Other Inventory Models

1
Other Inventory Models
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Continuous Review or Q System
The EOQ model is based on several assumptions,
one being that there is a constant demand. This
may not be realistic. Next we consider some
models that allow demand to occur more at random.
In the EOQ model once the best order size, Q,
was determined, the firm would order at a regular
interval that divided the period (year) up into
D/Q times. So, the same amount was ordered at
the same interval. In the Q system we will study
the same amount will be ordered at different
intervals of time. Then, in the P system (I say
more later) different amounts will be ordered at
a fixed time interval. The Q system relies on
the normal distribution, so we turn there next.
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Q System
Say daily demand is normal with mean 200 and
standard deviation 150. Then we might have a
question like, what is the probability daily
demand will be 350, or less?
200 350 daily demand

To answer the question we see the normal
distribution and we have to find the area under
the curve to the left of 350. We resort to a z
calculation (value (like 350) minus mean)/st
dev. Our z (350 200)/150 1.00 (usually we
round z to 2 decimals). Then we go to a z table
and see the value with z 1.00 of .8413 and we
say the probability is 84.13.
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Q System
Page 344 has a table that rounds z to 1 decimal
and the service level is 84.1 (84.13 rounded to 1
decimal). So this table really has an
abbreviated version of the z table. Now, when
orders are placed to replenish inventory, it
takes some number of days for the order to
arrive. If demand during this lead time is
higher than our trigger reorder level R (the
level such that when our inventory reaches this
level an order of Q Will be made), the demand
will not be met and we say there is a stockout.
If demand is less than this trigger reorder level
then demand will be met and we can calculate the
fill rate or service level as the percentage of
customer demand satisfied by inventory.
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Q System
Before we had an example of daily demand being
normal with mean 200 and standard deviation
150. Now, if the lead time is 4 days, then the
demand during the 4 days of lead time will be
normal with mean 4 times daily mean demand of
200 800 and standard deviation sqrt(4
days)times daily standard deviation of 150
300. On the next slide I show a normal
distribution with mean of 800 and a standard
deviation of 300. We will use this graph and
related ideas to help us determine what the
trigger reorder amount R should be.
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Q System
800 950 1100 1400 demand over
lead time
Say in general the mean demand over the lead time
period is m, which equals 800 here. For a while
I am just going to play a hypothetical game. I
am going to ask what would happen if our trigger
order amount R is various amounts. In fact I
will look at the cases where R 800, R 950, R
1100 and R 1400. Lets do this next, but
refer back to this slide to see what is going
on.
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Q System
Say we make R 1400, meaning that if our stock
position reaches R we will reorder some amount (I
will say how much to order later). Since our
lead time here is 4 days this will mean that over
the next 4 days if actual demand is 1400 or less
than we will have enough inventory to meet the
demand. But, if actual demand is over 1400 we
will not have enough on hand and there will be a
stockout. (Have you ever gone to a store to buy
something, maybe it was even advertised, and the
store ran out? How do you feel at that point?
Are you bummed out, upset or just plain seething
with anger? Stores do not what to bum you out,
but schtuff happens!)
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Q System
Since demand is random, and here assumed normal,
we can calculate what percentage of the time
demand will be above or below the trigger amount
R, here picked to be 1400. The z for 1400 is
(1400 800)/300 2.0. The table on page 344
tells us in this case we would meet demand 97.7
percent of the time. Thus our service level or
fill rate would be that we meet 97.7 percent of
customer demand from inventory. Similarly, 100
97.7 2.3 percent of the time we would have a
stockout. If R 1100, z (1100 800)/300 1.0
and the service level will be 84.1 and the
stockout will be 15.9. If R 950, z (950
800)/300 .5 and the service level will be 69.1
and the stockout will be 30.9. If R 800, z
(800 800)/300 0 and the service level will
be 50 and the stockout will be 50
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Q System
What I have done here is talk about hypothetical
R values, levels of the stock position that would
trigger an order be made. With different R
values we see different service levels and
stockout s. We could work in reverse to what I
have presented. If demand over lead time has
mean 800 and standard deviation 300, what
should R be to make the service level 97.7? The
z there is 2.0. Thus 2.0 (R 800)/300 and
solving for R we get R 800 2.0(300) 1400.
In general, R m zs mean over lead time
safety stock. Thus, we need to think about how
serious our customers become if there is a stock
out. The more serious the higher the service
level should be and thus the higher the z we pick.
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Q System
I mentioned before I would say how much should be
ordered. Just order the EOQ on a yearly average
demand basis. Thus, if average demand is 200 per
day, and say we are open 5 days a week for 50
weeks then annual average demand is 250(5)(50)
50,000 units per year. If S 20 per order, i
20 per year and C 10 per unit, the the EOQ Q
sqrt2(20)(50000)/.2(10) sqrt(1000000)
1000. So, 1000 units would be ordered when R is
reached. Since annual demand has an average of
50,000 and we order in lots of 1000 we will make
an average of 50 orders per year. Since there
are 250 working days in the year orders will be
made on average every 250/50 5 days.
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Q System
What should R be if you want a service level of
95.0? Note 95.0 is not in the table on page 344,
but it is in the middle of 94.5 and 95.5 so we
take the z in the middle of the associated zs
for a z 1.65. R 800 1.65(300) 1295. The
order amount R depends of the service level
desired. The Q amount to order still is picked
by the EOQ method, but using annual average
demand.