The Interaction of Location and Inventory in Designing
Distribution Systems
Stephen J. Erlebacher and Russell D. Meller
Presented By:
Hakan Gultekin
Aim:
• # of DCs
• Their location
• Which customers
they serve
Many DCs:
• Reduces the cost of
transporting product to
retailers
• Provide better service
Few DCs:
• Reduces the cost of
holding inventory via
pooling effects
• Reduces the fixed costs
associated with operating
DCs
The Location-Inventory Problem: Assumptions
• Unit-square grid structure with C columns and R rows
C= 5
1
2
3
4
5
1
2
3
4
R= 4
The Location-Inventory Problem: Assumptions
• Uniform customer demand across any grid
19
...
10
11
9
7
The Location-Inventory Problem: Assumptions
• Rectilinear distances between plants and DCs and between
DCs and continuously represented customer locations
.
(x,y)
.
(a,b)
d  x  a  y b
The Location-Inventory Problem: Assumptions
• Continuous review inventory system at DCs
• Plant locations and capacities are known in advance
and fixed
The Location-Inventory Problem: The Model
Minimize Total Cost
where,
Total Cost = (Operating Cost) + (DC Inventory Cost)
+ (Transportation Cost)
The Location-Inventory Problem: The Model
Operating Cost:
F = Annual cost of operating a DC
1 if DCi is opened,
zi  
0 otherwise
 = Upper bound on the number of DCs

 F  zi
i 1
The Location-Inventory Problem: The Model
Total Inventory Costs:
A= Order cost
z= Safety stock parameter
h= Holding cost
s Std. dev. of demand during lead time
For DC i, order cost + holding cost:
ADi
Q
2 ADi
 h(  zs L ) , Q 
Q
2
h
M
Di   d j wij
j 1
dj= Avg. demand for customer j
1 if DCi serves customer grid j,
wij  
0 otherwise
The Location-Inventory Problem: The Model
Total Inventory Costs:
For 1 DC:
zs

 2 Ah 
D


h

M
d w
j 1
j
ij
For all DCs:
zs 

zi  2 Ah 
h

D 

i 1

M
d w
j 1
j
ij
The Location-Inventory Problem: The Model
Transportation cost from plants to DCs:
= Unit plant to DC transportation cost
upi = Demand shipped from plant p to DC i
qpi = Distance from plant p to DC i  xi  a p  yi  b p
From plant p to DC i:
zi u pi q pi
From all plants to all DC s:
P

 z u
p 1 i 1
i
pi
q pi
The Location-Inventory Problem: The Model
Transportation cost from DCs to customers:
tij = Avg distance from DCi to customer grid j
1
1
( x  (c j  1))(c j  x)
2
2
.
(c -1,y)
j
.
.
(x,y) (c ,y)
j
1
1
( x  (c j  1)) ( x  (c j  1))  (c j  x) (c j  x)
2
2
The Location-Inventory Problem: The Model
Transportation cost from DCs to customers:
tij = Avg distance from DCi to customer grid j
1
(x  c j ) 
2
1
2
. (c ,y)
j
1
( x  c j(() c j 1)  x)
2
.
.
(c ,y)
(x,y)
1
1
x  (c j  1)  c j  x
2
2
j
1
((c j  1)  x) 
2
The Location-Inventory Problem: The Model
Transportation cost from DCs to customers:
s = Unit DC to customer transportation cost
From DC i customer j:
szi (t ij  t ij )d j wij
x
y
For all DCs and all customers:

M
x
y
sz
(
t

t
 i ij ij )d j wij
i 1 j 1
The Location-Inventory Problem: The Model
Constraints ::
Each customer must be assigned to an open DC:
M
w
ij
j 1
 Mzi ,
i
One customer can be assigned to one DC:

w
i 1
ij
 1,
j
The Location-Inventory Problem: The Model
Constraints :
Each DC must be fully supplied:
P
M
 u  d w ,
p 1
pi
j 1
j
ij
i
Capacity constraint for plants :
P
u
p 1
pi
 vp ,
p
The Location-Inventory Problem: The Model
zs

F  zi +  zi  2 Ah 
D

i 1
i 1


Min
P

+  zi u pi q pi
p 1 i 1
s.t.
M
w
ij
j 1

w
i 1
ij

+

h

 1,
j
i
d w
j
j 1
ij
M
x
y
sz
(
t

t
 i ij ij )d j wij
i 1 j 1
P
 Mzi ,
M
M
 u  d w ,
p 1
pi
P
u
p 1
pi
j 1
 vp ,
j
ij
p
zi {0,1}, i; wij {0,1}, i, j; u pi  0, p, i
i
The Location-Inventory Problem: Solution Method
• Find the number of DCs, N
• Find the location of these DCs and allocation of the
customers to these DCs
The Location-Inventory Problem: Solution Method
Finding N: Stylized model
• Customer demand is
entirely homogeneos
• Any amount of demand
can be assigned to any DC
• Each DC serves an
“optimally shaped region”
(for rectilinear, diamond
shaped region)
• Ignores different customer
demands
• Discrete nature of the
customer grid structure
• Impossible to have each
DC serve an “optimally
shaped region”
The Location-Inventory Problem: Solution Method
Finding N: Stylized model
Lemma 1: Given a number of DCs, N, any DCs that
serve positive demand must serve the same size demand.
Di= D/N
The Location-Inventory Problem: Solution Method
Finding N: Stylized model
Let I   2 ADh  zs h be the inventory parameter,
2sD  M

T  

 3  2
V  uD
be the transportation parameter and
be the inbound logistics costs.
The Location-Inventory Problem: Solution Method
Finding N: Stylized model
Lemma 2: Optimal N for stylized model can be found by
(i) 4F 2 N 3  I 2 N 2  2(T   V )I N  (T   V )2  0
(ii) If N  P, then N*  N; Stop.
(iii) 4[ F  (V / P3/ 2 )]2 N 3  I 2 N 2  2T I N  T 2  0
(iv) If N  P, then N*  P; else N *  N.
Optimal number of DCs
The Location-Inventory Problem: Solution Method
Actual
Stylized
Inventory Parameter
Optimal number of DCs
The Location-Inventory Problem: Solution Method
Actual
Stylized
Transportation Parameter
The Location-Inventory Problem: Solution Method
Optimal number of DCs
Actual
Stylized
Fixed Cost
The Location-Inventory Problem: Solution Method
Location Problems:
• N-facility location problem
NP-hard
• N independent single facility location
• Rectilinear mini-sum location problem
The Location-Inventory Problem: Solution Method
Allocation Heuristics v1:
The Location-Inventory Problem: Solution Method
Allocation Heuristics v2:
Lower bound
Relaxations:
Separate inventory and transportation decisions
Relax the actual customer locations
Lemma 3: Lower bound on inventory costs are obtained
by assigning the N-1 lowest demand customer grids to the
first N-1 DCs and the remaining M-N+1 customer girds to
DCN
Sort customers from highest demand to lowest demand and
assign them one at a time to a DC.
Highest demand customers have more influence on the
location of the DC which they are assigned
Computational Results & Managerial Insight
Two datasets considered:
• Set I consists of 12 customers (on a 3X4 grid)
• Set II consists of 16 customers (on a 4X4 grid)
• 4 different ABC customer curves:
(80/20), (70/30), (60/40), (50/50)
• v2 performed better than v1 in both sets
Computational Results & Managerial Insight
• Lower bound was between 4 and 36 % lower than the
optimal solution
• Neither of the heuristics are guaranteed to terminate at a
local optimum.
• Pairwise-exchange improvement procedure is added (v2+).
• For dataset I, v2 found the optimum in 10/75
while v2+ found in 62/75
• For 600 customers v2 solved in 2 minutes,
v1 solved in 30 hours and v2+ solved in 117 hours.
Computational Results & Managerial Insight
• As the skewness of ABC curve increases, N either stays the
same or decreases, since larger demand is concentrated in
fewer and fewer customers.
• The customer layout also affects the optimal number of DCs.
• Geary Ratio, is an autocorrelation factor that quantifies
spatial correlations
• Tends to decrease when similarly-sized customer
demands are adjacent
Computational Results & Managerial Insight
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0.60 0.33 .
41.5 1.76 .
0.29 0.74 .
1.08 0.41
0.23 411.9
0.31 0.63
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0.25
0.27
0.28
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. 0.23
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. 0.26
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. 0.39
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. .
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0.60 0.33 .
0.23 1.76 .
0.29 0.74 .
1.08 0.410
41.5 411.9
0.31 0.63
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• Higher Geary Ratio
• Smaller Geary Ratio
• Higher number of DCs
• Smaller number of DCs
Future Research
• Capacity limitations at the DCs
• Different type of inventory policies
• Multi-product environment
The Interaction of Location and Inventory in Designing
Distribution Systems
Stephen J. Erlebacher and Russell D. Meller
Presented By:
Hakan Gultekin