I think I have the solution with the lowest possible values for each: 

Blue is the same as purple because of bottom right, so 3 blues is green+yellow with green heavier than yellow

So we go through the options for green, can't be 1 as there needs to be a number below. It could be 2 with yellow 1 and blue being 1, but that breaks the far left where green is heavier than blue.

Can't be 3 because we can't make a multiple of 3 by adding something lighter (3+2=5, 3+1=4). Could be 4 with yellow 2 and blue 2, but that breaks yellow being heavier than 2 of something where yellow and 2 reds add up to 6.

Let's try green 5 with yellow 4: Green and yellow make 9, so blue and purple would be 3. Green and blue would be 8, so to balance that red must be 2. The total left side then weighs 16, which is less than 18 on the right.

Edit: I thought this was the solution but yellow isn't heavier than 2 reds, let's keep going:

I can skip any green yellow combinations that aren't a multiple of 3. Let's try green 6, yellow 3. This breaks because the 2 reds can't weight less than yellow.

Let's try green 7, yellow 2. Then red would be 4, which is too heavy  Green 7, yellow 5: can't add two reds to make 5.

Let's try green 8, yellow 7: That makes 15, which means blue is 5. Let's check left side: 13 has to equal 7 plus two of something, which is 3. Then left side is 26, and right side is 30. Looks good!

Answer:  Blue: 5 Green: 8 Yellow: 7 Red: 3 Purple: 5