Elena Arcari
Supervisor : Francesco Martinelli
Engineering Sciences
Universit degli Studi di Roma Tor Vergata
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2
One-dimensional supply-chain
0 1 +1
+11 2 - Suppliers b 0, ,- Consumers +1- Feeding rate b 0, , +
1
- Dynamics inventory :()
= +1
Rate receivedproducts
Rate deliveredproducts
- Dynamics delivery rate :
=
= adaptation time interval
() = desired delivery rate
3
Management function
=
= =
(+ + +
)
= 0 if + < 0 + >
=
= 1
= forecast time horizon
Normalization condition
Our strategy
Weighted mean value
= ( () , ( + ) ) where + +
General strategy
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Stability condition
< +1
012
+ =
Stationary solutionsInventory : 0
Delivery rate : 0 = (0)
Dynamic equations in the vicinityof the stationary state :
= 0
= (0)
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Dynamical solution in the vicinity of the stationary state
(l)
Deviation of the inventory:
= 0
() =
+1
Deviation of the delivery rate:
= (0)
() =
1
0 +
Deriving and substituting
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we obtain: + + = ()
2 =1 + (0)
02 =
(0) =
(0)
+1 + +1
Damped forcedharmonic oscillator
model
Solution at steady state under the effect of a periodic
oscillation of the form = 0 cos (u is the last supplier)
() = 0 cos( )
where tan = 2
2 02 =
12 02 2 + 422
Dynamical solution in the vicinity of the stationary state
(ll)
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=(0)
+1 +
+1
For b = u-1
1 =(0)
+ () = 0 cos( )
1 = 10 cos( 1)
Dynamical solution in the vicinity of the stationary state
(lll)
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= 0 cos
10 =(0)
0 1 + 2
(With tan 1 = )
1 = 10 cos( 1)
If10
0> 1 OSCILLATIONS INVENTORIES
WILL INCREASE
Bullwhip effect
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Optimal velocity model
-Dynamics distance between vehicles:
()
= +1
-Dynamics velocity vehicles:
()
=( )
The choice of corresponds to the choice of the control
strategy
= +1
=
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Control strategies (l)- Effects due to choice of adaptation time
and forecast time horizon (considering the
stability condition).
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- Adaptation of the consumption rate to either downstream
suppliers ( pull strategies ) or upstream suppliers ( push
strategies ).
Control strategies (ll)
= 0 = 0.4
Supplier b Supplier b
() ()t t
Push strategy Pull strategy
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- Taking in account the inventories of other suppliers by
choosing specific weights in the management function.
Control strategies (lll)
b+2
b+1
u+1
1 0
= 0 + (1 0)()
With a = b+1, b+2, u+1
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Conclusions
It should be possible to obtain the stabilizing effects due to
the choice of a specificcontrol strategy in more complex systems
than the one-dimensional supply chain
Limited buffers Limited feedingrates
Supply networks
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Thanks for your attention!
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Modelling of supply chains inspired by traffic
dynamicsDiapositiva numero 2One-dimensional supply-chain
Diapositiva numero 4Diapositiva numero 5Dynamical solution in the
vicinity of the stationary state (l) Dynamical solution in the
vicinity of the stationary state (ll) Dynamical solution in the
vicinity of the stationary state (lll)Bullwhip effectOptimal
velocity modelControl strategies (l)Diapositiva numero
12Diapositiva numero 13Diapositiva numero 14Diapositiva numero
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