From: mailbox@dmitry-kazakov.de   
      
   On Sat, 16 Oct 2004 16:21:41 -0700, user@domain.invalid wrote:   
      
   > Dmitry A. Kazakov wrote:   
   >> On 15 Oct 2004 09:28:59 -0700, Ethan Seng wrote:   
   >>   
   >>>Thank you for your kind assistance and advice. Yes, you are right. My   
   >>>problem is about determining the membership functions of input   
   >>>variables (specifically C and V) for which the crisp output contains   
   >>>(t1, y1), (t2,y2), (t3,y3),..., (tN,yN)  pairs of numerical data. I   
   >>>also know beforehand that all variables are related by   
   >>>y=(30/V)*exp(-C*t/V). Can you suggest some technical reference from   
   >>>which I may find the methodology for determining membership functions   
   >>>of C and V using Sup-norm? Once again, thank you in advance for your   
   >>>attention.   
   >>   
   >> I would say first look for a good book on numerical methods. Unfortunately   
   >> it is not my field so I can only tell that you have a non-linear   
   >> optimization problem. It is also ill defined, because obviously you can fix   
   >> C=0 and get Vmin=30/ymax, Vmax=30/ymin, which I presume is definitely not   
   >> what you want. So you have to add some additional constraints on how to   
   >> vary V and C. Like Vmax-Vmin=Cmax-Cmin or (Vmax-Vmin)*(Cmax-Cmin)->min etc.   
   >>   
   >> A very brutal and crude method could be to evaluate V and C for each pair   
   >> (ti,yi), (tk, yk) and then get max and min for both over all pairs. But I   
   >> think there should be better, iterative methods. For example, you fix Cmin,   
   >> Cmax then find Vmin, Vmax, then attune C's find new V's etc, a kind of   
   >> Newton's process. It is pure numerical methods...   
   >   
   > Thanks for the reply. Does that mean that there will not be a unique   
   > solution to the problem?   
      
   Yes. It is ill-defined. So the solution will vary depending on which   
   additional assumptions one could take.   
      
   > In addition, can I conclude that there is no   
   > fuzzy logic component to the solution of this problem?   
      
   Fuzziness may still come in consideration by two ways:   
      
   1. The problem is formulated in terms of fuzzy sets.   
   2. The solution is fuzzy due to lack of knowledge how to solve that.   
      
   (1) is not the case, but it could be, if for example the error risk   
   criteria will be defined in terms of a fuzzy measure.   
      
   (2) is also well possible. There are many examples of how uncertainty   
   appears where there should be no one. The most notorious coming in mind   
   ones are Monte Carlo methods, interval computations, fuzzy control.   
      
   ... and, well, fuzzy logic is just a nice abstraction. In the end   
   everything is just a numerical problem! (:-)) Membership function is a   
   function as any other. It is at our discretion to attribute some semantics   
   to it. But it is fuzzy no longer we think it is.   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
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