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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
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
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