reorder point R?
R = Average demand during lead time + Safety stock
R =
3(10) + 8 = 38 units
What is the total annual cost?
520
62 C =
($12) + ($45) + 8($12) = $845.42
62
2

61. Application 12.5
Suppose that the current policy is Q = 80 and R = 150. What will be the
changes in average cycle inventory and safety stock if your EOQ and R
values are implemented?
Reducing Q from 80 to 62
Cycle inventory reduction = 40 – 31 = 9 units
Safety stock reduction = 120 – 8 = 112 units
Reducing R from 150 to 38

62. Periodic Review System (P)
Order placed for variable amount after fixed passage
of time
 Four of the original EOQ assumptions
maintained
 No constraints are placed on lot size
 Holding and ordering costs only
 Independent demand
 Lead times are certain
 Order is placed to bring the inventory position
up to the target inventory level, T, when the
predetermined time, P, has elapsed

63. How Much to Order in a P System
EXAMPLE 12.6
A distribution center has a backorder for five 36-inch color TV sets.
No inventory is currently on hand, and now is the time to review.
How many should be reordered if T = 400 and no receipts are
scheduled?
SOLUTION
IP = OH + SR – BO
= 0 + 0 – 5 = –5 sets
T – IP = 400 – (–5) = 405 sets
That is, 405 sets must be ordered to bring the inventory
position up to T sets.

64. Application 12.6
The on-hand inventory is 10 units, and T is 400. There are no back orders,
but one scheduled receipt of 200 units. Now is the time to review. How
much should be reordered?
SOLUTION
IP = OH + SR – BO
= 10 + 200 – 0 = 210
T – IP = 400 – 210 = 190
The decision is to order 190 units

65. Periodic Review System
Selecting the time between reviews, choosing P
and T
 Selecting T when demand is variable and
lead time is constant
 IP covers demand over a protection interval of
P + L
 The average demand during the protection
interval is d(P + L), or
T = d(P + L) + safety stock for protection interval
Safety stock = zσP + L , where σP + L = P L d  

66. Calculating P and T
EXAMPLE 12.7
Again, let us return to the bird feeder example. Recall
that demand for the bird feeder is normally distributed
with a mean of 18 units per week and a standard
deviation in weekly demand of 5 units. The lead time is 2
weeks, and the business operates 52 weeks per year.
The Q system developed in Example 12.4 called for an
EOQ of 75 units and a safety stock of 9 units for a cycle-service
level of 90 percent. What is the equivalent P
system?

67. Calculating P and T
SOLUTION
We first define D and then P. Here, P is the time between reviews,
expressed in weeks because the data are expressed as demand per
week:
D =(18 units/week)(52 weeks/year) = 936 units
EOQ
D
P = (52) =
(52) = 4.2 or 4 weeks
75
936
With d = 18 units per week, an alternative approach is to
calculate P by dividing the EOQ by d to get 75/18 = 4.2 or 4
weeks. Either way, we would review the bird feeder
inventory every 4 weeks.

68. Calculating P and T
We now find the standard deviation of demand over the protection
interval (P + L) = 6:
   5 6 12.25 units  P L P L d  
Before calculating T, we also need a z value. For a 90
percent cycle-service level z = 1.28. The safety stock
becomes
Safety stock = zσP + L 1=. 28(12.25) = 15.68 or 16 units
We now solve for T:
T = Average demand during the protection interval + Safety stock
= d(P + L) + safety stock
= (18 units/week)(6 weeks) + 16 units = 124 units

69. Comparative Advantages
• Primary advantages of P systems
– Convenient
– Only need to know IP when review is made
 Primary advantages of Q systems
 Review frequency may be individualized
 Fixed lot sizes can result in quantity
discounts
 Lower safety stocks

70. Unsolved Problems

71. Unsolved # Qn 3
Yellow pages Inc. buys paper in 1500 pound
rolls for printing. Annual demand is 2500
rolls. The cost per roll is $8,00 and the
annual holding cost is 15% of the cost. Each
order costs $50.
A) How many rolls should Yellow press order
at a time
B) What is the time between orders

72. Unsolved # Qn 7
Sam’s CAT hotel operates 52 weeks per year, 6 days per week, and uses a
continuous review inventory system. It purchases kitty litters for $ 11.70
per bag. The following information is available about these bags:
- Demand = 90 bags / week; Order cost = $54 per order; Annual holding
cost = 27 percent of cost; Desired cycle service level = 80%; Lead T= 3
weeks (18 working days); SD of weekly demand = 15 bags; Current on
hand inventory is 320 bags, with no open orders or backorders.
A) What is the EOQ? what would be the average time between orders (in
weeks)
B) What should R be
C) An inventory withdrawl of 10 bags was just made. Is it time to reorder
D) The store currently uses a lot size of 500 bags (Q=500). What is the
annual holding cost of this policy? Annual ordering cost? Without
calculating the EOQ, how can you conclude from these 2 calculations that
the current lot size is too large
E) What should be the annual cost saved by shifting from the 500 bag lot
size to the EOQ

73. Unsolved # Qn 8
Consider again the Kitty litter ordering policy for Sam’s CAT hotel in previous
problem.
A) Suppose that the weekly demand forecast of 90 bags is incorrect and
actual demand averages only 60 bags per week. How much higher will
total costs be, owing to the distorted EOQ caused by this forecast error
B) Suppose that actual demand is 60 bags but that ordering costs are
cut to only $6 by using internet to automate order placing. However, the
buyer does not tell anyone and the EOQ is not adjusted to reflect this
reduction in S. How much higher will total costs be, compared to what
they could be, if EOQ were adjusted

74. Unsolved # Qn 24
The farmer’s wife is a country store specializing in knickknacks suitable for a
farm house décor. One item experiencing a considerable buying frenzy is a
miniature Holstein cow. Average weekly demand is 30 cows, with a SD of 5
cows. The cost to place a replenishment order is $15 and the holding cost is
$.75/ cow /year. The supplier however is in China. The LT for new orders is 8
weeks, with a SD of 2 weeks. The farmer’s wife which is open only 50 weeks
a year wants to develop a continuous review system for this item with a
cycle service level of 90 percent.
A. Specify the continuous review system for cows. Explain how it would
work in practice
B. What is the total annual cost for the system you develop

75. Solved Problem 2
Nelson’s Hardware Store stocks a 19.2 volt cordless drill that is a popular
seller. Annual demand is 5,000 units, the ordering cost is $15, and the
inventory holding cost is $4/unit/year.
a. What is the economic order quantity?
b. What is the total annual cost for this inventory item?
SOLUTION
a. The order quantity is
2DS
H
EOQ = =
2(5,000)($15)
$4
= 37,500 = 193.65 or 194 drills
b. The total annual cost is
Q
2
D
Q
C = (H) + (S) =
($4) + ($15) = $774.60
194
2
5,000
194

76. Solved Problem 3
A regional distributor purchases discontinued appliances from
various suppliers and then sells them on demand to retailers in the
region. The distributor operates 5 days per week, 52 weeks per year.
Only when it is open for business can orders be received.
Management wants to reevaluate its current inventory policy, which
calls for order quantities of 440 counter-top mixers. The following
data are estimated for the mixer:
Average daily demand (d) = 100 mixers
Standard deviation of daily demand (σd) = 30 mixers
Lead time (L) = 3 days
Holding cost (H) = $9.40/unit/year
Ordering cost (S) = $35/order
Cycle-service level = 92 percent
The distributor uses a continuous review (Q) system

77. Solved Problem 3
a. What order quantity Q, and reorder point, R, should be used?
b. What is the total annual cost of the system?
c. If on-hand inventory is 40 units, one open order for 440 mixers is
pending, and no backorders exist, should a new order be placed?

78. Solved Problem 3
SOLUTION
a. Annual demand is
D = (5 days/week)(52 weeks/year)(100 mixers/day)
= 26,000 mixers/year
The order quantity is
2DS
H
EOQ = =
2(26,000)($35)
$9.40
= 193,167 = 440.02 or 440 mixers

79. Solved Problem 3
The standard deviation of the demand during lead time distribution is
σdLT = σd L = 30 3 = 51.96
A 92 percent cycle-service level corresponds to z = 1.41
Safety stock = zσdLT = 1.41(51.96 mixers) = 73.26 or 73 mixers
Average demand during lead time = dL = 100(3) = 300 mixers
Reorder point (R) = Average demand during lead time + Safety stock
= 300 mixers + 73 mixers = 373 mixers
With a continuous review system, Q = 440 and R = 373

80. Solved Problem 3
b. The total annual cost for the Q systems is
Q
2
D
Q
C = (H) + (S) + (H)(Safety stock)
440
2
26,000
440
C = ($9.40) + ($35) + ($9.40)(73) = $4,822.38
c. Inventory position = On-hand inventory + Scheduled receipts
– Backorders
IP = OH + SR – BO = 40 + 440 – 0 = 480 mixers
Because IP (480) exceeds R (373), do not place a new order

81. Solved Problem 4
Suppose that a periodic review (P) system is used at the distributor in
Solved Problem 3, but otherwise the data are the same.
a. Calculate the P (in workdays, rounded to the nearest day)
that gives approximately the same number of orders per
year as the EOQ.
b. What is the target inventory level, T? Compare the P system
to the Q system in Solved Problem 3.
c. What is the total annual cost of the P system?
d. It is time to review the item. On-hand inventory is 40 mixers;
receipt of 440 mixers is scheduled, and no backorders exist.
How much should be reordered?

82. Solved Problem 4
SOLUTION
a. The time between orders is
EOQ
D
P = (260 days/year) =
(260) = 4.4 or 4 days
440
26,000
b. Figure 12.12 shows that T = 812 and safety stock
= (1.41)(79.37) = 111.91 or about 112 mixers. The
corresponding Q system for the counter-top mixer requires
less safety stock.

83. Solved Problem 4
c. The total annual cost of the P system is
dP
2
D
dP
C = (H) + (S) + (H)(Safety stock)
100(4)
C = ($9.40) + ($35) + ($9.40)(1.41)(79.37)
2
26,000
100(4)
= $5,207.80
d. Inventory position is the amount on hand plus scheduled
receipts minus backorders, or
IP = OH + SR – BO = 40 + 440 – 0 = 480 mixers
The order quantity is the target inventory level minus the
inventory position, or
Q = T – IP =
812 mixers – 480 mixers = 332 mixers
An order for 332 mixers should be placed.

84. Solved Problem 5
Grey Wolf Lodge is a popular 500-room hotel in the North Woods.
Managers need to keep close tabs on all room service items, including a
special pine-scented bar soap. The daily demand for the soap is 275 bars,
with a standard deviation of 30 bars. Ordering cost is $10 and the
inventory holding cost is $0.30/bar/year. The lead time from the supplier is
5 days, with a standard deviation of 1 day. The lodge is open 365 days a
year.
a. What is the economic order quantity for the bar of soap?
b. What should the reorder point be for the bar of soap if
management wants to have a 99 percent cycle-service level?
c. What is the total annual cost for the bar of soap, assuming a
Q system will be used?

85. Solved Problem 5
SOLUTION
a. We have D = (275)(365) = 100,375 bars of soap; S = $10; and H = $0.30.
The EOQ for the bar of soap is
2DS
H
EOQ = =
2(100,375)($10)
$0.30
= 6,691,666.7 = 2,586.83 or 2,587 bars

86. Solved Problem 5
b. We have d = 275 bars/day, σd = 30 bars, L = 5 days,
and σLT = 1 day.
σdLT = Lσd
2 + d2σLT
2 = (5)(30)2 + (275)2(1)2 = 283.06 bars
Consult the body of the Normal Distribution appendix for
0.9900. The closest value is 0.9901, which corresponds to
a z value of 2.33. We calculate the safety stock and reorder
point as follows:
Safety stock = zσdLT = (2.33)(283.06) = 659.53 or 660 bars
Reorder point = dL + Safety stock = (275)(5) + 660 = 2,035 bars

87. Solved Problem 5
c. The total annual cost for the Q system is
Q
2
D
Q
C = (H) + (S) + (H)(Safety stock)
2,587
C = ($0.30) + ($10) + ($0.30)(660) = $974.05
2
100,375
2,587

88. Solved Problem 6
Zeke’s Hardware Store sells furnace filters. The cost to place an order to the
distributor is $25 and the annual cost to hold a filter in stock is $2. The
average demand per week for the filters is 32 units, and the store operates
50 weeks per year. The weekly demand for filters has the probability
distribution shown on the left below.
The delivery lead time from the distributor is uncertain and has the
probability distribution shown on the right below.
Suppose Zeke wants to use a P system with P = 6 weeks and a cycle-service
level of 90 percent. What is the appropriate value for T and the associated
annual cost of the system?

89. Solved Problem 6
Demand Probability
24 0.15
28 0.20
32 0.30
36 0.20
40 0.15
Lead Time (wks) Probability
1 0.05
2 0.25
3 0.40
4 0.25
5 0.05

90. Solved Problem 6
SOLUTION
Figure 12.13 contains output from the Demand During the Protection
Interval Simulator from OM Explorer.

91. Solved Problem 6
Given the desired cycle-service level of 90 percent, the appropriate T
value is 322 units. The simulation estimated the average demand
during the protection interval to be 289 units, consequently the safety
stock is 322 – 289 = 33 units.
The annual cost of this P system is
6(32)
2
50(32)
6(32)
C = ($2) + ($25) + (33)($2)
= $192.00 + $208.33 + $66.00 = $466.33

92. Solved Problem 7
Consider Zeke’s inventory in Solved Problem 6. Suppose that he
wants to use a continuous review (Q) system for the filters, with an
order quantity of 200 and a reorder point of 140. Initial inventory is
170 units. If the stockout cost is $5 per unit, and all of the other
data in Solved Problem 6 are the same, what is the expected cost
per week of using the Q system?
SOLUTION
Figure 12.14 shows output from the Q System Simulator in
OM Explorer. Only weeks 1 through 13 and weeks 41
through 50 are shown in the figure. The average total cost
per week is $305.62. Notice that no stockouts occurred in
this simulation. These results are dependent on Zeke’s
choices for the reorder point and lot size. It is possible that
stockouts would occur if the simulation were run for more
than 50 weeks.