# Belief decision matrix

Similar to a decision matrix, a belief decision matrix is used to describe a multiple criteria decision analysis (MCDA) problem in the Evidential Reasoning Approach. If in an MCDA problem, there are M alternative options and each needs 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, each element in a belief decision matrix is a belief structure.

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 (e.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 (e.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 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