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© Princeton Modeling Resources 1
PLASMA Status report
Princeton Modeling ResourcesDecember 19, 2006
Warren PowellBelgacem Bouzaiene-Ayari
© 2006 Princeton Modeling Resources
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© Princeton Modeling Resources 2
Locomotive planning models
Applications: Strategic planning
• What is the impact of changes in fleet size and mix on train delay?• How do changes in shop locations affect maintenance routing?• How do changes in schedule affect train delay for a given locomotive
fleet?• How do changes in operating policies affect performance?
Tactical planning• How much power will you have at each terminal 1, 2, 3 days out?• Where do you anticipate being short power?
Real-time planning• What train should a locomotive be assigned to in order to get it to
shop on time?
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© Princeton Modeling Resources 3
Locomotive planning models
The PLASMA family of locomotive models PLASMA/OP – Short term operational planning and
forecasting• Projects locomotive movements 1-5 days into the future.
PLASMA/FS – Fleet sizing• Strategic planning model for:
– fleet sizing– Reliability of transit times– ….?
PLASMA/RT – Real-Time locomotive assignment model• Makes detailed locomotive to train assignments for a single
terminal over a short horizon.• Works real-time with user, responding to user overrides.
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© Princeton Modeling Resources 4
Locomotive planning models
PLASMA/OP – Short term operational planning and forecasting Uses “optimizing-simulator” technology to capture high
level of detail.• Combines flexibility of simulation• Intelligence of optimization
Projects locomotive movements 1-5 days into the future.
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© Princeton Modeling Resources 5
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© Princeton Modeling Resources 6
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© Princeton Modeling Resources 7
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© Princeton Modeling Resources 8
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• Model breaks the network into the “subproblems."
• A “subproblem” can be• The entire network at one
point in time.• A single yard.• The yards covered by a
single desk (by an ST).• We will generally break
the problem into regions that are covered by a single ST.
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© Princeton Modeling Resources 10
New features
Implementation of “nonlinear logic.” Old LRM approximated the value of an additional
locomotive in the future by using a constant numbers (five locomotives produced five times as much as one locomotive).
PLASMA uses “nonlinear logic” – The fifth locomotive is not worth as much as the first.
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• In the first generation model, we used a “linear” model to capture the value of locomotives in the future.
• Easier to estimate.• Easier to solve
• But…• Algorithm was slow• Solution could be unstable.
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• PLASMA uses a “nonlinear” function to estimate the value of locomotives in the future.
• More realistic (if we need five locomotives, the value of the sixth locomotive is not as much as the fifth).
• Hard to solve, but modern technologies have no trouble handling this.
• Model is faster and more stable.
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© Princeton Modeling Resources 13
New features
Nonlinear logic Initial testing suggests a dramatic improvement in the
rate at which the model reaches a solution. Solution appears to be much more stable than the older
“linear” logic.
38000000
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Iteration
Tota
l rew
ard
“Linear logic” “Nonlinear logic”
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Iterations
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© Princeton Modeling Resources 14
Coverage
90
91
92
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94
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100
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Iteration
% t
rain
s m
ove
d
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© Princeton Modeling Resources 15
“Objective function”
Total contribution
14000000
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
Iterations
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© Princeton Modeling Resources 16
Solving the “Subproblem” Old LRM model: Used heuristic logic to get a
“good” but not optimal solution.• From time to time, logic simply would return unusual
solutions not explained by the data. New logic: Uses Cplex to return “optimal”
solution.• Surprisingly fast.• Robust – if we do not like the solution, we cannot
blame the algorithm.• Handles leader logic, consist breakup, shop routing,
foreign power, …• Commercial software rather than custom algorithm.
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Influencing model behavior: Humans use a series of rules to govern how
they make decisions. Models use a series of costs and rewards to
penalize undesirable behavior:• Cost (negative benefit) for breaking up a consist.• Reward (positive benefit) for moving a train.• Higher rewards for more important trains.• Reward for moving a locomotive toward its shop.
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© Princeton Modeling Resources 18
Knowledge
Costs allow you to make tradeoffs:
Atlanta
Issue “cost”/“bonus”
Moving a train $2,000
Appropriate type -$500
Consist breakup -$800
Train delay $0
Routing to shop +$1,000
Total “cost” +$1,700
Baltimore
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Rule-based logic How it works:
• Code rules to determine which locomotives get assigned to each train.
Strengths:• Easy to understand.
Weaknesses:• Very hard to code all
the different situations that might arise.
• Very hard to make tradeoffs.
Cost-based logic How it works:
• Design costs and bonuses to penalize bad behavior and reward good behavior.
Strengths:• Easily handles tradeofs.• Can handle broad range
of very complex problems.
Weaknesses:• Can be difficult to “get
the model to do what you want.”
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Consist Breakup
Locomotives Trains
Consist C
z1
z2
z3
z4
z5
Destinations
Cost on assignment arcs influence behavior.
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© Princeton Modeling Resources 21
Train connection: reward the connections
680-xx/xx/xx
667-xx/xx/xx
681-xx/xx/xx
667-xx/xx/xx
666-xx/xx/xx
1. Add a reward on each arc going from a consist to a connected train (Foreign reward or Plan reward)
2. At each train node and for each connection coming from a consist add a constraint limiting the number of axles assigned to the train to the connection max axles.
3. Stop enforcing the connection if: the total hold time is greater than the connection time or we connected the required axles.
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LRM Model
Subproblem Model: (build nodes & Train Arcs):
Locomotives Trains
Duplicated Trains
Super Sink
Yard A
Yard B
Yard C
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© Princeton Modeling Resources 23
LRM Model
Subproblem Model:(add Hold arcs . . . and VFAs)
Locomotives Trains
Duplicated Trains
Super Sink
VFAs
Yard A
Yard B
Yard C
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© Princeton Modeling Resources 24
LRM Model
Subproblem Model:(add Assign arcs . . . And more VFAs)
Locomotives Trains
Duplicated Trains
VFAs
Yard A
Yard B
Yard C
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© Princeton Modeling Resources 25
LRM Model
Subproblem Model:(add VFA arcs)
Locomotives Trains
Duplicated Trains
VFAs
Yard A
Yard B
Yard C
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Outline
Fleet sizing
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Locomotive planning models
PLASMA/FS – Fleet sizing Strategic planning model for recommending fleet sizes Uses PLASMA-OP as core engine.
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Planning model
Classical fleet sizing model Elegant, but requires large LP model…
Time
Circulation arc
10,000
7,850
Fleet size = 2,150
550
720
660
220
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© Princeton Modeling Resources 29
Planning model
Classical fleet sizing model Elegant, but requires large LP model…
Time
Circulation arc
10,000
7,850
Fleet size = 2,150
550
720
660
220
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© Princeton Modeling Resources 30
Planning model
PLASMA/FS Use PLASMA/OP as core engine
Circulation arc
10,000
7,850
Fleet size = 2,150
550
720
660
220
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© Princeton Modeling Resources 31
Planning model
PLASMA/FS Nonlinear functions determine how many locomotives to
send to each location.
Circulation arc
10,000
Fleet size = 2,150
550
720
660
220
7,850
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© Princeton Modeling Resources 32
Features: All the calibration that is invested in the
operational model is immediately used in the fleet sizing model.
We can specify the fleet size• PLASMA determines how many to send to each
location.• Eliminates need for artificial construction of locomotive
snapshot. Model will delay or cancel trains if the fleet size
is too small.• Allows for fleet size vs. service tradeoffs.
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Fleet sizing
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Fleet sizing
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Fleet sizing
Additional considerations: PLASMA will allow us to consider:
• Foreign power• Train delays• Shop routing• Unscheduled trains
Fleet requirement from plasma will be adjusted to reflect
• Locomotives being taken out of service due to maintenance issues
• Retirements• leasing
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Outline
Real-time locomotive planning
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Locomotive planning models
PLASMA/RT – Real-Time locomotive assignment model Works interactively with planners. Makes detailed locomotive assignment recommendations. Real-time response to user overrides. Keeps the “human in the loop.”
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Tactical planning model User will interaction with Plasma/OP through an
intermediate data server. Several minute delay between user inputs and model
response.
Data server PLASMA/OP
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Real-time planning PLASMA/RT will sit on user’s workstation. Response to user inputs < 5 seconds.
Data server PLASMA/OP
PLASMA/RT
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Solving a single “subproblem” When a single ST is planning, he will inevitably want to
override the model. We can react by reoptimizing just his problem.
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Outline
Calibration
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Model calibration Operational planning
• Are we moving trains on time?• Do the assignments obey basic operating rules?• Is this what an ST would do?• How well can we project future supplies and deficits?
Fleet sizing• How well can we predict train delay for a given fleet
size? Real-time
• Is this what an ST would do “right now”.
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Fleet sizing
Current fleet
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LRM performance
LRM vs. history (August, 2000)
2521
3032
41
21
3
7.710.6 12
0
5
10
15
20
25
30
35
40
45
Setouts Swaps Nonpreferredconsists
Underpowered Overpowered
Perc
ent
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Features: Foreign cycles. Locomotive Top Plan. Locomotive hard assignments. Locomotive consist logic: This logic is coded as part of
the IP model. The model is formulated is way that will penalize the consist break-up proportionally to the number of peaces. This has an impact on the CPU time and I spent sometime tuning the CPLEX parameters to accelerate the solution time.
Shop routing: partially coded
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Real-World operations
A
B
C D
E
1
2
3
4
5
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Real-World operations
This is the mathematically guaranteed, global optimal solution!
A
B
C D
E
1
2
3
4
5
?
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© Princeton Modeling Resources 51
ManagementModeler Central dispatch
Question: When developing a model, who do you keep happy?
Observation: If the modeler builds a model that central dispatch does not use, who are you going to fire? The computer model or your operations staff?
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The implementation challenge
How do we measure user happiness?
“Width of smile”
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The implementation challenge
But who do we keep happy?
Single rational decisionmaker model . . .
The real world . . .
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Question: Which driver would you assign to the load?
A
B
60 miles
30 miles
You are told that driver B won’t arrive until laterin the afternoon. Now what do you do?
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The implementation challenge
Formulating a model is like fitting a line through a set of points:
But every point is a different person!
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The implementation challenge
We may need to customize the model to accommodate individual preferences!
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Management would like to think
that buying an optimization model
is like buying a TV...
Optimization is a process, not a product.
More often, it is like having a baby ...
It is fun creating it, but you often have
second thoughts right after you get it!
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Outline
Next steps
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Next steps What is timing for first deliverable? How do we decide when a deliverable is
finished?