5-1
PERSEDIAAN PROBABILISTIK:
PERIODIC REVIEW
Replenishment Policies

When and how much to order

Continuous review with (Q,R) policy:


Inv. is continuously monitored and when it drops
to R, an order of size Q is placed
Periodic review with (R,S) policy:

Inv. is reviewed at regular periodic intervals (R),
and an order is placed to raise the inv. to a
specified level (order-up-to level, S)
Periodic Review System:
Order-Up-To-Level (R, S) System
Periodic Review Systems

Continuous Review Systems



Often, we are constrained by WHEN we can order -and it may be periodically



Always knew level of on-hand inventory
Could place an order at any time
Train dispatched once a week
Delivery truck arrives each morning
Thus, we do not need to continuously review
inventory, just check periodically
Periodic Review Systems



If demand were known and constant, we would just
resort to our EOQ solution, possibly modifying it to
meet shipping date
Now: demand is random variable
Setting:





Place an order every T periods
Policy: Order up to S
The value of Q (order quantity) will now change periodically
Previous concern: demand exceeding supply during the
lead time
Now: demand exceeding supply during the period and
lead time, or T + 
Periodic Review
Order up to S
every T periods of
time.
I(t)
S
Order arrives.
Cycle continues.
Demand
Q
Lead Time passes…


T
Time
Expected cost function


Include expected holding, setup,
penalty and ordering (per unit) costs
Average Inventory Level:
S
At level R*,on average,
order Q = S-R* units.
τ periods later, units arrive.
Inventory level?
R*
τ
T
Expected cost function


Include expected: holding, setup,
penalty and ordering (per unit) costs
Average Inventory Level:
S
S-l units present when
Q arrive (expected) as
l units consumed over
leading time.
T
Expected cost function


Include expected: holding, setup,
penalty and ordering (per unit) costs
Average Inventory Level:
S - l
lT units removed
(expected) from
inventory over
time T.
S
T
Expected cost function


Include expected: holding, setup,
penalty and ordering (per unit) costs
Average Inventory Level:
S
S-l
lT
T
Average Inventory Level = S  l  lT  
S-l-lT
lT
2
 S  l 
lT
2
Expected cost function

Include expected:
holding, setup, penalty and ordering (per unit) costs

Average Holding Cost:

lT 
h S  l 

2 

K
Average Set-up Cost:
T
Expected cost function

Expected Shortage per Cycle:

f(x)dx = P(demand in T +  is between x and x + dx)
E (max(l (T   )  S ,0))

  ( x  S ) f ( x ) dx  n( S )
S

Expected Penalty Cost :
n(S)
p
T
Cost Minimization

Expected Cost Function:
lT
K pn(S)
G(S,T)  h(S  l 
) 
 lc
2
T
T

Derivative:

Recall that T and  are given:
dG
p
 h  n(S )
dS
T
Cost Minimization

Derivative:
dG
p
 h  n( S )  0
dS
T

Note : n( S )   ( x  S ) f ( x)dx
S
n( S )  (1  F ( S ))
hT
 1  F (S ) 
p

or:
p  hT
F(S) 
p
Example

A special control board is used in a version of a
product on the production line

The board cost is $122.50

The holding cost rate is 30% per year

Reorders are placed at the start of each week,
and the supplier delivers these parts in one week

The shortage cost is $100 per board due to
worker downtime

Weekly demand is N (μ=125, δ2=104.17)

Set up cost (K) is $120

Find S
Solution

Holding cost is:
h = Ic = .30 (122.50) = 36.75 / 52 = $.7067 per week

Compute:
p  hT 100  (.7067)1
F(S) 

 0.993
p
100

Demand Distribution is Normal


mean = 125
variance = 104.17
Z = 2.455 from Normal table
S    z

S = 125+(2.455)(104.17)1/2 = 150.06
Solution


If penalty cost drops to $10 per unit:
Compute:
p  hT 10  (.7067)1
F(S) 

 0.929
p
10


S = 125+(1.47)(104.17)1/2 = 140
p = 1?
S = 125+(-.54)(104.17)1/2 =
119.49
References

“Production & Operations Analysis” by S.Nahmias

“Factory Physics” by W.J.Hopp, M.L.Spearman

“Inventory Management and Production Planning and
Scheduling” by E.A. Silver, D.F. Pyke, R. Peterson