Median Location: 2‐D Rectilinear Distance 8 EFs
Transcript of Median Location: 2‐D Rectilinear Distance 8 EFs
Median Location: 2‐D Rectilinear Distance 8 EFs
5 15 60 70 90
15
25
60
70
95
1
2
3
4
5
6
7
8
X
Y
62
62 < 129
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81 < 129
48129 = 129
*
3939 < 12990
129 = 129*
wi : y:
1 1 2 1 2 1 2
2 22 1 2 1 2 1 2
( , )
( , )
d P P x x y y
d P P x x y y
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Ex 3: 2D Loc with Rect Approx to GC Dist• It is expected that 25, 42, 24, 10, 24, and 11 truckloads will be shipped
each year from your DC to six customers located in Raleigh, NC (36N,79W), Atlanta, GA (34N,84W), Louisville, KY (38N,86W), Greenville, SC (35N, 82W), Richmond, VA (38N,77W), and Savannah, GA (32N,81W). Assuming that all distances are rectilinear, where should the DC be located in order to minimize outbound transportation costs?
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136, 682i
WW w -86 -84 -82 -80 -78
32
33
34
35
36
37
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Raleigh
Atlanta
Louisville
Greenville
Richmond
Savannah
Optimal location (36N,82W)
(65 mi from opt great-circle location)
Answer :
24
10
42
11
25
2448
25
10
42
11
24 42 10 11 25 2424<68 66<68 76>68
*
11<68
53<68
63<68* 88<68
Logistics Network for a Plant
DDDDBBBB
CustomersDCsPlantTier One
Suppliers
Tier Two Suppliers
vs.
vs.
vs.
vs.
Distribution Network
Distribution
Outbound Logistics
Finished Goods
Assembly Network
Procurement
Inbound Logistics
Raw Materials
downstream
upstream
A = B + C
B = D + E
C = F + G
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Basic Production System
FOB (free on board)
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FOB and Location• Choice of FOB terms (who directly pays for transport) usually
does not impact location decisions:
– Purchase price from supplier and sale price to customer adjusted to reflect who is paying transport cost
– Usually determined by who can provide the transport at the lowest cost
• Savings in lower transport cost allocated (bargained) between parties
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Procurement Landed costcost at supplier
Production Procurement Local resource cost cost cost (labor, etc.)
Total delivered Production
Inbound transport cost
Outbound transport cocost cost
Transport
s
cos
t
t (T
Inbound transport Outbound transport C) cost cost
Monetary vs. Physical Weight
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in out
in out
(Montetary) Weight Gaining:
Physically Weight Losing:
w w
f f
1 1
min ( ) ( , ) ( , )
where total transport cost ($/yr)
monetary weight ($/mi-yr)
physical weight rate (ton/yr)
transport rate ($/ton-mi)
( , ) distance between NF at an
m m
i i i i ii i
i
i
i
i
iwTC X w d X P f r d X P
TC
w
f
r
d X P X
d EF at (mi)
NF = new facility to be located
EF = existing facility
number of EFs
i iP
m
Minisum Location: TC vs. TD• Assuming local input costs are
– same at every location or – insignificant as compared to transport costs,
the minisum transport‐oriented single‐facility location problem is to locate NF to minimize TC
• Can minimize total distance (TD) if transport rate is same:
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1 1
min ( ) ( , ) ( , )
where total transport distance (mi/yr)
monetary weight (trip/yr)
trips per year (trip/
transport rate =
yr)
( , ) per-trip distance between NF an E
1
d
m m
i i i i ii
i
i
i
i
i
iw
r
TD X w d X P f r d X P
TD
w
f
d X P
F (mi/trip)i
Ex 4: Single Supplier and Customer Location
• The cost per ton‐mile (i.e., the cost to ship one ton, one mile) for both raw materials and finished goods is the same (e.g., $0.10).1. Where should the plant for each product be located?2. How would location decision change if customers paid for distribution costs
(FOB Origin) instead of the producer (FOB Destination)?• In particular, what would be the impact if there were competitors located along I‐40
producing the same product?
3. Which product is weight gaining and which is weight losing?4. If both products were produced in a single
shared plant, why is it now necessary to know each product’s annual demand (fi)?
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-83 -82 -81 -80 -79 -78
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34.5
35
35.5
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36.5
AshevilleStatesville
Winston-Salem GreensboroDurham
Raleigh
Wilm
ington
50
150
190 220270
295
420
40
Wilmington Winston-Salem
rawmaterial
finishedgoods
ubiquitous inputs
1 ton 3 ton
2 ton
Product B
1
( ) ( , )m
i i ii iw
TC X f r d X P
Ex 5: 1‐D Location with Procurement and Distribution Costs
Assume: all scrap is disposed of locally
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Asheville unit offinished
good
1 tonProduction
System
Durham
A product is to be produced in a plant that will be located along I‐40. Two tons of raw materials from a supplier in Ashville and a half ton of a raw material from a supplier in Durham are used to produce each ton of finished product that is shipped to customers in Statesville, Winston‐Salem, and Wilmington. The demand of these customers is 10, 20, and 30 tons, respectively, and it costs $0.33 per ton‐mile to ship raw materials to the plant and $1.00 per ton‐mile to ship finished goods from the plant. Determine where the plant should be located so that procurement and distribution costs (i.e., transportation costs to and from the plant) are minimized, and whether the plant is weight gaining or weight losing.
Ex 5: 1‐D Location with Procurement and Distribution Costs
($/yr) ($/mi-yr) (mi)
monetary physicalweight weight
($/mi-yr) ($/ton-mi)(ton/yr)
i i
i i i
TC w d
w f r
in out
in out
(Montetary) Weight Gaining: 50 60
Physically Weight Losing: 150 60
w w
f f
1
:2
j
ii
Ww
:iw
Assume: all scrap is disposed of locally
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Asheville unit offinished
good
1 tonProduction
System
Durham
NF 4
3
5
1
2
out $1.00/ton-mir
3 3 3 out10, 10f w f r
4 4 4 out20, 20f w f r
5 5 5 out30, 30f w f r
in $0.33/ton-mir
1 1 out 1 1 in2 60 120, 40f BOM f w f r
2 2 out 2 2 in0.5 60 30, 10f BOM f w f r
2‐D Euclidean Distance
2 21 1,1 2 1,2
12 2
2 1 2,1 2 2,2
3 2 21 3,1 2 3,2
1 12 3 , 7 1
4 5
x p x pdd x p x pd
x p x p
x P
d
y
d 1
d 344
Minisum Distance Location
2 21 ,1 2 ,2
3
1
*
* *
1 17 14 5
( )
( ) ( )
arg min ( )
( )
i i i
ii
d x p x p
TD d
TD
TD TD
x
P
x
x x
x x
x
1 4 7x
1
2.73
5
1
d2
1 2
3
*
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120°
Fermat’s Problem (1629):Given three points, find fourth (Steiner point) such that sum to others is minimized(Solution: Optimal location corresponds to all angles = 120°)
Minisum Weighted‐Distance Location• Solution for 2‐D+ and
non‐rectangular distances:– Majority Theorem: Locate NF at EFj if
– Mechanical (Varigon frame)– 2‐D rectangular approximation– Numerical: nonlinear unconstrained optimization
• Analytical/estimated gradient (quasi‐Newton, fminunc)
• Direct, gradient‐free (Nelder‐Mead, fminsearch)
1
*
* *
number of EFs
( ) ( )
arg min ( )
( )
m
i ii
m
TC w d
TC
TC TC
x
x x
x x
x
Varignon Frame
1, where
2
m
j ii
Ww W w
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Convex vs Nonconvex Optimization
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Gradient vs Direct Methods• Numerical nonlinear unconstrained optimization:
– Analytical/estimatedgradient
• quasi‐Newton• fminunc
– Direct, gradient‐free• Nelder‐Mead• fminsearch
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Nelder‐Mead Simplex Method
• AKA amoeba method
• Simplex is triangle in 2‐D (dashed line in figures)
reflection expansion
outsidecontraction
insidecontraction a shrink
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Feasible Region
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( ), if is in , if is true( , , ) , otherwise , otherwiseTCb aiff a b c c
x x Rif a is true
return belse
return cend