MS 401
Production and Service Systems Operations
Spring 2009-2010
Inventory Control – I
Deterministic Demand: EOQ Model
Slide Set #3
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Office Hours
• Ceyda Sol: Thursday 14:00-15:30
• Özlem Çoban: Wednesday 15:00-16:30
• Mahir Umman Yıldırım: Monday 14:30-16:00
• The assistants' office hours will be conducted at L068,
unless announced otherwise
• Murat Kaya: Tuesday 14:40-15:40, Thursday 13:40-14:40,
Office: G020
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Whom to Email? The Coordinators
• Each graduate assistant coordinates some aspect of the
course. Please email the assistants on administrative issues
before emailing the instructor (Dr. Kaya). Your questions
should be directed as follows:
• Assignment-related questions to CEYDA
• Project-related questions to OZLEM
• Recitation and Exam-related questions to MAHIR
(including make-up exam issues)
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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
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Total Investment in Inventories in US (1999)
Copyright © 2001 by The McGraw-Hill Companies, Inc
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Motivation for Holding Inventories
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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
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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
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Inventory Turnover Ratios for Different Industries
Industry
Upper quartile
Median
Lower quartile
Electronic
components and
accessories
8.1
4.9
3.3
Electronic computers
22.7
7.0
2.7
Household audio and
video equipment
6.3
3.9
2.5
Paper Mills
11.7
8.0
5.5
Industrial chemicals
14.1
6.4
4.2
Bakery products
39.7
23.0
12.6
Books: Publishing
and printing
7.2
2.8
1.5
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The Economic Order Quantity (EOQ) Model
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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
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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
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EOQ - Notation
Recall the notation we introduced earlier:
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)),
λ = demand rate, units / (unit time)
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h= I c
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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
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EOQ Model - 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
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EOQ Model - Intuition
• “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
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Economic Order Quantity Model
Decision Variable: Q, the order size
Inventory
Slope= -λ
Q
T
Time
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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
Costs per T units of time T otalcost in a cycle

T 
T
Cyclelength
lim
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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 =
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Deriving the EOQ: The Holding Cost
• Holding cost per unit time =
T
Q

2
Q
h  Q  t dt h
holdingcost over one cycle
Q
2

0


h
Q
Q
cyclelength
2
T


• This can also be calculated as
Q
hAverage inventory level   h
2
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Deriving the EOQ
G(Q) 
G' (Q)   K

Q2

h
2
G' ' (Q) 
2 K
Q 
h
*
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2 K
0
3
Q
=Ic
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The Average Annual Cost Curve
cost
unit time
Annual fixed ordering
and holding cost
The minimum
G(Q) hQ
2
K
Q
Q*
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Q
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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?
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Solution
• λ=20*52 = 1040 boxes annually
• K (fixed ordering cost) = $30 each time
• h (holding cost)
• Q*=
T: cycle length =
Annual holding cost =
Annual fixed ordering cost =
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Inventory Turnover Ratio with EOQ Model
• Turnover Ratio: A measure of effective inventory control
– “how many times I sold my inventory?”
demand rate

2h


avg. inventory Qopt / 2
K
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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
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