1. Agenda
● Industry Problem
● Demand forecast construction
● Forecast Pattern
● Calculations
● Profit maximizing Order Quantity calc
● Example
2. Industry problem
● Ecommerce companies that sells items for whom customer taste changes very fast.
For example latest toys, gadgets etc.
● Another example could be any technology product with a long lead time to source
components and only a short life before better technology becomes available.
● Purchase too many components and you risk having to sell off obsolete technology.
Purchase too few and you may forgo sizable profits. Juniper netowrks, cisco etc are
companies that can relate to these issues.
3. Demand forecast construction
● What is the probability demand will be <= Q
units for Q value we desire
In statistics we define every random variable
by its distribution function
● Let F(Q) = Prob{ Demand is less than or equal
to Q}
Q F(Q)
2200 0.25
3200 0.75
4200 1
8. Calculations
● Expected actual demand =
Expected A/F ratio X Forecast
● Standard deviation of demand =
Standard deviation of A/F ratios X Forecast
● Average(μ) A/F ratio is = Σ of all A/F ratio / total
SKUs
9. Calculations
With discrete find F(Q) using above two tables.
With continuous distribution
● The avergae A/F ratio = .9976
● μ = .9976*3200 =3192
Similarly sigma (σ) = .369 * 3200 =1181 (use std
dev function in excel).
● Use excel Normdist(Q, 3192, 1181, 1) function
11. Balancing expected loss with gain
● Co is the loss incurred when unit is ordered but not sold.
● Cu is the opportunity cost of not ordering a unit that
could have been sold.
Now that we have defined the overage and underage
costs, we need to choose Q to
● strike the balance between them that results in the
maximum expected profit. Based on our previous
reasoning, we should keep ordering additional units until
the expected loss equals the expected gain.
12. Balancing expected loss with gain
The expected loss on a unit is the cost of having the unit in
inventory (the overage cost) times the probability it is left in
inventory. For the Qth unit, that probability is F(Q): It is left in
inventory if demand is less than Q. Therefore, the expected
loss is Co X F(Q).
● The expected gain on a unit is the benefit of selling a unit (the
underage cost) times the probability the unit is sold, which in
this case occurs if demand is greater than Q.The probability
demand is greater than Q is ( 1 - F(Q)). Therefore, the
expected gain is Cu X (1 - F(Q)) .
●
13. Balancing expected loss with gain
● Co* F(Q) = Cu * (1-F(Q))
So F(Q) = Cu/ (Co+Cu)
Qrder quantity that maximizes expected profit
is the order qty Q such that demand is <= Q
with probability F(Q) or Cu/(Co+Cu)
14. Example Calculation
We need to find order quanity Q such that
there is a 77.78 % probability that demand is
Q or lower.
Suppose we use discrete distribution function
from previous table as our demand FCST then
reading the table F(4064) = .75 and F(4160)
=.78. Therefore , rounding up larger qty 4160
is the “Profit maximizing order quantity”.
15. References
● Zipkin, P . Foundations of Invenltory
Mallagement.
● Cachon and Terwiesch
● Porteus, E. Stochastic lnventory Theory
