Inventory Management
(Session 3,4)
1
Outline
Elements of Inventory Management
Inventory Control Systems
Economic Order Quantity Models
The Basic EOQ Model
The EOQ Model with Non-Instantaneous Receipt
The EOQ Model with Shortages
Quantity Discounts
Reorder Point
Determining Safety Stocks Using Service Levels
Order Quantity for a Periodic Inventory System
2
Elements of Inventory Management
Role of Inventory
„
Inventory is a stock of items kept on hand used to meet
customer demand.
„
A level of inventory is maintained that will meet anticipated
demand.
„
If demand not known with certainty, safety (buffer) stocks
are kept on hand.
„
Additional stocks are sometimes built up to meet seasonal
or cyclical demand.
3
Elements of Inventory Management
Role of Inventory
„
Large amounts of inventory sometimes purchased to take
advantage of discounts.
„
In-process inventories maintained to provide independence
between operations.
„
Raw materials inventory kept to avoid delays in case of
supplier problems.
„
Stock of finished parts kept to meet customer demand in
event of work stoppage.
4
Elements of Inventory Management
Demand
„
Inventory exists to meet the demand of customers.
„
Customers can be external (purchasers of products) or
internal (workers using material).
„
Management needs accurate forecast of demand.
„
Items that are used internally to produce a final product are
referred to as dependent demand items.
„
Items that are final products demanded by an external
customer are independent demand items.
5
Elements of Inventory Management
Inventory Costs
„
Carrying costs - Costs of holding items in storage.
Vary with level of inventory and sometimes with length
of time held.
Include facility operating costs, record keeping,
interest, etc.
Assigned on a per unit basis per time period, or as
percentage of average inventory value (usually
estimated as 10% to 40%).
6
Elements of Inventory Management
Inventory Costs
„
Ordering costs - costs of replenishing stock of inventory.
Expressed as dollar amount per order, independent of
order size.
Vary with the number of orders made.
Include purchase orders, shipping, handling,
inspection, etc.
7
Elements of Inventory Management
Inventory Costs
„
Shortage, or stockout costs - Costs associated with
insufficient inventory.
Result in permanent loss of sales and profits for items
not on hand.
Sometimes penalties involved; if customer is internal,
work delays could result.
8
Inventory Control Systems
An inventory control system controls the level of inventory
by determining how much (replenishment level) and when
to order.
Two basic types of systems -continuous (fixed-order
quantity) and periodic (fixed-time).
In a continuous system, an order is placed for the same
constant amount when inventory decreases to a specified
level.
In a periodic system, an order is placed for a variable
amount after a specified period of time.
9
Inventory Control Systems
Continuous Inventory System
A continual record of inventory level is maintained.
Whenever inventory decreases to a predetermined level,
the reorder point, an order is placed for a fixed amount to
replenish the stock.
The fixed amount is termed the economic order quantity,
whose magnitude is set at a level that minimizes the total
inventory carrying, ordering, and shortage costs.
Because of continual monitoring, management is always
aware of status of inventory level and critical parts, but
system is relatively expensive to maintain.
10
Inventory Control Systems
Periodic Inventory System
Inventory on hand is counted at specific time intervals and
an order placed that brings inventory up to a specified level.
Inventory not monitored between counts and system is
therefore less costly to track and keep account of.
Results in less direct control by management and thus
generally higher levels of inventory to guard against
stockouts.
System requires a new order quantity each time an order is
placed.
Used in smaller retail stores, drugstores, grocery stores and
offices.
11
Economic Order Quantity Models
Economic order quantity, or economic lot size, is the
quantity ordered when inventory decreases to the reorder
point.
Amount is determined using the economic order quantity
(EOQ) model.
Purpose of the EOQ model is to determine the optimal
order size that will minimize total inventory costs.
Three model versions to be discussed:
Basic EOQ model
EOQ model without instantaneous receipt
EOQ model with shortages
12
Economic Order Quantity Models
Basic EOQ Model
A formula for determining the optimal order size that
minimizes the sum of carrying costs and ordering costs.
Simplifying assumptions and restrictions:
Demand is known with certainty and is relatively
constant over time.
No shortages are allowed.
Lead time for the receipt of orders is constant.
The order quantity is received all at once and
instantaneously.
13
Economic Order Quantity Models
Basic EOQ Model
Figure The Inventory Order Cycle
14
Basic EOQ Model
Carrying Cost
Carrying cost usually expressed on a per unit basis of time,
traditionally one year.
Annual carrying cost equals carrying cost per unit per year
times average inventory level:
Carrying cost per unit per year = Cc
Average inventory = Q/2
Annual carrying cost = CcQ/2.
15
Basic EOQ Model
Carrying Cost
Figure: Average Inventory
16
Basic EOQ Model
Ordering Cost
Total annual ordering cost equals cost per order (Co) times
number of orders per year.
Number of orders per year, with known and constant
demand, D, is D/Q, where Q is the order size:
Annual ordering cost = CoD/Q
Only variable is Q, Co and D are constant parameters.
Relative magnitude of the ordering cost is dependent on
order size.
17
Basic EOQ Model
Total Inventory Cost
Total annual inventory cost is sum of ordering and carrying
cost:
TC = C D + C Q
c2
oQ
TC = Total Inventory Cost
Co = Ordering Cost
Cc = Inventory Carrying Cost
Q = Order Quantity
D = Demand
18
Basic EOQ Model
Total Inventory Cost
Figure: The EOQ Cost Model
19
Basic EOQ Model
EOQ and Min. Total Cost
EOQ occurs where total cost curve is at minimum value
and carrying cost equals ordering cost:
Q
C D
= o + C opt
TC
c 2
min Qopt
2C D
o
Q =
opt
C
c
The EOQ model is robust because Q is a square root and
errors in the estimation of D, Cc and Co are dampened.
20
Example 1
I-75 Carpet Discount Store, Super Shag carpet sales.
Given following data, determine number of orders to be
made annually and time between orders given store is open
every day except Sunday, Thanksgiving Day, and
Christmas Day.
Model parameters :
C = $0.75, C = $150, D = 10,000yd
c
o
Optimal order size :
2C D
o = 2(150)(10,000) = 2,000 yd
Q =
opt
C
(0.75)
c
21
Total annual inventory cost :
Q
= C D + C opt = (150)10,000 + (0.75) (2,000) = $1,500
TC
c 2
oQ
2,000
2
min
opt
Number of orders per year :
D = 10,000 = 5
2,000
Q
opt
Order cycle time = 311 days = 311 = 62.2 store days
5
D/Q
opt
22
Basic EOQ Model
EOQ Analysis over time
For any time period unit of analysis, EOQ is the same.
Shag Carpet example on monthly basis:
Model parameters :
C = $0.0625 per yd per month
c
C = $150 per order
o
D = 833.3 yd per month
Optimal order size :
2C D
o = 2(150)(833.3) = 2,000 yd
Q =
opt
C
(0.0625)
c
23
Basic EOQ Model
EOQ Analysis over time
Total monthly inventory cost :
Q
= C D + C opt = (150) (833.3) + (0.0625) (2,000)
TC
c 2
oQ
2,000
2
min
opt
= $125 per month
Total annual inventory cost = ($125)(12) = $1,500
24
EOQ Model
Non-Instantaneous Receipt Description
In the non-instantaneous receipt model the assumption that
orders are received all at once is relaxed. (Also known as
gradual usage or production lot size model.)
The order quantity is received gradually over time and
inventory is drawn on at the same time it is being
replenished.
25
EOQ Model
Non-Instantaneous Receipt Description
Figure: The EOQ Model with Non-Instantaneous Order Receipt
26
EOQ Model
Non-Instantaneous Receipt Description
p = daily rate at which the order is received over time
d = daily rate at which inventory is demanded
Model
⎛
⎞
d
⎜
Maximum inventory level = Q⎜1 − ⎟⎟
Formulation
p⎠
⎝
⎛
⎞
Average inventory level = Q ⎜⎜1 − d ⎟⎟
2 ⎝ p⎠
⎛
⎞
Q
d
⎜
Total carrying cost = C ⎜1− ⎟⎟
c 2 ⎝ p⎠
⎛
⎞
Total annual inventory cost = C D + C Q ⎜⎜1 − d ⎟⎟
o Q c 2 ⎝ p⎠
27
EOQ Model
Non-Instantaneous Receipt Description
⎛
⎞
C Q ⎜⎜1 − d ⎟⎟ = C D at lowest point of total cost curve
c 2 ⎝ p⎠ o Q
2C D
o
Optimal order size : Q =
opt
C (1 − d / p)
c
Equation shown above is the optimal order
size under non-Instantaneous receipt
28
Example
Super Shag carpet manufacturing facility:
C = $150
o
C = $0.75 per unit
c
D = 10,000 yd per year = 10,000/311 = 32.2 yd per day
p = 150 yd per day
2C D
o
Optimal order size : Q =
= 2(150)(10,000)
opt
⎛
⎞
⎛
⎞
32
.
2
d
⎜
⎟
⎜
⎟⎟
0
.
75
1
−
C ⎜1 − ⎟
⎜
c⎝ p ⎠
150 ⎠
⎝
= 2,256.8 yd
29
Example
⎛
⎞
Total minimum annual inventory cost = C D + C Q ⎜⎜1 − d ⎟⎟
c 2 ⎝ p⎠
oQ
⎛
⎞
= (150 ) (10,000 ) + (.075 ) (2,256 .8) ⎜⎜1 − 32 .2 ⎟⎟ = $1,329
150 ⎠
2
(2,256 .8)
⎝
Production run length = Q = 2,256 .8 = 15 .05 days
p
150
Number of orders per year (productio n runs) = D
Q
= 10,000 = 4.43 runs
2,256 .8
⎛
⎞
⎛
⎞
Maximum inventory level = Q ⎜⎜1 − d ⎟⎟ = 2,256 .8⎜⎜1 − 32 .2 ⎟⎟ = 1,772 yd
p⎠
150 ⎠
⎝
⎝
30
EOQ Model
With Shortages
In the EOQ model with shortages, the assumption that
shortages cannot exist is relaxed.
Assumed that unmet demand can be backordered with
all demand eventually satisfied.
31
EOQ Model
With Shortages
Figure: The EOQ Model with Shortages
32
EOQ Model
With Shortages
2
S
Total shortage costs = C
s 2Q
2
Q
S
(
)
−
Total carrying costs = C
c 2Q
Total ordering cost = C D
oQ
2
2
Q
S
(
)
−
S
Total inventory cost = C
+C
+C D
s 2Q
c 2Q
oQ
2Co D ⎛⎜ Cs + Cc ⎞⎟
Optimal order quantity = Q
=
opt
Cc ⎜ Cs ⎟
⎝
Shortage level = S
opt
⎛
C
⎜
c
=Q ⎜
opt ⎜ C + C
s
⎝ c
⎠
⎞
⎟
⎟
⎟
⎠
33
EOQ Model
With Shortages
Figure: Cost Model with Shortages
34
Example
I-75 Carpet Discount Store allows shortages; shortage cost
Cs, is $2/yard per year.
C = $150
o
C = $0.75 per yd
c
C = $2 per yd
s
D = 10,000 yd
Optimal order quantity :
2C D ⎛⎜ C + C ⎞⎟
s c = 2(150)(10,000) ⎛⎜ 2 + 0.75 ⎞⎟ = 2,345.2 yd
o
Q =
⎜
⎟⎟
opt
0.75
2 ⎟⎠
C ⎜⎜ C
⎝
s ⎠
c ⎝
35
Example
Shortage level:
S
opt
⎞
⎛ C
⎜
c ⎟ = 2,345.2⎛⎜ 0.75 ⎞⎟ = 639.6 yd
=Q ⎜
⎜ 2 + 0.75 ⎟
opt ⎜ C + C ⎟⎟
⎝
⎠
s⎠
⎝ c
Total inventory cost :
2
2
Q
S
(
)
−
S
TC = C
+C
+C D
s 2Q c 2Q
oQ
2 (0.75)(1,705.6)2 (150)(10,000)
(
2
)(
639
.
6
)
=
+
+
2,345.2
2(2,345.2)
2(2,345.2)
= $174.44 + 465.16 + 639.60 = $1,279.20
36
Example
Number of orders = D = 10,000 = 4.26 orders per year
Q 2,345.2
Maximum inventory level = Q − S = 2,345.2 − 639.6 =1,705.6 yd
Time between orders = t =
days per year = 311 = 73.0 days
number of orders 4.26
Time during which inventory is on hand
= t1 = Q − S = 2,345.2 -639.6 = 0.171 or 53.2 days
D
10,000
Time during which there is a shortage
= t 2 = S = 639.6 = 0.064 year or 19.9 days
D 10,000
37
Quantity Discounts
Price discounts are often offered if a predetermined number of
units is ordered or when ordering materials in high volume.
Basic EOQ model used with purchase price added:
TC = C D + C Q + PD
oQ c 2
where: P = per unit price of the item
D = annual demand
Quantity discounts are evaluated under two different
scenarios:
With constant carrying costs
With carrying costs as a percentage of purchase price
38
Quantity Discounts with Constant Cc
Optimal order size is the same regardless of the discount
price.
The total cost with the optimal order size must be compared
with any lower total cost with a discount price to determine
which is the lesser.
39
Example
University bookstore: For following discount schedule
offered by Comptek, should bookstore buy at the discount
terms or order the basic EOQ order size?
Quantity
1 – 49
50-89
90 +
Price
$ 1400
1100
900
Determine optimal order size and total cost:
C = $2,500
o
C = $190 per unit
c
2C D
o = 2(2,500)(200) = 72.5
Q
=
opt
C
190
c
D = 200
40
Example
Compute total cost at eligible discount price ($1,100):
Q
C D
TC
= o + C opt + PD
c 2
min Qopt
= (2,500 )( 200 ) + (190 ) (72 .5) + (1,100 )( 200 ) = $233,784
2
(72 .5)
Compare with total cost of with order size of 90 and price of
$900:
C D
TC = o + C Q + PD
c2
Q
= (2,500)(200) + (190)(90) + (900)(200) = $194,105
2
(90)
Because $194,105 < $233,784, maximum discount price
should be taken and 90 units ordered.
41
Quantity Discounts with Cc
% of Price
University Bookstore example, but a different optimal order
size for each price discount.
Optimal order size and total cost determined using basic
EOQ model with no quantity discount.
This cost then compared with various discount quantity
order sizes to determine minimum cost order.
This must be compared with EOQ-determined order size for
specific discount price.
Data:
Co = $2,500
D = 200 computers per year
42
Quantity Discounts with Cc
% of Price
Quantity
Price
Carrying Cost
0 - 49
50 - 89
90 +
$1,400
1,100
900
1,400(.15) = $210
1,100(.15) = 165
900(.15) = 135
Compute optimum order size for purchase price without
discount and Cc = $210:
2C D
o = 2(2,500)(200) = 69
Q =
opt
C
210
c
Compute new order size:
Q = 2(2,500)(200) = 77.8
opt
165
43
Quantity Discounts with Cc
% of Price
Compute minimum total cost:
C D
TC = o + C Q + PD = (2,500)(200) + 165 (77.8) + (1,100)(200)
c2
77.8
2
Q
= $232,845
Compare with cost, discount price of $900, order quantity of 90:
TC = (2,500)(200) + (135)(90) + (900)(200) = $191,630
90
2
Optimal order size computed as follows:
Q = 2(2,500)(200) = 86.1
opt
135
Since this order size is less than 90 units , it is not feasible,thus
optimal order size is 90 units.
44
Reorder Point
The reorder point is the inventory level at which a new
order is placed.
Order must be made while there is enough stock in place to
cover demand during lead time.
Formulation:
R = dL
where d = demand rate per time period
L = lead time
For Carpet Discount store problem:
R = dL = (10,000/311)(10) = 321.54
45
Reorder Point
Figure: Reorder Point and Lead Time
46
Reorder Point
Inventory level might be depleted at slower or faster rate
during lead time.
When demand is uncertain, safety stock is added as a
hedge against stockout.
Figure: Inventory Model with Uncertain Demand
47
Reorder Point (4 Reorder
of 4)
Point
Figure: Inventory model with safety stock
48
Determining Safety Stock by Service
Level
Service level is probability that amount of inventory on hand
is sufficient to meet demand during lead time (probability
stockout will not occur).
The higher the probability inventory will be on hand, the
more likely customer demand will be met.
Service level of 90% means there is a .90 probability that
demand will be met during lead time and .10 probability of a
stockout.
49
Reorder Point with Variable Demand
R = d L + Zσ d L
where:
R = reorder point
d = average daily demand
L = lead time
σ d = the standard deviation of daily demand
Z = number of standard deviations corresponding
to service level probability
Zσ d L = safety stock
50
Reorder Point with Variable Demand
Figure: Reorder Point for a Service Level
51
Example
I-75 Carpet Discount Store Super Shag carpet.
For following data, determine reorder point and safety stock
for service level of 95%.
d = 30 yd per day
L = 10 days
σ = 5 yd per day
d
For 95% service level, Z = 1.65 (Table A -1, appendix A)
R = d L + Zσ
= 326.1 yd
d
L = 30(10) + (1.65)(5)( 10 ) = 300 + 26.1
Safety stock is second term in reorder point formula : 26.1.
52
Reorder Point with Variable Lead Time
For constant demand and variable lead time:
R = d L + Zdσ L
where:
d = constant daily demand
L = average lead time
σ L = standard deviation of lead time
dσ L = standard deviation of demand during lead time
Zdσ L = safety stock
53
Example
Carpet Discount Store:
d = 30 yd per day
L = 10 days
σ L = 3 days
Z = 1.65 for a 95% service level
R = d L + Zdσ L = (30)(10) + (1.65)(30)(3) = 300 +148.5 = 448.5 yd
54
Reorder Point with Variable Demand
and Lead Time
When both demand and lead time are variable:
2
2 2
R = d L + Z (σ d ) L + (σ L ) d
where:
d = average daily demand
L = average lead time
2
2 2
(σ d ) L + (σ L) d = standard deviation of demand during lead time
2
2 2
Z (σ d ) L + (σ L ) d = safety stock
55
Example
Carpet Discount Store:
d = 30 yd per day
σ d = 5 yd per day
L = 10 days
σ L = 3 days
Z = 1.65 for 95% service level
2
2 2
R = d L + Z (σ d ) L + (σ L) d
= (30)(10) + (1.65) (5)(5)(10) + (3)(3)(30)(30)
= 300 + 150.8
= 450.8 yds
56
Order Quantity for a Periodic Inventory
System
A periodic, or fixed-time period inventory system is one in
which time between orders is constant and the order size
varies.
Vendors make periodic visits, and stock of inventory is
counted.
An order is placed, if necessary, to bring inventory level
back up to some desired level.
Inventory not monitored between visits.
At times, inventory can be exhausted prior to the visit,
resulting in a stockout.
Larger safety stocks are generally required for the periodic
inventory system.
57
Order Quantity for Variable Demand
For normally distributed variable daily demand:
Q = d (tb + L) + Zσ d tb + L − I
where:
d = average demand rate
tb = the fixed time between orders
L = lead time
σ d = standard deviation of demand
Zσ d tb + L = safety stock
I = inventory in stock
58
Order Quantity for Variable Demand
Corner Drug Store with periodic inventory system.
Order size to maintain 95% service level:
d = 6 bottles per day
σ d = 1.2 bottles
tb = 60 days
L = 5 days
I = 8 bottles
Z = 1.65 for 95% service level
Q = d (tb + L) + Zσ d tb + L − I
= (6)(60 + 5) + (1.65)(1.2) 60 + 5 −8
= 398 bottles
59
Example 1 (Electronic Village Store)
Electronic Village store stocks and sells a particular
brand of PCs. It costs the store $450 each time it
places an order with the manufacturer for the PCs.
The annual cost of carrying the PCs in inventory is $
170. the store manager estimates the annual
demand for the PCs will be 1200 units.
a) Determine the optimal order quantity and the total
min. inventory cost.
b) Assume that shortages are allowed and shortage
cost is $600 per unit per year. Compute the optimal
order quantity and the total minimum inventory cost.
60
Example 1
For data below determine:
Optimal order quantity and total minimum inventory cost.
Assume shortage cost of $600 per unit per year,
compute optimal order quantity and minimum inventory
cost.
Step 1 (part a): Determine the Optimal Order Quantity.
D = 1,200 personal computers
C = $170
c
C = $450
o
2C D
o = 2(450)(1,200) = 79.7 personal computers
Q=
C
170
c
61
Example 1
⎛
⎞
⎛
⎞
Total cost = C Q + C D = 170⎜⎜ 79.7 ⎟⎟ = 450⎜⎜1,200 ⎟⎟
c2
oQ
⎝ 2 ⎠
⎝ 79.7 ⎠
= $13,549.91
Step 2 (part b): Compute the EOQ with Shortages.
Cs = $600
2C D ⎛⎜ C + C ⎞⎟
o
s c = 2(450)(1200) ⎛⎜ 600 + 170 ⎞⎟
Q=
⎜
⎟⎟
C ⎜⎜ C
170
600 ⎟⎠
⎝
c ⎝
s ⎠
= 90.3 personal computers
62
Example 1
S
⎛ C
⎞
⎛
170 ⎞⎟ = 19.9 personal computers
⎜
⎟
c
⎜
= Q⎜
=
90
.
3
⎜170 + 600 ⎟
⎜ C + C ⎟⎟
⎝
⎠
s⎠
⎝ c
2 C D
C S2
−
(
)
Q
S
+ o
Total cost = s + C
c 2Q
2Q
Q
2
2
⎛
⎞
−
(
600
)(
19
.
9
)
(
90
.
3
19
.
9
)
=
+170
+ 450⎜⎜1,200 ⎟⎟
2(90.3)
2(90.3)
⎝ 90.3 ⎠
= $11,960.98
63
Example 2 (Computer Product Store)
A computer products store stocks color graphics
monitors, and the daily demand is normally
distributed with a mean of 1.6 monitors and a
standard deviation of 0.4 monitors. The lead time to
receive an order from the manufacturer is 15 days.
Determine the reorder point that will achieve a 98%
service level.
64
Example 2
Sells monitors with daily demand normally distributed with
a mean of 1.6 monitors and standard deviation of 0.4
monitors. Lead time for delivery from supplier is 15 days.
Determine the reorder point to achieve a 98% service level.
Step 1: Identify parameters.
d = 1.6 monitors per day
L =15 days
σ d = 0.4 monitors per day
Z = 2.05 (for a 98% service level)
65
Example 2
Step 2: Solve for R.
R = d L + Zσ d L = (1.6)(15) + (2.05)(.04) 15
= 24 + 3.18 = 27.18 monitors
66
ABC Classification System
„
Items kept in inventory are not of equal
importance in terms of:
„
dollars invested
„
profit potential
„
sales or usage volume
„
stock-out penalties
60
% of
$ Value 30
0
% of
Use
30
A
B
C
60
So, identify inventory items based on percentage of total
dollar value, where “A” items are roughly top 15 %, “B”
items as next 35 %, and the lower 65% are the “C” items
67
Inventory Accuracy and Cycle Counting
Defined
„
Inventory accuracy refers to how well the
inventory records agree with physical
count
„
Cycle Counting is a physical inventorytaking technique in which inventory is
counted on a frequent basis rather than
once or twice a year
68
End of Session
69