BOSTON BOUND

TEAM 6: Charlene Lieu, Tom Heaps-Nelson, Chris McFadden,

Jeff Cerilles, Chris Kilburn-Peterson, Maria Claudia Sonnet, Derrick Lay

Optimization of LFM Student Summer Travel to Boston

Map courtesy of http://www.theodora.com/maps used with permission.

Introduction

•Problem Definition•Assumptions•Non-Linear Model•Linear Column Gen. Model•Comparative Results•Implementation•Questions

•Most students moved themselves•The cost of moving trucks is substantial, and in some cases prohibitive•Objective to improve/optimize the total cost of moving for LFMers.

•Financial Burden: $50,613•Total Distance Traveled: 58,258 miles

Map courtesy of http://www.theodora.com/maps used with permission.

•Problem Definition

C

1) Students are willing to share U-HAUL trucks

2) Used average U.S. gasoline price

3) Max number of students for a trip is 3

4) Int’l and local students excluded

5) Traveling ‘as the crow flies’, always moving closer to Boston

6) Volume of shipped goods is proportional to number of students per trip

•Assumptions

Map courtesy of http://www.theodora.com/maps used with permission.

Decision Variables: for i = student, j = route

Constraints:

must be binary

Objective Function: minimize

ijx

ijx

•Model Equationsix

jij 1

jxi

ij 3

j

jtruckj

jfuelj

j ccc ,,

2

107.56 - fd*0.7647*3*2* j

,i

iji

iji

ijjtruck xxxc

1

107.33 - fd*0.7899*3*1* j

iij

iij

iij xxx

2

168.1 - fd*0.1735 fd*0.0002*2*1* j

2j

iij

iij

iij xxx

efficiency fuel

cost fuel*, jjfuel dc

14’ Truck, 1 Person

17’ Truck, 2 Person

24’ Truck, 3 Person

1 and only 1 route per person

max 3 people per route

a person is included in a route, or not

Simplified Version:

Name City St Lat Long X Y Dist R1 R2 R3 R4

Naughton Rumford RI 41.73 71.43 -27 43 52 1 0 0 0 1

McKenney Falmouth MA 41.78 70.5 36 40 55 0 1 0 0 1

Raphel Dedham MA 41.4 70.62 28 66 72 0 0 1 0 1

McFadden Windsor CT 41.93 70.68 114 30 118 0 0 0 1 1

1 1 1 1

•Non-Linear Modelijx

ixj

ij 1jxi

ij 3

Simplified Version:

Name City St Lat Long X Y Dist R1 R2 R3 R4

Naughton Rumford RI 41.73 71.43 -27 43 52 1 0 0 0 1

McKenney Falmouth MA 41.78 70.5 36 40 55 0 1 0 0 1

Raphel Dedham MA 41.4 70.62 28 66 72 0 1 0 0 1

McFadden Windsor CT 41.93 70.68 114 30 118 1 0 0 0 1

2 2 0 0

•Non-Linear Model

Simplified Version:

X Y Dist

R1

RX RY

RDist

-27 43

52 1 -27 43 50.8

36 40

55 0 -27 43 0

28 66

72 0 -27 43 0

114

30

118 1 114

30 141.6

2 192.3

•Non-Linear Model

RX =Xij*Current X value + (1-Xij)* Previous X value

RDist = SQRT (( Current RX – Previous RX)^2 + (Current RY – Previous RY)^2 )

A

ArK

C

XX

A

K

Ar

C

Map courtesy of http://www.theodora.com/maps used with permission.

Zone DefinitionZone 2

Zone 3

Zone 4

Zone 1

Much of the spirit and infrastructure is shared with the Non-Linear Model

Decision Variables: binary with j = route

Constraints:

Objective Function: minimize

where

and the truck and fuel costs are the same

•Column Generation Model

jx

i 1 jij xm

jmi

ij 3

1 and only 1 route per person

ensured by column generation

jtruckjfuelj ccc ,,

j

jjcx

Simplified Version:

Name …… R1 R2 R3 R4 R5 R6 R7 R8 R9 R10

Naughton …… 1 0 0 0 1 1 1 0 0 0 1

McKenney …… 0 1 0 0 1 0 0 1 1 0 1

Raphel …… 0 0 1 0 0 1 0 1 0 1 1

McFadden …… 0 0 0 1 0 0 1 0 1 1 1

1 1 1 1 0 0 0 0 0 0

•Column Generation Model

Simplified Version:

Name …… R1 R2 R3 R4 R5 R6 R7 R8 R9 R10

Naughton …… 1 0 0 0 1 1 1 0 0 0 1

McKenney …… 0 1 0 0 1 0 0 1 1 0 1

Raphel …… 0 0 1 0 0 1 0 1 0 1 1

McFadden …… 0 0 0 1 0 0 1 0 1 1 1

0 0 0 0 0 0 1 1 0 0

•Column Generation Model

Routes Before Optimization

•Results

Routes After Optimization

•Results

Routes After Optimization

•Results

Optimization demonstrates significant cost and distance reduction

Current Non-Linear Program LP - CG Improvement

Run w/ Manhattan

Run w/out cost

Run w/cost

Distance (miles)

58,000 (19333=LB)

22,219 22211 29,680 24,773 35,789

Fuel ($) 9936 ----- 3788 4287 4225 6,148

Truck ($) 40,676 ----- 16660 18999 17,298 24,016

Total Cost ($)

50,612 ----- 20448 23,286 21,523 $30,164$30,164Time ---- 15 x 5 min 3 x 5 min 24 hrs 3 sec ----

•Results

Optimization demonstrates significant cost and distance reduction

Current Non-Linear Program LP - CG Improvement

Run w/ Manhattan

Run w/out cost

Run w/cost

Distance (miles)

58,000 (19333=LB)

22,219 22211 29,680 24,773 35,789

Fuel ($) 9936 ----- 3788 4287 4225 6,148

Truck ($) 40,676 ----- 16660 18999 17,298 24,016

Total Cost ($)

50,612 ----- 20448 23,286 21,523 $30,164$30,164Time ---- 15 x 5 min 3 x 5 min 24 hrs 3 sec ----

•Results

Model is simple to setup and run:• Interested students must be contacted• Latitude & longitude for each city must be determined• Sufficiently powerful solver must be available (LS-GRG)• Occasionally, averages for fuel cost and truck rental should be

recalculated.

Possible Complications/Developments

• Timing of students’ moves may conflict• Truck size volume is NOT necessarily proportional to number of

students• How to determine savings distribution per student• How to solve LFM financial woes• Expanding the NL program to run for a population of thousands.

•Implementation

•Questions?

Map courtesy of http://www.theodora.com/maps used with permission.

•Non-Linear Model

Manhattan & Pythagorean Models

20000

21000

22000

23000

24000

25000

26000

27000

28000

29000

30000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15Run Number (Manual)

To

tal

Dis

tan

ce

Pythagorean

Manhattan