Evaluation methods for multi-class classification
In https://www.sciencedirect.com/science/article/pii/S004896971831163X?via%3Dihub, we have used the product of the sensitivity of class i (i.e., the ratio of data correctly classified for class i) obtained for each class. This summarizes these values into a single index, which ranges between 0 and 1 and to some extent is irrespective to data imbalance. This approach induces a good balance between the error comitted to each category because any value appreciably below 1 reduces significantly the performance. To relax this allowing a greater error in 1 o more categories you can simply use (in increasing order): the min, the geometric mean or the arithmetic mean. In https://ieeexplore.ieee.org/document/6940273/ and https://ieeexplore.ieee.org/abstract/document/5428802/ you can find these alternatives.
Good luck.
Rafael.