df <- cbind(expand.grid(shop=1:3,supply=1:2),distance=c(2.8,5.4,1.4,4.2,3.0,6.3))

df["Entry"] <- 1:dim(df)[[1]]

shop.mat <- table(df$shop,df$Entry)
shop.mat

    1 2 3 4 5 6
  1 1 0 0 1 0 0
  2 0 1 0 0 1 0
  3 0 0 1 0 0 1

supply.mat <- table(df$supply,df$Entry)
supply.mat

    1 2 3 4 5 6
  1 1 1 1 0 0 0
  2 0 0 0 1 1 1

N_supply <- dim(supply.mat)[[1]]
N_shop <- dim(shop.mat)[[1]]
N_entry <- dim(df)[[1]]

constr.mat <- NULL
dir <- NULL
rhs <- NULL

constr.mat <- rbind(constr.mat,cbind(shop.mat,matrix(0,N_shop,N_supply)))
dir <- c(dir,rep("==",N_shop))
rhs <- c(rhs,rep(1,N_shop))

obj <- c(aggregate(distance~Entry,df,c)[["distance"]],rep(0,N_supply))

constr.mat <- rbind(constr.mat,cbind(supply.mat,-diag(table(df$supply))))
dir <- c(dir,rep("<=",N_supply))
rhs <- c(rhs,rep(0,N_supply))

constr.mat <- rbind(constr.mat,c(rep(0,N_entry),rep(1,N_supply)))
dir <- c(dir,"<=")
rhs <- c(rhs,2)

require(lpSolve)
sol <- lp("min", obj, constr.mat, dir, rhs, all.bin = TRUE,num.bin.solns = 1, use.rw=FALSE, transpose.constr=TRUE)

sol$solution
[1] 1 0 1 0 1 0 1 1

sol$objval
[1] 7.2

selected_Entry <- dimnames(shop.mat)[[2]][as.logical(sol$solution[1:N_entry])]
selected_Entry
[1] "1" "3" "5"

df[df$Entry %in% selected_Entry,]
  shop supply distance Entry
1    1      1      2.8     1
3    3      1      1.4     3
5    2      2      3.0     5

constr.mat %*% sol$solution
  [,1]
1    1
2    1
3    1
1   -1
2   -2
     2

constr.mat %*% c(1,1,1,0,0,0,1,1)
  [,1]
1    1
2    1
3    1
1    0
2   -3
     2

constr.mat <- rbind(constr.mat,cbind(supply.mat,-diag(table(df$supply))),cbind(supply.mat,-diag(table(df$supply))))
dir <- c(dir,rep("<=",N_supply),rep(">=",N_supply))
rhs <- c(rhs,rep(0,N_supply),1-table(df$supply))

paste(t(unlist(constr.mat %*% sol$solution)),dir,rhs)
 [1] "1 == 1"   "1 == 1"   "1 == 1"   "-1 <= 0"
 [5] "-2 <= 0"  "2 <= 2"   "-1 <= 0"  "-2 <= 0"
 [9] "-1 >= -2" "-2 >= -2"

paste(t(unlist(constr.mat %*% c(1,1,1,0,0,0,1,1))),dir,rhs)
 [1] "1 == 1"   "1 == 1"   "1 == 1"   "0 <= 0"
 [5] "-3 <= 0"  "2 <= 2"   "0 <= 0"   "-3 <= 0"
 [9] "0 >= -2"  "-3 >= -2"