Reliability and Importance Discounting of Neutrosophic Masses

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Reliability and Importance Discounting of Neutrosophic Masses Florentin Smarandache Originally published in Smarandache, F. - Neutrosophic Theory and its Applications, Collected Papers, Vol. I. 2014. <hal-01092887v2>, and reprinted with permission.

„•–”ƒ…–Ǥ In this paper, we introduce for the first time the discounting of a neutrosophic mass in terms of reliability and respectively the importance of the source. We show that reliability and importance discounts commute when dealing with classical masses. ͳǤ Â?–”‘†—…–‹‘Â?Ǥ Let ÎŚ = {ÎŚ1 , ÎŚ2 , ‌ , ÎŚn } be the frame of discernment, where đ?‘› ≼ 2, and the set of ˆ‘…ƒŽ elemeÂ?–•: đ??š = {đ??´1 , đ??´2 , ‌ , đ??´đ?‘š }, for đ?‘š ≼ 1, đ??š ⊂ đ??ş đ?›ˇ . (1) Let đ??ş đ?›ˇ = (đ?›ˇ,âˆŞ,∊, đ?’ž) be the ˆ—•‹‘n •’ƒ…‡. A Â?‡—–”‘•‘’Š‹… Â?ƒ•• is defined as follows: đ?‘šđ?‘› : đ??ş → [0, 1]3 for any đ?‘Ľ ∈ đ??ş, đ?‘šđ?‘› (đ?‘Ľ) = (đ?‘Ą(đ?‘Ľ), đ?‘–(đ?‘Ľ), đ?‘“(đ?‘Ľ)), (2) where

đ?‘Ą(đ?‘Ľ) = believe that đ?‘Ľ will occur (truth); đ?‘–(đ?‘Ľ) = indeterminacy about occurence; and đ?‘“(đ?‘Ľ) = believe that đ?‘Ľ will not occur (falsity).

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Simply, we say in neutrosophic logic: đ?‘Ą(đ?‘Ľ) = believe in đ?‘Ľ; đ?‘–(đ?‘Ľ) = believe in neut(đ?‘Ľ) [the neutral of đ?‘Ľ, i.e. neither đ?‘Ľ nor anti(đ?‘Ľ)]; and đ?‘“(đ?‘Ľ) = believe in anti(đ?‘Ľ) [the opposite of đ?‘Ľ]. Of course, đ?‘Ą(đ?‘Ľ), đ?‘–(đ?‘Ľ), đ?‘“(đ?‘Ľ) ∈ [0, 1], and ∑đ?‘Ľâˆˆđ??ş [đ?‘Ą(đ?‘Ľ) + đ?‘–(đ?‘Ľ) + đ?‘“(đ?‘Ľ)] = 1, (3) while đ?‘šđ?‘› (Ń„) = (0, 0, 0). (4) It is possible that according to some parameters (or data) a source is able to predict the believe in a hypothesis đ?‘Ľ to occur, while according to other parameters (or other data) the same source may be able to find the believe in đ?‘Ľ not occuring, and upon a third category of parameters (or data) the source may find some indeterminacy (ambiguity) about hypothesis occurence. An element đ?‘Ľ ∈ đ??ş is called ˆ‘…ƒŽ if đ?‘›đ?‘š (đ?‘Ľ) ≠(0, 0, 0), (5) i.e. đ?‘Ą(đ?‘Ľ) > 0 or đ?‘–(đ?‘Ľ) > 0 or đ?‘“(đ?‘Ľ) > 0. Any …Žƒ••‹…ƒl mƒ••: đ?‘š âˆś đ??ş Ń„ → [0, 1] (6) can be simply written as a neutrosophic mass as: đ?‘š(đ??´) = (đ?‘š(đ??´), 0, 0). (7)

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ʹǤ ‹•…‘—Â?–‹Â?‰ ƒ Neu–”‘•‘’Š‹… ƒ•• due –‘ Rel‹ƒ„‹Ž‹–› of –Š‡ ‘—”…‡Ǥ Let đ?›ź = (đ?›ź1 , đ?›ź2 , đ?›ź3 ) be the reliability coefficient of the source, đ?›ź ∈ [0,1]3 . Then, for any đ?‘Ľ ∈ đ??ş đ?œƒ ∖ {đ?œƒ, đ??źđ?‘Ą }, where đ?œƒ = the empty set and đ??źđ?‘Ą = total ignorance, đ?‘šđ?‘› (đ?‘Ľ)đ?‘Ž = (đ?›ź1 đ?‘Ą(đ?‘Ľ), đ?›ź2 đ?‘–(đ?‘Ľ), đ?›ź3 đ?‘“(đ?‘Ľ)), (8) and đ?‘šđ?‘› (đ??źđ?‘Ą )đ?›ź = (đ?‘Ą(đ??źđ?‘Ą ) + (1 − đ?›ź1 )

đ?‘Ą(đ?‘Ľ) ,

∑ đ?‘Ľâˆˆđ??ş đ?œƒ ∖{đ?œ™,đ??źđ?‘Ą }

đ?‘–(đ??źđ?‘Ą ) + (1 − đ?›ź2 )

đ?‘–(đ?‘Ľ), đ?‘“(đ??źđ?‘Ą ) + (1 − đ?›ź3 )

∑ đ?‘Ľâˆˆđ??ş đ?œƒ ∖{đ?œ™,đ??źđ?‘Ą }

∑

đ?‘“(đ?‘Ľ))

đ?‘Ľâˆˆđ??ş đ?œƒ ∖{đ?œ™,đ??źđ?‘Ą }

(9),

and, of course, đ?‘šđ?‘› (đ?œ™)đ?›ź = (0, 0, 0).

The missing mass of each element đ?‘Ľ, for đ?‘Ľ ≠đ?œ™, đ?‘Ľ ≠đ??źđ?‘Ą , is transferred to the mass of the total ignorance in the following way: đ?‘Ą(đ?‘Ľ) − đ?›ź1 đ?‘Ą(đ?‘Ľ) = (1 − đ?›ź1 ) ∙ đ?‘Ą(đ?‘Ľ) is transferred to đ?‘Ą(đ??źđ?‘Ą ), (10) đ?‘–(đ?‘Ľ) − đ?›ź2 đ?‘–(đ?‘Ľ) = (1 − đ?›ź2 ) ∙ đ?‘–(đ?‘Ľ) is transferred to đ?‘–(đ??źđ?‘Ą ), (11) and đ?‘“(đ?‘Ľ) − đ?›ź3 đ?‘“(đ?‘Ľ) = (1 − đ?›ź3 ) ∙ đ?‘“(đ?‘Ľ) is transferred to đ?‘“(đ??źđ?‘Ą ). (12)

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;Ǥ ‹•…‘—Â?–‹Â?‰ ƒ ‡—–”‘•‘’Š‹… ĥs †—e –‘ –Š‡ Â?’‘”–ƒÂ?ce ‘f –Š‡ ‘—”…‡Ǥ Let đ?›˝ ∈ [0, 1] be the importance coefficient of the source. This discounting can be done in several ways. a. For any đ?‘Ľ ∈ đ??ş đ?œƒ ∖ {đ?œ™}, đ?‘šđ?‘› (đ?‘Ľ)đ?›˝1 = (đ?›˝ ∙ đ?‘Ą(đ?‘Ľ), đ?‘–(đ?‘Ľ), đ?‘“(đ?‘Ľ) + (1 − đ?›˝) ∙ đ?‘Ą(đ?‘Ľ)), (13) which means that đ?‘Ą(đ?‘Ľ), the believe in đ?‘Ľ, is diminished to đ?›˝ ∙ đ?‘Ą(đ?‘Ľ), and the missing mass, đ?‘Ą(đ?‘Ľ) − đ?›˝ ∙ đ?‘Ą(đ?‘Ľ) = (1 − đ?›˝) ∙ đ?‘Ą(đ?‘Ľ), is transferred to the believe in đ?‘Žđ?‘›đ?‘Ąđ?‘–(đ?‘Ľ). b. Another way: For any đ?‘Ľ ∈ đ??ş đ?œƒ ∖ {đ?œ™}, đ?‘šđ?‘› (đ?‘Ľ)đ?›˝2 = (đ?›˝ ∙ đ?‘Ą(đ?‘Ľ), đ?‘–(đ?‘Ľ) + (1 − đ?›˝) ∙ đ?‘Ą(đ?‘Ľ), đ?‘“(đ?‘Ľ)), (14) which means that đ?‘Ą(đ?‘Ľ), the believe in đ?‘Ľ, is similarly diminished to đ?›˝ ∙ đ?‘Ą(đ?‘Ľ), and the missing mass (1 − đ?›˝) ∙ đ?‘Ą(đ?‘Ľ) is now transferred to the believe in đ?‘›đ?‘’đ?‘˘đ?‘Ą(đ?‘Ľ). c. The third way is the most general, putting together the first and second ways. For any đ?‘Ľ ∈ đ??ş đ?œƒ ∖ {đ?œ™}, đ?‘šđ?‘› (đ?‘Ľ)đ?›˝3 = (đ?›˝ ∙ đ?‘Ą(đ?‘Ľ), đ?‘–(đ?‘Ľ) + (1 − đ?›˝) ∙ đ?‘Ą(đ?‘Ľ) ∙ đ?›ž, đ?‘“(đ?‘Ľ) + (1 − đ?›˝) ∙ đ?‘Ą(đ?‘Ľ) ∙ (1 − đ?›ž)), (15) where đ?›ž ∈ [0, 1] is a parameter that splits the missing mass (1 − đ?›˝) ∙ đ?‘Ą(đ?‘Ľ) a part to đ?‘–(đ?‘Ľ) and the other part to đ?‘“(đ?‘Ľ). For đ?›ž = 0, one gets the first way of distribution, and when đ?›ž = 1, one gets the second way of distribution.

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͜Ǥ ‹•…‘—Â?–‹Â?‰ ‘f ‡Ž‹ƒ„‹Ž‹–y ƒÂ?† Â?’‘”tance ‘f ‘—”ces ‹n ‡Â?‡”ƒŽ ‘ Not CoÂ?Â?—–‡Ǥ a. Reliability first, Importance second. For any đ?‘Ľ ∈ đ??ş đ?œƒ ∖ {đ?œ™, đ??źđ?‘Ą }, one has after reliability Îą discounting, where đ?›ź = (đ?›ź1 , đ?›ź2 , đ?›ź3 ): đ?‘šđ?‘› (đ?‘Ľ)đ?›ź = (đ?›ź1 ∙ đ?‘Ą(đ?‘Ľ), đ?›ź2 ∙ đ?‘Ą(đ?‘Ľ), đ?›ź3 ∙ đ?‘“(đ?‘Ľ)), (16) and

đ?‘šđ?‘› (đ??źđ?‘Ą )đ?›ź = (đ?‘Ą(đ??źđ?‘Ą ) + (1 − đ?›ź1 ) ∙

∑

đ?‘Ą(đ?‘Ľ) , đ?‘–(đ??źđ?‘Ą ) + (1 − đ?›ź2 )

đ?‘Ľâˆˆđ??ş đ?œƒ ∖{đ?œ™,đ??źđ?‘Ą }

∙

∑

đ?‘–(đ?‘Ľ) , đ?‘“(đ??źđ?‘Ą ) + (1 − đ?›ź3 ) ∙

đ?‘Ľâˆˆđ??ş đ?œƒ ∖{đ?œ™,đ??źđ?‘Ą }

∑

đ?‘“(đ?‘Ľ))

đ?‘Ľâˆˆđ??ş đ?œƒ ∖{đ?œ™,đ??źđ?‘Ą }

â‰? (đ?‘‡đ??źđ?‘Ą , đ??źđ??źđ?‘Ą , đ??šđ??źđ?‘Ą ).

(17)

Now we do the importance β discounting method, the third importance discounting way which is the most general: đ?‘šđ?‘› (đ?‘Ľ)đ?›źđ?›˝3 = (đ?›˝đ?›ź1 đ?‘Ą(đ?‘Ľ), đ?›ź2 đ?‘–(đ?‘Ľ) + (1 − đ?›˝)đ?›ź1 đ?‘Ą(đ?‘Ľ)đ?›ž, đ?›ź3 đ?‘“(đ?‘Ľ) + (1 − đ?›˝)đ?›ź1 đ?‘Ą(đ?‘Ľ)(1 − đ?›ž))

(18)

and đ?‘šđ?‘›(đ??źđ?‘Ą)đ?›źđ?›˝3 = (đ?›˝ ∙ đ?‘‡đ??źđ?‘Ą , đ??źđ??źđ?‘Ą + (1 − đ?›˝)đ?‘‡đ??źđ?‘Ą ∙ đ?›ž, đ??šđ??źđ?‘Ą + (1 − đ?›˝)đ?‘‡đ??źđ?‘Ą (1 − đ?›ž)). (19) b. Importance first, Reliability second. For any đ?‘Ľ ∈ đ??şđ?œƒ ∖ {đ?œ™, đ??źđ?‘Ą}, one has after importance β discounting (third way): đ?‘šđ?‘›(đ?‘Ľ)đ?›˝3 = (đ?›˝ ∙ đ?‘Ą(đ?‘Ľ), đ?‘–(đ?‘Ľ) + (1 − đ?›˝)đ?‘Ą(đ?‘Ľ)đ?›ž, đ?‘“(đ?‘Ľ) + (1 − đ?›˝)đ?‘Ą(đ?‘Ľ)(1 − đ?›ž)) (20) and đ?‘šđ?‘› (đ??źđ?‘Ą )đ?›˝3 = (đ?›˝ ∙ đ?‘Ą(đ??źđ??źđ?‘Ą ), đ?‘–(đ??źđ??źđ?‘Ą ) + (1 − đ?›˝)đ?‘Ą(đ??źđ?‘Ą )đ?›ž, đ?‘“(đ??źđ?‘Ą ) + (1 − đ?›˝)đ?‘Ą(đ??źđ?‘Ą )(1 − đ?›ž)). (21)

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Now we do the reliability đ?›ź = (đ?›ź1 , đ?›ź2 , đ?›ź3 ) discounting, and one gets: đ?‘šđ?‘› (đ?‘Ľ)đ?›˝3đ?›ź = (đ?›ź1 ∙ đ?›˝ ∙ đ?‘Ą(đ?‘Ľ), đ?›ź2 ∙ đ?‘–(đ?‘Ľ) + đ?›ź2 (1 − đ?›˝)đ?‘Ą(đ?‘Ľ)đ?›ž, đ?›ź3 ∙ đ?‘“(đ?‘Ľ) + đ?›ź3 ∙ (1 − đ?›˝)đ?‘Ą(đ?‘Ľ)(1 − đ?›ž)) (22) and đ?‘šđ?‘› (đ??źđ?‘Ą )đ?›˝3đ?›ź = (đ?›ź1 ∙ đ?›˝ ∙ đ?‘Ą(đ??źđ?‘Ą ), đ?›ź2 ∙ đ?‘–(đ??źđ?‘Ą ) + đ?›ź2 (1 − đ?›˝)đ?‘Ą(đ??źđ?‘Ą )đ?›ž, đ?›ź3 ∙ đ?‘“(đ??źđ?‘Ą ) + đ?›ź3 (1 − đ?›˝)đ?‘Ą(đ??źđ?‘Ą )(1 − đ?›ž)). (23) emark. We see that (a) and (b) are in general different, so reliability of sources does not commute with the importance of sources. ͡Ǥ ƒ”–‹…—Žƒr ƒse ™Šen ‡Ž‹ƒ„‹Ž‹ty ƒÂ?d Â?’‘”–ƒÂ?…‡ ‹•…‘—Â?–‹Â?g ‘ˆ ƒ••‡• CoÂ?Â?—–‡Ǥ Let’s consider a classical mass and the focal set đ??š ⊂ đ??ş đ?œƒ ,

đ?‘š: đ??ş đ?œƒ → [0, 1] (24) đ??š = {đ??´1 , đ??´2 , ‌ , đ??´đ?‘š }, đ?‘š ≼ 1, (25)

and of course đ?‘š(đ??´đ?‘– ) > 0, for 1 ≤ đ?‘– ≤ đ?‘š. Suppose đ?‘š(đ??´đ?‘– ) = đ?‘Žđ?‘– ∈ (0,1]. (26) a. Reliability first, Importance second. Let đ?›ź ∈ [0, 1] be the reliability coefficient of đ?‘š (∙). For đ?‘Ľ ∈ đ??ş đ?œƒ ∖ {đ?œ™, đ??źđ?‘Ą }, one has

đ?‘š(đ?‘Ľ)đ?›ź = đ?›ź ∙ đ?‘š(đ?‘Ľ), (27)

and đ?‘š(đ??źđ?‘Ą ) = đ?›ź ∙ đ?‘š(đ??źđ?‘Ą ) + 1 − đ?›ź. (28) Let đ?›˝ ∈ [0, 1] be the importance coefficient of đ?‘š (∙). Then, for đ?‘Ľ ∈ đ??ş đ?œƒ ∖ {đ?œ™, đ??źđ?‘Ą }, đ?‘š(đ?‘Ľ)đ?›źđ?›˝ = (đ?›˝đ?›źđ?‘š(đ?‘Ľ), đ?›źđ?‘š(đ?‘Ľ) − đ?›˝đ?›źđ?‘š(đ?‘Ľ)) = đ?›ź ∙ đ?‘š(đ?‘Ľ) ∙ (đ?›˝, 1 − đ?›˝), (29) considering only two components: believe that đ?‘Ľ occurs and, respectively, believe that đ?‘Ľ does not occur. Further on, đ?‘š(đ??źđ?‘Ą )đ?›źđ?›˝ = (đ?›˝đ?›źđ?‘š(đ??źđ?‘Ą ) + đ?›˝ − đ?›˝đ?›ź, đ?›źđ?‘š(đ??źđ?‘Ą ) + 1 − đ?›ź − đ?›˝đ?›źđ?‘š(đ??źđ?‘Ą ) − đ?›˝ + đ?›˝đ?›ź) = [đ?›źđ?‘š(đ??źđ?‘Ą ) + 1 − đ?›ź] ∙ (đ?›˝, 1 − đ?›˝). (30) 262

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b. Importance first, Reliability second. For đ?‘Ľ ∈ đ??ş đ?œƒ ∖ {đ?œ™, đ??źđ?‘Ą }, one has đ?‘š(đ?‘Ľ)đ?›˝ = (đ?›˝ ∙ đ?‘š(đ?‘Ľ), đ?‘š(đ?‘Ľ) − đ?›˝ ∙ đ?‘š(đ?‘Ľ)) = đ?‘š(đ?‘Ľ) ∙ (đ?›˝, 1 − đ?›˝), (31) and đ?‘š(đ??źđ?‘Ą )đ?›˝ = (đ?›˝đ?‘š(đ??źđ?‘Ą ), đ?‘š(đ??źđ?‘Ą ) − đ?›˝đ?‘š(đ??źđ?‘Ą )) = đ?‘š(đ??źđ?‘Ą ) ∙ (đ?›˝, 1 − đ?›˝). (32) Then, for the reliability discounting scaler Îą one has: đ?‘š(đ?‘Ľ)đ?›˝đ?›ź = đ?›źđ?‘š(đ?‘Ľ)(đ?›˝, 1 − đ?›˝) = (đ?›źđ?‘š(đ?‘Ľ)đ?›˝, đ?›źđ?‘š(đ?‘Ľ) − đ?›źđ?›˝đ?‘š(đ?‘š)) (33) and đ?‘š(đ??źđ?‘Ą )đ?›˝đ?›ź = đ?›ź ∙ đ?‘š(đ??źđ?‘Ą )(đ?›˝, 1 − đ?›˝) + (1 − đ?›ź)(đ?›˝, 1 − đ?›˝) = [đ?›źđ?‘š(đ??źđ?‘Ą ) + 1 − đ?›ź] ∙ (đ?›˝, 1 − đ?›˝) = (đ?›źđ?‘š(đ??źđ?‘Ą )đ?›˝, đ?›źđ?‘š(đ??źđ?‘Ą ) − đ?›źđ?‘š(đ??źđ?‘Ą )đ?›˝) + (đ?›˝ − đ?›źđ?›˝, 1 − đ?›ź − đ?›˝ + đ?›źđ?›˝) = (đ?›źđ?›˝đ?‘š(đ??źđ?‘Ą ) + đ?›˝ − đ?›źđ?›˝, đ?›źđ?‘š(đ??źđ?‘Ą ) − đ?›źđ?›˝đ?‘š(đ??źđ?‘Ą ) + 1 − đ?›ź − đ?›˝ − đ?›źđ?›˝). (34) Hence (a) and (b) are equal in this case. ͸Ǥ šƒÂ?’Ž‡•Ǥ 1. Classical mass. The following classical is given on đ?œƒ = {đ??´, đ??ľ} âˆś

m

A

B

AUB

0.4

0.5

0.1

(35)

Let � = 0.8 be the reliability coefficient and � = 0.7 be the importance coefficient. a. Reliability first, Importance second. A

B

AUB

��

0.32

0.40

0.28

���

(0.224, 0.096)

(0.280, 0.120)

(0.196, 0.084) (36)

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We have computed in the following way: đ?‘šđ?›ź (đ??´) = 0.8đ?‘š(đ??´) = 0.8(0.4) = 0.32, (37) đ?‘šđ?›ź (đ??ľ) = 0.8đ?‘š(đ??ľ) = 0.8(0.5) = 0.40, (38) đ?‘šđ?›ź (đ??´đ?‘ˆđ??ľ) = 0.8(AUB) + 1 − 0.8 = 0.8(0.1) + 0.2 = 0.28, (39) and đ?‘šđ?›źđ?›˝ (đ??ľ) = (0.7đ?‘šđ?›ź (đ??´), đ?‘šđ?›ź (đ??´) − 0.7đ?‘šđ?›ź (đ??´)) = (0.7(0.32), 0.32 − 0.7(0.32)) = (0.224, 0.096), (40) đ?‘šđ?›źđ?›˝ (đ??ľ) = (0.7đ?‘šđ?›ź (đ??ľ), đ?‘šđ?›ź (đ??ľ) − 0.7đ?‘šđ?›ź (đ??ľ)) = (0.7(0.40), 0.40 − 0.7(0.40)) = (0.280, 0.120), (41) đ?‘šđ?›źđ?›˝ (đ??´đ?‘ˆđ??ľ) = (0.7đ?‘šđ?›ź (đ??´đ?‘ˆđ??ľ), đ?‘šđ?›ź (đ??´đ?‘ˆđ??ľ) − 0.7đ?‘šđ?›ź (đ??´đ?‘ˆđ??ľ)) = (0.7(0.28), 0.28 − 0.7(0.28)) = (0.196, 0.084). (42) b. Importance first, Reliability second. A

B

AUB

m

0.4

0.5

0.1

đ?‘šđ?›˝

(0.28, 0.12)

(0.35, 0.15)

(0.07, 0.03)

���

(0.224, 0.096

(0.280, 0.120)

(0.196, 0.084) (43)

We computed in the following way: đ?‘šđ?›˝ (đ??´) = (đ?›˝đ?‘š(đ??´), (1 − đ?›˝)đ?‘š(đ??´)) = (0.7(0.4), (1 − 0.7)(0.4)) = (0.280, 0.120), (44) đ?‘šđ?›˝ (đ??ľ) = (đ?›˝đ?‘š(đ??ľ), (1 − đ?›˝)đ?‘š(đ??ľ)) = (0.7(0.5), (1 − 0.7)(0.5)) = (0.35, 0.15), (45) đ?‘šđ?›˝ (đ??´đ?‘ˆđ??ľ) = (đ?›˝đ?‘š(đ??´đ?‘ˆđ??ľ), (1 − đ?›˝)đ?‘š(đ??´đ?‘ˆđ??ľ)) = (0.7(0.1), (1 − 0.1)(0.1)) = (0.07, 0.03), (46) and đ?‘šđ?›˝đ?›ź (đ??´) = đ?›źđ?‘šđ?›˝ (đ??´) = 0.8(0.28, 0.12) = (0.8(0.28), 0.8(0.12)) = (0.224, 0.096), (47) đ?‘šđ?›˝đ?›ź (đ??ľ) = đ?›źđ?‘šđ?›˝ (đ??ľ) = 0.8(0.35, 0.15) = (0.8(0.35), 0.8(0.15)) = (0.280, 0.120), (48)

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đ?‘šđ?›˝đ?›ź (đ??´đ?‘ˆđ??ľ) = đ?›źđ?‘š(đ??´đ?‘ˆđ??ľ)(đ?›˝, 1 − đ?›˝) + (1 − đ?›ź)(đ?›˝, 1 − đ?›˝) = 0.8(0.1)(0.7, 1 − 0.7) + (1 − 0.8)(0.7, 1 − 0.7) = 0.08(0.7, 0.3) + 0.2(0.7, 0.3) = (0.056, 0.024) + (0.140, 0.060) = (0.056 + 0.140, 0.024 + 0.060) = (0.196, 0.084). (49) Therefore reliability discount commutes with importance discount of sources when one has classical masses. The result is interpreted this way: believe in đ??´ is 0.224 and believe in đ?‘›đ?‘œđ?‘›đ??´ is 0.096, believe in đ??ľ is 0.280 and believe in đ?‘›đ?‘œđ?‘›đ??ľ is 0.120, and believe in total ignorance đ??´đ?‘ˆđ??ľ is 0.196, and believe in non-ignorance is 0.084.

͚Ǥ ƒÂ?‡ šƒÂ?’Ž‡ ™‹–h ‹ˆˆ‡”‡Â?– ‡†‹•–”‹„—–‹‘Â? ‘f ƒ••‡• ‡Žƒ–‡† –‘

Â?’‘”–ƒÂ?…‡ of ‘—”…‡•Ǥ Let’s consider the third way of redistribution of masses related to importance coefficient of sources. đ?›˝ = 0.7, but đ?›ž = 0.4, which means that 40% of đ?›˝ is redistributed to đ?‘–(đ?‘Ľ) and 60% of đ?›˝ is redistributed to đ?‘“(đ?‘Ľ) for each đ?‘Ľ ∈ đ??ş đ?œƒ ∖ {đ?œ™}; and đ?›ź = 0.8.

a. Reliability first, Importance second. A

B

AUB

m

0.4

0.5

0.1

��

0.32

0.40

0.28

���

(0.2240, 0.0384,

(0.2800, 0.0480,

(0.1960, 0.0336,

0.0576)

0.0720)

0.0504). (50)

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We computed đ?‘šđ?›ź in the same way. But: đ?‘šđ?›źđ?›˝ (đ??´) = (đ?›˝ ∙ đ?‘šđ?›ź (đ??´), đ?‘–đ?›ź (đ??´) + (1 − đ?›˝)đ?‘šđ?›ź (đ??´) ∙ đ?›ž, đ?‘“đ?›ź (đ??´) + (1 − đ?›˝)đ?‘šđ?›ź (đ??´)(1 − đ?›ž)) = (0.7(0.32), 0 + (1 − 0.7)(0.32)(0.4), 0 + (1 − 0.7)(0.32)(1 − 0.4)) = (0.2240, 0.0384, 0.0576). (51) Similarly for đ?‘šđ?›źđ?›˝ (đ??ľ) and đ?‘šđ?›źđ?›˝ (đ??´đ?‘ˆđ??ľ).

b. Importance first, Reliability second. A

B

AUB

m

0.4

0.5

0.1

đ?‘šđ?›˝

(0.280, 0.048,

(0.350, 0.060,

(0.070, 0.012,

0.072)

0.090)

0.018)

(0.2240, 0.0384,

(0.2800, 0.0480,

(0.1960, 0.0336,

0.0576)

0.0720)

0.0504).

�� �

(52) We computed đ?‘šđ?›˝ (∙) in the following way: đ?‘šđ?›˝ (đ??´) = (đ?›˝ ∙ đ?‘Ą(đ??´), đ?‘–(đ??´) + (1 − đ?›˝)đ?‘Ą(đ??´) ∙ đ?›ž, đ?‘“(đ??´) + (1 − đ?›˝)đ?‘Ą(đ??´)(1 − đ?›ž)) = (0.7(0.4), 0 + (1 − 0.7)(0.4)(0.4), 0 + (1 − 0.7)0.4(1 − 0.4)) = (0.280, 0.048, 0.072). (53) Similarly for đ?‘šđ?›˝ (đ??ľ) and đ?‘šđ?›˝ (đ??´đ?‘ˆđ??ľ). To compute đ?‘šđ?›˝đ?›ź (∙), we take đ?›ź1 = đ?›ź2 = đ?›ź3 = 0.8, (54) in formulas (8) and (9).

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đ?‘šđ?›˝đ?›ź (đ??´) = đ?›ź ∙ đ?‘šđ?›˝ (đ??´) = 0.8(0.280, 0.048, 0.072) = (0.8(0.280), 0.8(0.048), 0.8(0.072)) = (0.2240, 0.0384, 0.0576). (55) Similarly đ?‘šđ?›˝đ?›ź (đ??ľ) = 0.8(0.350, 0.060, 0.090) = (0.2800, 0.0480, 0.0720). (56) For đ?‘šđ?›˝đ?›ź (đ??´đ?‘ˆđ??ľ) we use formula (9): đ?‘šđ?›˝đ?›ź (đ??´đ?‘ˆđ??ľ) = (đ?‘Ąđ?›˝ (đ??´đ?‘ˆđ??ľ) + (1 − đ?›ź)[đ?‘Ąđ?›˝ (đ??´) + đ?‘Ąđ?›˝ (đ??ľ)], đ?‘–đ?›˝ (đ??´đ?‘ˆđ??ľ) + (1 − đ?›ź)[đ?‘–đ?›˝ (đ??´) + đ?‘–đ?›˝ (đ??ľ)], đ?‘“đ?›˝ (đ??´đ?‘ˆđ??ľ) + (1 − đ?›ź)[đ?‘“đ?›˝ (đ??´) + đ?‘“đ?›˝ (đ??ľ)]) = (0.070 + (1 − 0.8)[0.280 + 0.350], 0.012 + (1 − 0.8)[0.048 + 0.060], 0.018 + (1 − 0.8)[0.072 + 0.090]) = (0.1960, 0.0336, 0.0504). Again, the reliability discount and importance discount commute.

ͺǤ ‘Â?…Ž—•‹‘Â?Ǥ In this paper we have defined a new way of discounting a classical and neutrosophic mass with respect to its importance. We have also defined the discounting of a neutrosophic source with respect to its reliability. In general, the reliability discount and importance discount do not commute. But if one uses classical masses, they commute (as in Examples 1 and 2).

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Ǥ The author would like to thank Dr. Jean Dezert for his opinions about this paper.

Ǥ 1.

F. Smarandache, J. Dezert, J.-M. Tacnet, Fusion of Sources of

Evidence with Different Importances and Reliabilities, Fusion 2010 International Conference, Edinburgh, Scotland, 26-29 July, 2010. 2.

Florentin Smarandache, Neutrosophic Masses & Indeterminate

Models. Applications to Information Fusion, Proceedings of the International Conference on Advanced Mechatronic Systems [ICAMechS 2012], Tokyo, Japan, 18-21 September 2012.

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