Inventory Control
Known Demand
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Contents
Types of Inventories
Motivation for Holding Inventories;
Characteristics of Inventory Systems;
Relevant Costs;
The EOQ Model;
EOQ Model with Finite Production Rate
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Introduction
Definition: Inventory is the stock of any item or resource
used in an organization.
An inventory system is the set of policies and controls that
monitors levels of inventory or determines what levels
should be maintained.
Generally, inventory is being acquired or produced to meet
the need of customers;
Dependant demand system - the demand of components and
subassemblies (lower levels depend on higher level) -MRP;
The fundamental problem of inventory management :
When to place order for replenishing the stock ?
How much to order?
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Introduction
Inventory: plays a key role in the logistical behavior of
virtually all manufacturing systems.
The classical inventory results: are central to more
modern techniques of manufacturing management, such
as MRP, JIT, and TBC.
The complexity of the resulting model depends on the
assumptions about the various parameters of the system -
the major distinction is between models for known demand
and random demand.
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Introduction
The current investment in inventories in USA is enormous;
It amounted up to $1.37 trillion in the last quarter of 1999;
It accounts for 20-25% of the total annual GNP (general net product);
There exists enormous potential for improving the efficiency of
economy by scientifically controlling inventories;
Breakdown of total investment in inventories 5
Types of Inventories
A natural classification is based on the value added
from manufacturing operations
Raw materials: Resources required in the production
or processing activity of the firm.
Components: Includes parts and subassemblies.
Work-in-process (WIP): the inventory either waiting
in the system for processing or being processed.
The level of WIP is taken as a measure of the
efficiency of a production scheduling system.
JIT aims at reducing WIP to zero.
Finished good: also known as end items or the final
products.
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Why Hold Inventories (1)
For economies of scale
It may be economical to produce a relatively large
number of items in each production run and store
them for future use.
Coping with uncertainties
Uncertainty in demand
Uncertainty in lead time
Uncertainty in supply
For speculation
Purchase large quantities at current low prices and
store them for future use.
Cope with labor strike
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Why Hold Inventories (2)
Transportation
Pipeline inventories is the inventory moving from
point to point, e.g., materials moving from suppliers
to a plant, from one operation to the next in a plant.
Smoothing
Producing and storing inventory in anticipation of
peak demand helps to alleviate the disruptions
caused by changing production rates and workforce
level.
Logistics
To cope with constraints in purchasing, production,
or distribution of items, this may cause a system
maintain inventory.
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Characteristics of Inventory Systems
Demand (patterns and characteristics)
Constant versus variable
Known versus random
Lead Time
Ordered from the outside
Produced internally
Review
Continuous: e.g., supermarket
Periodic: e.g., warehouse
Excess demand
demand that cannot be filled immediately from stock
backordered or lost.
Changing inventory
Become obsolete: obsolescence
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Relevant Costs - Holding Cost
Holding cost (carrying or inventory cost)
The sum of costs that are proportional to the amount
of inventory physically on-hand at any point in time
Some items of holding costs
Cost of providing the physical space to store the
items
Taxes and insurance
Breakage, spoilage, deterioration, and obsolescence
Opportunity cost of alternative investment
Inventory cost fluctuates with time
inventory as a function of time
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Relevant Costs - Holding Cost
Inventory as a Function of Time
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EOQ History
Introduced in 1913 by Ford W. Harris, “How Many Parts
to Make at Once”
Product types: A, B and C
A-B-C-A-B-C: 6 times of setup
A-A-B-B-C-C: 3 times of setup
A factory producing various products and switching
between products causes a costly setup (wages, material and
overhead). Therefore, a trade-off between setup cost and
production lot size should be determined.
Early application of mathematical modeling to Scientific
Management
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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.
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Notation
demand rate (units per year)
c proportional order cost at c per unit ordered (dollars per unit)
K fixed or setup cost to place an order (dollars)
h holding cost (dollars per year); if the holding cost consists
entirely of interest on money tied up in inventory, then h = ic
where i is an annual interest rate.
Q the unknown size of the order or lot size
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Inventory vs Time in EOQ Model
slope = -
Inventory
Q
Q/ 2Q/ 3Q/ 4Q/
Time
Order cycle: T=Q/
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Costs
Q
Holding Cost: average inventory
2
hQ
annual holding cost
2
hQ
unit holding cost
2
K K
Setup Costs: K per lot, so unit setup cost , annual setup cost
Q Q
hQ K
The average annual cost: G (Q)
2 Q
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MedEquip Example
Small manufacturer of medical diagnostic equipment.
Purchases standard steel “racks” into which components
are mounted.
Metal working shop can produce (and sell) racks more
cheaply if they are produced in batches due to wasted
time setting up shop.
MedEquip doesn't want to hold too much capital in
inventory.
Question: how many racks should MedEquip order at
once?
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MedEquip Example Costs
= 1000 racks per year
c = $250
K = $500 (estimated from supplier’s pricing)
h = i*c + floor space cost = (0.1)($250) + $10 =
$35 per unit per year
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Costs in EOQ Model
20.00
18.00
16.00
14.00
Cost ($/unit)
12.00
10.00 Y(Q)
Q* =169
8.00
6.00 hQ/2D
4.00
2.00 c A/Q
0.00
0 100 200 300 400 500
Order Quantity (Q)
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Economic Order Quantity
hQ K
G (Q)
2 Q
dG (Q) h K Solution (by taking derivative
G ' (Q) 2 0
dQ 2 Q and setting equal to zero):
2 K
G ' ' (Q) 3 0 if Q 0 Since Q”>0，G(Q) is convex
Q function of Q
2 K
Q
*
EOQ Square Root Formula
h
2(500)(1000)
Q *
169 MedEquip Solution
35
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Another Example
Example 2
Pencils are sold at a fairly steady rate of 60 per week;
Pencils cost 2 cents each and sell for 15 cents each;
Cost $12 to initiate an order, and holding costs are based on
annual interest rate of 25%.
Determine the optimal number of pencils for the book store to
purchase each time and the time between placement of orders
Solutions
Annual demand rate =6052=3,120;
The holding cost is the product of the variable cost of the
pencil and the annual interest-h=0.02 0.25=0.05
2K 2 12 3,120 T
Q
3,870
1.24 yr
Q *
3,870
h 0.05 3,120
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The EOQ Model-Considering Lead Time
Since there exists lead time (4 moths for Example 2), order
should be placed some time ahead of the end of a cycle;
Reorder point R-determines when to place order in term of
inventory on hand, rather than time.
Reorder Point Calculation for Example 2 22
The EOQ Model-Considering Lead Time
Determine the reorder point when the lead time
exceeds a cycle. Computing R for placing order
Example: 2.31 cycles ahead is the same as
Q=25; that 0.31 cycle ahead.
=500/yr;
=6 wks;
T=25/500=2.6
wks;
/T=2.31---2.31
cycles are included
in LT.
Action: place
every order 2.31
cycles in advance.
Reorder Point Calculation for Lead Times
Exceeding One Cycle
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EOQ Modeling Assumptions
1. Production is instantaneous – there is no capacity constraint
and the entire lot is produced simultaneously.
relax via EOQ Model for Finite Production Rate
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.
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The EOQ Model for Finite Production Rate
The EOQ model with finite production rate is a
variation of the basic EOQ model
Inventory is replenished gradually as the order is
produced (which requires the production rate to be
greater than the demand rate)
Notice that the peak inventory is lower than Q since
we are using items as we produce them
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Notation – EOQ Model for Finite Production Rate
demand rate (units per year)
P production rate (units per year), where P>
c unit production cost, not counting setup or inventory costs
(dollars per unit)
K fixed or setup cost (dollars)
h holding cost (dollars per year); if the holding cost is consists
entirely of interest on money tied up in inventory, then h = ic
where i is an annual interest rate.
Q the unknown size of the production lot size decision variable
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Inventory vs Time
1. Production run of Q takes Q/P time units
slope = -
(P-)(Q/P)
Inventory
- P-
(P-)(Q/P)/2
slope = P-
Time
2. When the inventory reaches 0, production begins until
Q products are produced (it takes Q/P time units). During
the Q/P time units, the inventory level will increases to (P-)(Q/P)
Time
Inventory
increase rate
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Solution to EOQ Model with Finite Production Rate
Annual Cost Function:
K h(1 / P)Q
G (Q)
Q 2
setup holding
Solution (by taking first dG Q d 2G Q
derivative and setting equal 0 2
0
dQ dQ
to zero):
• tends to EOQ as P
2 K
Q*
h(1 / P ) • otherwise larger than EOQ
because replenishment takes
longer
2 K
Q* EOQ model
h
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