Why Are Cities Located Where They Are? 9 Taxonomy of Location - - PowerPoint PPT Presentation

Why Are Cities Located Where They Are?

9

Taxonomy of Location Problems

Location Decision Cooperative Location Competitive Location Minisum Location “Nonlinear” Location Resource Oriented Location Market Oriented Location Transport Oriented Location Local-Input Oriented Location

Minimax Cost Maximin Cost Center of Gravity Minimize Sum of Costs Sum of Costs = SC = TC +LC LC > TC Local Input Costs = LC = labor costs, ubiquitous input costs, etc. Minimize Individual Costs PC > DC Procurement Costs = PC “Weight-losing” activities DC > PC Distribution Costs = DC “Weight-gaining” activities Minimize System Costs TC > LC Transport Costs = TC = PC + DC

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Hotelling's Law

1

34

1

34

1

1 2

1

34 1 4

11

1-D Cooperative Location

30

1 2

w1 = 1 w2 = 2

Durham Raleigh US-70 (Glenwood Ave.)

Min

k i i

TC w d  Min

i i

TC w d 

2

Min

i i

TC w d 

12

   

1 2 2 2 *

0, 30 2 1(0) 2(30) 20 1 2

i i i i i i i i i i i i

a a TC w d w x a dTC w x a dx x w w a w a x w                

      

Squared−Euclidean Distance  Center of Gravity:

“Nonlinear” Location

k i i

TC w d  

13

Minimax and Maximin Location

• Minimax

– Min max distance – Set covering problem

• Maximin

– Max min distance – AKA obnoxious facility location

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2-EF Minisum Location

30 10

• 8

+8 +5

• 3

+2

• 5

+3

1 2

25

x TC

90

+w1 +w2 +w1+w2

• w2
• w1
• w1-w2

+ w

1

• w

2

1 1 2 2 1 2

, if ( ) ( ) ( ), where , if (25) (25 10) ( )(25 30) 5(15) ( 3)( 5) 90

i i i i i i i

w x x TC x w d x x x x w x x TC w w                        



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Median Location: 1-D 4 EFs

wi

• 5-3-2-4 = -14

+5-3-2-4 = -4 +5+3+2-4 = +6

Minimum at point where TC curve slope switches from (-) to (+)

5

TC

3 2 4

1 2 3 4

• 14
• 4

+2 +6 +14 +5+3-2-4 = +2 +5+3+2+4 = +14 5 < W/2 5+3=8 > W/2 4 < W/2 4+2=6 < W/2 4+2+3=9 > W/2

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Median Location: 1-D 7 EFs

• 83
• 82
• 81
• 80
• 79
• 78

34 34.5 35 35.5 36 36.5

Asheville Statesville Winston-Salem Greensboro Durham Raleigh Wilmington 50 150 190 220 270 295 420

40

3 2 4 3 5

6

1 13>12 10<12 6<12

1

: 2

j i i

W w







:

i

w

14>12 11<12 5<12 9<12 8<12

*

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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

19 53 82 42 9 8 39 6 101 101 < 129 50 151 > 129 157 > 129

*

48 107 < 129 53 59 < 129 6 6 < 129 62 62 < 129 19 81 < 129 48 129 = 129

*

39 39 < 129 90 129 = 129

*

Optimal location anywhere along line

: wi

: x

wi :

y :

   

1 1 2 1 2 1 2 2 2 2 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?

19

136, 68 2

i

W W w   



Optimal location (36N,82W) (65 mi from opt great-circle location) Answer :

24 10 42 11 25 24 48 25 10 42 11 24 42 10 11 25 24 24<68 66<68 76>68

*

11<68 53<68 63<68

* 88<68

Logistics Network for a Plant

D D D D EEEE FFFF GGGG CCCC BBBB A A A A A A A A

Customers DCs Plant Tier One Suppliers Tier Two Suppliers Resource Market

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

Supplier Customer

raw material finished goods ubiquitous inputs scrap 4 ton 3 ton 1 ton 2 ton

Production System Inbound FOB Origin

title transfer Seller you pay Buyer Seller

FOB Destination

supplier pays title transfer

FOB Destination

title transfer you pay Buyer

FOB Origin

customer pays title transfer

Outbound

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 cost cost at supplier Production Procurement Local resource cost cost cost (labor, etc.) Total delivered Production Inbound transport cost Outbound transport co cost cost Transport s cos t t (T       Inbound transport Outbound transport C) cost cost  

Monetary vs. Physical Weight

23

in

• ut

in

• ut

(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 i i i i i i i

i

w

TC 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 i

P 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:

24

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 i i i i i i i

i

w

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)?

25

Asheville Durham

raw material finished goods scrap

2 ton 1 ton 1 ton Product A

• 83
• 82
• 81
• 80
• 79
• 78

34 34.5 35 35.5 36 36.5

Asheville Statesville Winston-Salem Greensboro Durham Raleigh Wilmington 50 150 190 220 270 295 420 40

Wilmington Winston- Salem

raw material finished goods ubiquitous inputs

1 ton 3 ton 2 ton Product B

1

( ) ( , )

m i i i i

i

w

TC X f r d X P





Ex 5: 1-D Location with Procurement and Distribution Costs

Assume: all scrap is disposed of locally

26

Asheville

unit of finished good

1 ton Production 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 physical weight weight ($/mi-yr) ($/ton-mi) (ton/yr) i i i i i

TC w d w f r    



in

• ut

in

• ut

(Montetary) Weight Gaining: 50 60 Physically Weight Losing: 150 60 w w f f          

20 10 30

40

10 70>55 50<55 40<55

1

: 2

j i i

W w







:

i

w

60>55 30<55 40<55

*

Asheville Durham Statesville Winston-Salem Wilmington

Assume: all scrap is disposed of locally

27

Asheville

unit of finished good

1 ton Production System Durham

NF

4 3 5 1 2

• ut

$1.00/ton-mi r 

3 3 3 out

10, 10 f w f r   

4 4 4 out

20, 20 f w f r   

5 5 5 out

30, 30 f w f r   

in

$0.33/ton-mi r 

 

1 1

• ut

1 1 in

2 60 120, 40 f BOM f w f r     



 

2 2

• ut

2 2 in

0.5 60 30, 10 f BOM f w f r     

