When to re-order with EOQ Ordering
EOQ: How much to order; ROP: When to order
Re-Order Point (ROP) in periodic inventory system is the start of
the period. Where is ROP in a perpetual system?
ROP in Perpetual System: Quantity on hand drops to a
predetermined amount.
If there were no variations in demand, and demand was really
constant; 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 receiving
the order.
When to re-order with EOQ Ordering
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 demand
during 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 prefer
90%, 95%, or 99%.
Safety Stock (SS): serves to reduce the probability of a stockout
during the lead time.
ROP in perpetual inventory control is the inventory level equal to
the average demand during the lead time + a safety stock
(to cover demand variability)
Example 1
Average demand for an inventory item is 200 units per
day, lead time is three days, there is no variation in
demand and lead time and therefore, safety stock is
zero. What is the reorder point?
In a perpetual inventory system, the ROP is the point at which
inventory on hand is equal to the [average] demand in
lead time plus safety stock.
ROP = 3(200)
Whenever inventory level drops to 600 units we place an
order.
We receive the order in 3 days. Because there is no variations
in demand, there is no stock out.
Example 2
Average demand for an inventory item is 200 units per
day, lead time is three days, and safety stock is 100
units. What is the reorder point
In a perpetual inventory system, the ROP is the point at which
inventory on hand is equal to the average demand in lead
time plus safety stock.
ROP = 3(200)+100
Whenever inventory level drops to 700 units we place an
order.
Since there is variation in demand, during the next three days,
we may have a demand of 500, 600, 700, or even more.
Inventory
Demand During Lead Time
Time
ROP when demand during lead time is fixed
LT
Inventory
Demand During Lead Time
Time
Demand During Lead Time is Variable
Demand During Lead Time is Variable
Inventory
Time
Quantity
Safety Stock
A large demand
during lead time
Average demand
during lead time
ROP
Safety stock reduces risk of
stockout during lead time
Safety stock
LT
Time
Quantity
Safety Stock
ROP
LT
Time
When to re-order (ROP)
Demand during lead time has Normal distribution.
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 stock
out
We can accept some risk of being out of stock,
but we usually like a risk of less than 50%.
Safety Stock and ROP
Risk of a
stockout
Service level
Probability of
no stockout
RO
P
Average
demand
Quantity
Safety
stock
0
z
z-scale
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 any probability of not exceeding z. What is the value of z
Given any value for z. What is the probability of not exceeding z
Common z Values
Risk of a
stockout
Service level
Probability of
no stockout
RO
P
Average
demand
Quantity
Safety
stock
0
z
Risk
Service level
0.1
0.05
0.01
0.9
0.95
0.99
z-scale
z value
1.28
1.65
2.33
Relationship between z and Normal Variable x
Risk of a
stockout
Service level
Probability of
no stockout
RO
P
Average
demand
Quantity
Safety
stock
0
z
z-scale
z = (x-Average x)/(Standard Deviation of x)
x = Average x +z (Standard Deviation of x)
μ = Average x
 x = μ +z σ
σ = Standard Deviation of x
Risk
0.1
0.05
0.01
Service z value
level
0.9
1.28
0.95
1.65
0.99
2.33
Relationship between z and Normal Variable ROP
Risk of a
stockout
Service level
Probability of
no stockout
RO
P
Average
demand
Quantity
Safety
stock
0
z
z-scale
LTD = Lead Time Demand
ROP = Average LTD +z (Standard Deviation of LTD)
ROP = Average LTD +ss
ss = z (Standard Deviation of LTD)
Safety Stock and ROP
Risk of a
stockout
Service level
Probability of
no stockout
RO
P
Average
demand
Safety
stock
0
Risk
0.1
0.05
0.01
Service level
0.9
0.95
0.99
Quantity
z
z-scale
z value
1.28
1.65
2.33
ss = z × (standard deviation of demand during lead time)
Demand During Lead Time is Variable N(μ,σ)
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
μ and σ of demand during lead time
What is ROP?
ROP = Average demand during lead time + ss
ROP = 50 + 1.65(5) = 58.25
Example 2; total demand during lead time is variable
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
0.1
0.9
0.05
0.95
0.01
0.99
z value
1.28
1.65
2.33
Example 2; total demand during lead time is variable
z = 1.28
What 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
μ and σ of demand per period and fixed LT
If demand is variable and Lead time is fixed
LT : Lead time
d : Demand
d : Average Demand
Average demand during lead time : ( LT )d
 d : Standard deviation of demand
 dLT : Standard deviation of demand during lead time
 dLT : LT d
μ and σ of demand per period and fixed LT
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
LT : 2 weeks
d : 50 tons
ROP  d  LT  z LT  d
ROP  50  2  1.28 2  3
ROP  100  5.43
 d : 3 tons
μ and σ of Lead Time and fixed demand
If lead time is variable and demand is fixed
d : Demand
LT : Lead time
LT : Average Lead time
Average demand during lead time : ( LT )d
 LT : Standard deviation of Lead Time
 dLT : d LT
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
d : 50 tons
LT : 2 weeks
ROP  d  LT  zd LT
ROP  100  32
 LT : .5 week
ROP  50  2 1.28  50  .5
Assignment 12.d
(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. Compute
ROP 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.