I work at a publishing house and I am setting up one of our presses for "ganging", in other words, printing multiple jobs simultaneously. Given that different print jobs can have different quantities, and anywhere from 1 to 20 jobs might need to be considered at a time, the problem would be to determine which jobs to group together to minimize waste (waste coming from over-printing on smaller-quantity jobs in a given set, that is).
Given the following stable data:
- All jobs are equal in terms of spatial size--placement on paper doesn't come into consideration.
- There are three "lanes", meaning that three jobs can be printed simultaneously.
- Ideally, each lane has one job. Part of the problem is minimizing how many lanes each job is run on.
- If necessary, one job could be run on two lanes, with a second job on the third lane.
- The "grouping" waste from a given set of jobs (let's say the quantities of them are x, y and z) would be the highest number minus the two lower numbers. So if x is the higher number, the grouping waste would be (x - y) + (x - z). Otherwise stated, waste is produced by printing job Y and Z (in excess of their quantities) up to the quantity of X. The grouping waste would be a qualifier for the given set, meaning it could not exceed a certain quantity or the job would simply be printed alone.
So the question is stated: how to determine which sets of jobs are grouped together, out of any given number of jobs, based on the qualifiers of 1) Three similar quantities OR 2) Two quantities where one is approximately double the other, AND with the aim of minimal total grouping waste across the various sets.
(Edit) Quantity Information: Typical job quantities can be from 150 to 350 on foreign languages, or 500 to 1000 on English print runs. This data can be used to set up some scenarios for an algorithm. For example, let's say you had 5 jobs:
1000, 500, 500, 450,