Inventory Management
MD707 Operations Management
Professor Joy Field
Types of Inventory

Cycle inventory

Safety stock

Anticipation inventory

Pipeline inventory (WIP, finished goods, goods-in-transit)

Replacement parts, tools, and supplies
2
Functions of Inventory

To meet anticipated customer demand

To smooth seasonal requirements

To decouple operations

To protect against stockouts

To take advantage of order cycles

To hedge against price increases

As a result of operations cycle and throughput times

To take advantage of quantity discounts
3
Managing Independent Demand Inventory
Managing independent demand inventory involves answering
two questions:


How much to order?
When to order?
When answering these questions, a manager needs to consider
costs (holding or carrying costs, ordering costs, shortage costs,
unit costs) and the tradeoff between costs and customer service.
Inventory management efforts may be allocated based on the
relative importance of an item, determined through a
classification system such as an A-B-C approach.
Customer satisfaction and inventory turnover (i.e., the ratio of
annual cost of goods sold to average inventory investment) are
two measures of inventory management effectiveness.
4
EOQ Assumptions

Item independence

Demand is known and constant

Lead time does not vary

Each order is received in a single delivery

There are no quantity discounts

Only two relevant costs (holding and ordering costs)
5
EOQ Inventory Cycle
Demand is Known and Constant
Profile of Inventory Level Over Time
Q
Usage
rate
Quantity
on hand
Reorder
point
Receive
order
Place
order
Receive
order
Time
Place
order
Receive
order
Lead time
12-6
Total Annual Cost

Annual carrying cost
Q
(H )
 Annual carrying cost =
2

Annual ordering cost
D
(S )
 Annual ordering cost =
Q

Total annual cost: TC 
Q
D
( H )  (S )
2
Q
7
Derivation of Economic Order Quantity (EOQ)
and Time Between Orders (TBO)
Total annual cost: TC =
Q
D
( H )  (S )
2
Q
Take the first derivative of cost with respect to
quantity: dTC H D
  2 (S )
dQ
2 Q
2 DS
dTC
EOQ
,
Q

 0 and solving for Q:
Setting
o
H
dQ
Time between orders: TBOQ 
Q
D
8
Annual Cost
EOQ Cost Curves
The Total-Cost Curve is U-Shaped
Q
D
TC  H  S
2
Q
Holding Costs
Ordering Costs
Q* (optimal order quantity)
Order Quantity
(Q)
12-9
Overland Motors Example
Overland Motors uses 25,000 gear assemblies each year
(i.e. 52 weeks) and purchases them at $3.40 per unit. It
costs $50 to process and receive each order, and it costs
$1.10 to hold one unit in inventory for a whole year. Assume
demand is constant. The purchasing agent has been
ordering 1,000 gear assemblies at a time, but can adjust his
order quantity if it will lower costs.

What is the annual cost of the current policy of using a 1,000unit lot size?

What is the order quantity that minimizes cost?

What is the time between orders for the quantity in part b?

If the lead time is two weeks, what is the reorder point?
10
Economic Production Quantity (EPQ)
Similar to the EOQ but used for batch production. A complete
order is no longer received at once and inventory is replenished
gradually (i.e., non-instantaneous replenishment).

Maximum Cycle Inventory

Q
p u
I max  ( p  u )  Q(
)
p
p
Total cost = Annual holding cost + Annual ordering cost
I max
D
Q p u
D
TC 
( H )  (S )  (
)( H )  ( S )
2
Q
2 p
Q

Economic Production Quantity (EPQ)
2 DS
p
Qo 
H
p u
11
EPQ Inventory Cycle
Noninstantaneous Replenishment
Q
Q*
Production
and usage
Usage
only
Production
and usage
Usage
only
Production
and usage
Cumulative
production
Imax
Amount
on hand
Time
12-12
EPQ Example
A domestic automobile manufacturer schedules 12 two-person teams to
assemble 4.6 liter DOHC V-8 engines per work day. Each team can
assemble five engines per day. The automobile final assembly line
creates an annual demand for the DOHC engine at 10,080 units per
year. The engine and automobile assembly plants operate six days per
week, 48 weeks per year. The engine assembly line also produces
SOHC V-8 engines. The cost to switch the production line from one type
of engine to the other is $100,000. It costs $2,000 to store one DOHC V8 for one year.

What is the economic production quantity?

How long is the production run?

What is the average quantity in inventory?

What are the total annual costs associated with the EPQ?
13
Quantity Discounts

In the case of quantity discounts (price incentives to purchase
large quantities), the unit price, P, is relevant to the calculation
of total annual cost (since the price is no longer fixed).

Total cost = Annual holding cost + Annual ordering cost +
Annual cost of materials
TC 
Q
D
( H )  ( S )  PD
2
Q
14
Quantity Discounts
Two-Step Procedure

Step 1


Beginning with lowest price, calculate the EOQ for each price
level until a feasible EOQ is found. It is feasible if it lies in the
range corresponding to its price.
Step 2

If the first feasible EOQ found is for the lowest price level, this
quantity is best. Otherwise, calculate the total cost for the first
feasible EOQ and for the larger price break quantity at each lower
price level. The quantity with the lowest total cost is optimal.
15
Total Cost Curves with Quantity Discounts
12-16
Quantity Discounts Example
Order Quantity
1-99
100 or more
Price Per Unit
$50
$45
If the ordering cost is $16 per order, annual holding cost is 20% of
the per unit purchase price, and annual demand is 1,800 items,
what is the best order quantity?

Step 1
EOQ45.00 =
EOQ50.00 =

Step 2
TC ___ =
TC ___ =
17
Perpetual (Continual) Inventory Review System

A perpetual (continual) inventory review system
tracks the remaining inventory of an item each time a
withdrawal is made, to determine if it is time to reorder.

Decision rule: Whenever a withdrawal brings the
inventory down to the reorder point (ROP), place an
order for Q (fixed) units.
18
Variations of the Perpetual Inventory System
Based on the characteristics of the lead time demand and the lead time, the perpetual inventory
system is implemented by ordering the EOQ, Qo , at the ROP as follows:
Lead time demand (dLT)
Known and constant
Variable, normally
distributed, average lead
time demand and  dLT
known
Unknown, but average
daily or weekly demand
and  d known, normally
distributed
Unknown, but daily or
weekly demand known
and constant
Unknown, but average
daily or weekly demand
and  d known, normally
distributed
Lead time (LT)
Known and constant
Variable, normally
distributed,  dLT
known
Approach
Order Qo when ROP is equal to the lead time demand.
Calculate the safety stock for a given service level
(= z  dLT ) using the table on p.576 or pp.882-3 to determine
z.*
Known and constant
Calculate the expected lead time demand by multiplying the
average daily or weekly demand by the lead time. Calculate
the safety stock for a given service level (= z LT  d ) with z
determined as above.*
Calculate the expected lead time demand by multiplying the
daily or weekly demand by the average lead time. Calculate
the safety stock for a given service level (= zd  LT ) with z
determined as above.*
Calculate the expected lead time demand by multiplying the
average daily or weekly demand by the average lead time.
Calculate the safety stock for a given service level
Variable, normally
distributed, average
lead time and  LT
known
Variable, normally
distributed, average
lead time and  LT
known
2
2
(= z LT d2  d  LT
) with z determined as above.*
*Order Qo when ROP is equal to the expected lead time demand plus safety stock.
19
Reorder Point
The ROP based on a normal
distribution of lead time demand
Risk of
stockout
Service level
Expected
demand
ROP
Quantity
Safety
stock
0
z
z-scale
12-20
Shortages and Service Levels
The ROP calculation relates the probability of being able to
satisfy demand during the lead time for ordering. In order to
determine the expected amount of units to be short during this
period, calculate:
E (n)  E ( z ) dLT
where: E(n) = Expected number of units short per order cycle,
E(z) = Standardized number of units short (obtained from Table
12.3, p.576),  dLT = Standard deviation of lead time demand
To calculate the expected number of units short per year:
D
E ( N )  E ( n)
Q
where: E(N) = Expected number of units short per year, D =
Yearly demand, Q = Order size
21
Perpetual Inventory System Example
You are reviewing the company’s current inventory policies for its perpetual
inventory system, and began by checking out the current policies for a sample of
items. The characteristics of one item are:
Average demand = 10 units/wk (assume 52 weeks per year)
Ordering cost (S) = $45/order
Holding cost (H) = $12/unit/year
Mean lead time demand = 30 units
Standard deviation of lead time demand = 17 units
Service-level = 70%




What is the EOQ for this item?
What is the desired safety stock?
What is the desired reorder point?
What is the expected number of units short each cycle and per year?
If instead of the above situation, suppose the lead time is known and constant at
2 weeks and the standard deviation of lead time demand is unknown. However,
we do know the standard deviation of weekly demand to be 10 units. How do
your answers change?
22
The Single-Period Model
Used to handle ordering of perishables and items that have a limited
useful life
Analysis of single-period situations generally focuses on two costs:
shortage costs (i.e., unrealized profit per unit) and excess costs (the
cost per unit less any salvage cost)



Calculate the shortage and excess costs
 Cshortage = Cs = Revenue per unit – Cost per unit
 Cexcess = Ce = Original cost per unit – Salvage value per unit
Calculate the service level, which is the probability that demand
will not exceed the stocking level
Cs
 Service level =
C s  Ce
Determine the optimal stocking level, S o  d  z d , using the
service level and demand distribution information
23
Single-Period Problem
The concession manager for the college football stadium must
decide how many hot dogs to order for the next game. Each hot
dog is sold for $2.25 and makes a profit of $0.75. Hot dogs left
over after the game are sold to the student cafeteria for $0.50
each. Based on previous games, the demand is normally
distributed with an average of 2000 hot dogs sold per game and
a standard deviation of 400. Find the optimal stocking level for
hot dogs.
24