Ch12d
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Transcript of Ch12d
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EOQ: How much to order; ROP: When to order
Re-Order Point (ROP) in periodic inventory system is the start ofthe period. Where is ROP in a perpetual system?
ROP in Perpetual System: Quantity on hand drops to apredetermined amount.
If there were no variations in demand, and demand was reallyconstant; ROP is when inventory on hand is equal to the[average] demand during lead time.
Lead Time is the time interval from placing an order until receivingthe order.
When to re-order with EOQ Ordering
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The greater the variability of demand during lead time, the greater
the need for additional inventory to reduce the risk of shortage(i.e., the greater the safety stock).
When variability exists in demand or lead time, actual demandduring lead time may exceed average demand during lead time.
If ROP is equal to average demand during lead time, there is 50%probability of satisfying all the demand.
But we want this probability to be greater than 50%. We prefer90%, 95%, or 99%.
Safety Stock (SS): serves to reduce the probability of a stockoutduring the lead time.
ROP in perpetual inventory control is the inventory level equal tothe average demand during the lead time+ a safety stock(to cover demand variability)
When to re-order with EOQ Ordering
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Example 1
Average demand for an inventory item is 200 units per
day, lead time is three days, there is no variation indemand and lead time and therefore, safety stock iszero. What is the reorder point?
In a perpetual inventory system, the ROP is the point at whichinventory on hand is equal to the [average] demand inlead time plus safety stock.
ROP = 3(200)
Whenever inventory level drops to 600 units we place anorder.
We receive the order in 3 days. Because there is no variations
in demand, there is no stock out.
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Example 2
Average demand for an inventory item is 200 units per
day, lead time is three days, and safety stock is 100units. What is the reorder point
In a perpetual inventory system, the ROP is the point at whichinventory on hand is equal to the average demand in leadtime plus safety stock.
ROP = 3(200)+100
Whenever inventory level drops to 700 units we place anorder.
Since there is variation in demand, during the next three days,we may have a demand of 500, 600, 700, or even more.
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Time
Inve
ntory
Demand During Lead Time
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LT
ROP when demand during lead time is fixed
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Time
Inve
ntory
Demand During Lead Time
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Demand During Lead Time is Variable
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Inventory
Time
Demand During Lead Time is Variable
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Average demand
during lead time
A large demand
during lead time
ROP
Time
Quantity
Safety stock reduces risk of
stockout during lead time
Safety Stock
Safety stock
LT
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ROP
Time
Quantity
Safety Stock
LT
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When to re-order (ROP)
Demand during lead time has Normal distribution.
We can accept some risk of being out of stock,
but we usually like a risk of less than 50%.
If we order when the inventory on hand is equal
to the average demand during the lead time;
then there is 50% chance that the demand during
lead time is less than our inventory.
However, there is also 50% chance that the
demand during lead time is greater than our
inventory, and we will be out of stock for a while.
We usually do not like 50% probability of stockout
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RO
P
Risk of a
stockout
Service level
Probability of
no stockout
Safety
stock
0 z
Quantity
z-scale
Safety Stock and ROP
Each Normal variable x is associated with a standard Normal Variable z
x is Normal (Average x , Standard Deviation x) z is Normal (0,1)
There is a table for z which tells us
Given anyprobability of not exceeding z. What is the value of z
Given anyvalue forz. What is the probability of not exceeding z
Average
demand
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Common z Values
Risk Service level z value0.1 0.9 1.28
0.05 0.95 1.65
0.01 0.99 2.33
RO
P
Risk of a
stockout
Service level
Probability of
no stockout
Safety
stock
0 z
Quantity
z-scale
Average
demand
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Relationship between z and Normal Variable x
RO
P
Risk of a
stockout
Service level
Probability of
no stockout
Safety
stock
0 z
Quantity
z-scale
Average
demand
z = (x-Average x)/(Standard Deviation of x)
x = Average x +z (Standard Deviation of x)
= Average x
= Standard Deviation of x
Risk Service z value
level
0.1 0.9 1.28
0.05 0.95 1.650.01 0.99 2.33
x = +z
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Relationship between z and Normal Variable ROP
RO
P
Risk of a
stockout
Service level
Probability of
no stockout
Safety
stock
0 z
Quantity
z-scale
Average
demand
LTD = Lead Time Demand
ROP = Average LTD +z (Standard Deviation of LTD)
ROP = Average LTD +ss
ss = z (Standard Deviation of LTD)
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Safety Stock and ROP
RO
P
Risk of a
stockout
Service level
Probability of
no stockout
Safety
stock
0 z
Quantity
z-scale
Risk Service level z value
0.1 0.9 1.28
0.05 0.95 1.65
0.01 0.99 2.33
Average
demand
ss = z (standard deviation of demand during lead time)
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Demand During Lead Time is Variable N(,)
Average Demand of sand during lead time is 50 tons.
Standard Deviation of demand during lead time is 5 tons.Assume that management is willing to accept a risk no
more that 5%.
What is the service level?Service level = 1-risk of stockout = 1-.05 = 0.95
What is the z value corresponding to this service level?
z = 1.65
What is safety stock (ss)?
ss = z (standard deviation of demand during lead time)
ss = 1.65 (5) = 8.25
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What is ROP?
ROP = Average demand during lead time + ss
and of demand during lead time
ROP = 50 + 1.65(5) = 58.25
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Average Demand of sand during lead time is 75 tons.
Standard deviation of demand during lead time is 10 tons.Assume that the management is willing to accept a risk no
more that 10%.
What is the service level?
Service level = 1-risk of stockout = 1-0.1 = 0.9
Risk Service level z value0.1 0.9 1.28
0.05 0.95 1.65
0.01 0.99 2.33
Example 2;total demand during lead time is variable
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z= 1.28What is safety stock?
ss = 1.28(10) = 12.8
ROP = Average demand during lead time + ss
ROP = 75 + 12.8 = 87.8
The general relationship between service level, risk, and
safety stock: Service level increases
Risk decreases
ss increases
Example 2;total demand during lead time is variable
f f
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: Demandd
: Average Demandd
:Standard deviation of demandd
: Lead timeLT
Ifdemand is variable and Lead time is fixed
Average demand during lead time : ( )LT d
: LTdLT d
:Standard deviation of demand during lead timedLT
and of demand per period and fixed LT
d f d d i d d fi d LT
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Average Demand of sand is 50 tons per week.
Standard deviation of the weekly demand is 3 tons.Lead time is 2 weeks.
Assume that the management is willing to accept a risk
no more that 10%. z = 1.28
: 50 tonsd: 2 weeksLT : 3 tonsd
dROP d LT z LT
50 2 1.28 2 3ROP
100 5.43ROP
and of demand per period and fixed LT
d f L d Ti d fi d d d
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: Lead timeLT
: Average Lead timeLT
:Standard deviation of Lead TimeLT
Iflead time is variable and demand is fixed
Average demand during lead time : ( )LT d
:dLT LT d
: Demandd
and of Lead Time and fixed demand
L d Ti V i bl D d fi d
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Lead Time Variable, Demand fixed
Demand of sand is fixed and is 50 tons per week.
The average lead time is 2 weeks.Standard deviation of lead time is 0.5 week.
Assuming that the management is willing to accept a risk
no more that 10%. Compute ROP and SS.
Acceptable risk; 10% z = 1.28
:50 tonsd : 2 weeksLT : .5 weekLT
LTROP d LT zd 50 2 1.28 50 .5ROP
100 32ROP
A i t 12 d
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(a) Average demand per day is 20 units and lead time is 10 days.
Assume zero safety stock (ss=0). Compute ROP.
(b) Average demand per day is 20 units and lead time is 10 days.
Assume 70 units of safety stock. ss=70. Compute ROP.
(c) If average demand during the lead time is 200 and standard
deviation of demand during lead time is 25. Compute ROP at
90% service level. Compute ss.
(d) If average demand per day is 20 units and standard deviation
of demand is 5 per day, and lead time is 16 days. ComputeROP at 90% service level. Compute ss.
(e) If demand per day is 20 units and lead time is 16 days and
standard deviation of lead time is 4 days. Compute ROP at
90% service level Compute ss
Assignment 12.d