data.table non-equi join

library(data.table)

judgeDF <- data.table(x_lb = c(1, 4, 7), x_ub = c(3, 6, 9),
                      y_cate = paste0("M", 1:3), output = paste0("K", 1:3))
judgeDF
#    x_lb x_ub y_cate output
# 1:    1    3     M1     K1
# 2:    4    6     M2     K2
# 3:    7    9     M3     K3

input <- unique(data.table(X = sample(1:10, 100, TRUE),
                           Y = sample(paste0("M", 1:3), 100, TRUE)))
head(input)
#     X  Y
# 1:  3 M2
# 2: 10 M3
# 3:  4 M3
# 4:  3 M1
# 5:  6 M3
# 6:  6 M2

judgeDF[input, .(X, Y, output), on = .(x_lb <= X, x_ub >= X, y_cate == Y)]
#     X  Y output
# 1:  3 M2     NA
# 2: 10 M3     NA
# 3:  4 M3     NA
# 4:  3 M1     K1
# 5:  6 M3     NA
# 6:  6 M2     K2

judgeDF <- data.frame(x_lb = c(1, 4, 7), x_ub = c(3, 6, 9),
                      y_cate = paste0("M", 1:3), output = paste0("K", 1:3),
                      stringsAsFactors = FALSE)
input <- unique(data.frame(X = sample(1:10, 100, TRUE),
                           Y = sample(paste0("M", 1:3), 100, TRUE),
                           stringsAsFactors = FALSE))
output <- rep(NA_character_, nrow(input))
for (i in 1L:nrow(input)) {
  loc <- which(input[i, 1] >= judgeDF[, 1] & input[i, 1] <= judgeDF[, 2] &
                 input[i, 2] == judgeDF[, 3])
  if (length(loc) > 0L) {
    output[i] <- judgeDF[loc, 4]
  } else {
    output[i] <- NA_character_
  }
}