Optimization Halp!

From: Rishi Tandon (tandon.rishi_at_gmail.com)
Date: 03/21/05 

Date: 21 Mar 2005 10:23:18 -0800

This is a quite interesting (though somewhat tedious to explain)
problem:

The System:

...[Tank 1]............(-a)........(0).........(a)............[Tank
2]...

We have two fuel tanks, initially each with point mass (m1) at a
distance (x) on either side of the origin (0). A point mass (m0) is at
the origin.

Initially, the centre of gravity (c.g.) is at the origin (0).

(-a) and (+a) are limits to the c.g. travel in the 1-D problem.

I wish to consume the fuel in the two tanks, taking fuel out from only
one tank at a time without violating the constraints on c.g. travel.

The Problem:

When I change to taking fuel from one tank to the other, i call it a
"switch". When a switch occurs, the position of the c.g. is anywhere
within the permitted 1-d travel region.

Now to empty out the tanks, there will be a sequence of such switches
at points lying in the constraint region. *I wish to find that sequence
which requires the MINIMUM number of switches between tanks to EMPTY
out the fuel in the tanks!*

How do I go about finding this optimal sequence?

Intuition tells us that that sequence which always switches at the
extrema will yield the minimum number of switches needed. How do I
prove or disprove this?

Let me define this problem better:

1. R(n) is a value between 1 and -1.
2. a*R(n) gives us the Nth switching point on the 1-d constraint
segment. (a) is defined above.
3. R(0)=0 the origin.
4. R(1),R(2),R(3),....R(k) is a sequence of (k) steps (say) that is
required to remove empty the tanks.
5. What such sequence needs the minimum number of steps (k) to empty
the tanks? And is the same as the 1,-1,1,-1,1,-1,...... sequence that
would result if the extrema were taken?

More definitions:
1. Dm(n) is the mass removed from the appropriate tank at when the c.g.
moves from a*R(n-1) to a*R(n).
2. Thus successive masses removed are... Dm(1),Dm(2),Dm(3).....Dm(k).
3. Is the problem of minimizing (k) <the no. of steps> the same as
maximizing the sum of these masses removed for (k) steps? If so, what
sequence of R(1),R(2)...etc does this resilt in?

*******************Appendix*********************
More detail. A reduction formula was derived....

          R(n)-R(n-1) [x/a-R(1)] [x/a+R(2)] [x/a-R(3)]
[x/a-R(n-2)]
Dm(n)=-----------------------x----------x----------x----------x..------------
       [x/a +R(n)][x/a-R(n-1) [x/a+R(1)] [x/a-R(2)] [x/a+R(3)]
[x/a+R(n-2)]

Here, we assume that
1. (n) is an odd number. If you want it otherwise, reverse the signs of
R(n,n-1,n-2) in equation above.
2. This formulation assumes that Fuel is initially taken from Tank 1
such that c.g. moves towards (+a).
******************************************************

Question: How should I *approach* this problem to prove/disprove that
the extreme R(n) sequence is optimal in terms of no. of switches (k)?

Thank you for persevering enough to read this. Any further quries may
be addressed to tandon.rishi [AT] gmail DOT com.

Any help or directions regarding this will be greatly appreciated!

Rishi

Relevant Pages

  • Re: Other JSF options
    ... It gets a touch trickier than that, if only because you need more loggie assets to keep up with faster tanks which chew quickly into the available LIMs. ... that fuel ... OPORDS for brigade and battalion level operations, let me tell you that you ... unit charged and destroyed the Iraqi tanks at 73 Easting at close range... ...
    (rec.aviation.military)
  • Optimization Halp!
    ... I wish to consume the fuel in the two tanks, ... Now to empty out the tanks, there will be a sequence of such switches ...
    (sci.math.num-analysis)
  • Re: Optimization Halp!
    ... have as an optimization constraint the number of flippings of the switch. ... If, say, the tanks are on left and right wing of an airplane, I would ... Given the local status of fuel ... > which requires the MINIMUM number of switches between tanks to EMPTY ...
    (sci.math.num-analysis)
  • Re: Mustang & Spitfire
    ... They were so thick that the Mustang could carry as much fuel in one ... wing as the earlier spitfires could in their entire fuselage. ... gallon tanks added well behind the ...
    (rec.aviation.military)
  • Re: 2003 Highlaner V6 ---> Propane?
    ... replace the fuel injectors with ones ... Conversions are normally offered for current model-year new cars ... No. Propane has less heat energy per unit of volume than gasoline. ... LPG is in heavy steel tanks, and has a long proven history ...
    (alt.autos.toyota)
Quantcast