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Combining sources of evidence with reliability and importance for decision making

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Abstract

The combination of sources of evidence with reliability has been widely studied within the framework of Dempster-Shafer theory (DST), which has been employed as a major method for integrating multiple sources of evidence with uncertainty. By the fact that sources of evidence may also be different in importance, for example in multi-attribute decision making (MADM), we propose the importance discounting and combination method within the framework of DST to combine sources of evidence with importance, which is composed of an importance discounting operation and an extended Dempster’s rule of combination. Three evidence combination axioms are proposed and explored to uncover the differences between reliability and importance in evidence reasoning. Furthermore, a general scheme is proposed for combination of sources of evidence with both reliability and importance. An example of car performance evaluation is studied to show the efficiency of the new general scheme.

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Authors and Affiliations

  1. School of Automation, Northwestern Polytechnical University, Xi’an , 710072, People’s Republic of China

    Lianmeng Jiao, Quan Pan, Yan Liang, Xiaoxue Feng & Feng Yang

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  1. Lianmeng Jiao
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  2. Quan Pan
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  3. Yan Liang
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  4. Xiaoxue Feng
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  5. Feng Yang
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Corresponding author

Correspondence to Lianmeng Jiao.

Additional information

This work is partially supported by China Natural Science Foundation (No. 61135001, 61075029) and the Doctorate Foundation of Northwestern Polytechnical University (No. CX201319).

Appendices

Appendix A: Proof of Lemma 1

According to Definition 5, \(\hbox {m}_{12}^{\hbox {ED}}(\emptyset ) = 0\) is satisfied directly. Besides,

$$\begin{aligned} \sum \limits _{A \in {2^{{\varTheta ^ + }}}} {\hbox {m}_{12}^{\hbox {ED}}(A)}&= \sum \limits _{A \in {2^{{\varTheta ^ + }}},A \ne \emptyset } {\hbox {m}_{12}^{\hbox {ED}}(A)} + \hbox {m}_{12}^{\hbox {ED}}(\emptyset ) = \sum \limits _{A \in {2^{{\varTheta ^ + }}},A \ne \emptyset } {\hbox {m}_{12}^{\hbox {ED}}(A)}\nonumber \\&= \frac{{\sum \nolimits _{A \in {2^{{\varTheta ^ + }}},A \ne \emptyset } {\left( {\sum \nolimits _{B,C \in {2^{{\varTheta ^ + }}};B \cap C = A} {\hbox {m}_1^{{\beta _1}}(B)\hbox {m}_2^{{\beta _2}}(C)} } \right) } }}{{1 - \sum \nolimits _{B,C \in {2^{{\varTheta ^ + }}};B \cap C = \emptyset } {\hbox {m}_1^{{\beta _1}}(B) \hbox {m}_2^{{\beta _2}}(C)} }}. \end{aligned}$$
(10)

Because,

$$\begin{aligned}&\sum \limits _{A \in {2^{{\varTheta ^ + }}},A \ne \emptyset } {\left( {\sum \limits _{B,C \in {2^{{\varTheta ^ + }}};B \cap C = A} {\hbox {m}_1^{{\beta _1}}(B)\hbox {m}_2^{{\beta _2}}(C)} } \right) + } \sum \limits _{B,C \in {2^{{\varTheta ^ + }}};B \cap C = \emptyset } {\hbox {m}_1^{{\beta _1}}(B) \hbox {m}_2^{{\beta _2}}(C)}\\&\quad \quad = \sum \limits _{A \in {2^{{\varTheta ^ + }}}} {\left( {\sum \limits _{B,C \in {2^{{\varTheta ^ + }}};B \cap C = A} {\hbox {m}_1^{{\beta _1}}(B)\hbox {m}_2^{{\beta _2}}(C)} } \right) }\\&\quad \quad =\sum \limits _{B \in {2^{{\varTheta ^ + }}}} {\left( {\hbox {m}_1^{{\beta _1}}(B)\sum \limits _{C \in {2^{{\varTheta ^ + }}}} {\hbox {m}_2^{{\beta _2}}(C)} } \right) }= \sum \limits _{B \in {2^{{\varTheta ^ + }}}} {\hbox {m}_1^{{\beta _1}}(B)} = 1, \end{aligned}$$

so, Eq. (10) equals 1. Therefore, the combination result \(\hbox {m}_{12}^{\hbox {ED}}( \cdot )\) with the extended Dempster’s rule of combination is an IBBA.

Appendix B: Proof of Theorem 1

Suppose \({\hbox {m}_i}( \cdot )\) (\(i = 1, \ldots ,L\)) are \(L\) basic sources of evidence’s BBAs on the same frame of discernment \(\varTheta \) with reliability factors \({\alpha _i} \in [0,1]\) (\(i = 1, \ldots ,L\)). Denote \({\hbox {m}} ( \cdot )\) the integrated BBA with the reliability discounting and combination method.

As for the independence axiom, \(\forall S \in {2^\varTheta }{\setminus }\varTheta \), suppose \(\forall {S^ + } \supseteq S,\,{\hbox {m}_i}({S^ + }) = 0\) for all \(i = 1, \ldots ,L\). Discount all the \(L\) BBAs with their corresponding reliability factors \({\alpha _i}\) using Shafer’s discounting operation displayed as Eq. (3), we can get the reliability discounted BBAs assigned to \({S^ + }\)

$$\begin{aligned} {\hbox {m}_i}^{{\alpha _i}}(A) = \left\{ {\begin{array}{ll} 0, &{}\quad {\hbox {for }}A \in {2^\varTheta },A \supseteq S,A \ne \varTheta \\ 1 - {\alpha _i},&{}\quad {\hbox {for }}A = \varTheta \\ \end{array}} \right. \quad {\hbox {for }}i = 1, \ldots ,L. \end{aligned}$$
(11)

Then, the Dempster’s rule of combination displayed as Eq. (4) will be used to get the integrated BBAs assigned to \(S\)

$$\begin{aligned} \hbox {m}(S) = {{\sum \limits _{{X_i} \in {2^\varTheta };\bigcap \limits _{i = 1}^L {{X_i}} = S} {\prod \limits _{i = 1}^L {\hbox {m}_i^{{\alpha _i}}({X_i})} } } \mathord {\Bigg / {} } {\left( {1 - \sum \limits _{{X_i} \in {2^\varTheta };\bigcap \limits _{i = 1}^L {{X_i}} = \emptyset } {\prod \limits _{i = 1}^L {\hbox {m}_i^{{\alpha _i}}({X_i})} } } \right) }}. \end{aligned}$$
(12)

Since \(\bigcap _{i = 1}^L {{X_i}} = S\), so \({X_i} \supseteq S\). According to Eq. (11), if \({X_i} \ne \varTheta ,\,\hbox {m}_i^{{\alpha _i}}({X_i}) = 0\) for all \(i = 1, \ldots ,L\). Because it’s impossible for all \({X_i}\) to take \(\varTheta \) satisfying \(\bigcap _{i = 1}^L {{X_i}} = S\), so, \(\prod _{i = 1}^L {\hbox {m}_i^{{\alpha _i}}({X_i})}\) will equal 0 in any cases. Hence, \({\hbox {m}} (S)\) in Eq. (12) equals 0. That is, the reliability discounting and combination method satisfies the independence axiom.

As for the consensus axiom, \(\forall S \in {2^\varTheta }{\setminus }\varTheta \), suppose \({\hbox {m}_i}(S) = 1\) for all \(i = 1, \ldots ,L\). Discount all the \(L\) BBAs with their corresponding reliability factors \({\alpha _i}\) using Shafer’s discounting operation displayed as Eq. (3), we can get the reliability discounted BBAs

$$\begin{aligned} {\hbox {m}_i}^{{\alpha _i}}(A) = \left\{ {\begin{array}{l@{\quad }l} {\alpha _i}, &{}{\hbox {for }}A = S\\ 0, &{}{\hbox {for }}A \in {2^\varTheta },A \ne S,A \ne \varTheta {\hbox { for }}i = 1, \cdots ,L.\\ 1 - {\alpha _i}, &{}{\hbox {for }}A = \varTheta \end{array}} \right. \end{aligned}$$

Then, the Dempster’s rule of combination displayed as Eq. (4) will be used to integrate the reliability discounted BBAs

$$\begin{aligned} \hbox {m}(A) = \left\{ {\begin{array}{l@{\quad }l} 0, &{}{\hbox {for }}A \in {2^\varTheta },A \ne S,A \ne \varTheta \\ \prod \nolimits _{i = 1}^L {(1 - {\alpha _i})}, &{}{\hbox {for }}A = \varTheta . \end{array}} \right. \end{aligned}$$

So,

$$\begin{aligned} \hbox {m}(S) = 1 - \sum \limits _{A \in {2^\varTheta },A \ne S,A \ne \varTheta } {\hbox {m}(A)} - \hbox {m}(\varTheta ) = 1 - \prod \limits _{i = 1}^L {(1 - {\alpha _i})}. \end{aligned}$$

Thus, the reliability discounting and combination method only satisfies the consensus axiom when at least one source of evidence takes full reliability (\(\exists k \in \{ 1, \ldots ,L\} , {\alpha _k} = 1\)).

As for the completeness axiom, \(\forall S \in {2^\varTheta }{\setminus }\varTheta \), suppose \(\sum \nolimits _{{S^ - } \subseteq S} {{\hbox {m}_i}({S^ - })} = 1\) for all \(i = 1, \ldots ,L\). We can know that \(\forall {S^{ + + }} \supset S,\,{\hbox {m}_i}({S^{ + + }}) = 0\). Discount all the \(L\) BBAs with their corresponding reliability factors \({\alpha _i}\) using Shafer’s discounting operation displayed as Eq. (3), we can get the reliability discounted BBAs assigned to \({S^{ + + }}\)

$$\begin{aligned} {\hbox {m}_i}^{{\alpha _i}}(A) = \left\{ {\begin{array}{l@{\quad }l} 0, &{}{\hbox {for }}A \in {2^\varTheta },A \supset S,A \ne \varTheta \\ 1 - {\alpha _i}, &{}{\hbox {for }}A = \varTheta \end{array}} \right. {\hbox { for }}i = 1, \ldots ,L. \end{aligned}$$
(13)

Then, the Dempster’s rule of combination displayed as Eq. (4) will be used to get the integrated BBAs assigned to \({S^{ + + }}\)

$$\begin{aligned} \hbox {m}({S^{ + + }}) = {{\sum \limits _{{X_i} \in {2^\varTheta };\bigcap \limits _{i = 1}^L {{X_i}} = {S^{ + + }}} {\prod \limits _{i = 1}^L {\hbox {m}_i^{{\alpha _i}}({X_i})} } } \mathord {\Bigg / {} } {\left( {1 - \sum \limits _{{X_i} \in {2^\varTheta };\bigcap \limits _{i = 1}^L {{X_i}} = \emptyset } {\prod \limits _{i = 1}^L {\hbox {m}_i^{{\alpha _i}}({X_i})} } } \right) }}. \end{aligned}$$
(14)

Since \(\bigcap _{i = 1}^L {{X_i}}= {S^{ + + }}\), so \({X_i} \supseteq {S^{ + + }} \supset S\). Now, we consider it for two cases. If \({S^{ + + }} \ne \varTheta \), according to Eq. (13), \(\hbox {m}_i^{{\alpha _i}}({X_i}) = 0\) for all \(i = 1, \ldots , L\). So, \(\hbox {m}({S^{ + + }})\) in Eq. (14) equals 0. If \({S^{ + + }} = \varTheta \), it’s easy to get \(\hbox {m}({S^{ + + }}) = \hbox {m}(\varTheta ) = {{\prod _{i = 1}^L {(1 - {\alpha _i})} } \mathord {\big / {} } {(1 - k)}}\). So,

$$\begin{aligned} \sum \limits _{{S^ - } \subseteq S} {\hbox {m}({S^ - })}&= 1 - \sum \nolimits _{{S^{ + + }} \supset S} {\hbox {m}({S^{ + + }})} \!=\! 1 - \sum \nolimits _{{S^{ + + }} \supset S,{S^{ + + }} \ne \varTheta } {\hbox {m}({S^{ + + }})} - \hbox {m}(\varTheta )\\&= 1 - \hbox {m}(\varTheta ) = 1 - {{\prod \limits _{i = 1}^L {(1 - {\alpha _i})} } \mathord {\Big / {} } {(1 - k)}}. \end{aligned}$$

Therefore, the reliability discounting and combination method only satisfies the completeness axiom when at least one source of evidence takes full reliability (\(\exists k \in \{ 1, \ldots ,L\} ,{\alpha _k} = 1\)).

Appendix C: Proof of Theorem 2

Suppose \({\hbox {m}_i}( \cdot )\) (\(i = 1, \ldots ,L\)) are \(L\) basic sources of evidence’s BBAs on the same frame of discernment \(\varTheta \) with importance factors \({\beta _i} \in [0,1]\) (\(i = 1, \ldots ,L\)). Denote \({\hbox {m}} ( \cdot )\) the integrated BBA with the proposed importance discounting and combination method.

As for the independence axiom, \(\forall S \in {2^\varTheta }{\setminus } \varTheta \), suppose \(\forall {S^ + } \supseteq S,\,{{\hbox {m}} _i}({S^ + }) = 0\) for all \(i = 1, \ldots ,L\). Discount all the \(L\) BBAs with their corresponding importance factors \({\beta _i}\) using the importance discounting operation displayed as Eq. (5), we can get the importance discounted IBBAs assigned to \({S^ + }\)

$$\begin{aligned} {{\hbox {m}} _i}^{{\beta _i}}(A) = 0,\forall A \supseteq S, {\hbox { for }}i = 1, \ldots ,L. \end{aligned}$$
(15)

Then, the extended Dempster’s rule of combination displayed as Eq. (7) will be used to get the integrated IBBAs assigned to \(S\)

$$\begin{aligned} {{\hbox {m}} ^{{\hbox {ED}}}}(S) = {{\sum \limits _{{X_i} \in {2^{{\varTheta ^ + }}};\bigcap \limits _{i = 1}^L {{X_i}} = S} {\prod \limits _{i = 1}^L {{\hbox {m}} _i^{{\beta _i}}({X_i})} } } \mathord {\Bigg / {} } {\left( {1 - \sum \limits _{{X_i} \in {2^{{\varTheta ^ + }}};\bigcap \limits _{i = 1}^L {{X_i}} = \emptyset } {\prod \limits _{i = 1}^L {{\hbox {m}} _i^{{\beta _i}}({X_i})} } } \right) }}. \end{aligned}$$
(16)

As \(\bigcap _{i = 1}^L {{X_i}} = S\), so \({X_i} \supseteq S\). According to Eq. (15), \({\hbox {m}} _i^{{\beta _i}}({X_i}) = 0\) for all \(i = 1, \ldots ,L\). Hence, \({{\hbox {m}} ^{{\hbox {ED}}}}(S)\) in Eq. (16) equals 0. It’s straightforward that \({\hbox {m}} (S) = {{{{\hbox {m}} ^{{\hbox {ED}}}}(S)} \mathord {\big / {} } {\left( {1 - {{\hbox {m}} ^{{\hbox {ED}}}}(\Omega )} \right) }} = 0\). That is, the importance discounting and combination method satisfies the independence axiom.

As for the consensus axiom, \(\forall S \in {2^\varTheta }{\setminus } \varTheta \), suppose \({{\hbox {m}} _i}(S) = 1\) for all \(i = 1, \ldots ,L\). Discount all the \(L\) BBAs with their corresponding importance factors \({\beta _i}\) using the importance discounting operation displayed as Eq. (5), we can get the importance discounted IBBAs

$$\begin{aligned} {{\hbox {m}}_i}^{{\beta _i}}(A) = \left\{ {\begin{array}{l@{\quad }l} {\beta _i}, &{}{\hbox {for }}A = S\\ 0, &{}{\hbox {for }}A \in {2^\varTheta },A \ne S {\hbox { for }}i = 1, \ldots ,L.\\ 1 - {\beta _i}, &{}{\hbox {for }}A = \Omega \end{array}} \right. \end{aligned}$$

Then, the extended Dempster’s rule of combination displayed as Eq. (7) will be used to integrate the importance discounted IBBAs

$$\begin{aligned} {{\hbox {m}} ^{{\hbox {ED}}}}(A) = 0, \forall A \in {2^\varTheta },A \ne S. \end{aligned}$$

According to Lemma 1, it holds that

$$\begin{aligned} \sum \limits _{A \in {2^{{\varTheta ^ + }}}} {{{\hbox {m}} ^{{\hbox {ED}}}}(A)}&= \sum \nolimits _{A \in {2^\varTheta },A \ne S} {{{\hbox {m}} ^{{\hbox {ED}}}}(A)} + {{\hbox {m}} ^{{\hbox {ED}}}}(S) + {{\hbox {m}} ^{{\hbox {ED}}}}(\Omega )\\&= {{\hbox {m}} ^{{\hbox {ED}}}}(S) + {{\hbox {m}} ^{{\hbox {ED}}}}(\Omega ) = 1. \end{aligned}$$

Furthermore, via the normalization in Eq. (8), we obtain

$$\begin{aligned} {\hbox {m}} (S) = \frac{{{{\hbox {m}} ^{{\hbox {ED}}}}(S)}}{{1 - {{\hbox {m}} ^{{\hbox {ED}}}}(\Omega )}} = \frac{{{{\hbox {m}} ^{{\hbox {ED}}}}(S)}}{{{{\hbox {m}} ^{{\hbox {ED}}}}(S)}} = 1. \end{aligned}$$

So, the importance discounting and combination method satisfies the consensus axiom.

As for the completeness axiom, \(\forall S \in {2^\varTheta }{\setminus } \varTheta \), suppose \(\sum \nolimits _{{S^ - } \subseteq S} {{{\hbox {m}} _i}({S^ - })} = 1\) for all \(i = 1, \ldots ,L\). We can know that \(\forall {S^{ + + }} \supset S,\,{{\hbox {m}}_i}({S^{ + + }}) = 0\). Discount all the \(L\) BBAs with their corresponding importance factors \({\beta _i}\) using the importance discounting operation displayed as Eq. (5), we can get the importance discounted IBBAs assigned to \({S^{ + + }}\)

$$\begin{aligned} {{\hbox {m}}_i}^{{\beta _i}}(A) = 0,\forall A \supset S,{\hbox { for }}i = 1, \ldots ,L. \end{aligned}$$
(17)

Then, the extended Dempster’s rule of combination displayed as Eq. (7) will be used to get the integrated IBBAs assigned to \({S^{ + + }}\)

$$\begin{aligned} {{\hbox {m}}^{{\hbox {ED}}}}({S^{ + + }}) \!=\! {{\sum \limits _{{X_i} \in {2^{{\varTheta ^ + }}};\bigcap \limits _{i = 1}^L {{X_i}} = {S^{ + + }}} {\prod \limits _{i = 1}^L {{\hbox {m}} _i^{{\beta _i}}({X_i})} } } \mathord {\Bigg / {} } {\left( {1 - \sum \limits _{{X_i} \in {2^{{\varTheta ^ + }}};\bigcap \limits _{i = 1}^L {{X_i}} = \emptyset } {\prod \limits _{i = 1}^L {{\hbox {m}}_i^{{\beta _i}}({X_i})} } } \right) }}.\nonumber \\ \end{aligned}$$
(18)

As \(\bigcap _{i = 1}^L {{X_i}} = {S^{ + + }}\), so \({X_i} \supseteq {S^{ + + }} \supset S\). According to Eq. (17), \({\hbox {m}}_i^{{\beta _i}}({X_i}) = 0\) for all \(i = 1, \ldots ,L\). Hence, \({{\hbox {m}}^{{\hbox {ED}}}}({S^{ + + }})\) in Eq. (18) equals 0. It’s straightforward that \({\hbox {m}} ({S^{ + +}}) = {{{{\hbox {m}}^{{\hbox {ED}}}}({S^{ + + }})} \mathord {/ {} } {( {1 - {{\hbox {m}} ^{{\hbox {ED}}}}(\Omega )} )}} = 0\). So, \(\sum \nolimits _{{S^ - }\subseteq S} {{\hbox {m}} ({S^ - })} = 1 - \sum \nolimits _{{S^{+ + }}\supset S} {{\hbox {m}} ({S^{ + + }})} = 1\). That is, the importance discounting and combination method satisfies the completeness axiom.

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Jiao, L., Pan, Q., Liang, Y. et al. Combining sources of evidence with reliability and importance for decision making. Cent Eur J Oper Res 24, 87–106 (2016). https://doi.org/10.1007/s10100-013-0334-3

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