Transcript Why is Inventory Important?

Chapter 12: Inventory Control
Purposes of Inventory
1. To maintain independence of
operations
2. To meet variation in product demand
3. To allow flexibility in production
scheduling
4. To provide a safeguard for variation
in raw material delivery time
5. To take advantage of economic
purchase-order size
Inventory Costs
 Holding (or carrying) costs
Costs for capital, storage, handling,
“shrinkage,” insurance, etc
 Setup (or production change) costs
Costs for arranging specific
equipment setups, etc
 Ordering costs
Costs of someone placing an order,
etc
 Shortage costs
Costs of canceling an order, etc
Independent vs. Dependent Demand
Independent Demand (Demand for the final endproduct or demand not related to other items)
Finished
product
E(1
)
Component parts
Dependent
Demand
(Derived demand
items for
component parts,
subassemblies,
raw materials,
etc)
Inventory Systems
 Single-Period Inventory Model
 One time purchasing decision (Example:
vendor selling t-shirts at a football game)
 Seeks to balance the costs of inventory
overstock and under stock
 Multi-Period Inventory Models
 Fixed-Order Quantity Models
 Event triggered (Example: running out of
stock)
 Fixed-Time Period Models
 Time triggered (Example: Monthly sales
call by sales representative)
The Newsvendor Model:
Wetsuit example
 The “too much/too little problem”:
 Order too much and inventory is left over at the
end of the season
 Order too little and sales are lost.
 Example: Selling Wetsuits
Economics:
•
Each suit sells for p = $180
•
Seller charges
c = $110 per suit
•
Discounted suits sell for v = $90
“Too much” and “too little” costs
 Co = overage cost (i.e. order “one too many” --- demand < order amount)
 The cost of ordering one more unit than what you would have
ordered had you known demand – if you have left over inventory the
increase in profit you would have enjoyed had you ordered one fewer
unit.
 For the example Co = Cost – Salvage value = c – v = 110 – 90 = 20
 Cu = underage cost (i.e. order “one too few” – demand > order amount)
 The cost of ordering one fewer unit than what you would have
ordered had you known demand – if you had lost sales (i.e., you under
ordered), Cu is the increase in profit you would have enjoyed had you
ordered one more unit.
 For the example Cu = Price – Cost = p – c = 180 – 110 = 70
Newsvendor expected profit maximizing
order quantity
 To maximize expected profit order Q units so that the expected
loss on the Qth unit equals the expected gain on the Qth unit:
Co  F (Q)  Cu  1  F Q
 Rearrange terms in the above equation ->
Cu
Prob{Demand  Q}  F(Q) 
Co  Cu
 The ratio Cu / (Co + Cu) is called the critical ratio (CR).
 We shall assume demand is distributed as the normal distribution
with mean m and standard deviation s
 Find the Q that satisfies the above equality use NORMSINV(CR)
with the critical ratio as the probability argument.
 (Q-m)/s = z-score for the CR so
 Q=m+z*s
Note: where F(Q) = Probability Demand <= Q
Finding the example’s expected profit maximizing
order quantity
 Inputs:
 Empirical distribution function table; p = 180; c = 110; v =
90; Cu = 180-110 = 70; Co = 110-90 =20
Cu
70

 0.7778
Co  Cu 20  70
 Evaluate the critical ratio:
Find an order quantity Q
such that there is a 77.78%
prob that demand is Q or lower.
 NORMSINV(.7778) = 0.765
 Other Inputs: mean = m = 3192; standard deviation = s = 1181
 Convert into an order quantity
 Q=m+z*s
 = 3192 + 0.765 * 1181
 4095
Single Period Model Example
 A college basketball team is playing in a tournament
game this weekend. Based on past experience they
sell on average 2,400 tournament shirts with a
standard deviation of 350. They make $10 on every
shirt sold at the game, but lose $5 on every shirt not
sold. How many shirts should be ordered for the
game?
Cu = $10 and Co = $5; P ≤ $10 / ($10 + $5) = .667
Z.667 = .432 (use NORMSINV(.667) therefore we need 2,400 +
.432(350) = 2,551 shirts
Hotel/Airline Overbooking
 The forecast for the number of  Newsvendor setup:
customers that DO NOT SHOW
 Single decision when the
UP at a hotel with 118 rooms is
number of no-shows in
Normally Dist with mean of 10
uncertain.
and standard deviation of 5
 Underage cost if X > Y
 Rooms sell for $159 per night
(insufficient number of
 The cost of denying a room to
rooms overbooked).
the customer with a confirmed
 For example, overbook 10
rooms and 15 people do
reservation is $350 in ill-will and
not show up – lose revenue
penalties.
on 5 rooms
 Let X be number of people who
 Overage cost if X < Y
do not show up – X follows a
(too many rooms
probability distribution!
overbooked).
 How many rooms ( Y ) should be
 Overbook 10 rooms and 5
do not show up; pay
overbooked (sold in excess of
penalty on 5 rooms
capacity)?
Overbooking solution
 Underage cost:
 if X > Y then we could have sold X-Y more rooms…
 … to be conservative, we could have sold those rooms at the low rate, Cu =
rL = $159
 Overage cost:
 if X < Y then we bumped Y - X customers …
 … and incur an overage cost Co = $350 on each bumped customer.
 Optimal overbooking level:
 Critical ratio:
Cu
F (Y ) 
.
Co  Cu
Cu
159

 0.3124
Cu  Co 350  159
Optimal overbooking level
 Suppose distribution of “no-shows” is normally distributed with
a mean of 10 and standard deviation of 5
 Critical ratio is:
Cu
159

 0.3124
Cu  Co 350  159
 z = NORMSINV(.3124) = -0.4891
 Y = m + z s = 10 -.4891 * 5 = 7.6
 Overbook by 7.6 or 8
 Hotel should allow up to 118+8 reservations.
Multi-Period Models:
Fixed-Order Quantity Model Model
Assumptions (Part 1)
 Demand for the product is
constant and uniform throughout
the period
 Lead time (time from ordering to
receipt) is constant
 Price per unit of product is
constant
 Inventory holding cost is based
on average inventory
 Ordering or setup costs are
constant
 All demands for the product will
be satisfied (No back orders are
allowed)
Basic Fixed-Order Quantity Model and
Reorder Point Behavior
4. The cycle then repeats.
1. You receive an order quantity Q.
Number
of units
on hand
Q
Q
Q
R
2. Your start using
them up over time.
L
R = Reorder point
Q = Economic order quantity
L = Lead time
L
Time
3. When you reach down to a
level of inventory of R, you
place your next Q sized
order.
Cost Minimization Goal
By adding the item, holding, and ordering costs together, we
determine the total cost curve, which in turn is used to find
the Qopt inventory order point that minimizes total costs
Total Cost
C
O
S
T
Holding
Costs
Annual Cost of
Items (DC)
Ordering Costs
QOPT
Order Quantity (Q)
Deriving the EOQ
Using calculus, we take the first derivative of the total cost
function with respect to Q, and set the derivative (slope) equal
to zero, solving for the optimized (cost minimized) value of Qopt
Q OPT =
2DS
=
H
2(Annual Dem and)(Order or Setup Cost)
Annual Holding Cost
_
We also need a
reorder point to
tell us when to
place an order
R eo rd er p o in t, R = d L
_
d = average daily demand (constant)
L = Lead time (constant)
Basic Fixed-Order Quantity
(EOQ) Model Formula
Total
Annual =
Cost
Annual
Annual
Annual
Purchase + Ordering + Holding
Cost
Cost
Cost
D
Q
TC = DC +
S+
H
Q
2
TC=Total annual
cost
D =Demand
C =Cost per unit
Q =Order quantity
S =Cost of placing
an order or setup
cost
R =Reorder point
L =Lead time
H=Annual holding
and storage cost
per unit of
inventory
EOQ Class Problem 1
Dickens Electronics stocks and sells a particular
brand of PC. It costs the firm $450 each
time it places and order with the
manufacturer. The cost of carrying one PC
in inventory for a year is $170. The store
manager estimates that total annual demand
for computers will be 1200 units with a
constant demand rate throughout the year.
Orders are received two days after
placement from a local warehouse maintained
by the manufacturer. The store policy is to
never have stockouts. The store is open for
business every day of the year. Determine
the following:
 Optimal order quantity per order.
 Minimum total annual inventory costs (i.e.
carrying plus ordering – ignore item costs).
 The optimum number of orders per year (D/Q*)
Problem 1
Demand
Ordering Cost
Carrying Cost
Lead Time
1200
450
170
2
Q*
79.7
Ordering + Carrying= $ 13,549.91
Orders/year =
15.1
Rounding up …
$
80
13,550.00
EOQ Problem 2
The Western Jeans Company purchases denim from Cumberland
textile Mills. The Western Jeans Company uses 35,000 yards
of denim per year to make jeans. The cost of ordering denim
from the Textile Mills is $500 per order. It costs Western
$0.35 per yard annually to hold a yard of denim in inventory.
Determine the following:
a. Optimal order quantity per order.
b. Minimum total annual inventory costs (i.e. carrying plus
ordering).
c. The optimum number of orders per year.
Demand
Ordering Cost
Carrying Cost
a. Optimal order quantity per order.
Q*
35000
500
0.35
10000.0
b. Minimum total annual inventory costs (i.e. carrying plus ordering).
Ordering + Carrying=
$
3,500.00
c. The optimum number of orders per year.
Orders/year =
3.5
Problem 3
A store specializing in selling wrapping
paper is analyzing their inventory
system. Currently the demand for
paper is 100 rolls per week, where
the company operates 50 weeks
per year.. Assume that demand is
constant throughout the year. The
company estimates it costs $20 to
place an order and each roll of
wrapping paper costs $5.00 and
the company estimates the yearly
cost of holding one roll of paper to
be 50% of its cost.
a) If the company currently orders
200 rolls every other week (i.e., 25
times per year), what are its
current holding and ordering costs
(per year)?
D
S
c
i
H
per year
5000
$20.00
$5.00
0.5
$2.50
part a
Ordering
Holding
Total
$500.00
$250.00
$750.00
Problem 3
b)
The company is considering
c)
implementing an EOQ model. If
they do this, what would be the
new order size (round-up to the
next highest integer)? What is
the new cost? How much money in
ordering and holding costs would
be saved each relative to their
current procedure as specified in
part a)?
The vendor says that if they order
only twice per year (i.e., order 2500
rolls per order), they can save 10
cents on each roll of paper – i.e.,
each roll would now cost only $4.90.
Should they take this deal (i.e.,
compare with part b’s answer)
[Hint: For c]. calculate the item,
holding, and ordering costs in your
analysis.]
part c
part b
EOQ
282.8
Ordering
$353.55
Holding
$353.55
Total
$707.11
$$ Saved:
$42.89
283
Ordering
Holding
Total
$40.00
$3,062.50
$3,102.50
Item Cost
Holding
Ordering Net Total
Part b) option $
25,000.00
$353.55 $353.55
$25,707.11
Part c) option
$24,500.00
$3,062.50
Do not accept the deal
$40.00
$27,602.50
Safety Stocks
Suppose that we assume orders occur at a fixed review period and
that demand is probabilistic and we want a buffer stock to ensure
that we don’t run out
Cycle-service level = 85%
Probability of stockout
(1.0 - 0.85 = 0.15)
Average
demand
during
lead time
R
zsL
Safety Stock Formula
Reorder Point = Average demand + Safety stock
Reorder Point = Demand during Lead Time +
Safety Stock
Demand during lead time = daily demand * L
= d*L
Safety stock = Zservice~level * sL
Where sL = square root of L*s2, where s is
the standard deviation of demand for one
day
Problem 4
A large manufacturer of VCRs sells 700,000 VCRs
per year. Each VCR costs $100 and each time
the firm places an order for VCRs the
ordering charge is $500. The accounting
department has determined that the cost of
carrying a VCR for one year is 40% of the VCR
cost. If we assume 350 working days per
year, a lead-time of 4 days, and a standard
deviation of lead time of 20 per day, answer
the following questions.
a)
How many VCRs should the company order
each time it places an order?
b)
If the company seeks to achieve a 99%
service level (i.e. a 1% chance of being out of
stock during lead time), what will be the
reorder point? How much lower will be the
reorder point if the company only seeks a 90%
service level?
D
c $
S $
i
days
L
sigma(d)
part a
700,000.00
100.00
500.00
40%
350
4
20
4183.3
4184
part b
d(L)
For 99%
z(.99)
sigma(L)
Safety Stock
8000
2.33
40
93.05
ROP
8093.1
8094
For 90%
z(.90)
sigma(L)
Safety Stock
1.282
40
51.26
D ROP
41.79
42
Fixed-Time Period Model with
Safety Stock Formula
q = Average demand + Safety stock – Inventory currently on
hand
q = d (T + L) + Z s T +L - I
W here :
q = quanti ti y to be ordered
T = the number of days between revi ews
L = lead ti me i n days
d = forecast averagedai ly demand
z = the number of standard devi ati onsfor a speci fi edservi ceprobabi li ty
s T +L = standard devi ati onof demand over the revi ew and lead ti me
I = current i nventorylevel (i ncludesi tems on order)
Multi-Period Models: Fixed-Time Period Model:
Determining the Value of sT+L
s T+ L =
 s
T+ L
i 1
di

2
Since each day is independent and s d is constant,
s T+ L =
(T + L)s d 2
 The standard deviation of a sequence of random
events equals the square root of the sum of the
variances
Example of the Fixed-Time Period Model
Given the information below, how many units should be
ordered?
Average daily demand for a product is 20 units.
The review period is 30 days, and lead time is 10
days. Management has set a policy of satisfying
96 percent of demand from items in stock. At
the beginning of the review period there are 200
units in inventory. The daily demand standard
deviation is 4 units.
Example of the Fixed-Time Period Model: Solution
(Part 1)
s T+ L =
(T + L)s d =
2
 30 + 10  4  2 = 25.298
The value for “z” is found by using the Excel NORMSINV
function.
q = d(T + L) + Z s T + L - I
q = 20(30+ 10) + (1.75)(25.
298)- 200
q = 800  44.272- 200 = 644.272,or 645u n i ts
ABC Classification System
 Items kept in inventory are not of equal
importance in terms of:

dollars invested

profit potential

sales or usage volume

stock-out penalties
60
% of
$ Value 30
0
% of
Use
30
A
B
C
60
So, identify inventory items based on percentage of total dollar value, where
“A” items are roughly top 15 %, “B” items as next 35 %, and the lower 65% are
the “C” items