Define x as a multiple of a number as a constraint in the Maximize function

Question

I am trying to optimize a linearly constrained Production Function. I have three inputs and want them all to be multiples of 150. I can make all the inputs to belong to the Integer Domain but is there a way to specify as a constraint that they be multiples of 150 as well? TIA.

Maximize[4 a + 8.75 b + 8.3333 c,a + b + c == 3000 && 600 a + 1600 b + 1500 c \[LessSlantEqual] 3500000 && 2 a + 1.3333 b + 3 c \[LessSlantEqual] 6000 && c >= 150 &&  a \[GreaterSlantEqual] 300  && (a \[Element] Integers) && (b\[Element]Integers) && (c \[Element] Integers) , {a, b, c}]

Answer 1

What about this approach?

objective = 4 a + 8.75 b + 8.3333 c;
constraints = a + b + c == 3000 && 600 a + 1600 b + 1500 c <= 3500000 && 2 a + 1.3333 b + 3 c <= 6000 && c >= 150 && a >= 300;
domain = (a ∈ Integers) && (b ∈ Integers) && (c ∈ Integers);
repo = {a -> 150 a, b -> 150 b, c -> 150 c};
sol = Maximize[
 objective /. repo,
 Join[constraints /. repo, domain],
 {a, b, c}];

If you like, you can retranslate into the solutions of originale problem as follows:

sol[[2, All , 2]] *= 150;
sol

{19775., {a -> 1350, b -> 1500, c -> 150}}

Answer 2

I would tackle this by introducing 3 new variables a1, b1 and c1 as follows:

Maximize[4 a + 8.75 b + 8.3333 c, 
 a + b + c == 3000 && 
  600 a + 1600 b + 1500 c ⩽ 3500000 && 
  2 a + 1.3333 b + 3 c ⩽ 6000 && c >= 150 && 
  a ⩾ 
   300 && (a1 ∈ Integers) && (b1 ∈ 
    Integers) && (c1 ∈ Integers) && a == 150 a1 && 
  b == 150 b1 && c == 150 c1, {a, b, c, a1, b1, c1}]
(* {19775., {a -> 1350., b -> 1500., c -> 150., a1 -> 9, 
  b1 -> 10, c1 -> 1}} *)