Some general systems model.
A. Inventory model. Why are we interested in it?
What do we really study in such cases.
General strategy of matching two dissimilar processes, say,
matching a fast process with a slow one.
We need an inventory or a buffer or temporary storage.
Producer process
Consumer process
Stock on-hand
How much to stock?
How much to order? When to order?
First a simple model. A simple picture is needed. A singleitem model.
One complete cycle
profile
Stock
On-hand
time
An improvement on the basic model.
Place order
Now.
Safety level
Stock
On-hand
time
Let’s try the simplest model first. A typical profile
may look like (over multiple cycles) if the demand
is constant.
Stock
Volume
Q
T
Time
This is a simplest deterministic model.
Here D = demand rate (constant over time)
A = stock-ordering cost (once per cycle)
T = cycle length in days ..
v = inventory carrying cost per item per unit time
Q = optimum order amount
T
1
Average stock on hand in a cycle = I   I ( t )dt where
T0
I(t) is the amount of stock available at time t.
In our case, I ( t )  Q  Dt . Thus, I 
Q
2
Therefore, inventory carrying cost during a cycle =
The total relevant cost per cycle = A 
The relevant cost per unit cycle, C =
But T 
C
Q
vT
2
Q
vT
2
1
Q
( A  vT )
T
2
Q
. Therefore,
D
AD 1
 Qv
Q 2
dC
0
dQ Q*

. And at the optimum order amount Q* ,
AD
( Q* )2

2 AD
v
 Q* 
v
2
2 AD
is called Economic Order Quantity (EOQ).
v
This is EOQ model.
Q* 
Can we relax some of the assumptions? Make our model
more realistic?
Irregular but deterministic demand
Stock
On-hand
Irregular & constant
Demand D
Time
At time ti , demand is d i . Suppose at t  0 we’ve enough
inventory to last t units of time.
An extension. Stochastic demand rate with the probability
of demand distribution given.
Stock
On-hand
Stochastic
Demand D
Time
Allow occasional shortage! Pay in goodwill if shortage
occur.
Order if stock goes below the reorder point. Allow leadtime L for replenishment if L is stochastic.
Stochastic
Demand D
Stock
On-hand
Time
Typical Inventory profile under these assumptions.
We can expand the scope of the model progressively. If this
becomes mathematically intractable, try to simulate it.
Irregular
demand
But known
stock
H
Time
At time t i demand is d i . Other assumptions, as before.
At t  0 , we have sufficient stock to last t units of time.
Then the relevant cost per unit time is (note that H is the
area under the demand curve)
A Hv A v n
A v
C( t )  
  t  di    di
t
t
t 2t i 1
t 2
1
appearing in the formula? Consider the area of a
2
trapezium such as this
(Why
d
e
t
Its area =
1
t( d  e )
2
)
If C ( t ) were optimum, C ( t )  C ( t  1 ). This implies
A v t
A
v t 1
  di 
  di
t 2 i 1
t  1 2 i 1
 t ( t  1 )d t  1 
optimum t, when the inequality is satisfied.
Stochastic Inventory model
2A
v
Safety level
stock
y
s
Lead time =L
Assumptions.
a. Probabilistic demand. Avg. demand rate = D
b. Reorder when stock reaches level s. Order an
optimal Q
c. Lead time L is not constant
d. Replenishment at the end of lead time is immediate
e. Shortage cost incurred when inventory is negative.
It is C s per unit stock per unit time.
f. Order cost & inventory holding cost are as before,
namely A and v, respectively.
g. p( y ) is the probability that y unit is demanded
during lead time.
Some computations.
A. What is the average inventory during lead time L?
O
Safety level
y
s
C
A
B
Average volume of the inventory during L
1
s s y
= Average height of OABC = ( OA  BC )= 
2
2
2
Expected inventory during L =
s 1
  ( s  y ) p( y )
2 2
B. Next, we compute expected inventory level after a
replenishment until next reorder
After replenishment, inventory level = s  M L  Q
M L = average consumption during lead time
At reorder point the inventory = s
Therefore, the average inventory level =
1
1
( s  s  M L  Q )= ( 2 s  M L  Q )
2
2
C. Next, we compute time-durations.
s  ML  Q
T1
ML
T
T1 = expected time interval between order arrival and
reorder point
T = Cycle time, i.e. time interval between two consecutive
arrivals
( Q  s  ML )  s Q  ML

D
D
M
Q
T  T1  L Therefore, T 
D
D
Then, T1 
D. Next, we compute the expected shortage cost.
Shortage
Expected shortage during lead time =

 ( y  s ) p( y )  S L
y s
Note that
s


y 0
y 0
y s1
 ( s  y ) p( y )   ( s  y ) p ( y )   ( s  y ) p( y )
 s  M L  SL
Now we compute the relevant inventory cost X per cycle.
v
v
X  A  ( 2 s  M L  Q )T1  ( 2 s  M L  S L )( T  T1 ) 
2
2
Cs SL
Total relevant const per unit time is
C
X AD
Q
  vM L C s D 

 v   M L  s  

 SL
T
Q
Q 
2
  2Q
C
 0 and solving for it.
Q
2 DC s 
 2 AD
Q*  
 SL( M L 
)
v
 v

Optimum Q, Q* is obtained from
All these models pertain to a single-item inventory. Next
consider the possibility to two or more-items inventory
(multi-inventory) system, with the additional control
feature:
● Assume replenishments for all items from a single
source.
● If any one item triggers a replenishment order to go
out, check if any other item from this supplier could be
ordered now even if it hasn’t touched the reorder point.
e.g.
A
B
In this case, perhaps, replenishment orders for both A and
B could go out at the same time. But not in the following
case, perhaps!
A
B
Perhaps, we could carry out simulation to decide the order
quantity of each item given that they could often, but not
always, be launched earlier in advance.