Importance of Sources using the Repeated Fusion Method and the Proportional Conflict Redistribution Rules #5 and #6

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1 Iportance of Sources using the Repeated Fusion Method and the Proportional Conflict Redistribution Rules #5 and #6 Florentin Sarandache Math & Sciences Departent University of New Mexico, Gallup Capus, US Jean Dezert French erospace Research Lab. ONER/DTIM/SIF 9 venue de la Division Leclerc 930 Châtillon, France bstract. We present in this paper soe exaples of how to copute by hand the fusion rule for three sources, so the reader will better understand its echanis. We also take into consideration the iportance of sources, which is different fro the classical discounting of sources. 1. Introduction. Discounting of Sources. Discounting a source 1 (.) with the coefficient 0 α 1 and a source (.) with a coefficient 0 β 1 (because we are not very confident in the), eans to adjust the to 1 (.) and (.) such that: 1 () = α 1 () for (total ignorance), and 1 ( ) = α 1 ( ) 1-α, and () = β () for (total ignorance), and ( ) = β ( ) 1- β. Iportance of Sources using Repeated Fusion. But if a source is ore iportant than another one (since a such source coes fro a ore iportant person with a decision power, let s say an executive director), for exaple if source (.) is twice ore iportant than source 1 (.), then we can cobine 1 (.) with (.) and with (.), so we repeated (.) twice. Doing this procedure, the source which is repeated (cobined) ore ties than another source attracts the result towards its asses see an exaple below. Jean Dezert has criticized this ethod since if a source is repeated say 4 ties and other source is repeated 6 ties, then cobining 4 ties 1 (.) with 6 ties (.) will give a result different fro cobining ties 1 (.) with 3 ties (.), although 4/6 = /3. In order to avoid this, we take the siplified fraction n/p, where gcd(n, p) =1, where gcd is the greatest coon divisor of the natural nubers n and p. This ethod is still controversial since after a large nuber of cobining n ties 1 (.) with p ties (.) for np sufficiently large, the result is not uch different fro a previous one which cobines n 1 ties 1 (.) with p 1 ties (.) for n 1 p 1 sufficiently large but a little less than np, so the ethod is not well responding for large nubers.

2 ore efficacy ethod of iportance of sources consists in taking into consideration the discounting on the epty set and then the noralization (see especially paper [4] and also [1]).. Using for 3 Sources. Exaple calculated by hand for cobining three sources using fusion rule. Let s say that ( ) is ties ore iportant than 1 ( ) 1(.), (.), (.). B B B= Φ x1 1B B = = = = x1 = y1 B = z B = x B B = = = = = x = yb = z B = x3 3B 3 B = = = = x y3b z3 B x4 4B 4 B (0.4)(0.1)(0.) = = = = = x y4b z4 B

3 x5 5B 5 B = = = = x y5b z5 B x6 6B 6 B = = = = x y6b z6 B x7 y7b (0.1)(0.1)(0.1) = = = 0.1 (0.1)(0.1) x y6b x8 y8b (0.4)(0.7)(0.1) = = = = 0.4 (0.7)(0.1) x y8b x = x y = y B 8B x10 y10b (0.1)(0.4)(0.1) = = = = = (0.1)(0.4) x y8 B x = x y = y B 10B

4 x1 y1b (0.4)(0.4)(0.7) = = = = (0.1)(0.4) x y1b B B If we didn t double (.) in the fusion rule, we d get a different result. Let s suppose we only fusion 1 (.) with (.): B B B= Φ nd now we copare the fusion results: B B three sources(sec ond source doubled ); iportance of sources considered; two sources; iportance of sources not considered. The ore ties we repeat (.) the closer 1... () ()=0.4, 1... (B) (B)=0.1, and 1... ( B) ( B)=0.5. Therefore, doubling, tripling, etc. a source, the ass of each eleent in the frae of discernent tends towards the ass value of that eleent in the repeated source (since that source is considered to have ore iportance than the others). For the readers who want to do the previous calculation with a coputer, here it is the PCR 5 Forula for 3 Sources: 1( ) ( X) 3( Y) ( ) = 13 XY, G 1( ) ( X) 3( Y ) X Y X Y=Φ 1 Y 3 X 1 X Y 3 1 Y 3 X 1 X Y 3

5 1 X 3 X 1 X 3 X 1 X X 3 1 X 3 X 1 X 3 X 1 X X X 1 X 3 1 X X 1 X 3 1 X 3 3. Siilarly, let s see the PCR6 Forula for 3 Sources: 1( ) ( ) 3( ) X Y PCR6( ) = 13 X Y XY, G 1 3 X Y X Y=Φ 1 Y 3 X 1 X Y 3 1 Y 3 X 1 X Y 3 1( ) ( X) 3( X) 1( X) ( ) 3( X) 1( X) ( X) 3( ) 1( ) ( X) 3( X) 1( X) ( ) 3( X) 1( X) ( X) 3( ) ( ) ( ) ( X) X X ( X) ( ) 3( ) 1( X) ( ) 3( ) 1( X) ( ) 3( ) 1( ) ( X) 3( ) 1( ) ( X) 3( ) 1( ) ( X) 3( ) 4. General Forula for PCR 6 for s Sources. s 1 ( ) = ( ) ( )... ( ) PCR s i1 i ik X1, X,..., Xs 1 G k= 1 ( i1, i,..., is ) P(1,,..., s) Xi, i { 1,,..., s 1} s 1 Xi =Φ i= 1 i ( ) ( )... ( ) ( 1 i i 1 1)... ( ) k i X k i X s s k ( ) ( )... ( ) ( X )... ( X ) i1 i ik ik 1 1 is s k where P(1,,, s) is the set of all perutations of the eleents {1,,, s}.

6 It should be observed that X 1, X,, X s-1 ay be different fro each other, or soe of the equal and others different, etc. We wrote this PCR6 general forula in the style of, different fro rnaud Martin & Christophe Oswald s notations, but actually doing the sae thing. In order not to coplicate the forula of PCR6, we did not use ore suations or products after the third Siga. s a particular case: ( ) = PCR6 13 i ( )... ( ) ( )... ( ) ( 1 i 1 1 1)... ( 3 ) k i i k i X k i X ( )... ( ) ( X )... ( X ) X1, k= 1( i1, i, i3) P(1,,3) i1 ik ik 1 1 i3 X1, X X1 X1 =Φ where P(1,, 3) is the set of perutations of the eleents { 1,, 3 }. It should also be observed that X 1 ay be different fro or equal to X. Conclusion. The ai of this paper was to show how to anually copute for 3 sources on soe exaples, thus better understanding its essence. nd also how to take into consideration the iportance of sources doing the Repeated Fusion Method. We did not present the Method of Discounting to the Epty Set in order to ephasize the iportance of sources, which is better than the first one, since the second ethod was the ain topic of paper [4]. We also presented the forula for 3 sources (a particular case when n=3), and the general forula for PCR6 in a different way but yet equivalent to Martin-Oswald s PCR6 forula []. References: 1. Dezert J., Tacnet J.-M., Batton-Hubert M., Sarandache F., Multi-criteria Decision Making Based on DST-HP, in Proceedings of Workshop on the Theory of Belief Functions, pril 1-, 010, Brest, France (available at Martin,., Osswald, C., new generalization of the proportional conflict redistribution rule stable in ters of decision, in the book dvances and pplications of DST for Inforation Fusion,. Res. Press, Rehoboth, US, Vol., Chapter (pages 69-88), 006; online at: 3. Sarandache F., Dezert J., dvances and pplications of DST for Inforation Fusion, Vols. 1-3,. Res. Press, Rehoboth, 004, 006, 009; 4. Sarandache F., Dezert J., Tacnet J.-M., Fusion of Sources of Evidence with Different Iportances and Reliabilities, subitted to Fusion 010, International Conference, Edinburgh, U.K., July 010.