Belief decision matrix
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Similar to decision matrix
Decision Matrix
A decision matrix is a list of values in rows and columns that allows an ologist to systematically identify, analyze, and rate the performance of relationships between sets of values and information. Elements of a decision matrix show decisions based on certain decision criteria...

, a belief decision matrix is used to describe a multiple criteria decision analysis (MCDA) problem in the Evidential Reasoning Approach
Evidential Reasoning Approach
In decision theory, the evidential reasoning approach , is a generic evidence-based multi-criteria decision analysis approach for dealing with problems having both quantitative and qualitative criteria under various uncertainties including ignorance and randomness...

. If in an MCDA problem, there are M alternative options and each need to be assessed on N criteria, then the belief decision matrix for the problem has M rows and N columns or M X N elements, as shown in the following table. Instead of being a single numerical value or a single grade as in a decision matrix
Decision Matrix
A decision matrix is a list of values in rows and columns that allows an ologist to systematically identify, analyze, and rate the performance of relationships between sets of values and information. Elements of a decision matrix show decisions based on certain decision criteria...

, each element in a belief decision matrix is a belief structure
Belief structure
A belief structure is a distributed assessment with beliefs. It is used in the evidential reasoning approach for multi-criteria decision analysis to represent the performance of an alternative option on a criterion. For example, the quality of a car engine may be assessed to be “excellent” with a...

.

For example, suppose Alternative i is "Car i", Criterion j is "Engine Quality" assessed by five grades {Excellent, Good, Average, Below Average, Poor}, and "Car i" is assessed to be “Excellent” on "Engine Quality" with a high degree of belief (i.g. 0.6) due to its low fuel consumption, low vibration and high responsiveness. At the same time, the quality is also assessed to be only “Good” with a lower degree of belief (i.g. 0.4 or less) because its quietness and starting can still be improved. If this is the case, then we have Xij={ (Excellent, 0.6), (Good, 0.4)}, or Xij={ (Excellent, 0.6), (Good, 0.4), (Average, 0), (Below Average, 0), (Poor, 0)}.

A conventional decision matrix
Decision Matrix
A decision matrix is a list of values in rows and columns that allows an ologist to systematically identify, analyze, and rate the performance of relationships between sets of values and information. Elements of a decision matrix show decisions based on certain decision criteria...

is a special case of belief decision matrix when only one belief degree in a belief structure is 1 and the others are 0.
Criterion 1 Criterion 2 ... Criterion N
Alternative 1 x11 x12 ... x1N
Alternative 2 x21 x22 ... x2N
... ... ... Xij={ (Excellent, 0.6), (Good, 0.4)} ...
Alternative M xM1 xM2 ... xMN
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