Modeling Interplanetary Logistics: A Mathematical Model for Mission Planning

1
Modeling Interplanetary Logistics A
Mathematical Model for Mission Planning

• Christine Taylor
• Graduate Student, MIT
• Olivier de Weck
• Graduate Student, MIT
• SpaceOps 2006
• Rome, Italy
• June 21, 2006

Interplanetary Supply Chain Management and
Logistics Architectures
2
Overview

• Challenges
• Background
• Terrestrial vs. Space Networks
• Nomenclature
• Space Time Expanded Networks
• Mathematical Model
• Optimization Approach
• Lunar Outpost Scenario
• Future Work

3
Challenges

• The Space Exploration Initiative dictates the
  design of a sustainable space exploration system
  to reach the Moon, Mars, and beyond.
• Sustainability
• Affordability
• Uncertainty
• Achieving these goals requires a systems level
  approach to design
• View multiple missions as space network
• Analyze the logistics for exploration during the
  mission planning

4
Background

• Transportation networks (Ahuja, 93) (Bertsimas,
  97)
• Given a network determine optimal routing to
  satisfy demand
• Examination of a Hub-and-Spoke Network A Case
  Study Using Overnight Package Delivery (Yang, 03)
• Given a network and a set of vehicles, determine
  optimal routes and allocation of resources to
  satisfy demand
• The Logic of Logistics Theory, Algorithms, and
  Applications for Logistics and Supply Chain
  Management (Simchi-Levi, 03)
• Given a network for a school bus, determine the
  optimal routing of a fleet of busses to minimize
  cost and satisfy all timing constraints

Sustainability requires logistics planning
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Transportation Networks
Air Transportation Network Jet Blue route map
Space Transportation Network
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Comparison of Terrestrial and Space Networks
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Nomenclature

• Commodities
• Supply class
• Ex. Crew provisions
• Demand
• Origin node and time interval
• Destination node and time interval
• Physical characteristics

• Elements
• Element type
• Ex. CEV
• Propulsive Capability
• Structural Parameters
• Payload Capacity
• Number Available
• Cost

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Motivation

• Time is an important element in space networks
• Properties of the arcs can change with time
• Time discretization is small
• Capture orbital dynamic relations (dt 1 day)
• Time horizon is long
• Capture campaign evolution (t 20-40 years)
• Earth analogies break down in space
• Multiple arcs between nodes (multiple
  trajectories)
• DV not distance is the driving parameter
• Arc parameters, including existence, is time
  varying
• Everything in space must be delivered
• Vehicles must have been payloads
• Even ISRU resources require the delivery of ISRU
  production units
• Necessary to use a time expanded network

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Time Expanded Networks

• Static networks represent the physical nodes and
  connections
• Define the locations of interest and the feasible
  connections between them
• Time Expanded Networks
• Create a copy of each node at each time
  discretization
• Arcs represent two types of operations waiting
  and transfer
• Waiting arcs join the same physical node at
  different times
• Transfer arcs join two different physical nodes
  at different times
• All arc properties are constant in time expanded
  networks
• Arcs exist in the time expanded network if
• They exist in the static network
• They move forward in time
• They represent feasible orbital dynamics

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Fundamental Equations

• Mass balance Input Output
• Commodities can be stored at a depot
• Commodities can switch between vehicles at a node
• Need to account for gain/loss factor
• Need to allow for ISRU as well as in-space
  fueling
• Capacity
• Requires that the modules carrying payload can
  accommodate them (mass and volume)
• Capability
• Is the stack powerful enough to perform the
  transfer
• Can be defined as the thrust, fuel mass, flow
  rate, Isp, and/or power of the module
• Impulsive transfers are defined only by the fuel
  mass and Isp
• Restrictions for variable vs. constant parameters
  must be implemented
• Timing
• Are the payloads transferred such that all timing
  restrictions are met

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Assumptions

• In-Space Modeling
• The model considers all commodities and elements
  originate in LEO
• Launching requirements are handled separately by
  a different model
• Multiple burns
• Assume that an element can only burn on
  consecutive transfer arcs
• Fueling
• All propulsive elements begin fully fueled and
  all fuel is expelled after the element performs
  the last burn
• Staging
• All propulsive elements are staged after the
  final burn UNLESS they are carrying commodities

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Time Expanded Network Nomenclature

• Define the static network as
• NS is the set of static nodes
• AS is the set of static arcs
• Define the time expanded network as
• N is the set of nodes in the time expanded
  network
• A is the set of arcs in the time expanded network

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Nomenclature

• Define paths through the time expanded network
• A path is a defined sequence of arcs
• For a given path, the time of flight is known
• Define k items of supply
• Each item k is characterized by a set of
  parameters
• Origin node in static network
• Destination node in static network
• Availability time interval
• Delivery time interval
• Maximum travel time
• Mass
• Volume
• Define a loss factor for each commodity
• Define as an absolute loss per unit time

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Commodity Path Feasibility

• Define the set of feasible paths for each
  commodity k
• Commodity must not travel longer than the maximum
  travel time
• The amount of a commodity lost during the
  transfer cannot exceed the original amount

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Formulation

• The amount of commodity shipped must be equal to
  the supply
• The capacity of a vehicle cannot be exceeded

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Lunar Outpost Scenario

• Goal Find the optimal strategy for sending
  supplies
• Problem
• Multiple missions to the lunar polar surface
• Multiple types of commodities
• Three year time horizon
• Challenge Find the right mix of pre-deployment
  and re-supply

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Time (days)
1 6 7 8 9 10
11. 459 460 461 462 463 464
Crew operations 2130 kg Stowage 560 kg
Exploration 2170 kg Waste
260 kg Crew provisions 8590 kg
Crew operations 1000 kg
Low Earth Orbit
..
..
Low Polar Lunar Orbit
..
..
Lunar South Pole
..
..
Crew operations 1000 kg
LEGEND
LSAM DS
Lunar CEV CM
EDS
Lunar CEV SM
LSAM AS
CLV Boost
LSAM Cargo
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Time (days)
465 466 467 648 649 741 742
743 744 745 746 747 748
Crew operations 10250 kg Exploration
250 kg 4 Crew members 400 kg
Low Earth Orbit
..
..
Low Polar Lunar Orbit
..
..
Lunar South Pole
..
..
Crew operations 2130 kg Stowage 560 kg
Exploration 2170 kg Waste
260 kg Crew provisions 7920 kg
Crew operations 250 kg Exploration
250 kg 4 Crew members 400 kg
Crew provisions 670 kg
LEGEND
LSAM DS
Lunar CEV CM
EDS
Lunar CEV SM
LSAM AS
CLV Boost
LSAM Cargo
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Time (days)
749 830 831 924 925 926 927
928 929 1013 1014
Crew operations 1850 kg 4 Crew members 400
kg
Low Earth Orbit
..
..
Low Polar Lunar Orbit
..
..
..
Lunar South Pole
..
..
..
Crew operations 1850 kg 4 Crew members 400
kg
Crew operations 5000 kg
Crew operations 5000 kg
LEGEND
LSAM DS
Lunar CEV CM
EDS
Lunar CEV SM
LSAM AS
CLV Boost
LSAM Cargo
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Future Work

• Network optimization provides a powerful decision
  tool for exploration logistics planning
• Determine a push-pull boundary for commodities
• Defines if and where a depot should be located
• Indicates potential commonality requirements
  between elements
• Current implementation requires work
• Implement crew returns
• Consider non-zero loss/gain factors
• Expand the network to include multiple types of
  trajectories (i.e low thrust)

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Earth Surface Nodes
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Lunar Surface Nodes
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