How to more efficiently code integer constraints in LinearOptimization and Maximize

Question

I've written the following code and it works, but my code syntax is inefficient. This is a linear optimization with binary (0,1) integer constraints. Instead of writing each Ri as between 0 and 1 and each Ri as an integer, is the a better way to code this as a group of Ri's that must all be either 0 or 1 and hence an integer? Below is my code. It works but not tidy!!

Code Using LinearOptimization

Budget = 30 R1 + 150 R2 + 300 R3 + 25 R4 + 85 R5 + 95 R6 + 435 R7;
LinearOptimization[-(0.80 R1 + 0.75 R2 + 0.56 R3 + 0.32 R4 + 
    0.25 R5 + 0.86 R6 + 0.93 R7), {0 <= Budget <= 500, 0 <= R1 <= 1, 
  0 <= R2 <= 1, 0 <= R3 <= 1, 0 <= R4 <= 1, 0 <= R5 <= 1, 
  0 <= R6 <= 1, 0 <= R7 <= 1}, {R1 \[Element] Integers, 
  R2 \[Element] Integers, R3 \[Element] Integers, 
  R4 \[Element] Integers, R5 \[Element] Integers, 
  R6 \[Element] Integers, R7 \[Element] Integers}]

Code Using Maximize

Budget = 30 R1 + 150 R2 + 300 R3 + 25 R4 + 85 R5 + 95 R6 + 435 R7;
m = Maximize[{0.80 R1 + 0.75 R2 + 0.56 R3 + 0.32 R4 + 0.25 R5 + 
    0.86 R6 + 0.93 R7,
   0 <= Budget <= 500, 0 <= R1 <= 1, 0 <= R2 <= 1, 0 <= R3 <= 1, 
   0 <= R4 <= 1, 0 <= R5 <= 1, 0 <= R6 <= 1, 0 <= R7 <= 1},
  {R1, R2, R3, R4, R5, R6, R7} \[Element] Integers]
Budget /. Flatten[Last[Last[{m}]]]

Answer 1

VectorLessEqual is your friend, Element on lists works too:

With[{Rs = {R1, R2, R3, R4, R5, R6, R7},
  Budget = 30 R1 + 150 R2 + 300 R3 + 25 R4 + 85 R5 + 95 R6 + 435 R7},
 LinearOptimization[-(0.80 R1 + 0.75 R2 + 0.56 R3 + 0.32 R4 + 
     0.25 R5 + 0.86 R6 + 0.93 R7),
  {0 <= Budget <= 500,
   VectorLessEqual[{0, Rs, 1}],
   Element[Rs, Integers]},
  Rs]]

(* {R1 -> 1, R2 -> 1, R3 -> 0, R4 -> 1, R5 -> 1, R6 -> 1, R7 -> 0} *)

I'd probably write the constants with dot products, though:

With[{Rs = {R1, R2, R3, R4, R5, R6, R7}},
 LinearOptimization[-{0.80, 0.75, 0.56, 0.32, 0.25, 0.86, 0.93} . Rs,
  {0 <= {30, 150, 300, 25, 85, 95, 435} . Rs <= 500,
   VectorLessEqual[{0, Rs, 1}],
   Element[Rs, Integers]},
  Rs]]