Create an optimization model such that selecting start date and end date produces some outcome

The key to this question is how to subset the data frame based on row and column names. Here I designed a function to achieve this. To use this function, you can set start to the start date, end to the end date, and dt to your data frame.

outcome <- function(start, end, dt){
  out <- dt[rownames(dt) %in% start, colnames(dt) %in% end]
  return(out)
}

# Example:
outcome(start = "29-Apr", end = "28-Oct", dt = dat)
# [1] 100

Or as mentioned in the comment, you can do the following directly.

outcome <- function(start, end, dt){
  out <- dt[start, end]
  return(out)
}

DATA

dat <- read.table(text = "'22-Apr'   90     95      95    95    
'29-Apr'   95     100     100   100   
'6-May'    95     100     100   100   
'13-May'   90     100     100   95    
'20-May'   90     95      95    90    
'27-May'   80     85      85    90    
'3-Jun'    75     80      80    85", header = FALSE, row.names = 1)

names(dat) <- c("21-Oct", "28-Oct", "4-Nov", "11-Nov")