From: mailbox@dmitry-kazakov.de   
      
   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...   
      
   --   
   Regards,   
   Dmitry A. Kazakov   
   http://www.dmitry-kazakov.de   
      
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