1
Distribution Centers and Order Picking
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IFL
Multistage logistic process chain
• Topology => location and materials flow‘s relation(graph) (node) (arrows)
Procurement network Distribution network
SU
SU
SU
SU
SU
C1
P1 D1 C2
C3
Cn
TSC1 GKHGVRailwayShip
Aircraft
Location Problem
TSC2trans-shipment-center
2
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IFL
Definition: strategic decision about the geographic place and amount of locations (for production, warehousing, distribution, etc.)
Site-related factors:
• Basement and site
• Traffic and transport
• Infrastructure
• Labor force
• Procurement
• Turnover
• Tax
• etc.
Location Problem
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IFL
Selecting the number and location of distribution centers
Typical management questions:
• How many distribution centers should the company use, and where should they be located?
• What customers or market areas should be serviced from each distribution center?
• Which products should be stocked at each distribution center?
• What logistic channels should be used to source material and serve markets?
high complexity (number of locations * alternative location sites)
high data intensity (detailed demand and transportation information)
Location Problem
3
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IFL
models and methods to solve the Warehouse Location Problem (WLP)
• analytic techniques
• linear programming techniques
• Simulation
Objective: Minimize the total costs
Examples: • continuous model (Steiner-Weber) (1)
• discrete model (2)
• Break-Even-Analysis (3)
Location Problem
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Continuous Model: Steiner-Weber-model (1)Given: n customer locations (coordinates): (uj, vj) with j = 1,...,n)
demand of a customer: bj
costs per unit: cSought: location (coordinates) for a warehouse (x,y) with minimal transportation costs
(x,y)
),( jj vu
warehouse
customer
Euclidean distance customers
x
y
Location Problem
4
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IFL
objective function: minimize transportation costs
There‘s no known analytic solution for K(x,y)
iterative solution
partial derivation of K(x,y):
is the gradient of the plane K(x,y)
( ) ( )22
1),( ii
n
ii vyuxbcyxK −+−= ∑
=
( ) ⎥⎦
⎤⎢⎣
⎡∂∂
∂∂
=∇yK
xKyxK ,,
( )yxK ,∇
( )( ) ( )
01
22=
−+−
−=
∂∂ ∑
=
n
i ii
ii
vyux
uxbcxK
( )( ) ( )
01
22=
−+−
−=
∂∂ ∑
=
n
i ii
ii
vyux
uybcyK
Location Problem
Euclidean distance
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IFL
We have: and
x and y can’t be solved analyticallyiteration process
x0 and y0 are the focal point’s coordinates: and
Termination Condition: a determined limit value (if the difference between the solutioni+1 and i is small enough)
Result: storage location (x,y)
∑
∑
=
=
−+−
−+−
⋅
= n
i ii
i
n
i ii
ii
vyuxb
vyuxvb
y
122
122
)()(
)()(
∑
∑
=
=
−+−
−+−
⋅
= n
i ii
i
n
i ii
ii
vyuxb
vyuxub
x
122
122
)()(
)()(
∑
∑
=
=
⋅= n
ii
n
iii
b
ubx
1
10
∑
∑
=
=
⋅= n
ii
n
iii
b
vby
1
10
Location Problem
5
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IFL
Discrete-model (2)
given: n customer locations j = 1,...,ndemand of a customer: bjwarehouse location’s capacity: ai
storage location’s fixed cost: fi
m potential warehouses i = 1,…,mtransport costs from warehouse i to customer j: cij
sought: warehouse location/locations with minimal costs (transportation and fixed costs)
Location Problem
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IFL
objective function: minimize costsi
m
i
m
ii
n
jijij yfbcK ∑ ∑∑
= ==
+=1 11
0 usedt isn' 1 used is
location storage theif (0,1) iablebinary var location storage for the costs fixed
quantitytransport costs transport :with
==
==
=
=
i
i
i
i
ij
ij
yy
iyif
bc
Constraints:the delivered quantity from storage location i to all the customer j=1 m has to be:
• for all warehouses i = 1,…,m (warehouse capacity constraint)
• for all customers j = 1,…,n (customer demand constraint)
ii
n
jij yab ⋅≤∑
=1
j
m
iij bb =∑
=1
Location Problem
6
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IFL
Break-Even-Analysis (3)
Basis: simplified costs’ consideration• Linear, static cost’s function• Fixed costs specific for a storage location: fi• Variable costs specific for a storage location (gradient αi)
Goods‘ quantity(production quantity)
Costs
0=f
1α
2α2f
1f
Break-Even
Outsourcing: product oriented, no fixed cost
Storage location 2
Storage location 1
So far there is no economic storage location=> outsourcing
Location Problem
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IFL
Optimal amount of distribution centers (DC)
Costs
Transport costs: DC --> customer
Transport costs: production --> DC
Storage costs
Total costs
Amount of DC‘s1 432
Minimum
Location Problem
5
7
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IFL
DistributionCenter
suppliers/production
customers
material flowmaterial flow
information flow information flow
Distribution Centers within the supply chain
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IFL
Distribution Centers – Technical equipment
Technical EquipmentTechnical EquipmentFunktions
• discharging• control• store• transshipping • sorting• order picking• packing• loading
• ramps, fork lift truck• identification and control system• racks, automatic storage system• pallet transporter• sorter• order picking machines• packing equipment• ramps, fork lift truck, conveyor
Functionality
8
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IFL
Benefits:
material- or product-flow-related• consolidation
• Break Bulk operation
• Distribution assortment
• In-Transit-Mixing
• Manufacturing support
time-related• postponement
• stockpiling
• market presence
Distribution Centers – Benefits
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IFL
Receiving area
Bulk storagearea
Rack storagearea
Staging/Shipping area
Packing or unitizing area
Order picking area
Product flow
Distribution Centers – typical warehouse design
9
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IFL
a) b)
cost
s
timedeliveryb
K LK G
KBb
stoc
k
quantity delivered
Optimal size of a warehouse from the economic point of view ?
Dimensioning of a warehouse: a simple stock keeping model:“Economic Order Quantity Model” (EOQ)
Inventory Management
quantity delivered
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IFL
optimal order quantity:
(Andler formula)
LBG KKK +=bMkK bB ⋅= PbkK l
L ⋅⋅=2100
GK
bk
BK
lk
= total costs
= order costs
LK = stock keeping costs
= fix costs for order
= stock keeping costs in percent of the purchase costs resp. of the manufacturing costs,
PkMkb
l
b
⋅⋅⋅⋅
=1002
Inventory Management
P = purchase costs resp. manufacturing costs
10
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IFL
Stochastically distributed stocks:
warehousesource sinkz(t) a(t)
Stochastic input and output processFor an inventory time t = T :
∑ ∑= =
−+=T
tt
T
tt
tatztbtb0 0
)()()()( 0
inventory starting )( 0 =tb
Inventory Management
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IFL
Storage capacity:
Storage capacity b: maximum number of loading units that a warehouse can receive
with
If all storage bins are the same homogeneous warehouse
All following examples based on assumption:
tcompartmenper units loading bins storage / handon tscompartmen
capacity storage
===
lmblmb ⋅=
1=l
Inventory Management
11
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IFL
Fill factor :or
= number of storage bins that are filled (on average)
1≤=mmf
b
b
bbf
k
j j∑ === 1
m
t
m
0
not availablecapacity
shortfall
stoc
k
Inventory Management
m
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IFL
Warehouse planning task:
Dimensioning the warehouse in order to make it useable with the smallest possible capacity
Two types of of bin location management:
a) Definite allocation of items to storage bins
the required number of storage bins is the summation of the maximum expected stocks
b) Free choice of the storage bin
chaotic resp. dynamic bin location management
Supposition: the different item‘s inventory courses aren‘t correlated (statistical independence)
Offset between either large or small stocks of the different items
Inventory Management
12
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IFL
Warehouse planning task:
i. Stock analysis (observed values, forecast values)
ii. Stationary distribution of the stock referring to the items can be derived
iii. Analysis of the stock distributions for each item
- mean value, variance, variation coefficient
Planning the warehouse‘s dimension so that it is able to receive goods(with a given statistical security)
)( jbf
Inventory Management
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IFL
Warehouse dimensioning for normal distributed item’s stocks:
Stock’s processes of the separate items are statistically independent and are approximately normal distributed
)(tb j
bj
f(bj)t
bjbj
bj,max
bj
0
Process and density function of a normal distributed stock:
Inventory Management
13
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IFL
Stock level:
Minimum and maximum stock levels:
and
21 α−
⋅±= usbb jjj
96,1 :examplefor quantil son'distributi normal
item of variationstandard
975,0
j
==
=
uu
js
γ
21min, α−
⋅−= usbb jjj2
1max, α−
⋅+= usbb jjj
21max,min 20 α−
⋅⋅=⇒= usbb jjj,
Inventory Management
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IFL
Calculation of the maximum number of storage bins (m):
• Definite allocation of items to storage bins:
• Free choice of the storage bin:
Special case: all the item’s stocks are identical distributed (i.e.: and )
a)
b)
∑= −
⋅⋅=n
jj usm
1 21
2 α
∑∑ = −=
⋅+=n
j j
n
jj usbm
12
211
)( α
mbnm j ⋅=⋅⋅= 22
nmmbnbnm jj +=⋅+⋅=
Inventory Management
bb j = ss j =
14
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IFL
Warehouse dimensioning for any distributed item’s stocks:
Input process z(t), output process a(t), stock level b(t), density function f(b):
t
(t)
(t)
t
t
bj
bj
f(bj)
bj,max
0
aj
zj
(t)
bj,min
t
(t)
(t)
taj
zj
t
bjbj
f(bj)0
(t)
C-items: items, that are ordered in large
amounts and are consumed continuously
A-items: expensive items
reducing the storage time
Inventory Management
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IFL
a) Definite allocation of items to storage bins:
For all item j, an inventory value is determined, which can’t be exceeded due to a statistical security.
b) Free choice of the storage bin:
Overcapacity resp. under capacity of item groups can balance themselves
The cumulated storage stocks can be defined as a linear combination of the stocks that refer to the item
The density function of the cumulated storage stocks can be calculated by the convolution operator
jb^
∑=
=n
jjbm
1
^
∑ =
n
j jb1
jb
)()(1
bfbf n
j j =∑ =
Inventory Management
15
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IFL
Convolution: gives the probability distribution of a summation of 2 independent discrete random variables.
)2...(0for 210
−+=⋅= −=∑ kkkbac ik
k
iik
p (bx)
20 40 60 80
0,10,2
0,3
0,4
bx
0
p (by)
20 40 60 80
0,10,2
0,3
0,4
by
0
p (bz)
20 40 60 80
0,10,2
0,3
0,4
bz
0
Example (Probability diagrams for the items x,y,z):
Inventory Management
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IFL
Given: Items x,y,z
Sought : computation of warehouse capacity with a statistical security of 90%.
a) Definite allocation of items to storage bins:
n)1(1 βα −=− n1
)1(1 αβ −=−
1807060503
1
^=++==⇒ ∑
=jjbm
6 0
F (b x)
2 0 4 0 6 0 8 0
0 ,2
0 ,4
b x
0
0 ,6
0 ,8
1 ,0F (b y)
b y2 0 4 0 8 0
0 ,2
0 ,4
0
0 ,6
0 ,8
1 ,0F (b z)
b z2 0 4 0 6 0 8 0
0 ,2
0 ,4
0
0 ,6
0 ,8
1 ,0(1 -β )
b̂ = 50x b̂ = 60y b̂ = 70z
(1 -β )(1 -β )
Inventory Management
965,0)1,01( 31≈−
Statistical security for the warehouse
Statistical security for the individual items 1,…,n
Capacity of 180 storage bins guarantees that warehouse canreceive items with an overall statistical security of 90%
To get an overallstatisticalsecurity of 90%, the individualstatisticalsecurities haveto be 96,5%
= 0,965 = 0,965 = 0,965
16
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IFL
b) Free choice of the storage bin:
)()()()( zybb
x
z
xjj bPbPbPbbP
z
xj j
∑∑∑
===
==
)()( bpbpc z =⊗
cbpbp yx =
⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜
⎝
⎛
=
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
⊗
⎟⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜⎜
⎝
⎛
=⊗
01,002,004,010,015,016,022,022,008,0
1,01,01,03,04,0
1,01,02,04,02,0
)()(
Inventory Management
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IFL
Diagram and distribution function of the convoluted function
F (b)
0,2
0,4
0
0,6
0 ,8
1 ,0
P (b)
20 40 60 80
b
bm
(1- α )
100 120 160 180
0,200
0 20 40 60 80 100 120 140 160 180
0,020
0,100
140
a)
b)
)(∑ =
n
xj jbf
Inventory Management
= 0,9
Capacity of 140 storage binsguarantees that warehouse canreceive items with an overallstatistical security of 90%
Compare to 180 storage bins ifusing definite allocations of items to storage bins
17
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IFL
Order picking
Definition: Taking a subset from the stock to satisfy customer orders
Examples for order picking:
• in a warehouse: assembling of individual orders from a wide assortment of products
• in a production site: assembling of required items for supplying the production line resp. the
assembly stations
• in a supermarket: filling the shopping cart
Order Picking
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IFL
routstim e
assortm entinventory m odelam ounts
routstim e
assortm entam ounts (positions)tim e
z(t)
a(t)
info
rmat
ion
flow
orde
r pic
king
syst
em
stockb(t)
orderpicking
assem blingr = 1, 2,...n
transferto stock
E mE 2E 1 . . . . .
. . . . .A 1 A 2 A k
a(t) = output streamb(t) = stock levelz(t) = input streamA = pick orderE = input
Order picking: Global material and information flow
Order Picking
18
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IFL
Distinction between:
• static or dynamic supply
• one or two dimensional movement
• automatic or manual picking
• central or decentralize delivery
Goals:
• minimizing the order picking time
• maximizing the service level
• maximizing the picking order quality
• minimizing the stock level
Strategies:
• multistage order picking (item or order orientated)
• organization in zones (picking frequency, dangerous goods, etc.)
Order Picking
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IFL
Order Picking
• Static vs. dynamic supply
Source:Günthner
„Man to Goods“ „Goods to man“
19
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IFL
Order Picking
• One vs. two dimensional movement
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IFL
Order Picking
• Manual vs. automatic picking
20
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IFL
Order Picking
• central vs. decentralized delivery
Base Station
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IFL
Basic concepts for order picking systems
Basic concept A:
• static (man to goods): the order picker goes to the compartments, where the items are stored
• one dimensional movement
• manual picking
• decentralized delivery (at the base station, assembly station, etc.)
delivery
Order Picking
21
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IFL
Basic concept B:
• static (man to goods)
• two dimensional movement
• manual picking
• centralized delivery
Basic concept C:
• dynamic (goods to man)
• one dimensional movement
• manual picking
• decentralized delivery1 2 3
4
5678
9A SR S
delivery
conveyor
10
full
em ty
Order Picking
Source: Arnold, 1998
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IFL
Order Picking time:
desired is a fast processing of the picking orders
Parameters:• dimension of the warehouse, assortment
• amount and composition of the picking orders
• concept of the pick order system
• technical standard
• information flow
• strategies
Order Picking
22
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IFL
Calculating of the order picking time
given: a pick order with n positions
t0 basic time: time, that is independent from the amount of picking positionstt,r dead time: time, that is necessary at every picking place in addition to the physical
material movement (e.g. reading, searching,...)tg,r picking time: time for the physical material movement (picking or storing)tl,r travel time: time for the travel of the order picker or the picking machine
( )∑=
+++=n
rrlrgrtk ttttt
1,,,0
Order Picking
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IFL
Calculating the travel time for basic concept A:
1 2 3 4 5 6 massortment
Δx
startfinish
tl,1
xr=1 xrxr=n
tl,n
xr
xi
L0
{{
( )∑ ∑= =
− ++−
+−
+==n
r
nn
r
rrrll a
vnv
xLvxx
vxtt
1 2
11, 1
1,lt nlt , Time for acceleration
Order Picking
23
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IFL
and:
and
( ) i
m
i
it p
vxtE ⋅= ∑
=11, L
xm
piΔ
==1 ( )
Lx
vxtE
m
i
il
Δ= ∑
=11,
( )vLdx
vLxtE
L
l 21
01, =
⋅= ∫
( ) ij
m
j
jim
iijl p
vxx
tE ⋅−
= ∑∑== 11
, ( ) ( )j
jiij xLL
xxmm
pΔ−
Δ⋅Δ=
−=
11
( )vLdxdx
vxx
LtE
Lji
L
ijl 31'1
002, =
−= ∫∫
( ) ( ) ( )avn
vLn
vLtE l 1
311 ++−+=
Order Picking
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IFL
Travel time per position:
For a large amount of positions n:
Trivial-strategy: Picking in start-finish-direction
For a large amount of positions n:
( ) ( ) ( ) ( )av
nn
vL
nn
nvL
ntEtE l
rl1
311
,+
+−
+==
( )av
vLtE rl +≈
31
,
1−> ii xx
( )av
nn
vL
ntE S
rl11
,+
+=
( )avtE S
rl ≈,
Order Picking
24
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IFL
Sequence problem
given: amount of compartments
sought: sequence of compartments, so that the required travel time is minimal
Problems:
• distance measuring
• often several order pickers are in activity. This can lead to disruptions
Solution:
different Strategies (e.g. Mäander-Strategy, Largest-Gap-Strategy, diverse algorithms for routing, ...)
Order Picking
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IFL
ABC-inventory classification
a small percentage of the entities account for a large percentage of the volume
classification by sales:
the products are classified in descending order by sales volume, so that the fast movers (high-volume products) are listed first followed by slow moving items
100%
50%
80%
95%
A B C
percentage of sales
percentage of items100%
Order Picking
B-articles
Main aisle
C-articlesA-articlesBase station0
0