1

Inventory Theory

The Management of Idle Resources

A quantity of commodity held for some time to satisfy some future demand.

2

Why do we need inventory?

• Economies of batch production

• Unpredictable or unreliable vendors

• buffer for imbalanced production lines

• buffer for machine downtimes

• safety stock against random demands or uncertain lead-times

• 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

3

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

4

Types of Inventory

Raw material and purchased parts inventory

In-process inventory

Finished goods inventory

vendor vendor vendor

warehouse warehouse warehouse

production

production

customers

5

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

6

More Types of Inventory• Raw material

– 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

7

A Taxonomy

Inventory Models

Deterministic Stochastic

Single Multi-Static (EOQ) Dynamic period period

basic smoothingbackorder dynamic lot size continuous periodicfinite production review reviewfinite production

with backordersquantity discountmulti-item

Repairable

8

Fundamental Questions

• What to order?

• When to order?

• How much to order?

• When to review?

I knew that.

9

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

10

Measures of Effectiveness

• Monetary– Profit maximization

– Cost minimization

• Supply effectiveness– minimize expected backorders

– maximize probability of meeting demands

• Operational effectiveness– minimize expected downtime

– maximize system availability

I would like to see less inventory.

11

Conflicting objectivesMarketing: I can’t sell from an empty wagon. I can’t keep our customers if we continue to stockout andthere 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.

12

More conflicting objectives

Area Responsibility Inventory goal Desiredinventory level

Marketing Sell the product Good customersvc

High

Production Make the product Efficient lot sizes High

Purchasing Buy requiredmaterial

Low unit cost High

Finance Provide workingcapital

Efficient use ofcapital

Low

Warehousing Store the product Efficient use ofspace

Low

Engineering Design theproduct

Avoidobsolescence

low

13

Characteristic of Inventory• Demand

– Constant

– Variable

– Random variable

• Lead time– Constant (known)

– Random variable

• Review– Continuous

– Periodic

• Stockouts– Backordered

– Lost sales

We just don’t have that model in stock. It is

backordered but should arrive any day now.

14

Inventory costs• Purchasing costs

– material cost, unit cost

• Ordering costs– fixed cost of preparing and monitoring order

– receiving and handling

• Production cost– material and variable manufacturing cost

• Set-up cost– 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

Purchaseoption

Productionoption

15

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

16

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.

17

Assumptions – general model

• 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

18

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)

ParametersD = 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)

19

Basic EOQ ModelAdditional assumptions:1. No backorders2. Instantaneous arrivals

time

inventory

slope = -D

T

Q

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Total inventory cost = ordering cost + purchase cost + holding cost

order cost per cycle = K;purchase cost per cycle = cQaverage inventory per cycle = Q/2holding 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/QTotal inventory cost per year = (D/Q) x [ K + cQ + h(Q/D) (Q/2) ]

= DK/Q + cD + h (Q/2)

21

Cost Minimization

22

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*

DKQ

h

T* = Q*/D

23

Reorder Point

time

inventory

slope = -D

L

Q

R

L

R = lead-time demand = LD – mQ*where m = integer [L / T*]

24

2*

DKQ

h

G(Q) = DK/Q + cD + h (Q/2)

2( *)

22

22

2 2

2 2

2 2

2

DK h DKG Q cD

hDKh

DKDK

h DKh cDDK hh

h DK h DKcD

h h

DKh cD

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Example – Basic EOQMonthly demand for plastic bolts used to fasten the wing ofthe C-5A to the fuselage is 8,000. Each bolt cost $ .075. Thecost 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 8000)* 50,597

.0075* 50,597 / 96000 .53 year

(50,597) 2(100)(.0075)(96000) 96,000(.075)

309.88 10,800 11,109.88

xQ

T

G

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

time

On-handinventory

T1

T2 T3

Imax

T4

P-D -D

-b

Q

T

tp = T1 + T2

td = T3 + T4

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Total Annual Cost

Set-up purchase holding backordercost cost cost cost

TC Q b

Q

TC Q b

b

( , );

( , )0 0

To solve:

2

2

,2

'

2

DK h Q PTC Q b cD P D b

Q Q P P D

gb P g bD

Q P D Q

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Solution

bhQ g D D P

h g*

( * ' ) ( / )

1

22 ( ' )*

(1 / ) ( )

KD g D h gQ

h D P h h g g

29

r L D mQ for L mT td' * *

r L D mQ P t T L mT

for L mT t

p

d

' * ( * { *})

*

' *

*

**

R r b

Lm

T

QT

D

number complete cycles

cycle time

tp = T1 + T2

td = T3 + T4

Now I will tell you the story of how the re-order point is calculated.

case II

case I

30

Re-order Point

time

On-handinventory

-b

L

R

L = production lead-timeCase I.

0 : ' ( * )p dm r L D P t T L for L t

31

Re-order Point

time

On-handinventory

-b

L

R

L = production lead-timeCase II.

0 : ' dm r L D for L t

32

Note if g’ = 0 then Q* and b* will be positive. If g’ > 0 then theradicand 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 backorders2) or no inventory carried (i.e. produce-to-order):

D g’ < annual cost with no backorders

22 ( ' )*

(1 / ) ( )

KD g D h gQ

h D P h h g g

bhQ g D D P

h g*

( * ' ) ( / )

1

33

No backorders permitted

2

g g

2 ( ' )lim * lim

(1 / ) ( )

2 1 2

(1 / ) 1 /

KD g D h gQ

h D P h h g g

KD KD

h D P D P h

( * ' ) (1 / )lim * lim 0g g

hQ g D D Pb

h g

34

The Production Model with no Backorders

time

on-hand inventory

tp td

B

t = tp + td

tp = Q/Ptd = B/D where B = (Q/P) (P – D)G(Q) = K D / Q + (h /2) (Q /P)[P – D]

35

No backorders permitted (continued)

* ( * { *})

*

p

d

R L D mQ P t T L mT

for L mT t

* * dR L D mQ for L mT t

Re-order point:

36

lim * lim( * ' ) ( / )

( * ' )

P Pb

hQ g D D P

h g

hQ g D

h g

1

Purchase Optioninstantaneous arrivals

2

2

2 ( ' )lim * lim

(1 / ) ( )

2 ( ' )

( )

P P

KD g D h gQ

h D P h h g g

KD g D h g

h h h g g

37

The Backorder Model

time

slope = -D

S

-b

On-handinventory

t1 t2

t1 = b/D; t2 = (Q-b)/D

Q

t = t1 + t2

38

Purchase Optioninstantaneous arrivals (continued)

' *

' *

*

**

r L D b

R r mQ

Lm

T

QT

D

Re-order point:

On-hand + on-order

on-hand

number complete cycles

cycle time

39

Purchase option No backorders permitted

*R LD mQ

Lm

T

Say. This is just the simple EOQ model isn’t it?

1 2 2lim * lim

1 /P P

KD KDQ

D P h h

40

EOQ Homework

I always work extra problems – not only the ones that are assigned!

Text Chapter 4: 4,8,10,12,13,14,17,20,21,2224,26,27,28,29,30,33,35,40 + Handout