ESI 6912 Advanced Network Optimization

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ESI 6912Advanced Network Optimization

• Network Applications I

Ravindra K. Ahuja Professor, Industrial Systems
Engg. University of Florida Gainesville
ahuja_at_ufl.edu Office [phone] ext
2004 Cell [phone] www.ise.ufl.edu/ahuja
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Lecture Overview

• Me
• Arvind Kumar (arvind_at_InnovativeScheduling.com)
• Education PhD, Industrial Engineering,
  University of Florida
• Current Project Manager, Locomotive Scheduling
  Optimizer,
• Innovative Scheduling, Gainesville, FL.
• Problem Introduction
• Mathematical Formulation
• Solution Methodology
• Additional Issues

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Goals of this presentation

• An important point which I want to bring home
• Network representation is a good way to
  facilitate communication between you and the
  customer.
• In business, communication is everything. In this
  example, I will show you how for an OR
  professional, a picture is worth thousand words.
• Remember, even if you are not solving the problem
  as a network flow, it aids in understanding and
  explaining the formulation.
• It makes both your Left and Right side of brain
  work. ?

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Goals of this presentation (contd..)

• Introduce an important transportation problem
• Locomotive Planning
• Why is it important?
• Each locomotive costs almost 1 Million.
• A railroad owns a fleet of around 5,000
  locomotives.
• Every year, a railroad buys over 100 new
  locomotives.
• The problem we are dealing with has a potential
  of saving millions of dollars even if we reduce
  the expenditure by just 5.

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Locomotive Optimization
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Locomotive Optimization, Inputs

• Train Schedule
• Train origin, destination, arrival time,
  departure time
• Train power requirement
• Locomotive Availability
• Types of locomotives, pulling power
• Locomotive sets (CONSISTS) Allowed
• Example locomotive type SD40, CW40, CW60
• Example Allowed consists 2xSD40,
  SD40CW40, 2xCW40CW60

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Locomotive Optimization, Decision Variables

• Train Locomotive Assignment
• Assigning locomotive consists to trains which
  provide enough power to pull the train.
• The set of locomotives should meet some other
  basic requirements
• Lower bound on total number of locomotives on a
  train
• Upper bound on total number of locomotives on a
  train
• Train-locomotive preference
• Train-Train Connection
• Connecting locomotives on a train arriving at a
  station to a train departing at the station while
  making sure that
• The connection time should be in the range of
  min/max time allowed
• Some connections are preferred over another

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Locomotive Optimization, Decision Variables
Train 1
Train 4
Train 2
Time
Train 5
Train 3
Train 6
Ground Nodes
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Locomotive Optimization, Decision Variables

• The number of locomotives in/out of a station is
  dependent on the train moves and their power
  requirement.
• If the total power requirement on inbound trains
  is not equal to the total power requirements on
  outbound trains, the flow cannot be balanced.
• Locomotives are allowed to DEADHEAD on a train
  while meeting certain requirements.
• At most 12 locomotives allowed on a train.
• Not all locomotive types are allowed on a train.
• Deadheading causes extra fuel to burn, thus
  adding to the operational cost.

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Locomotive Optimization, Constraints

• Do not perform consist busting
• Do not bust any consist. Assign consists from one
  train to another without busting them.
• A locomotive based formulation versus consist
  based formulation.
• Locomotive assignment to be consistent
• A train should be assigned the same consist each
  day it runs. It is a soft requirement.
• Phase-wise approach.

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Locomotive Optimization, Constraints (contd.)

• Locomotives must be balanced
• The number of incoming locomotives of each type
  into a station must equal the number of outgoing
  locomotives of that type at that station.
• Flow balance constraints
• Plan should be repeatable
• The status of locomotives at the end of the plan
  should be same as the beginning of the plan.
• By connecting the trains on the last day to the
  trains on first day, and solving the flow problem
  in the cycle.

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Locomotive Optimization, Constraints (contd.)

• Honor fleet size requirements
• The number of assigned locomotives of each type
  used is at most the number of available
  locomotives of that type. A railroad can provide
  the number of locomotives available of each type.
  
• How will you find the total number of locomotives
  used over the week?
• Option 1 By summing up the locomotive-minutes
  spent by locomotives on all the arcs and divide
  it by number of minutes in week
• Option 2 By counting the number of locomotives
  in an appropriate cut. What is a cut here?

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Locomotive Optimization Model Summary
Objective Function

• Consist
• Assignment
• Active/Deadhead
• Light travel
• Shop power
• Run-through power

Innovative Locomotive Planning Optimizer
Train Schedules, Tonnages HP
Standard Consists
Cycle Reports
Shop, Servicing Fueling Locations
Locomotive Fleet Description

• Yard Reports
• Dwell time
• Daily supply-demand inventory
• Train-to-train connections

Constraints
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Locomotive Planning Engine

• Decision Variables
• Active consist assignment
• Deadhead consist assignment
• Light travel assignment
• Train-train connections
• Positioning of locomotives

• Objective Function
• Active locomotive cost
• Deadhead cost
• Light travel cost
• Single locomotive consist cost
• Consist-busting cost

Locomotive Planning Optimizer
Parameter Settings
User-Defined Policies

• Hard Constraints
• Meet horsepower tonnage req.
• Satisfy active axle requirements
• Balance locomotive flow
• Assign acceptable locos to trains
• Honor geographical constraints
• Assure cab signal requirements
• Meet foreign power constraints
• Satisfy shop power constraints

• Soft Constraints
• Consistency of the consist assignment
• Consistency of train-train connections
• Simplicity of the planning solution

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Mathematical formulation

• Multi-commodity integer program on the space-time
  network
• Let
• l a train
• k a locomotive type
• c a consist type
• Binary variable representing the number of
  active consists of type c on arc l
• Integer variable representing the number of
  non-active (idle or deadhead) consists of type c
  on arc l
• Integer variable indicating the number of
  unused locomotives of type k

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Mathematical Formulation (contd.)
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Isnt this slide easier to explain than the
previous one?
Plan should be repeatable
Train 1
Train 4
Train 2
Time
Train 5
Train 3
Train 6
Must get a compatible active consist. UB on total
locomotives
Flow should be balanced on all nodes.
Ground Nodes
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Solving complex real-life problems?

• Think decomposition !!
• Many real-life problems are very large scale in
  nature.
• It is tough to solve the whole problem in one
  shot.
• Use decomposition approaches.
• Decompose the problem in several tractable
  steps, where each of the steps add substantial
  value to the final solution.
• The decomposition should be done in a way so that
  you do not lose the global picture.

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Solution Methodology Two-Stage Optimization
One-Day Locomotive Optimizer
Seven-Day Locomotive Optimizer
Input Data
Solution

• Stage 1 considers only the trains with high
  frequencies (say, trains which run over four
  times a week) and creates a target locomotive
  plan for these trains.
• Stage 2 makes adjustment to the target locomotive
  plan to power trains with low frequencies.

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Two stage optimization Killing two birds with
an arrow

• Makes the problem tractable by making a
  justifiable assumption
• Consistency of the locomotive plan implies that a
  train gets the same consist type each day it
  runs.
• Obtaining a consistent locomotive plan when the
  train schedule is inconsistent is a computational
  challenge. We accomplish it using a two-stage
  approach.

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Solution Methodology

• One-day locomotive optimizer
• Constructs the one-day train network
• Solves the multi-commodity flow problem in this
  one-day network using mixed integer programming.
• Seven-day locomotive optimizer
• Constructs the seven-day train network.
• Uses the flow in one-day trains as the target in
  seven day.
• Solves the multi-commodity flow problem in this
  seven-day network using mixed integer programming.

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Homework A Research Issue

• Balancing locomotive over the network by
  deadheading
• Restricted by train schedule
• Locomotives deadheading on the train
  unnecessarily wait while train is performing work
  at intermediate stops.
• Resolution Allow LIGHT travel
• Allowing group of locomotive to travel together
  independently
• Faster travel time and flexible schedule
• Each Light travel must contain at least 4
  locomotives. (Fixed charge)
• What light travel to run and when?

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• Additional Issues.

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Outputs of the Locomotive Optimization

• Train locomotive assignments Which consist types
  are assigned to which train.
• Train-to-train connections How do consists
  connect between trains.

2?AC44
2?AC44
CW40CW66
CW40CW66
3?SD40
3?SD40
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A Potential Problem with our Solution

• Locomotives need to be fuelled and serviced
  regularly.
• Our plan ignores these two issues.
• The plan generated by planning optimizer is not
  guaranteed to be fueling and servicing compliant
  and might not be implementable.
• The solution does not account for fueling costs
  since fueling costs are location dependent.

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An Example

• Locomotive Optimization could generate
  connections of the following kind

A, 800 miles
F
B, 300 miles
F
Blue location Supports fueling, Red Does not
support fueling Suppose a locomotive cannot
travel more than 900 miles without fueling. Then,
in this solution, it runs out of fuel
Conclusion We need a method that generates
implementable planning solutions.
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Fueling and Servicing Friendly Routing

• Feasibility requirements
• Every locomotive is routed in such a way that it
  has at least one fueling opportunity for every
  900 miles of travel.
• Every locomotive is routed in such a way that it
  has at least one servicing opportunity for every
  3,000 miles of travel.
• Every locomotive is routed in a fueling and
  servicing friendly rotations or cycles.
• Cost requirements
• Minimize the total fueling cost
• Minimize the total servicing cost

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Fueling and Servicing Add-On Module
Locomotive Planning Optimizer
Fueling and Servicing Post-processor
Solution

• Fueling and Servicing Inputs
• Fueling locations
• Servicing locations
• Fueling time at each location
• Servicing time at each location
• Fueling and servicing capacities
• Fuel and service costs

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Post-Processor Module

• Take the output of the Locomotive Optimization as
  an input
• Fix the assignments of locomotives to trains as
  per the output and then re-structure
  train-to-train connections of locomotives to
  enforce fuel and service feasibility
• Decision problem in this module Connection of
  locomotives between trains!!

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Post-Processor Module (contd.)

• Importance of good connections

F
A, 500 miles
B, 300 miles
F
D, 300 miles
F
C, 500 miles
F
Blue nodes Fueling nodes, Red nodes
Non-fueling nodes Sequences A B, D B, D - C
Fueling feasible Sequence A - C Fueling
infeasible
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Solution Methodology
S
A, 500 miles
X 1
B, 300 miles
F
X 1
C, 300 miles
F
X 1
D, 500 miles
S
X 1
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Post-Processor Module Steps

• Enumerate the set of service (and fuel) feasible
  sequences of trains in the network using a
  dynamic programming enumeration algorithm.
• Decompose the output of the Locomotive
  Optimization into flows on service sequences
  while minimizing cost (decomposition).
• Assemble the service (and fuel) feasible
  sequences into fuel and service friendly cycles.

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What are Fuel and Service Sequences

• What is a fuel feasible sequence (or string)?
• It is a sequence of connected trains starting and
  ending at fuel locations
• Sequence is such that any locomotive that is
  assigned on the sequence will not run out of fuel
• What is a service feasible sequence (or string)?
• Sequence of connected trains starting and ending
  at maintenance locations
• Sequence is such that any locomotive that flows
  on the sequence can be serviced before it is due
• Every service feasible sequence is also fuel
  feasible gt Service feasible sequence is made up
  of fuel feasible sub-sequences

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Illustration of Fuel Service Sequences
A, 500 miles
S
F
B, 300 miles
C, 400 miles
F
S
D, 450 miles
Orange nodes Support servicing and fueling Blue
nodes Support only fueling Red nodes Support
neither Service sequence A B C D is hence
made up of two fuel friendly sub-sequences A B
and C D
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Solution Methodology Recap
S
A, 500 miles
X 1
B, 300 miles
F
X 1
C, 300 miles
F
X 1
D, 500 miles
S
X 1
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Post-processor String decomposition

• Definition Decomposition of a locomotive
  schedule into flow on service feasible sequences
• Decision variables Flow on service feasible
  sequences
• Strings are enumerated using dynamic programming
• Integer programming problem on space-time network
• Research issue
• How do we integrate fuel and service issues in
  the main locomotive optimization formulation?
• Post-processor does not guarantee a fuel-service
  feasible plan. What will you do if the plan is
  not feasible yet?

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Conclusion

• We solved a tough real life problem using a
  network based formulation.
• The formulation used decomposition methods to
  address many business requirements.
• Network representation are powerful tool to
  facilitate communication !!
• Dont forget your homework
• Light Travel
• Integrating fuellingservicing with locomotive
  optimization formulation.