boston bound team 6 : charlene lieu, tom heaps-nelson, chris mcfadden, jeff cerilles, chris...
TRANSCRIPT
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