The EOQ Model

THE EOQ MODEL
Syed Mohammed Sajl
Semester 6
B Tech – PS & E
Sunday, October 06, 2013 Industrial Management - THE EOQ MODEL 1

To a pessimist, the glass is half empty.
To an optimist, it is half full.
- Anonymous

SOME BASIC DEFINITIONS
• An INVENTORY is an accumulation of a
commodity that will be used to satisfy some
future demand.
• Inventories may be of the following form:
- Raw material
- Components (subassemblies)
- Work-in-process
- Finished goods
- Spare parts
- Pipeline

EOQ History
• Introduced in 1913 by Ford W. Harris, “How Many
Parts to Make at Once”
• Interest on capital tied up in wages, material and
overhead sets a maximum limit to the quantity of
parts which can be profitably manufactured at
one time; “set-up” costs on the job fix the minimum.
Experience has shown one manager a way to
determine the economical size of lots.
• Early application of mathematical modeling to
Scientific Management

EOQ MODELING ASSUMPTIONS
1. Production is instantaneous – there is no capacity constraint and the
entire lot is produced simultaneously.
2. Delivery is immediate – there is no time lag between production and
availability to satisfy demand.
3. Demand is deterministic – there is no uncertainty about the quantity
or timing of demand.
4. Demand is constant over time – in fact, it can be represented as a
straight line, so that if annual demand is 365 units this translates
into a daily demand of one unit.
5. A production run incurs a fixed setup cost – regardless of the size of
the lot or the status of the factory, the setup cost is constant.
6. Products can be analyzed singly – either there is only a single product
or conditions exist that ensure separability of products.

TOTAL INVESTMENT IN INVENTORIES IN US

CHARACTERISTICS OF INVENTORY
SYSTEMS
1. Demand
• constant versus variable
• known versus uncertain
2. Replenishment Lead Time
3. Review
• periodic or continuous
4. Replenishment
• periodic or continuous
5. Excess demand
• backordered or lost
5. Changing Inventory
• perishability or obsolescence

COST STRUCTURE
• Order costs
– fixed (Set-up cost, K)
– variable, per-unit (c)
• Holding costs (h)
• Penalty cost (p)
– if backordering allowed: loss of goodwill
– if demand lost: loss of goodwill + loss of profit

THE ECONOMIC ORDER
QUANTITY (EOQ) MODEL

ASSUMPTIONS LEADING TO EOQ
• Demand rate, λ, is constant and deterministic over
time (units/day, units/year, etc.)
• Shortages not permitted
• No replenishment lead time
• The cost structure includes
– fixed ordering cost, K, per order placed
– unit variable cost, c, independent of order size (no
discounts)
– holding cost, h
• Infinite planning horizon

WHY EOQ?
• Easy to compute
• Does not require data that is hard to obtain
• Policies are surprisingly robust
• Assumptions can be relaxed
• Gives a good overall idea
• Can be starting point for more complicated
models

EOQ - NOTATION
Q = order quantity
(we want to find the optimal Q)
K = fixed order cost
c = unit variable cost, $ / unit
h = holding cost, $ / (unit * (unit time))
h = I c
λ = demand rate, units / (unit time)

EOQ – THE OBJECTIVE
• Our aim is to determine the best
replenishment strategy (when and how
much to order) under the criterion that the
relevant costs (order and holding costs) will
be minimized over time.

EOQ – THE INTUITION
• “When” should we place a new order ?
– assume zero starting inventory
– demand is deterministic and at a constant rate
– lead time is negligible
– no backorders are allowed

• “How much” we have to order each time ?
– parameters do not change over time
– there is no reason for ordering different quantities
– so, order the same quantity Q in each order

Decision Variable: Q, the order size
T
Time
Q
Inventory
Slope= -λ

DERIVING THE EOQ: THE “CYCLE”
• The same picture occurs over and over again
– Why?
– any one of the triangles will be called a “cycle”
• Why not minimize the total cost in a cycle?
• Minimize the average cost per unit time (annual cost)
• Convert the order and holding costs to “average annual
cost”, using the total cost in a cycle
lengthCycle
cycleaincostTotal
T
timeofunitsTperCosts
lim
T

DERIVING THE EOQ: THE ORDER COST
• Cycle length T =Q / λ (slope, - λ, = -Q/T)
• Since Q / λ is the time between two orders
(the cycle time), λ /Q is the number of
orders per unit time
• In each order, we pay K+Qc
• Order cost per unit time

DERIVING THE EOQ: THE HOLDING COST
• Holding cost per unit time =
2
2
lengthcycle
cycleoneovercostholding
2
0 Q
h
Q
Q
h
Q
T
dttQh
Q
T
2
levelinventoryAverage
Q
hh
• This can also be calculated as

DERIVING THE EOQ
2
)(' 2
h
Q
KQG
0
2
)('' 3
Q
K
QG
h
K
Q
2*
= I c

THE AVERAGE ANNUAL COST CURVE
unit time
cost
Q
2
hQ
G(Q)
Q
K
Q*
Annual fixed
ordering and
holding cost
The minimum

AN EXAMPLE
• Suppose that you are working for a retail store and have
to determine how many boxes of detergent to order.
• Suppose every time you make an order you pay $20 for
transportation and $10 to prepare the ordering request.
• Suppose every dollar tied in the inventory would earn 10
cents annually, if you had invested it.
• Suppose there is a regular weekly demand of 20 boxes
and each box costs you $10.
• What is the order quantity?
• What is the trade-off here?

THE SOLUTION
• λ=20*52 = 1040 boxes annually
• K (fixed ordering cost) = $30 each time
• h (holding cost)

INVENTORY TURNOVER RATIO WITH EOQ
MODEL
• Turnover Ratio: A measure of effective inventory
control
– “How many times I sold my inventory?”
demand rate 2
avg. inventory / 2opt
h
Q K

EOQ IN A PRODUCTION ENVIRONMENT
• So far we have discussed a case in which we
“order” from an outside supplier
• If we produce in-house, the relevant problem is
called the “Economic Production Quantity”
• In this case, K denotes the “setup cost”
• Why do we have a setup cost? Examples include:
– cost of production setup independent of the number
produced: heating an oven, cost of dyes, etc.
– time required for the setup: lost production!
– first few units might need to be scrapped

THANK YOU! 

The EOQ Model

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