Kutting Edge.....first things first.....it's been a while since I've done any Linear Programming, but I know the first step is to determine your X's(decision variables) and Y(objective Function). After your decision variables and objective function have been established it is time to set-up some conditions (constraints) that have to satisfied.
Here's a few suggestions you might want to consider before attacking this problem.
1.Seems to me that a balanced work load is more of a constraint than an objective. Maybe the Objective should be to Minimize the work load thus, Labor cost(Z= Min Labor Cost)....period.....it's possible to have a balanced work load that is cost prohibitive. I assume the 10 Min cycle time is based on your current demand and is subject to change based on future demand.
2. You can define your model such that the work load (X1) is always balanced....so you could have a constraint equation that would state in effect(would have to set-up 5 or 6 conditions realistically) that the cycle time for a workstation = total standard cycle time for a product/10 min cycle time....and because your objective function states that you want to minimize the work load (well, the labor cost)....it will balance it based on a minimal number of operators. Seldomly is any job 100% efficient(perfectly balanced), so you will have to include an efficiency that you can live w/.
3. Without knowing more about the relationship of the tools to the conveyor line. It seems like the constraints for tool allocation (X2) will be established based upon what has to be done to min. and balance X1...certainly there are other other constraints based upon mfg. requirements etc. but to some extent minimizing X1 will probably depend on how you allocate the tools.
