optimal fueling strategies for locomotive fleets in railroad networks
DESCRIPTION
Optimal Fueling Strategies for Locomotive Fleets in Railroad Networks. William W. Hay Railroad Engineering Seminar February 17, 2012. Seyed Mohammad Nourbakhsh Yanfeng Ouyang. Outline. Background Model Formulation Optimality Properties and Solution Techniques Case Studies Conclusion. - PowerPoint PPT PresentationTRANSCRIPT
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Optimal Fueling Strategies for Locomotive Fleets in Railroad Networks
Seyed Mohammad Nourbakhsh
Yanfeng Ouyang
1
William W. Hay Railroad Engineering Seminar
February 17, 2012
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Outline
• Background
• Model Formulation
• Optimality Properties and Solution Techniques
• Case Studies
• Conclusion
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Fuel Price
3
• Railroad fuel consumption remains steady
• Crude oil price sharply increases in recent years
• Fuel-related expenditure is one of the biggest cost items in the railroad industry
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Fuel Price
• Fuel (diesel) price influenced by:– Crude oil price
– Refining
– Distribution and marketing
– Others
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Fuel Price
• Fuel price vary across different locations
• Each fuel station requires a long-term contractual partnership– Railroads pay a contractual
fee to gain access to the station
– Sometimes, a flat price is negotiated for a contract period
5
• US national fuel retail price, by county, 2009
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Locomotive Routes in a Network
6
Candidate fuel station
Contracted fuel station
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Motivation
• Usage of each fuel station requires a contractual partnership cost
• Hence, should contract stations and purchase fuel where fuel prices are relatively low (without significantly interrupting locomotive operations)
– In case a locomotive runs out of fuel, emergency purchase is available anywhere in the network but at a much higher price
– Each fueling operation delays the train
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The Challenge
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The Challenge
10
$ $
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$
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• Fuel cost vs. contract cost– Too few stations = high fueling cost (e.g., emergency purchase)
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The Challenge
11
• Fuel cost vs. contract cost– Too many stations = high contracting costs
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Problem Objective
• To determine:– Contracts for fueling stations– Fueling plan for all locomotives
• Schedule
• Location
• Quantity
• To minimize: – Total fuel-related costs:
• Fuel purchase cost
• Delay cost
• Fuel stations contract cost
12
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Outline
• Background
• Model Formulation
• Optimality Properties and Solution Techniques
• Case Studies
• Conclusion
13
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Notation
14
• Set of candidate fuel stations, N = {1,2,…, |N|}
• Set of locomotives, J = {1,2,…, |J|}
• Sequence of stops for locomotive j, Sj = {1,2,…, nj}, for all j J
i=1 i=2 i=3
i=|N|i=|N|–1
i
For any location i
ci = Unit fuel cost
a1 = Delay cost per fueling stop
a2 = Contract cost per fuel station per year
Mi = Maximum number of locomotives passing
p=Unit fuel cost for emergency purchase (p>ci for all i)
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Notation
15j=1 j=|J|j
• Set of candidate fuel stations, N = {1,2,…, |N|}
• Set of locomotives, J = {1,2,…, |J|}
• Sequence of stops for locomotive j, Sj = {1,2,…, nj}, for all j J
For any locomotive j
bj=Tank capacity
rj=Fuel consumption rate
nj=Number of stops
fj=Travel frequency
gj=Initial fuel
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Notation
• Set of candidate fuel stations, N = {1,2,…, |N|}
• Set of locomotives, J = {1,2,…, |J|}
• Sequence of stops for locomotive j, Sj = {1,2,…, nj}, for all j J
16
lsj = Distance between the
sth and (s+1)th fuel stations that locomotive j passes
jj=1 j=|J|1
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s+1
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Decision Variables
• For each station, contract or not?– zi = 1 if candidate fuel station i is contracted and 0 otherwise
• For each locomotive, where to stop for fuel?– xs
j = 1 if locomotive j purchases fuel at its sth station and 0 otherwise
– ysj = 1 if locomotive j purchases emergency fuel between its sth and
(s+1)th station and 0 otherwise
• How much to purchase?– ws
j =Amount of fuel purchased at stop s of locomotive j
– vsj =Amount of emergency fuel purchased between the sth and (s+1)th
stations of locomotive j
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Formulation
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Never run out of fuel
Must contract fuel stations for usage
Integrality constraints
Non-negativity constraints
Tank capacity never exceeded at emergency purchase
Fueling cost + delay cost + contract cost
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Problem Characteristics
• The MIP problem is NP hard…– Integration of facility location and production scheduling
• The problem scale is likely to be large
– of integer variables, of constraints
– For |J|=2500 locomotives each having nj=10 stops among |N|=50 fuel
stations, there are 50,050 integer variables and 100,050 constraints
• Commercial solver failed to solve the problem for real applications
• Hence, to solve this problem– Derive optimality properties to provide insights
– Develop a customized Lagrangian relaxation algorithm
20
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Outline
• Background
• Model Formulation
• Optimality Properties and Solution Techniques
• Case Studies
• Conclusion
21
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Theoretical Findings
22
Optimality Condition 1There exists an optimal solution in which a locomotive stops for emergency fuel only when the locomotive runs out of fuel.
Fuel station s1 Fuel station s2
Fuel Level
Optimal emergency fueling location
station
Tank Capacity
Not optimum
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Theoretical Findings
Station
Fuel level
1
Optimality Condition 2There exists an optimal solution in which a locomotive purchases emergency fuel only if the following conditions hold:
– Its previous fuel purchase (from either an emergency or fixed station) must have filled up the tank capacity
– If the next fuel purchase is at a fixed stations, then the purchased fuel should be minimum; i.e., the locomotive will arrive at the next station with an empty tank
23s+1s
Tank capacity
Not optimum
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Station
Fuel level
1 s+1s
Tank capacity
Theoretical Findings
24
Optimality Condition 3If a locomotive purchases fuel at two fixed fueling stations s1 and s2 (not necessarily adjacent along the route) but no emergency fuel in between, then there exists an optimal solution in which the locomotive either departs s1 with a full tank, or arrives at s2 with an empty tank.
Not optimal
Either of these two is optimal
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Lagrangian Relaxation
• Relax hard constraints:
• Then add them to the objective function with penalty:
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Structure of the constraints:
25
Constraints
Variables
Fueling plan for locomotive 1
Facility location and fueling constraints
Fueling plan for locomotive 2
Fueling plan for locomotive j
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Formulation of Relaxed Problem
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Relaxed Problem
• After relaxing hard constraints the remaining problem could be decomposed into sub-problems– Each sub-problem solves the fueling planning for each
locomotive
where zj(u) is optimal objective function of jth sub-problem
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Sub-problem for the jth Locomotive
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Sub-problem for Individual Locomotive• Three types of possible “optimal” fuel trajectory
– Type a: From one station to nonzero fuel at another station
– Type b: From one station to zero fuel at another station, without emergency purchase
– Type c: From one station to zero fuel at another station, after one or more emergency fuel purchases
Station
Fuel level
2 s–1
bj
1
0
s
(a)
(b)
(c)
rj
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Shortest Path Method
Station
State
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bj
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s
(a)
(b)
(c)
(s, 1)
(s,)
(s,)
(a)(a)
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• We find a way to apply a simple shortest path method to solve the sub-problem
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Outline
• Background
• Model Formulation
• Optimality Properties and Solution Techniques
• Case Studies
• Conclusion
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Test Case
($5)($3)
($3) ($5) ($2)
($4)($2.5)($2)
($2)
($2)
($3)
($3.5)
1000 1050 1100
1050 1500 1000 1050 1100
1050
1050 1100 1550
1050
1100 1600
1050
1000
1050 1000 1100
35
Network Information
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Test Case
36
Locomotive Route Information
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Test Case
• 12 nodes, 8 locomotives
• 1=100, 2=10,000
• Tank capacity=2500
• Different fuel price for fixed stations between $2 to $5
and $7 for emergency
• Frequency assumed 1 for all locomotives
• Consumption rate assumed 1 for all locomotives
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Optimal Fuel Stations
($5)($3)
($3) ($5) ($2)
($4)($2.5)($2)
($2)
($2)
($3)
($3.5)
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Optimal Fueling Plan: Locomotive 1
$5$3
$3 $2
$4$2.5$2
$2
$2
$3
$3.5
$5
(1000)
(1700) (2500)
(750)(2500)
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Optimal Fueling Plan: Locomotive 2
$5$3
$3 $2
$4$2.5$2
$2
$2
$3
$3.5
$5
(650)
(2100)
40
(1000) (2500)
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Optimal Fueling Plan: Locomotive 8
53
32
42.52
2
2
3
3.5
5
(0)(0)
(1050)
(1650)
(2500) (1050)
46
$5$3
$3 $2
$4$2.5$2
$2
$2
$3
$3.5
$5
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Total Fuel Consumption
($5)($3)
($3)($5) ($2)
($4)($2.5)($2)
($2)
($2)
($3)
($3.5)
8700 15150 4950
610012650
6700 9,550
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Real World Case Study
• Full railroad network of a Class-I railroad company
• 50 potential fuel stations• Thousands of predetermined locomotive
trips (per week)• Fuel price from $1.9 - $3.0 per gallon with
average $2.5 per gallon• Tank capacity 3,000 - 5,000 gallons• Consumption rate 3 - 4 gallons per mile• Contracting cost of fuel stations $1 - $2
billion per year
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Real World Case Study
• Algorithm converges after 500 iterations in 1200 CPU
seconds
• The optimality gap was less than 6%
• This model can efficiently reduce the total cost of the
system
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Real World Case Study
• Solution 1: Bench-mark (current industry practice)
• Solution 2: Optimal fueling schedule using all current stations
• Solution 3: Global optimum (using an optimal subset of stations)
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Outline
• Background
• Model Formulation
• Optimality Properties and Solution Techniques
• Case Studies
• Conclusion
52
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Conclusion
• A Mixed Integer Programming (MIP) model
– Integrates fuel schedule problem and station (location) selection
problem
– Considers fuel cost, delay, and fuel station contracting costs
• LR and other heuristic methods are developed for large-scale
problem with good computational performance
• We developed a network representation and shortest path method
for solving scheduling sub-problems
• This problem was later used as a competition problem at
INFORMS Railway Applications Section (RAS)
53