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

19

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

34

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?

35

136, 682i

WW w -86 -84 -82 -80 -78

32

33

34

35

36

37

38

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

36

Basic Production System

FOB (free on board)

37

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

38

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

39

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:

40

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)?

41

-83 -82 -81 -80 -79 -78

34

34.5

35

35.5

36

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

42

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

43

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

*

45

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

46

Convex vs Nonconvex Optimization

47

Gradient vs Direct Methods• Numerical nonlinear unconstrained optimization:

– Analytical/estimatedgradient

• quasi‐Newton• fminunc

– Direct, gradient‐free• Nelder‐Mead• fminsearch

48

Nelder‐Mead Simplex Method

• AKA amoeba method

• Simplex is triangle in 2‐D (dashed line in figures)

reflection expansion

outsidecontraction

insidecontraction a shrink

49

Feasible Region

50

( ), if is in , if is true( , , ) , otherwise , otherwiseTCb aiff a b c c

x x Rif a is true

return belse

return cend