I'm trying to work out how to allocate work to different teams, and my brain is struggling with the logic.
I have three different teams who arrive in a geographic area at different dates. There's a set number of jobs in the area, and each team has a set productivity (the number of jobs they can complete per day). I'm trying to work out what percentage of the jobs each team needs to complete in order for all of the jobs to be completed, and for the teams to leave the area on (roughly, give or take 1 or 2) the same date.
Some sample:
Area A has a total of 3,030 jobs which need to be completed.
Team 1 arrives in Area A on 2014-09-30 and has a productivity of 33.25 jobs/day (fractions of a job are okay!). Team 2 arrives on 2014-10-10 and has a productivity of 28.5 Team 3 arrives on 2014-10-23 and has a productivity of 42.75
By pure trial and error I've worked out that if each team does the following % of work, they'll all finish up on 2014-11-21 and all the jobs will be completed:
Team 1: 42% of the jobs Team 2: 28% Team 3: 30%
Does anyone have any clues on a formula for calculating these percentages?
Let $P_A, P_B, P_C$ is the productivity of team $A, B, C$ respectively.
Let $D_A, D_B, D_C$ is the number of working days for team $A, B, C$ respectively.
$\therefore P_AD_A+P_BD_B+P_CD_C=\text{Total number of jobs}$
let all the jobs be done in $x$ days from the arrival of team $A$
$D_A=x$
$D_B=x-(\text{number of days elapsed since arrival of team A})$
$D_C=x-(\text{number of days elapsed since arrival of team A})$