Inventory Theory
The Management of Idle Resources
A quantity of commodity held for some
time to satisfy some future demand.
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Why do we need inventory?
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•
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•
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Economies of batch production
Unpredictable or unreliable vendors
buffer for imbalanced production lines
buffer for machine downtimes
safety stock against random demands or uncertain leadtimes
hedge against poor quality
bi-product from production smoothing
avoid loss of sales or high cost of backorders
fill logistics pipeline - resupply time
display goods to potential customers
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More of Why Inventory?
• A buffer between supply and demand
– Accounts for differences in rates and timing between
supply and demand
• Internal (matter of policy)
– Economies of scale
– Production smoothing
– Customer service
• External (uncontrollable)
– Uncertainty
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Types of Inventory
vendor
vendor
vendor
Raw material and purchased parts inventory
production
In-process inventory
production
Finished goods inventory
warehouse
warehouse
customers
warehouse
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Demands
• Independent – demand not related to any
other item and primarily influenced by
market conditions
• Dependent – demand for an item is
influenced by the demand of another item
A demanding manager
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More Types of Inventory
• Raw material
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–
–
–
Material needing further processing
Components that go into the product as is
Supplies such as glue, screws, ink, thread
Dependent demand
• Work in process (WIP)
– Inventory in the production system waiting to be processed or
assembled and may include semi-finished products
– Dependent demand
• Finished goods
– Output of the production process or end items
– Demand is usually independent
– Finished goods from one manufacturing plant may be raw material
for another
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A Taxonomy
Inventory Models
Deterministic
Static (EOQ)
Repairable
Dynamic
basic
smoothing
backorder
dynamic lot size
finite production
finite production
with backorders
quantity discount
multi-item
Stochastic
Single
period
continuous
review
Multiperiod
periodic
review
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Fundamental Questions
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What to order?
When to order?
How much to order?
When to review?
I knew that.
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Some Inventory Policies
• Continuous review - order Q when inventory
reaches r
• Continuous review - one for one reordering
• Periodic review - review inventory every T time
periods. If inventory is less than r, order up to R
• Periodic review - review inventory every T time
periods. Order up to R
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Measures of Effectiveness
• Monetary
– Profit maximization
– Cost minimization
I would like
to see less
inventory.
• Supply effectiveness
– minimize expected backorders
– maximize probability of meeting demands
• Operational effectiveness
– minimize expected downtime
– maximize system availability
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Conflicting objectives
Marketing: I can’t sell from an empty wagon.
I can’t keep our customers if we continue to stockout and
there is not sufficient product variety.
Production: If I can produce in larger lot sizes, I can reduce per
unit cost and function efficiently.
Purchasing: I can reduce our per unit cost if I buy large quantities in bulk.
Finance: Where I am going to get the funds to pay for the inventory?
The levels should be lower.
Warehousing: I am out of space. I can’t fit anything else in this building.
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More conflicting objectives
Area
Responsibility
Marketing
Sell the product
Production
Purchasing
Finance
Warehousing
Engineering
Inventory goal
Desired
inventory level
High
Good customer
svc
Make the product Efficient lot sizes High
Buy required
material
Provide working
capital
Store the product
Design the
product
Low unit cost
High
Efficient use of
capital
Efficient use of
space
Avoid
obsolescence
Low
Low
low
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Characteristic of Inventory
• Demand
– Constant
– Variable
– Random variable
• Lead time
– Constant (known)
– Random variable
We just don’t have that
model in stock. It is
backordered but should
arrive any day now.
• Review
– Continuous
– Periodic
• Stockouts
– Backordered
– Lost sales
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Inventory costs
• Purchasing costs
– material cost, unit cost
• Ordering costs
Purchase
option
– fixed cost of preparing and monitoring order
– receiving and handling
• Production cost
– material and variable manufacturing cost
• Set-up cost
Production
option
– fixed cost to prepare for manufacture
• Holding costs
–
–
–
–
opportunity cost
storage and handling costs
taxes and insurance
pilferage, damage, spoilage, obsolescence, etc.
• Backorder and lost sales costs
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Representative Holding Costs
Costs proportional to the quantity of inventory held.
Includes:
a) Physical Cost of Space (3%)
b) Taxes and Insurance (2 %)
c) Breakage Spoilage and Deterioration (1%)
d) Opportunity Cost of alternative investment. (10%)
(Total: 16%)
holding costs = 16 % x unit cost
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Shortage Costs
• Loss of revenue for lost demand
• Costs of bookkeeping for backordered
demands
• Loss of goodwill for being unable to satisfy
demands when they occur.
• Generally assume cost is proportional to
number of units of excess demand.
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Assumptions – general model
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Demand is known and constant
order quantity is not restricted to integers
unit cost does not depend upon the order quantity
no change in unit cost over time (inflation)
each item can be treated independently
lead-time is known and constant
backorders are permitted
infinite planning horizon
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Notation
Decision variables:
Q = order quantity or lot size (units)
r = reorder point (units)
T = time between orders or production runs (cycle time) (yr)
b = maximum backorders per cycle (units)
Parameters
D = demand rate (units per yr)
(note that book uses )
P = production rate (units per yr) (P > D)
c = unit purchase or production cost ($/unit)
K = order or set-up cost ($/order)
h = holding cost = ic ($/unit per yr)
g = backorder cost ($/unit per yr)
g’ = cost per backorder ($/unit)
L = lead-time (yr)
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Basic EOQ Model
Additional assumptions:
1. No backorders
2. Instantaneous arrivals
inventory
slope = -D
Q
T
time
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Total inventory cost = ordering cost + purchase cost + holding cost
order cost per cycle = K;
purchase cost per cycle = cQ
average inventory per cycle = Q/2
holding cost per cycle = (hT) (Q/2)
length of cycle = T = Q/D
Total inventory cost per cycle = K + cQ + h(Q/D) (Q/2)
Cycles per year = D/Q
Total inventory cost per year = (D/Q) x [ K + cQ + h(Q/D) (Q/2) ]
= DK/Q + cD + h (Q/2)
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Cost Minimization
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Total inventory cost per year = G(Q) = DK/Q + cD + h (Q/2)
Find Q that Minimizes G(Q):
dG(Q)/dQ = - DK/Q2 + h/2 = 0
solving:
h/2 = DK/Q2
Q2 = 2DK/h
2 DK
Q* 
h
T* = Q*/D
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Reorder Point
inventory
slope = -D
Q
R
time
L
L
R = lead-time demand = LD – mQ*
where m = integer [L / T*]
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G(Q) = DK/Q + cD + h (Q/2)
G (Q*) 
DK
h 2 DK
 cD 
2
h
2 DK
h
2 DK
Q* 
h
2 DK
DK
h 2 DK
h

 cD 
2 DK
2
h
h
h 2 DK
h 2 DK

 cD 
2
h
2
h
 2 DKh  cD
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Example – Basic EOQ
Monthly demand for plastic bolts used to fasten the wing of
the C-5A to the fuselage is 8,000. Each bolt cost $ .075. The
cost of ordering including delivery charges is $100 per shipment.
An annual holding cost of ten percent of the purchase cost is to
be used. That is h = .10 (.075) = .0075.
2(100)(12 x8000)
Q* 
 50,597
.0075
T *  50,597 / 96000  .53 year
G (50,597)  2(100)(.0075)(96000)  96, 000(.075)
 309.88  10,800  11,109.88
If L = 2 months = 2/12 = .1667 years, then R = .1667 (96,000) = 16,003
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The Finite Production Model with Backorders
tp = T1 + T2
td = T3 + T4
On-hand
inventory
Imax
-D
P-D
Q
T4
T1
T2
T3
time
-b
T
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Total Annual Cost
Set-up
cost
purchase
cost
holding
cost
backorder
cost
2
DK
h Q
  P 
TC  Q, b  
 cD 
P

D

b

  


Q
2Q  P
  PD
gb 2  P  g ' b D



2Q  P  D 
Q
To solve:
TC(Q, b)
0 ;
Q
TC(Q, b)
0
b
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Solution
2
2 KD
( g ' D)
Q* 

h(1  D / P) h(h  g )
h g
g
(hQ *  g ' D) (1  D / P)
b* 
h g
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Now I will tell you the story
of how the re-order point is
calculated.
r '  L  D  mQ *
for L  mT *  td
case II
r '  L  D  mQ *  P t p  (T * { L  mT *})
for L  mT *  t d
R  r ' b *
L
m
T*
Q*
T* 
D
number complete cycles
case I
tp = T1 + T2
td = T3 + T4
cycle time
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Re-order Point
On-hand
inventory
Case I.
L = production lead-time
m  0 : r '  L  D  P t p  (T *  L)  for L  td
R
time
-b
L
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Re-order Point
On-hand
inventory
Case II.
L = production lead-time
m  0 : r '  L  D for L  td
R
time
-b
L
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2 KD
( g ' D) 2
Q* 

h(1  D / P) h(h  g )
h g
g
(hQ *  g ' D) (1  D / P)
b* 
h g
Note if g’ = 0 then Q* and b* will be positive. If g’ > 0 then the
radicand may be negative. In this case no backorders should be
permitted and b* = 0.
Note if g = 0 and g’ > 0 then
1) either no backorders should be permitted:
D g’ > annual cost with no backorders
2) or no inventory carried (i.e. produce-to-order):
D g’ < annual cost with no backorders
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No backorders permitted
2 KD
( g ' D) 2
lim Q*  lim

g 
g 
h(1  D / P) h(h  g )

2 KD
h(1  D / P)
1

1 D / P
hg
g
2 KD
h
(hQ *  g ' D) (1  D / P)
lim b*  lim
0
g 
g 
h g
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The Production Model with no Backorders
on-hand inventory
t = tp + td
tp = Q/P
td = B/D where B = (Q/P) (P – D)
G(Q) = K D / Q + (h /2) (Q /P)[P – D]
B
time
tp
td
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No backorders permitted (continued)
Re-order point:
R  L  D  mQ *
for L  mT *  td
R  L  D  mQ *  P t p  (T * {L  mT *}) 
for L  mT *  td
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Purchase Option
instantaneous arrivals
2 KD
( g ' D) 2
lim Q*  lim

P 
P 
h(1  D / P ) h(h  g )
2 KD ( g ' D) 2


h
h( h  g )
h g
g
h g
g
(hQ *  g ' D) (1  D / P)
lim b*  lim
P 
P 
h g
(hQ *  g ' D)

h g
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The Backorder Model
On-hand
inventory
t = t 1 + t2
slope = -D
t1 = b/D;
t2 = (Q-b)/D
S
Q
time
-b
t1
t2
37
Purchase Option
instantaneous arrivals (continued)
Re-order point:
r '  LD  b*
On-hand + on-order
R  r ' m Q *
on-hand
L
m
T*
Q*
T* 
D
number complete cycles
cycle time
38
Purchase option
No backorders permitted
Say. This is just the
simple EOQ model isn’t
it?
lim Q*  lim
P 
P 
1
1 D / P
2 KD
2 KD

h
h
R  L D  mQ*
L
m
T
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EOQ Homework
I always work
extra problems –
not only the ones
that are assigned!
Text Chapter 4: 4,8,10,12,13,14,17,20,21,22
24,26,27,28,29,30,33,35,40 + Handout
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