# Load the required library library$($lpSolve$)$ ## Warning: package 'lpSolve' was built under R version $4\mathrm{.}3\mathrm{.}3$ # Step $1$: Generate Random Data for $12$ Students set.seed$(123)$ # For reproducibility students $<-$ data.frame$($ GPA $=$ runif$(12,2\mathrm{.}0,4\mathrm{.}0),$ # Random GPA between $2\mathrm{.}0$ and $4\mathrm{.}0$ Skills $=$ sample$(1$:$10,12,$ replace $=$ TRUE$),$ # Random Skills Relevance $(1$ to $10)$ Collaboration $=$ sample$(1$:$10,12,$ replace $=$ TRUE$)$ # Random Collaboration Score $(1$ to $10))$ # Step $2$: Calculate Success Score students$SuccessScore $<-0\mathrm{.}4*$ students$GPA $+0\mathrm{.}35*$ students$Skills $+0\mathrm{.}25*$ students$Collaboration # Step $3$: Define Optimization Problem # Number of students and groups num$\_$students $<-$ nrow$($students$)$ num$\_$groups $<-4$ # Objective coefficients: Repeat success scores for each group objective$\_$coefficients $<-$ rep$($students$SuccessScore, each $=$ num$\_$groups$)$ # Group size constraints group$\_$size$\_$constraints $<-$ matrix$(0,$ nrow $=$ num$\_$groups, ncol $=$ num$\_$students $*$ num$\_$groups$)$ for $($j in $1$:num$\_$groups$)\{$ group$\_$size$\_$constraints$[$j$,(($j $-1)*$ num$\_$students $+1)$:$($j $*$ num$\_$students$)]<-1\}$ # Student assignment constraints student$\_$assignment$\_$constraints $<-$ matrix$(0,$ nrow $=$ num$\_$students, ncol $=$ num$\_$students $*$ num$\_$groups$)$ for $($i in $1$:num$\_$students$)\{$ student$\_$assignment$\_$constraints$[$i$,$ seq$($i$,$ by $=$ num$\_$students, length.out $=$ num$\_$groups$)]<-1\}$ # Combine constraints constraints $<-$ rbind$($group$\_$size$\_$constraints, student$\_$assignment$\_$constraints$)$ # Right$-$hand side of constraints rhs $<-$ c$($rep$(3,$ num$\_$groups$),$ rep$(1,$ num$\_$students$))$ # Constraint directions constraint$\_$directions $<-$ rep$("=",$ length$($rhs$))$ # Debugging Outputs cat$("$Debugging Outputs:

$")$ ## Debugging Outputs: cat$("$Dimensions of constraints matrix:", dim$($constraints$),"$

$")$ ## Dimensions of constraints matrix: $1648$ cat$("$Length of rhs:$",$ length$($rhs$),"$

$")$ ## Length of rhs: $16$ cat$("$

Constraints Matrix:

$")$ ## ## Constraints Matrix: print$($constraints$)$ ## $[,1][,2][,3][,4][,5][,6][,7][,8][,9][,10][,11][,12][,13]$ ## $[1,]1111111111110$ ## $[2,]0000000000001$ ## $[3,]0000000000000$ ## $[4,]0000000000000$ ## $[5,]1000000000001$ ## $[6,]0100000000000$ ## $[7,]0010000000000$ ## $[8,]0001000000000$ ## $[9,]0000100000000$ ## $[10,]0000010000000$ ## $[11,]0000001000000$ ## $[12,]0000000100000$ ## $[13,]0000000010000$ ## $[14,]0000000001000$ ## $[15,]0000000000100$ ## $[16,]0000000000010$ ## $[,14][,15][,16][,17][,18][,19][,20][,21][,22][,23][,24][,25]$ ## $[1,]000000000000$ ## $[2,]111111111110$ ## $[3,]000000000001$ ## $[4,]000000000000$ ## $[5,]000000000001$ ## $[6,]100000000000$ ## $[7,]010000000000$ ## $[8,]001000000000$ ## $[9,]000100000000$ ## $[10,]000010000000$ ## $[11,]000001000000$ ## $[12,]000000100000$ ## $[13,]000000010000$ ## $[14,]000000001000$ ## $[15,]000000000100$ ## $[16,]000000000010$ ## $[,26][,27][,28][,29][,30][,31][,32][,33][,34][,35][,36][,37]$ ## $[1,]000000000000$ ##