74 J. Abellan and A. Masegosa Liu et al.(2007). Although the computational cost of every TU in dST is clearly different the calculation of every tU in DST is simple RB2: MI can be separated coherently in conflict and non-specificity by definition. Here, is used as a conflict measure and function I as a non-specificity measure. Recently Abellan et al.(2006)separated s into two parts that coherently quantify conflict and nor specificity for more general theories than DST. Here, S,(minimum of entropy) is used as a conflict measure and S-S, is used as a non-specificity function. In order to obtain these parts, Abellan and Moral (2005b) present a branch and bound algorithm to obtain S, on more general theories than DST. Only the AM function has no clear separation between conflict and non-specificity. In Jousselme et al.(2006), AM is presented as a special case of the function 8S+(1-8), for an unknown S E(0, 1). Therefore, when value AM(m)is used it is impossible to know what quantity corresponds to conflict and what to non-specificity RB3: In this point, we want to review the analysis presented in Jousselme et al.(2006) about the sensitivity of s in DST using an example by Klir and Smith(1999): Example 2. Suppose there are two elements in the given frame of discernment X=(1, 2) and we know n({1})=m1,m=({2}=m2,som({1,2})=m12=1-m1-m2.At this point, we should mention that by definition (Yager 1983), the non-specificity part of m depends only on the 12 value and the conflict part of m depends on the interaction between mi and m2 values In Klir and Smiths original example(1999), S was identified as highly insensitive to changes in evidence, an'unsatisfactory situation. S gives the same value for all bodies of evidence for which both m and m2 fall into the range [0, 0.5]. When m2 C[ 0.5, 1, the S measure is entirely independent of the value of and vice versa. Jousselme et al.(2006) proved that the AM measure does not behave in the same way and AM is neither independent of m, or m2 by considering the above example. We will use the example and will apply it to every Tu considered in this paper. Without loss of generality, it is supposed that mi is known. They then consider that three cases appear m1>0.5,m1=0.5,andm1<0.5 (1)m1>0.5,e.g.m1=0.6.Here,m12=0.4-m2. We have S"m)=S(0.6,0.4);S.(m)=S(1-m2,m2); (m)=(0.4-m2)log2 AM(m)=S(08 0.2+ MI: The conflict part of this function(S )is constant, and does not vary when m2 changes No distinction is made between the values of m2 and this does not make any sense because of the definition of the conflict part of a b.p.a S: The variations of the conflict part of this function (S)make sense; if m2 increases then so does the conflict part. Similarly, the non-specificity part behaves in a similar way: a decrease in m2 leads to an increase in m12 and as we can see via its non-specificity part S-S.=S0.6,0.4)-S(1-m2,m2)Liu et al. (2007). Although the computational cost of every TU in DST is clearly different, the calculation of every TU in DST is simple. RB2: MI can be separated coherently in conflict and non-specificity by definition. Here, S * is used as a conflict measure and function I as a non-specificity measure. Recently, Abella´n et al. (2006) separated S * into two parts that coherently quantify conflict and non￾specificity for more general theories than DST. Here, S* (minimum of entropy) is used as a conflict measure and S* 2 S* is used as a non-specificity function. In order to obtain these parts, Abella´n and Moral (2005b) present a branch and bound algorithm to obtain S* on more general theories than DST. Only the AM function has no clear separation between conflict and non-specificity. In Jousselme et al. (2006), AM is presented as a special case of the function dS* þ ð1 2 dÞI, for an unknown d [ ð0; 1Þ. Therefore, when value AMðmÞ is used, it is impossible to know what quantity corresponds to conflict and what to non-specificity. RB3: In this point, we want to review the analysis presented in Jousselme et al. (2006) about the sensitivity of S * in DST using an example by Klir and Smith (1999): Example 2. Suppose there are two elements in the given frame of discernment X ¼ {1, 2}, and we know m({1}) ¼ m1, m ¼ ({2}) ¼ m2, so mð{1; 2}Þ ¼ m12 ¼ 1 2 m1 2 m2. At this point, we should mention that by definition (Yager 1983), the non-specificity part of m depends only on the m12 value and the conflict part of m depends on the interaction between m1 and m2 values. In Klir and Smith’s original example (1999), S * was identified as ‘highly insensitive to changes in evidence’, an ‘unsatisfactory situation’. S * gives the same value for all bodies of evidence for which both m1 and m2 fall into the range [0, 0.5]. When m2 # ½0:5; 1
, the S * measure is entirely independent of the value of m1 and vice versa. Jousselme et al. (2006) proved that the AM measure does not behave in the same way and AM is neither independent of m1 or m2 by considering the above example. We will use the example and will apply it to every TU considered in this paper. Without loss of generality, it is supposed that m1 is known. They then consider that three cases appear: m1 . 0.5, m1 ¼ 0.5, and m1 , 0.5. (1) m1 . 0.5, e.g. m1 ¼ 0.6. Here, m12 ¼ 0.4 2 m2. We have S*ðmÞ ¼ Sð0:6; 0:4Þ; S*ðmÞ ¼ Sð1 2 m2; m2Þ; IðmÞ¼ð0:4 2 m2Þlog 2; AMðmÞ ¼ S 0:8 2 m2 2 ; 0:2 þ m2 2 : MI: The conflict part of this function (S * ) is constant, and does not vary when m2 changes. No distinction is made between the values of m2 and this does not make any sense because of the definition of the conflict part of a b.p.a. S* : The variations of the conflict part of this function (S*) make sense; if m2 increases then so does the conflict part. Similarly, the non-specificity part behaves in a similar way: a decrease in m2 leads to an increase in m12 and as we can see via its non-specificity part S* 2 S* ¼ Sð0:6; 0:4Þ 2 Sð1 2 m2; m2Þ; 740 J. Abella´n and A. Masegosa Downloaded by [New York University] at 12:09 08 November 2011