Modeling a situation of task dependencies

[danago]

An interesting problem has come up in a project I'm working on and I'm looking to build a model to simulate what would happen in particular scenarios. I'll describe the situation in generic terms.

I have x objects {1,2,3,...,x}, a pieces of type-A equipment, b pieces of type-B equipment and c pieces of type-C equipment (each object is identical to all other objects; the same can be said of each piece of equipment). The objects all start in state S₀, and the goal is to use each piece of equipment sequentially on the objects (it must be sequential) to take them from state S₀ to state S_A by using a type-A piece of equipment, then to state S_B, and then finally to state S_C. Then, once an object is in state S_C, it must wait until a particular time (e.g. midnight), at which point a a counter increases by 1. The object then reverts to state S₀ and the process starts again.

There are some constraints here. Each type of equipment takes a certain amount of time to complete its task - t_A, t_B, and t_C respectively, and a piece of equipment can only operate on one object at any given point in time. After it has finished a task, the piece of equipment can then move on to any other object that is waiting and at the appropriate state (remembering that the equipment must act on each object in the order A, B and C).

I am trying to set up a simulation of this so we can see what the impact is of changing various parameters. Eventually I'd also like to treat the times taken for each task as random variables (probably normally distributed). I was going to write something in visual basic to effectively simulate this, however I first wanted to ask the advice of this forum. My initial thought is that this is a fairly well-defined problem with seemingly many applications - is there a name for this type of problem? Are there any articles that look at modelling this?

Thanks in advance - any thoughts are much appreciated! More than happy to clarify anything if needed.

 

[fresh_42]

Sounds like an application of, e.g. MS Project or similar tools. Mathematically it can range from linear programming to Markov chains and every combination of it. I'd probably look out for the tools to play around with them. As I entered "MS Project" into Google to check whether it is still an available program I found several alternatives. The (economic) literature on this fills probably libraries.

 

[danago]

Sounds like an application of, e.g. MS Project or similar tools. Mathematically it can range from linear programming to Markov chains and every combination of it. I'd probably look out for the tools to play around with them. As I entered "MS Project" into Google to check whether it is still an available program I found several alternatives. The (economic) literature on this fills probably libraries.

Thanks for the reply. I did think about MS Project, though it's maybe a little rigid for what I'm looking for (though I haven't used it much, so it may be a lot more flexible than I'm thinking - I'll check it out again).

 

[mfb]

With fixed times, it should be possible to find an expression for the total processing time, maybe with a few cases to go through. Apart from starting up and finishing, one type of equipment will limit processing time, the other two equipment types will just determine the speed of the start and end - and I would expect this delay to be just the sum of the two processing times unless I miss some special case.
With random times, you'll need a simulation.