“Centralized Ordering Policies in a
Multi-Warehouse System with Lead
Times and Random Demand”
A paper by Gary Eppen and Linus Schrage
Presentation by Tor Schoenmeyr
This is a summary presentation based on: Eppen, Gary, and Linus Schrage. “Centralized Ordering Policies in a
Multi-Warehouse System with Lead times and Random Demand.” TIMS Studies in the Management Sciences, Vol.
16: Multi-Level Production/Inventory Control Systems, Theory and Practice. Edited by Leroy B. Schwarz. 1981.
System and Problem Description
The Allocation Assumption
Policy 1: Order up to y every period
Policy 2: Order up to y every m periods
System Description and Assumptions
Total Depot N warehouses Demand
inventory in (no inventory) Transportation (with inventories) (random)
system: y lead time l
Supplier
lead time L Z1 e1 = N ( µ1 , σ 1 )
Supplier Z2 e2 = N ( µ2 , σ 2 )
Z3 e3 = N ( µ3 , σ 3 )
Costs to be minimized: Decisions to be made every period:
•Holding cost h per unit in inventory •How much, if anything, should be
ordered from the supplier
•Penalty cost p per unit of unmet
demand (placed in backlog) •How should we distribute the
incoming orders at the Depot
•Fixed cost K for every order placed
Why have a Depot?
(with no inventory)
Problem Depot Benefit
• Separate warehouses have • Exploit quantity discounts from
little purchasing power the supplier
• Demand fluctuates for the • Fluctuations in different
individual warehouse warehouses even out, and you
gain “statistical economies of
scale”
• It is expensive/ impractical to • Depot need not to be a
build a depot physical entity (the point is that
goods are allocated after
orders completed)
• (Demand can vary also in the • (Maybe a depot with inventory
aggregate) can do even better)
Applicability of model
Good application: Questionable application:
Steel for conglomerate Coca-Cola for 7-Eleven
Production Long Short
lead times:
Inventory Holding costs Cheap, not to say
surplus: (expensive) desirable (up to shelf
capacity)
Inventory Order placed Customer walks (or
shortfall: on “backlog” at buys a substitute)
some penalty
System and Problem Description
The Allocation Assumption
Policy 1: Order up to y every period
Policy 2: Order up to y every m periods
Allocation Assumption
(for every period ordering, normal demand)
“Every period t, we can “Every period, we can find a
make an allocation (at the constant v, such that the
depot) such that the total inventory at and in
probability of running out at transit to the i th warehouse
each warehouse is the same is:
at period t+l”
(l + 1) µi + vσ i l + 1
Transportation
lead time l
Depot Warehouses
Supplier
Example when Allocation Assumption holds (identical warehouses)
Period t Ware- Demand Period t+1
houses Warehouses
L=1 10 6 L=1 10-6+5=9
Supplier 8 l=0 Supplier ? l=0
10 4 10-4+3=9
Example when Allocation Assumption is violated (identical warehouses)
Period t Ware- Demand Period t+1
houses Warehouses
L=1 10 9 L=1 10-9+8=9
Supplier 8 l=0 Supplier ? l=0
10 0 10-0+0=10
The Allocation Assumption holds for high
µ/σ and low N
Probability of A.A. being true according to experiment
Theoretical Result presented in paper (my experiment in parenthesis) Percent
Eppen and Schrage µ/σ ½ 1 3/2 2 5/2
derive a good N
theoretical 2 32.6 66.3 85.8 95.2 98.8
approximation (35.9) (66.3) (86.5) (95.5) (98.8)
formula for the 3 20.1 54.7 79.8 93.0 98.1
probability of A.A. (19.8) (53.2) (79.0) (93.5) (98.2)
being true. 4 11.4 43.1 73.3 90.1 97.3
(10.0) (41.0) (72.0) (90.5) (97.4)
5 7.6 36.5 68.6 88.3 96.5
(4.9) (30.6) (65.8) (88.0) (96.9)
The
The paper
paper does
does not
not explain
explain
how
how “negative
“negative demand”
demand” 6 4.6 29.9 63.2 86.4 96.1
should
should be be interpreted.
interpreted. This
This (2.5) (22.3) (60.7) (85.2) (95.8)
happens
happens frequently
frequently in in the
the
lower 7 2.8 24.5 59.1 84.1 95.5
lower left corner where my
left corner where my
experiments
experiments gavegave different
different (1.2) (15.8) (53.5) (83.2) (95.5)
results
results than
than those
those of
of the
the paper
paper 1.6 20.4 54.3 82.0 94.6
8
(0.5) (11.0) (46.6) (80.5) (95.3)
System and Problem Description
The Allocation Assumption
Policy 1: Order up to y every period
Policy 2: Order up to y every m periods
Policy 1: Order Every Period
(fixed ordering costs K = 0)
Problem: Intuitive answer: But…
How should we We should distribute But is this always
distribute the goods goods so that total possible?
that come in to the goods at and en If we make the
depot every period? route to every A.A., then yes!
factory is “the same”
How much should We should order so What should be
we order from the that the same total the value of y?
factory every inventory level y is
period? achieved every period.
(=order last period’s
demand)
Eppen and Schrage find an analytical expression
for the inventory at each warehouse
We
We know
know howhow much
much is is We
We know
know how
how the
the We
We know
know the
the (random
(random
ordered
ordered every
every period
period incoming
incoming goods
goods are
are split
split function
function for)
for) demand
demand
(as
(as aa function of y)
function of y) up
up at
at the
the Depot
Depot at
at each
each warehouse
warehouse
Eppen and Schrage derive this expression for the inventory S at each warehouse
(simplified form for the case of identical warehouses):
∑ ∑
L N
y e
− ∑ t = L +1 e jt
L +l
S j = (l + 1) µ + − (l + 1) µ − t =1 i =1 it
N N
Fixed component Random component
The problem is now equivalent to the
newsboy problem, and can be solved
analytically
Newsboy problem
“The newsboy buys i newspapers, at a cost c each. He sells
what is demanded d (random variable), or all he has got i,
whichever is less, at a price r. Any surplus is lost.”
Cost of Cost of
Deterministic Random surplus shortage
inventory inventory (per unit) (per unit)
Newsboy i -d c r-c
∑t =1 ∑i =1 it
L N
y e
− ∑t =L+1 e jt
Our system L +l
(at warehouse)
(l + 1)µ + − (l + 1)µ − h p
N N
System and Problem Description
The Allocation Assumption
Policy 1: Order up to y every period
Policy 2: Order up to y every m periods
Policy 2: Order Up to level y every
m periods
+ We can select m and y to minimize total costs,
including fixed ordering costs K
+ Periodic ordering policy easy to implement in
practice
+ Authors claim that theoretical results on this
policy has wider applicability
– Certain approximations have to be made to find
the best m and y
– Even if the best m and y were to be found, the
periodical ordering policy isn’t necessarily
optimal
Several new assumptions lead to an analytical
solution for the periodic ordering policy
2.
2. The
The allocation
allocation rule
rule
(l + m) µi + vσ i
is
is used,
used, but
but not
not proven
proven
to
to be
be optimal
optimal
3. As before,
1. “Stock can we make the
only run out Analytical expression a.a.
in the last for optimal y and cost, – it is always
period before given m possible
order arrivals” to make this
allocation
In
In typical
typical cases,
cases, we
we can
can
then
then find
find the
the best
best solution
solution
by optimizing yy
by optimizing
for m=1,2,3…
for m=1,2,3…
