Is there any algorithm to address the longest common subsequence problem with different weights for each character?

Question

I'm looking for an algorithm that addresses the LCS problem for two strings with the following conditions:

Each string consists of English characters and each character has a weight. For example:

sequence 1 (S1): "ABBCD" with weights [1, 2, 4, 1, 3]

sequence 2 (S2): "TBDC" with weights [7, 5, 1, 2]

Suppose that MW(s, S) is defined as the maximum weight of the sub-sequence s in string S with respect to the associated weights. The heaviest common sub-sequence (HCS) is defined as:

HCS = argmin(MW(s, S1), MW(s, S2))

The algorithm output should be the indexes of HCS in both strings and the weight. In this case, the indexes will be:

I_S1 = [2, 4] --> MW("BD", "ABBCD") = 7

I_S2 = [1, 2] --> MW("BD", "TBDC") = 6

Therefore HCS = "BD", and weight = min(MW(s, S1), MW(s, S2)) = 6.