Inventory

Stock of items or resources used in an organization.

Manufacturing inventory refers to materials that contribute or become part of a firm’s product output.

raw materials component parts or supplies work-in-process finished goods

Purpose of Inventory

meet variation in product demand

provide safeguard for variation in raw material delivery

maintain independence of operations

allow flexibility in production scheduling

take advantage of quantity discounts

Inventory Costs– storage

– handling

– insurance and taxes

– pilferage and breakage

– obsolescence and depreciation

– opportunity cost of capital

– order processing

– shipping and receiving

– production setup and changeover

– lost sales

– backorders

– lateness penalties

Inventory SystemSet of policies and controls that

monitor levels of inventory (for all items)..

. determine what levels should be

maintained when stock should be replenished how large an order should be

keep track of orders sent and received reconcile records with physical

inventory

Economic Order Quantity (EOQ) Model

Assumptions:– demand rate constant– instantaneous replenishment– price per item independent of

order size– no backorders allowed

Questions:– When to order?– How much to order?

EOQ Model

Notation:

D = demand rate per unit time (year)

C = cost per unit item

S = setup/order cost

H = holding cost per unit item per unit time

I = inventory holding percentage

(H = IC)

Q = quantity to be ordered

Inventorylevel

Time

Total Cost (per unit time)

= purchase cost + ordering cost

+ holding cost

TC = DC + SD/Q + HQ/2

Best Q?

or

EOQ formula

( )TC

Q

SD

Q

H 0

202

QDS

H

2

Annual Product Costs,Based on Size of the Order

TC (total cost)

HQ/2(holding cost)

DC (annualcost of items)

SD/Q(ordering cost)

d TC

dQ

DS

Q

H( )

0

202

QDS

Hopt 2

TC DCD

QS

QH

2

Qopt Order quantity size (Q)

Cost

Basic Fixed-Order Quantity Model with Lead Time

Order when inventory drops to

R = DL where L = lead time

L L L

Q Q Q Q

Numberof unitson hand

R

Time

Inventory position =on-hand + on-order inventory

inventoryposition

on-handinventory

R

L

EOQ with Usage p = replenishment rate d = demand rate

Buildup =Productionrate minususage rate(p-d)

Productiontaking place

Usagerate d Q

No production;usage only

L

Numberof unitson hand

R

L

TC DCD

QS p d

Q

PH

TC

QQ

DS

H

p

p dopt

1

2

02

( )

EOQ vs. JIT EOQ:

– order quantity set according to EOQ formula

JIT:– reduce inventory as much as

possible Who is right? Who is wrong? What is the “right” order

quantity? What is the “true” cost of

inventory?

Inventory Pooling

What happens to the optimal order quantity when the demand rate doubles? Triples?

Consequences? Large supermarkets vs. mom-&-pop

stores “backup” agreements between stores

Q Dopt

Global Supply/Distribution Chains

Decentralized System:

Centralized System:

FactoryRetailers

Factory Depot

Retailers

Inventory Management-Single Period Model

order placed (and delivered) before demand is known unmet demand is lost unsold inventory at the end of the period is discard (or

salvaged at lower value)

How much to order?How much to order?

Newsboy ModelNewsboy Model

Newsvendor ModelMarginal Profit = MP= profit resulting from last unit soldMarginal Loss = ML= loss from last unit unsold

Stock one unit if ...MP Pr(D > 0) > ML Pr(D < 0)

Stock 2 units if ...Stock 1 Stock 2

D = 0

D = 1

D = 2

D = 3

D = 4

Newsvendor Model

Increase order to n+1 if Prob(Demand > n) < MP

ML + MP

keep order size at 1

order 2instead of 1

1 more unsold

1 fewer lost sale

0

-ML

MP

Order 2 ifPr(D<1) (-ML) + Pr(D>1) MP > 0

or Pr(D<1) (-ML) + [1-Pr(D<1)] MP > 0or

Pr(D<1) < MP ML + MP

AdditionalBenefit

EXAMPLE / Salvage Value A product is priced to sell at $100 per unit, and its cost is constant at $70 per unit. Each unsold unit has a salvage value of $30. Demand is expected to range between 35 and 40 units for the period: 35 units definitely can be sold and no units over 40 will be sold. The demand probabilities and the associated cumulative probability distribution (P) for this situation are shown below.

The marginal profit if a unit is sold is the selling price less the cost, or MP = $100 - $70 = $30.

The marginal loss incurred if the unit is not sold is the cost of the unit less the salvage value, or ML = $70 - $30 = $40.

How many units should be ordered?

SOLUTION The optimal probability of the last unit being sold is

CPMP

MP MLn

30

30 400 43.

According to the cumulative probability table (the last column in table below, 37 units should be stocked. The net benefit from stocking the 37th unit is the expected marginal profit minus the expected marginal loss.

EXHIBIT 14.12 Demand and Cumulative Probabilities

(p) CPn

Number of Units Probability of

Demanded This Demand

35 0.10 1 to 35 0.10

36 0.15 36 0.25

37 0.25 37 0.50

38 0.25 38 0.75

39 0.15 39 0.90

40 0.10 40 1.00

41 0 41 or more 1.00

EXHIBIT 14.13

Marginal Inventory Analysis for Units Having Salvage Value(N) (p) (P) (MP) (ML)

Units of Probability CPn Expected Marginal Expected Marginal

Demand of Demand Profit of n-th Unit Loss of n-th Unit (Net)

(100-70)(1- CPn-1) (70-30)CPn-1 (MP)-(ML)

35 0.10 0.10 $30 $0 $30.00

36 0.15 0.25 27 4 23.00

37 0.25 0.50 22.50 10 12.50

38 0.25 0.75 15 20 (5.00)

39 0.15 0.90 7.50 30 (22.50)

40 0.10 1.00 3 36 (33.00)

41 0 1.00 (40.00)

Note: Expected marginal profit is the selling price of $100 less

the unit cost of $70 times the probability the unit will be sold.

Expected marginal loss is the unit cost of $70 less the salvage

value of $30 times the probability the unit will not be sold.

Net = (MP)(1 - CPn-1 ) - (ML) CPn-1

= (1 - 0.25)($100 - $70) - (0.75) ($70 - $30)

= $22.50 - $10.00 = $12.50

For the sake of illustration, Exhibit 14.13 shows all possible

decisions. From the last column, we can confirm that the

optimum decision is 37 units.

Newsvendor Model-Demand Distribution Continuous

Order y such that

Prob(Demand < y) = MP .

ML+MP

y

Summary

Uses and Costs of inventory EOQ Model

– lead time OK– non-instantaneous replenishment

OK– constant demand rate– no backorders

Demand Stochastic? Newsvendor model