peter lohmander 2008-10-26
DESCRIPTION
A two stage raw material stock optimization model To be presented at Swedish University of Agricultural Sciences, Umea, 2008-10-27. Peter Lohmander 2008-10-26. Questions. How should we optimize the stock level until the next period when we consider the following things: - PowerPoint PPT PresentationTRANSCRIPT
A two stage raw material stock optimization model
To be presented at Swedish University of Agricultural Sciences, Umea, 2008-10-27
Peter Lohmander2008-10-26
Questions
• How should we optimize the stock level until the next period when we consider the following things:
• The external deliveries, next period, are stochastic.• We may make ”last minute purchases” in the raw
material market next period if we have to. That is however expensive.
• The value of a possible surplus in the raw material stock in the next period may decrease because of volume and quality losses.
• The interest rate may change.
f(x)
x
_X-m
_X+m
_X
The probability density function of external deliveries in the next period based on the information available during this period.
f(x)
x
_X-m
_X+m
_X k
k is the amount of raw material needed in the industrial processin the next period.
f(x+q)
X+q
_X-m+q
_X+m+qk
q=0
We may purchase raw material alreadynow, and store it until the next period. We buy q units during this period for thatpurpose.
f(x+q)
X+q
_X-m+q
_X+m+qk
q=20
__
0 0
1 1
2 2
k x m qx m q k
C pq hq d y dy y dym m
Present purchasequantity
Present price
Storage cost per unit and period
Discounting factor (one period)
Price of ”last minute” purchase next period
Value per unit of surplus in thenext period stored from the present periodExpected present value
of the raw material cost.
Observation:The costs of the stochatic externaldeliveries are considered as exogenous in this context.
__
0 0
1 1
2 2
k x m qx m q k
C pq hq d y dy y dym m
_ _
2 2
0 02 2 2
k x m q x m q kd y y
C p h qm
2 2_ _
4
dC p h q k x m q x m q k
m
2_2 2 2
_ _ _
2_2 2 2
_ _ _
2 2 2 2 2 2
4
2 2 2 2 2 2
k x m q
k x km kq xm xq mqd
C p h qm
k x m q
xm xq x k mq mk qk
2_2 2 2
_ _
_
2 2 24
2 2 2
k x m q
dC p h q k x km xm
m
kq x q mq
2_2 2 2
_ _
_
4
2
dC p h q k x m q
m
k x km xmd
mkq x q mq
_
02 2
dC dp h q k x m
q m m
In order to minimize the total raw material cost, we let the derivative of the total cost with respect to the quantity that we now will purchase and store until the next period be zero.
2
20 min
2
dCUnique
q m
_
02 2
dC dp h q k x m
q m m
_2
0mp h q k x m
d
_20
mp h q k x m
d
_ 2mq k x m p h
d
_ 2 p hm
q k xd
The optimal quantity to store until the next period!
_ 2 p hm
q k xd
1q
k
_ 1q
x
_ 2 p hm
q k xd
2
0q m
p d
20
q m
h d
_ 2 p hm
q k xd
2
21 1
p h
dqm
2
22
p hq dm
2
2
0
q mp h d
dq
p h d
_ 2 p hm
q k xd
2
21 1
p h
dqm
2
22
0
p hq mp h d
d
q
_ 2 p hm
q k xd
21 p hq
m d
2
2
qp h d
m d
_
12m
q k x p h d
2
20
m p hq
d d
Answers
• Now we have the answers to these questions!• How should we optimize the stock level until the next
period when we consider the following things:• The external deliveries, next period, are stochastic.• We may make ”last minute purchases” in the raw
material market next period if we have to. That is however expensive.
• The value of a possible surplus in the raw material stock in the next period may decrease because of volume and quality losses.
• The interest rate may change.
Questions?
You are welcome to contact me.
Peter Lohmander, Swedish University of Agricultural Sciences, Faculty of Forest Sciences, Dept. of Forest Economics, SE-901 83 Umea, Sweden
• http://www.Lohmander.com