Operations
Management
Inventory Management
Chapter 12 - Part 2
12-1
Outline
 Functions of Inventory.
 ABC Analysis.
 Inventory Costs.
 Inventory Models for Independent Demand.

Economic Order Quantity (EOQ) Model.

Production Order Quantity (POQ) Model.

Quantity Discount Models.
 Probabilistic Models for Varying Demand.
 Fixed Period Systems.
12-2
Production Order Quantity Model
 Material is not received instantaneously.

For example, it is produced in-house.
 Other EOQ assumptions apply.
 Model provides production lot size (like EOQ
amount) for one product.
 Similar to EOQ with setup cost rather than
order cost.
12-3
Production Order Quantity Model
 Consider one product at a time.
 Produce Q units in a production run; then
switch and produce other products.
 Later produce Q more units in 2nd production
run (Q units of product of interest).
 Later produce Q more units in 3rd production
run, etc.
12-4
POQ Model Inventory Levels
Inventory Level
Production portion of cycle
Demand portion of cycle with no
production (of this product)
Time
Production
Begins
Production
Run Ends
12-5
POQ Model Inventory Levels
Inventory Level
Production rate = p = 20/day
Demand rate = d = 7/day
Slope = p-d = 13/day
Slope = -d = -7/day
Time
Production
Begins
Production
Run Ends
Note: Not all of production goes into
inventory
12-6
POQ Model Inventory Levels
Inventory Level
Production rate = p = 20/day
Demand rate = d = 7/day
Slope = p-d = 13/day
Inventory increases by 13 each day
while producing
Slope = -d = -7/day
Inventory decreases by 7/day
after producing
Time
Production
Begins
Production
Run Ends
Note: 1-(d/p) = fraction of production
that goes into inventory
12-7
POQ Model Equations
D = Annual demand (relatively constant)
S = Setup cost per setup
H = Holding (carrying) cost per unit per year
d = Demand rate (units per day, units per week, etc.)
p = Production rate (units per day, units per week, etc.)
Given
Determine: Q = Production run size (number of items per production run)
Number of Production Runs per year =
Setup Cost per year =
D
S
Q
D
Q
Holding Cost per year = (average inventory level)  H
12-8
POQ Model Inventory Levels
Inventory Level
Maximum Inventory
= Q(1-(d/p))
Production
Portion of Cycle
Time
Demand portion of cycle
with no supply
12-9
POQ Model Equations
D = Annual demand (relatively constant)
S = Setup cost per setup
H = Holding (carrying) cost per unit per year
d = Demand rate (units per day, units per week, etc.)
p = Production rate (units per day, units per week, etc.)
Given
Determine: Q = Production run size (number of items per production run)
Number of Production Runs per year =
Setup Cost per year =
D
S
Q
D
Q
Q
Holding Cost per year = (ave. inventory level)  H = H [1-(d/p)]
2
12-10
POQ Model Equations
D = Annual demand
S = Setup cost per setup
H = Holding (carrying) cost per unit per year
d = Demand rate
p = Production rate
Optimal Production Run Size = Q* =
Maximum inventory level = Q [1- (d/p)]
Total Cost =
D
Q
S+
H [1-(d/p)]
Q
2
12-11
2 ×D ×S
=
H[1-(d/p)]
Given
2DS
H
p
p-d
Run Length & Cycle Length
Inventory Level
Production Run length (time) = Q /p
Time
Cycle length (time) = Q /d
12-12
POQ Example
Demand = 1000/year (of product A)
Setup cost = $100/setup
Holding cost = $20 per year per item
Production rate = 10/day
365 working days per year
Qp* =
Demand rate = d = 1000/365
= 2.74/day
2 ×1000×100 = 117.36 units/run
20×[1-(2.74/10)]
Maximum inventory level = 117.36 [1- (2.74/10)] = 85.2 units
Total Cost =
1000
117.36
= 852.08
100 +
+
117.36
2
20 [1-(2.74/10)]
852.03
12-13
= $1704.11/year
POQ Example
Demand = 1000 units/year
Production rate = 10 units/day
Qp* = 117.36 units per run
42.8
Demand rate = d = 1000/365
= 2.74/day
11.74
Production run length = 117.36/(10/day) = 11.74 days
Cycle length = 117.36/(2.74/day) = 42.8 days
Number of production runs per year = 1000/117.36 = 8.52
12-14
Robustness of POQ
 POQ is robust (like EOQ):
Can adjust production run size.
 Useful even when parameters are uncertain.
 A large (20%) change in parameters or operations
will cause a small (~2%) change in total costs.

 Production run size (Q) and run length (Q/p) can
be adjusted to fit normal business cycles.
12-15
POQ Robustness Example
 Set production run length to 14 days (2 weeks)
rather than 11.74 days (as was optimal).
 Q/p = 14 days means that: Q = 10x14 = 140 units

Q = 140 is 19% over optimal value of 117.4 units.
 Cycle length = Q/d = 140/2.74 = 51.1 days.
 Total cost = $1730.68

Only 1.6% over minimum cost with optimal Q!
12-16
POQ & Multiple Products
 POQ computes a production run size for a single
product.
 For multiple products made on the same
equipment:
1.
Compute POQ, run time, and cycle time for each product.
2.
Find a common cycle time for all products.
3.
Recalculate run time and cycle time, so the common cycle
time is a multiple of each product’s cycle time.
4.
Fit production runs into largest cycle time.
12-17
Multiple Products Example
 Example: Company makes 3 products: A, B, C
A: Optimal run time = 3 days; Optimal cycle time = 10 days
B: Optimal run time = 6 days; Optimal cycle time = 18 days
C: Optimal run time = 10 days; Optimal cycle time = 33 days
A
A
3
7
3
A
7
3
7
B
B
6
6
12
C
23
10
12-18
12
Multiple Products Example
 Optimal run time and cycle time:
A: Run time = 3 days; Cycle time = 10 days (1 run/10 days)
B: Run time = 6 days; Cycle time = 18 days (1 run/18 days)
C: Run time = 10 days; Cycle time = 33 days (1 run/33 days)
Use 30 days as a common cycle; adjust run & cycle times:
A: Run time = 3 days; Cycle time = 10 days (3 runs/30 days)
B: Run time = 5 days; Cycle time = 15 days (2 runs/30 days)
C: Run time = 9 days; Cycle time = 30 days (1 run/30 days)
A
B
C
A
B
A
3
5
9
3
5
3
2 days idle time
12-19
Quantity Discount Model
 Variation of EOQ (not POQ).
 Allows quantity discounts.

Reduced price for purchasing larger quantities.

Other EOQ assumptions apply.
 Trade-off lower price to purchase item &
increased holding cost from more items.
 Total cost must include annual purchase cost.
Total Cost = Order cost + Holding cost + Purchase cost
12-20
Quantity Discount Model - Holding
Cost
 Holding cost:

Depends on price.

Usually expressed as a % of price per unit time.

20% of price per year, 2% of price per month, etc.
 I = Holding cost percent of price per year
 P = Price per unit
 H = Holding cost = IP
12-21
Quantity Discount Equations
D = Annual demand
S = Order cost per order
H = Holding (carrying) cost = IP
I = Inventory holding cost % per year
P = Price per unit
Order Quantity = Q* =
Total Cost ($/yr) =
2 ×D ×S
IP
D
Q
S+
IP + PD
Q
2
12-22
Annual purchase cost
Quantity Discount Model
Q
<500
500-1000
 1000
D = 1000/year
S = $100/order
I = 20% per year
P
$100
$ 95
$ 90
IP
$20
$19
$18
To solve:
1. Find EOQ amount for each discount level.
2. If EOQ is not in range for discount level, adjust to the nearest
end of range.
3. Calculate total cost for each discount level.
4. Select lowest cost and corresponding Q.
12-23
Quantity Discount Example
Q
<500
500-1000
 1000
D = 1000/year
S = $100/order
I = 20% per year
P
$100
$ 95
$ 90
IP
$20
$19
$18
1. P = $100 IP = $20
EOQ = 100
in range!
Total Cost = 1,000 + 1,000 + 100,000 = $102,000/year
2. P = $95 IP = $19
EOQ = 102.6
not in range (500-1000)!
Adjust to Q = 500
Total Cost = 200 + 4,750 + 95,000 = $99,950/year
12-24
Quantity Discount Example - cont.
Q
<500
500-1000
 1000
D = 1000/year
S = $100/order
I = 20% per year
P
$100
$ 95
$ 90
3. P = $90 IP = $18
EOQ = 105.4
not in range (>1000)!
Adjust to Q = 1000
Total Cost = 100 + 9,000 + 90,000 = $99,100/year
Q
Total costs
<500
$102,100
500-1000 $ 99,950
 1000
$ 99,100
Lowest cost, so order 1000
12-25
IP
$20
$19
$18
Stockouts
 In basic EOQ model, demand and lead time are
known and constant, so there should never be a
stockout.
 If demand or lead time vary, then may have a
stockout:

Due to larger than expected demand.

Due to longer than expected lead time.
12-26
Probabilistic Models
Inventory Level
Average demand
Reorder
Point
(ROP)
Place
order
Lead Time
12-27
Receive
order
Time
Probabilistic Models - Stockout
Inventory Level
If demand is greater than
average - then stockout
Reorder
Point
(ROP)
Place
order
Lead Time
12-28
Receive
order
Time
Safety Stock to Reduce Stockouts
Inventory Level
Safety stock
New ROP
Old ROP
Place
order
Lead Time
12-29
Receive
order
Time
Safety Stock & Service Level
 Safety stock is inventory held to protect against
stockout.
 Service level = 1 - Probability of stockout

Service level of 95% means 5% chance of stockout.

Higher service level means more safety stock.
 More safety stock means higher ROP.

ROP = Expected demand during lead time + Safety stock
12-30
Probabilistic Models
 Demand follows normal distribution.

d = Average demand rate per day.

 = Standard deviation of demand.
 ROP = d L + safety stock
 safety stock = ss = Z 
 Z is from Standard Normal Table in Appendix I.
12-31
EOQ-based Models
 Order same amount every time = Q.
 Time between orders varies.
ROP
Time between
1st & 2nd order
Time between 2nd
& 3rd order
12-32
Time
Fixed Period Model
 Order at fixed intervals (e.g., every 2 weeks).
 Order different amounts each time, based on
amount on hand.

If large amount on hand, then order small amount.

If small amount on hand, then order large amount.
 Useful when vendors visit routinely.

Example: P&G representative calls every 2 weeks.
12-33
Fixed Period Model
 Compute optimal order interval, T (equation is
similar to EOQ).

For example, 27.35 days
 Compute maximum inventory level, M (equation is
similar to ROP).
 Adjust order interval to a convenient length.

For example, one month.

Then, adjust M correspondingly.
 Order M - inventory on hand every T time units.
12-34
Fixed Period Models
 Order at constant interval.
 Order amount Q varies: M - amount on hand.
On-hand
for order 2
On-hand
for order 1
1st
order
2nd
order
12-35
3rd
order
Time