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Safeguarding Transport Systems

Safeguarding Transport Systems. Frank Horowitz Peter Hornby CSIRO Exploration & Mining. With Thanks To:. Fabio Boschetti, CSIRO David Batten, CSIRO Hugh Barkley, DSTO John Finnigan, CSIRO John M. Blatt, late of UNSW. Problem Specification.

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Safeguarding Transport Systems

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  1. Safeguarding Transport Systems Frank Horowitz Peter Hornby CSIRO Exploration & Mining

  2. With Thanks To: • Fabio Boschetti, CSIRO • David Batten, CSIRO • Hugh Barkley, DSTO • John Finnigan, CSIRO • John M. Blatt, late of UNSW

  3. Problem Specification • Transportation of weapons systems needs to be done safely • Orders of merit (to be "optimised") • Systemic Risk • Contracts satisfied? (time, cost) • Transport reliable? ("public liability") • Police & Emergency Services • contacted? response needed? resources? timing? • Political Issues • how to "prove" to government system is O.K. • ...

  4. Mobile Unit 1 Mobile Unit 2 Mobile Unit 3 Mobile Unit 4 Central Depot Branch Depot Mobile Unit 5 Boiling Down to Tractability • Model transport from Central Depot to Mobile Unit via Branch Depot • Appears to be a Network Flow problem, but there's more to it than that...

  5. Background Scientific Questions • Put this/these (Multi-Agent?) System(s) on firm mathematical basis? • Can we begin to describe dynamics of global system? • Leading in to our companion project • Dual Behavioural State Based Modelling of Multi-Agent Systems • "Optimal" vs. "Adaptive" system? • "Innovation" comes from where?

  6. Chosen Model • Input/Output System • Matrix representation appealing from a number of standpoints • Optimal Control • Each agent optimising locally vs. "Fat Controller" optimising globally.

  7. I/O Model Structure • System evolution is represented by • Subject to the constraints • Perhaps better represented via a "partitioned input/output matrix":

  8. Element: One Arc Problem

  9. Constraints: One Arc Problem • We need to "solve" constraint equation • C is rectangular leading to consideration of its nullspace (one dimensional in this case)

  10. One Arc Constraints (contd.) • Numerically • But i is bounded by considering "do nothing" and "transport everything" policies • Agents are left with 1 D.O.F. control instead of 3 D.O.F. control • good for optimisation

  11. One Arc optimal control • Bellman equation • Final time step: • Need to know functional form of h

  12. Functional Forms • Boundary minima • Bang-Bang controller for many families of cost function

  13. Element: Disassembly Process • Some Defence systems must not be transported assembled

  14. Disassembled Constraints • Similarly to One Arc Problem, we find • With 1 D.O.F. control bounded between "do nothing" and "disassemble everything" • Similar for reassembly...

  15. Transport Network?Multiple Time-Steps? • We treat a "road junction" as a commodity (without an inventory process) that must be exhausted at each time step • an alternative would be an inventory with a very high cost to allow for the possibility that the transport breaks down at the junction • different choices of route from junction are treated as distinct composite transport processes with control decision already made • implication: no decisions made by driver at junction • if you want the driver able to make decision, either commodity or inventory process at junction must always be exhausted in one time step by one out of the many available "destination transport" processes • Bang-bang still optimal whenever it was previously • Similar treatment for multiple time step processes

  16. Junction/Multiple Timestep

  17. Constructing a Bigger System • Analyze each Element to find reduced D.O.F. control and constraints • Assemble each Element into Global C/B system • Global controls and constraints follow from Elemental ones (with appropriate bookkeeping) • Pick some subset and assign controls to "Agent" • Global cost function • linear if Elemental cost functions are linear • Deliberate analogies with Finite Element Method

  18. One arc disassembled transport...

  19. ...and its constraints

  20. Goal state? • Formal end state "free" • Add to cost function • may be a function of time • Allows machinery to treat long time systematics in a meaningful way. • Run the Belman equation (or it's equivalents from either Pontryagin theory or Blatt's sufficient condition)...

  21. Stochastic control? • Instead of treating risk as a simple time varying function, and minimizing it, we could treat it as a random variable of known distribution and minimize the expectation value • Leads to nonsensical results of the class: "go-for-broke" at all times • Probability of bad outcome approaches unity with time • Partial solution: define a utility function that has infinite cost for a bad outcome, and minimize expectation value of the utility (due to D. Bernoulli, 1738) • John Blatt published a nice paper arguing against an axiom underlying using this as a general policy entitled "The Utility of Being Hanged on the Gallows" (1979)

  22. Adaptive Control? • What if we don't know distribution of stochastic shocks a priori? • One strategy is to send empty "probe" trucks down each route simply to sample probability of being attacked (Blatt, 1981) • Another similar one might be to continually send empty trucks down all routes, and only load some randomly (Grisogono, personal communication) • Still probes probability distribution as a function of time • expen$ive • A third strategy might be to allow each agent to be running something like a suite of Kalman filters (effectively predicting stochastic shock distributions) simultaneously with controlling processes (Hornby; Dual Behavioural State... project) • choosing one that has highest recent success rate to feedback into controller • what does "success rate" mean for extremely rare events???

  23. Innovation? • Could try "dumb" A.I. approaches • e.g. building random C/B matrices, writing some cost function for "usefulness" and running an optimizer over them, deploying the "best" few • Might the entire class under-perform humans in dreaming up new strategies? • Personally, I think I'd bet on Bernoulli, Blatt, Grisogono, Hornby, et al. for better innovation than a machine • But it's a bet on human intelligence and pattern recognition vs. machine state-space exploration, so it is not a sure thing by any stretch!

  24. Summary • Input/Output framework for optimally controlled transport system • Quite a general framework • Reduction in D.O.F.s of control system available from structure of problem elements • Construction of composite system from analyzable simple elements • Physical constraints on state implementable via analysis of elements • "Optimal" is slippery concept: defined w.r.t. specified cost function. • specify that wrong, and "optimal policy" is likely thundering nonsense • Optimality might not be what you actually want: • Adaptability? • Innovation? • Robustness? • Sustainability? • We're running out of time before we're running out of problem!

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