Inventories with Multiple Supply Sources and Networks of Queues with Overflow Bypasses Jing-Sheng Song Paul Zipkin Duke University WLU, 25 April, 2008
Jan 21, 2016
Inventories with Multiple Supply Sourcesand Networks of Queues with Overflow Bypasses
Jing-Sheng SongPaul Zipkin
Duke University
WLU, 25 April, 2008
Why multiple sources?
Extensive global supply chainsprone to disruptions
Shipment tracking technology (GPS, RFID)allows us to use more information
Review: Minner 2003
Two sources, constant leadtimesMoinzadeh & Schmidt 1989
One itemPoisson demand with rate No scale economiesStockouts are backlogged (for now)Two supply sources
with different (constant) leadtimesT1 + T2 and T2
Demand
T1 T2Stock
State variables
IN = net inventoryNk = units in transit in node k, k
= 1,2IP2 = inventory position = IN +
N2
IP1 = inventory position = IP2 + N1N1 N2
IN
IP2IP1
Heuristic policy
Policy variables sk , s1 s2
Maintain IPk sk
So, a standard base-stock policy generates orders.
An order uses the fast source, when too many units are too far away.
Equivalent rule
Start with (and keep) IP1 = s1
So, every demand generates an order
Define u1 = s1 – s2
Route units to keep N1 u1
Network of queueswith overflow bypass
N1 N2Poisson
N1 u1
SolutionDefine
p(n1,n2) = Pr{ N1 = n1 , N2 = n2 } (steady state)
k(n) = exp(-Tk) (Tk)n / n!
N(u1) = feasible states = { (n1,n2) : n1 u1 }
G(u1) = normalizing constant
Then,p(n1,n2) = 1(n1)2(n2) / G(u1) , (n1,n2) N(u1)
IN = s1 – ( N1 + N2 )
From here on, it’s easy to evaluate a policy and optimize
Lost salesMinor adjustment: Only N and G change
Two sources, stochastic leadtimes
Expand each node to a Jackson sub-network
N1 N2Poisson
N1 u1
Solution
Reinterpret notationNk = total number in sub-network k, k = 1,2
k(n) = equilibrium distribution of Nk for sub-network k alone, with Poisson input
Then, the solution is the samep(n1,n2) = 1(n1)2(n2) / G(u1) , (n1,n2) N(u1)
IN = s1 – ( N1 + N2 )
LiteratureJackson 1963, Lam 1977, van Dijk 1989,
Serfozo 1989
Example
M/M/1 = 8
ConstantT2 = 1/2
N1 u1
stockoutprobability
1.E-07
1.E-06
1.E-05
1.E-04
1.E-03
1.E-02
1.E-01
1.E+005 10 15 20
One source
Two sourcesu1 = 8
s1
Exogenous leadtimes and other demand processes
Idea: A queue with many other inputsOur orders have negligible effectsTk(t) = virtual leadtime for sub-
network kTime required for a unit arriving at
tThis is a given stochastic process
Tk = virtual leadtime in equilibrium
Analysis
DenoteD(s,t] = demands in (s,t]D(s,t] = slow orders in (s,t]
Then,IN(t+T2(t)) = IP2(t) – D(t,t+T2(t)]
IP2(t+T1(t)) = IP1(t) – D(t,t+T1(t)]
Combine these and take the limitIN = s1 – ( N1 + D2 )
Here, N1 is the content of a loss system in equilibrium,and D2 = D(0,T2]
The solution has the same form as before
Markov-modulated demand
Suppose X is a Markov chainWhile X = x, demand is Poisson with rate
(x)
The analysis above goes throughIN = s1 – ( N1 + D2 )
Now we need the joint distribution of (N1,X)
D2 = D(0,T2], conditional on X(0)
All this can be done numerically
Batch orders
Special case – equal batch sizes qReorder points rk
Use (r,q) policy, to maintain IPk > rk
This works just like the non-Poisson demand systemIN = s1 – ( N1 + D2 )
s1 = r1 + q
X = IP1
N1/q is a the content of a loss system with non-Poisson (Erlang-q) input
More sources
Demand
T1Stock
T2 T3
Heuristic policy
K = number of sourcesPolicy variables sk , s1 … sK
Maintain IPk sk
EquivalentDefine uk = s1 – sk+1
Route units to keep N1 + … + Nk uk
Network of queues
Poisson
N1 u1
N1 + N2 u2
N1 N2 N3
Solution
n = (n1,…,nK)
u = (u1,…,uK)
p(n) = 1(n1)…K(nK) / G(u) , n N(u)
IN = s1 – ( N1 + … + NK )
Discussion
This is quite a flexible modeling approach
What does it not cover?Independent sources
The sources here are on a line, in effect
Asia –> west coast –> mid-westBut what if the fast source is on the east coast?
Dynamic expediting
Optimal policy?