I'm stuck on modeling this problem...
You own a machine shop with 4 machines which all have the same processing speeds. You also have a set of 15 jobs. No job can be interrupted once it starts nor can a job be simultaneously processed on two machines. The processing time for job j = $P_j$. Your objective is to minimize the total completion time, that is, the sum of the completion time for every job.
I tried using an $x_{ij}$ decision variable where $x_{ij}$ $= 1$ when machine i processes job j. I got stuck on that because I had no way to determine the order that jobs were processed. I then tried to make an $x_{ijk}$ variable where $x_{ijk}$ $= 1$ if job i is the kth job processed on machine j. I can't figure out the constraints when I do it that way, though.
Note: I know how to solve this problem. Simply process the shortest jobs first on each machine. I am concerned about being able to model it as an integer program.
UPDATE:
Here is what I have come up with: $C_{i,j}$ = the completion time of job in slot i on machine j. $x_{i,j}$ = 1 if machine j does job i.
Minimize $$\sum_{i,j} C_{i,j}$$
s.t. $$C_{i,j}= C_{i-1,j}+x_{i,j}*P_j, \forall{i} \forall{j}$$ $$\sum_{j} x_{i,j}= 1, \forall{i} $$ $$\sum_{i} x_{i,j}= 1, \forall{j} $$
The sum of the completion times is a very odd objective function. I wonder if perhaps you're misreading the problem (or perhaps it is misstated).
At any rate suppose that you create 45 dummy jobs with 0 durations. Set up 15 slots on each of the four machines, and assign your sixty jobs (15 actual, 45 dummy) to those slots. The completion time of the job in slot $i$ on machine $j$ is the completion time of the job in slot $i-1$ (0 if $i=1$) plus the processing time of the job in slot $i$.
With a little bit of algebra, you can directly express the cost of assigning any job to any slot as a function of the processing time $P_j$ and the slot index $i$. From there, I think you can solve it as a transportation problem (or an assignment problem if you don't lump all the dummy jobs into one category), meaning you should not need integer programming to solve it.