a workshop honoring gideon weiss, 2012, yonit barron & david perry
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A Workshop honoring Gideon Weiss, 2012, Yonit Barron & David Perry.
The Make-To-Stock concept was very common in the past. Companies
manufactured inventory in order to feel safe and secure. But, inventory
cost a lot of money and was very expensive.
Therefore, the new trend is JIT. In the JIT concept you manufacture a
minimum amount of inventory, and wish to be in a situation where you
don't have any backorders or lost sales. It is not always simple.
The combination between these two concepts (MTS and JIT) leads us to
build our model, in the sense that when the inventory level is positive, the
production rate is low and when the inventory level is negative, the
production rate is high.
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A Workshop honoring Gideon Weiss, 2012, Yonit Barron & David Perry.
A Jump Fluid Production-
Inventory Model with a Double
Band Control
Yonit Barron, David Perry.
Haifa University
Israel
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The Inventory Process.
N(t) - the demand number process- Poisson().
I(t) -continuous time inventory level process.
Vi - the ith demand quantity, V ~ PH(,G)
1, 2 - the production rates.
1< E(V), 2 > E(V).
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The storage capacity is finite with level b.
Backordering to a level (-a) is allowed. Any backlog that
exceeds level (-a) is lost.
1 2
T1 T2
Ĭ(t)
t
b
0
-a
Ĭ+(t) - positive process
Ĭ-(t) - negative process
holding cost
idle cost
shortage costlost sale cost
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Tn - recovery points.
( Tn-Tn-1 ) - a cycle.
a renewal process.
n - the times of switching rates, zero points.
Total Cost (β – the discounted factor).
TC(β) = HC(β) + KC(β) + SC(β) + UC(β)
(t)I
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Lost Sales Cost.
Ø -the cost for an unsatisfied unit.
- the ith time of down crossing level (-a).
Yi - the ith lost sale quantity.
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Long Run Average Cost.
The process in the absence of control.
b→∞, a→∞ Ĭ(t) is a stable process.
Main Tools:
Matrix analytic approach and fluid model (Ramaswami, Ahn,
2003-2007).
Optional Sampling Theorem (OST) to the Kella- Whitt
Martingale (1992).
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The fluid model (F(t), K(t)).
- replacing the downwards jump (demand) of size v by
a line with slope (-).
State space of the fluid process – S1 S2.
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S1 - the process increases (state 0).
S2 - the process decreases (states {1,2,…,n}) .
Three variants of F(t):
bF - Finite buffer fluid flow.
Fr - Reverse fluid flow.
bFr- Reverse finite buffer fluid flow.
Q - transition rate matrix for the fluid model.
GGQQ
QQQ
o 2221
1211
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(s) - Laplace transform of the first return time to level 0.
- The LST of the first passage time of the
fluid process from (x, Si) to (y, Sj) while
avoiding a visit to levels in [0,a], [z,].
The Basic Martingale.
X(t) - a Levy process with no negative jumps.
Y(t) - an adapted process Z(t)=X(t)+Y(t)
),,(ˆ, syxf ji
za
15
The Discounted Model
The Positive Inventory Process
L(t) - the local time. Z(t) = X(t) + L(t) →
Y(t) = L(t) + (/)t , W(t)=X(t)+Y(t)Ĭi(t)
t
b
0
Zi(t)
Z() = k
k ~ PH(ek,G)
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- the time until zero point.
- Laplace transform vector of
Ahn & Ramaswami (2007) obtain the expression:
(1)
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Notice that:
-the expected discounted time of the idle period due to a
finite storage b.
)2( )(
•L(t) increases, if and only if , Z(t) = -b
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The Negative Inventory Process
1 2
T1 T2
Ĭ(t)
t
b
0
-a
•Backordering to a level (-a) is allowed.•Any backlog that exceeds level (-a) is lost and the inventory restarts from level (-a).
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Assume the demand hits level 0 with state iS2:
X(t) - a Levy process with no negative jumps.
i ~PH(ei, G) Zi(t) = X(t) + i → Zi(t)=- Ĭ(t).
Two stopping times:
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(2)
Using martingale and the optional theorem, and fluid
flow analysis )1(
In a similar way, we obtain ().
Derive () with respect to , set =0
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The lost sales cost.
-the expected discounted value of an unsatisfied demand.
Similar to a renewal equation:
T
-a
Zi(t)
-u
0
Yi
E(Yi)
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