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Inventory Policy Decisions

What is Inventory?

Customer
service goals
•The product
•Logistic service
•Order processing &
information systems
Location Strategy
•Location Decisions
•The network planning process

Transport Strategy
•Transport fundamentals
•Transport decisions

CONTROLING

Inventory Strategy
•Forecasting
•Inventory Decisions
•Purchasing and supply scheduling
decisions
•Storage fundamentals
•Storage decisions

ORGANIZING

Inventories are stockpiles of raw materials, supplies, components,
work in process, and finished goods that appear at numerous points
throughout a firm’s production and logistics channel
...
Consider why a firm might
want inventories at some level in its operations, and why that firm
would want to keep them to a minimum
...


Reduce Costs




It encourage economies of production by allowing larger, longer and more
level production runs
...

Forward buying

Appraisal of Inventories
Arguments against Inventories


Being overstocked is much more defensible from criticism than being
short of supplies
...




Inventories can mask quality problems
...


about

the

Product Availability


A primary objective of inventory management is to ensure that
product is available at the time and in the quantities desired
...
This probability or item fill rate is referred to as
the service level, and, for a single item, can be defined as

Service Level=1-

Expected number of units out of stock annually
Total annual demand



Service level is expressed as a value between 0 and 1
...


Weighted average fill rate (WAFR)


A number of orders from many customers will show that a mixture of
items can appear on any one order
...


Example:
A specialty chemical company receives orders for one of its paints
products
...
From a sampling of orders
over time, the items appear on orders in seven different
combinations with frequencies as noted in table below
...
95; SLB = 0
...
80
...

ITEM
COMBINATION
ON OREDR
FREQUENCY
OF ORDER

A

B

C

A, B

A, C

B, C

A, B, C

0
...
1

0
...
2

0
...
1

0
...
1

0
...
2

0
...
1

0
...
2

PROBABILITY OF
FILLING ORDER
COMPLETE (2)

0
...
900

0
...
95)(0
...
855

(0
...
80)
= 0
...
90)(0
...
720

(0
...
90) (0
...
684

(3) = (1) X (2)
MARGINAL
VALUE

0
...
090

0
...
171

0
...
072

0
...
095 + 0
...
160 + 0
...
076 + 0
...
132 = 0
...
For determining the order quantity to replenish an
item in inventory, these relevant costs trade-off are shown in figure
below
...


Carrying Costs


1
...
These costs can be collected into four classes:
Space Costs
Space costs are charges made for the use of the volume inside the storage
building
...
If the space is privately owned or contracted,
space costs are determined by allocating space-related operating costs,
such as heat and light, as well as fixed costs, such as building and storage
equipment costs, on a volume-stored basis
...


Capital Costs
Capital costs refer to the cost of the money tied up in inventory
...


Carrying Costs
3
...


4
...


Out-of-Stock Costs


Out-of-stock costs are incurred when an order is placed but can not
be filled from the inventory to which the order is normally assigned
...
The cost is the profit that
would have been made on this particular sales and may include additional cost for
the negative effect that the stockout may have on future sales
...


Push Inventory Control


This method is appropriate where production or purchase quantities
exceed the short-term requirements of the inventories into which the
quantities are to be shipped
...




Push is always a good approach to inventory control where
production or purchasing is the dominant force in determining the
replenishment quantities in the channel
...





How much inventory should be maintained at each stocking point?
How much of purchase order or production run should be allocated to each
stocking point?
How should the excess supply over requirements be apportioned among the
stocking points?

Push Inventory Control


A method of pushing quantities into stocking points involves the
following steps:
1
...


2
...


3
...


4
...


5
...


6
...


7
...


Push Inventory Control


Example: When the tuna boats are sent to the fishing grounds, a packer of
tuna products must process all the tuna caught since storage is limited and,
for competitive reasons, the company does not want to sell the excess of this
valued product to other packers
...
There is only enough storage at the plant
for one month’s demand
...




For upcoming month, the needs of each warehouse were forecasted, the
current stock level checked, and desired stock availability level noted for
each warehouse
...

Total Requirements = Forecast + ( z x Forecast error )
where, z = number of standard deviations on the normal distribution curve beyond the forecast (the
distribution mean) to the point where 90 percent of the area under the curve is represented
...
28
hence, total requirements for warehouse 1 = 10,000 + (1
...




Summing the net requirements (110,635) shows that 125,000 – 110,635 = 14,365, which is the
excess production that needs to be prorated to the warehouses
...

WAREHOUSE

(1) TOTAL
REQUIREMENTS

(2) ON HAND

(3) = (1) – (2)
NET
REQUIREMENTS

(4) PRORATED
EXCESS

(5) = (3) + (4)
ALLOCATION

1

12,560 lb

5,000

7,560 lb

1,105 lb

8,665 lb

2

52, 475

15,000

37
...

Single-Order Quantity:


Many practical inventory problems exist where the product involved are perishable or
the demand for them is a one-time event
...
have a short and defined shelf life, and they are not available
for subsequent selling periods
...

Only one order can be placed for these products to meet such demand
...

That is, Q* is found at the point where the marginal profit on the next unit sold
equals the marginal loss of not selling the next unit
...
That is,
CPn (Loss) = (1 – CPn) (Profit)
where CPn represents the cumulative frequency of selling at least n units of the product
...


Single-Order Quantity


Example: A grocery store estimates that it will sell 100 pounds of its specially
prepared potato salad in the next week
...
The supermarket can sell
the salad for $5
...
It pays $2
...

Since no preservatives are used, any unsold salad is given to charity at no
cost
...

Solution:
CPn =

Profit
(5
...
50)

 0
...
99  2
...
50

From the normal distribution curve, the optimum Q* is at the point of 58
...

This is a point where z = 0
...
21(20 lb) = 104
...


Single-Order Quantity


When demand is discrete, the order quantity may be between whole values
...




Example: An equipment repair firm wishes to order enough spare parts to keep a
machine tool running throughout a trade show
...
He pays $70 for each part
...
The demand for
the parts is estimated according to the following distribution
...
10

0
...
15

0
...
20

0
...
30

0
...
20

0
...
05

1
...
555
Profit + Loss (95  70)  (70  50)

1
...

Rounding up, we choose Q* = 3
...
Inventory replenishment orders repeat over time
and may be supplied instantaneously in their entirety, or the items in the
orders may be supplied over time
...


The quantity that will be used to replenish the inventory on a periodic basis

2
...


This is a problem of balancing conflicting cost pattern
...

Ford Harris recognized this problem as early as 1913
...


Instantaneous Resupply


The basic EOQ formula is developed from a total cost equation involving
procurement cost and inventory carrying cost
...
For a particular part, the annual demand is expected
to be 750 units
...
Find EOQ and
optimum time between orders
...
58 or 93units
IC
(0
...
58
T =
=
= 0
...
12 x 52 (weeks per year) = 6
...




Reorder point is the quantity to which inventory is allowed to drop before a replenishment
order is placed
...

The reorder point (ROP) is,
ROP = d x LT
where,
ROP = reorder point quantity, units
d = demand rate, in time units
LT = average lead time, in time units
The demand rate (d) and the average lead time (LT) must be expressed in the same time
dimension
...




In some manufacturing and resupply processes, output is continuous for a
time, and it may take place simultaneously with demand
...

Quantity on hand



Production
Production
less
demand

LT



Demand only

Time

The order quantity now becomes the production run, or production lot size quantity
(POQ) labeled Qp*
...
Computing Qp* only makes sense when the output
rate p exceeds the demand rate d
...




Therefore, we must plan for the situation where not enough stock may be on
hand to fill customer requests
...




The amount of this safety, or buffer, stock sets the level of stock availability
provided to customers by controlling the probability of a stock out occurring
...
Reorder point method
2
...




When inventory is depleted to the point where its level is equal to or less
than a specified quantity called the reorder point, an EOQ of Q* is placed on
the supplying source to replenish the inventory
...




The entire quantity Q* arrives at a point in time offset by the lead time
...




The probability of this occurring is controlled through raising or lowering the
reorder point and by adjusting Q*
...




The values of x’ and s’d are usually not known directly, but they can be easily
estimated by summing a single period demand distribution over the length of
the lead time
...
Lead
time is 3 weeks
...
3
x’ = 300

The mean of the DDLT distribution is simply the demand rate d times LT, or
x’ = d x LT



The varience of DDLT distribution is found by adding the variances of the
weekly demand distributions
...
The following data have been
collected for this item held in inventory
...
error of forecast, sd

3,099 units



Replenishment lead time, LT

1
...
11/unit



Cost for processing vendor order, S

$10/order



Carrying cost, I

20% per year



In –stock probability during lead time, P

75%

A Reorder Point Model with Uncertain Demand



Solution:
The reorder quantity,

Q* 

2 DS
2(11,107)(10)

 11, 008 units
IC
(0
...
11)

'
sd  sd LT  3, 099 1
...
67 for area under the normal distribution curve is equal to 0
...
5) + (0
...
795) = 19,203 units

So, when the effective inventory level drops to 19,203 units, place a replenishment order for 11,008
units
...
67 x 3,795) = 8,047 units

A Reorder Point Model with Uncertain Demand


Solution:



Total Relevant Cost (TC)
Total cost = Order cost + Carrying cost, regular stock + Carrying cost, safety stock + Stockout cost
D
Q
D '
'
TC  S  IC  ICzsd  ksd E( z )
Q
2
Q
11,107(12)(10)
11, 008
11,107(12)
=
 0
...
11)(
)  0
...
11)(0
...
01)(3, 795)(0
...
008
2
11, 008
= $367
...
67) = 0
...
150)
= 1
 0
...




The optimum balance between service and cost can be calculated
...
Approximate the order quantity from the basic EOQ formula,

2DS
IC

Q* 

2
...
Find the z value that corresponds to P in the normal distribution table
...


A Reorder Point Method with Known Stockout Costs
3
...


Repeat steps 2 and 3 until there is no change in P or Q
...


5
...


A Reorder Point Method with Known Stockout Costs


Example: Repeating the tie bar example, with the known stockout cost of $0
...




Solution:



Estimate Q,



Estimate P,

Q

2 DS
2(11,107)(12)(10)

 11, 008 units
IC
0
...
11)



Q/C
11, 008(0
...
11)
 1
 0
...
01)
From Normal distribution table, for P = 0
...
92, and from unit normal loss integral table



Revise Q
...
Now,

P  1

Q

'
2 D  S  ksd E( z ) 





E(z=0
...
0968
...
01(3, 795)(0
...
20(0
...
20) /(0
...
79
11,107(12)(0
...
79 = 0
...
81) = 0
...
01(3, 795)(0
...
20(0
...




The Results are, P = 0
...


A Reorder Point Method with Demand and Lead Time
Uncertainty


Accounting for uncertainty in the lead time can extend the realism of the
reorder point model
...




Adding the demand variance to the lead time variance gives a revised
formula as,
'
2
2
sd  LTsd  d 2 sLT
Where, sLT is the lead time standard deviation
...
A reorder point is the method
for inventory control
...
The distribution of these times
that form the order replenishment lead time are shown in figure
...

We also know that,
I = 10% / year

S = $10 / order
C = $5 / unit
P = 0
...


(Multiple time elements throughout a supply channel)

A Reorder Point Method with Demand and Lead Time
Uncertainty


Solution:
'
2
2
sd  LTsd  d 2 sLT
2
2
sLT  s 2  si2  so  0
...
0  0
...
35 days
p

LT  x p  xi  xo  1  4  2  7 days
Now,
'
sd  7 x102  1002 x1
...
16 units

and,

where,

Q*
'
AIL 
 zsd
2
Q* 

2(100)(10)
 63 units
0
...
33(119
...


2
...


4
...
This might be done on a
cycle count basis, in which a portion of stock is reviewed each day or week,
perhaps on an ABC basis
...

Items ordered have a significant effect on the supplying plant’s production
output, and order predictability is desirable
...


A Periodic Review Model with Uncertain Demand:
Single Item Control


One important difference in the periodic review model is that demand
fluctuations during order interval and the lead time must be protected
against, whereas only demand fluctuations during the lead time are
important in calculating safety stock using reorder point method
...


A Periodic Review Model with Uncertain Demand:
Single Item Control


The inventory level for an item is audited at predetermined intervals (T)
...




Thus, inventory is controlled through the setting of T* and M*
...
That is,
2DS
Q* 

IC

Order Quantity Q*
T 

Annual Demand D



The review interval is,



The point of maximum level (M*) may be calculated as,

*

'
M *  d (T *  LT )  z(sd )



Here, s’d is the standard deviation of the DD(T*+LT) distribution
...


A Periodic Review Model with Uncertain Demand:
Single Item Control


Example: Buyers Products Company distributes an item known as a tie bar,
which is a U-bolt used on truck equipment
...
Develop a periodic review policy for
it
...
error of forecast, sd

3,099 units



Replenishment lead time, LT

1
...
11/unit



Cost for processing vendor order, S

$10/order



Carrying cost, I

20% per year



In –stock probability during lead time, P

75%

A Periodic Review Model with Uncertain Demand:
Single Item Control


Solution:


The optimum order quantity, Q* 

2 DS
2(11,107)(10)

 11, 008 units
IC
(0
...
11)

Q* 11, 008
T 

 0
...
991  1
...
75 is,
'
M *  d (T *  LT )  z (sd )

= 11,107(0
...
5)  0
...


A Periodic Review Model with Uncertain Demand:
Single Item Control


Solution:


Average inventory level,

dT *
11,107(0
...
67(4,891)  8, 780 units
2
2


Total Cost,
D
Q
D '
'
TC  S  IC  ICzsd  ksd E( z )
Q
2
Q
11,107(12)(10)
11, 008
11,107(12)
=
 0
...
11)(
)  0
...
11)(0
...
01)(4,891)(0
...
008
2
11, 008
= $ 403
...
150)
= 1
 0
...



An inventory joint ordering policy involves determining a common inventory
review time for all jointly ordered items, and then finding each item’s
maximum level (M*) as dictated from its particular costs and service level
...




The maximum level for each item is,
'
M i*  di (T *  LT )  zi (sd )i



Total relevant cost is,
Total cost = Order cost + Regular stock carrying cost + Safety stock carrying cost +
Stockout cost

TC 

O   Si
i

T



TI  Ci Di
i

2

'
 I  Ci zi ( sd )i 
i

1
 ki (sd' )i ( E( z ) )i
T i

A Periodic Review Model with Uncertain Demand:
Joint Ordering


Example: Two items are to be jointly ordered from the same vendor
...

ITEM

A

B

Demand Forecast, units/day

25

50

Error of the forecast, units/day

7

11

Lead time, days

14

14

Inventory carrying cost, % / year

30

30

Procurement cost, $/order/item

10

10

with a common cost of, $/order

30

In-stock probability during order
cycle plus lead time

70%

75%

Product value, $/unit

150

75

Stockout cost, $/unit

10

15

Selling days per year

365

365

A Periodic Review Model with Uncertain Demand:
Joint Ordering


Solution:



The common review time for jointly ordered items is,

T 
*

2(O   Si )
I  Ci Di



2 30  (10  10)
 4
...
30 / 365150(25)  75(50)

'
(sd ) A  (sd ) A T *  LT  7 4  14  29
...
52(29
...
67 units



The maximum level for item B is,
*
'
M B  dB (T *  LT )  zB (sd ) B  50(4  14)  0
...
67)  931 units

A Periodic Review Model with Uncertain Demand:
Joint Ordering


Solution:



The Average inventory level for item A is,
d AT *
4
'
AILA 
 z A ( sd ) A  25  0
...
70)  65 units
2
2



The Average inventory level for item B is,
d BT *
4
'
AILB 
 z B ( sd ) B  50  0
...
67)  131 units
2
2



Total relevant cost is,
O   Si TI  Ci Di
1
'
'
i
i
TC 

 I  Ci zi ( sd )i   ki ( sd )i ( E( z ) )i
T
2
T i
i
TC 

30  2(10)  4 / 365 0
...
30 150(0
...
70)  75(0
...
67) 
1
10(29
...
1917)  15(46
...
1503)
4 / 365
 $25, 258 per year


A Periodic Review Model with Uncertain Demand:
Joint Ordering


Solution:



The Service level for item A is,

SLA  1 
= 1

'
sd A ( E( z ) ) A
*
QA

29
...
1917)
 0
...
03 (25) =101 units)


The Service level for item B is,
SLB  1 
= 1

'
sdB ( E( z ) ) B
*
QB

46
...
1503)
 0
...
03(50)



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