Resource constraint project scheduling

Question

I need your help to solve this problem. I have a set of tasks, each task has its execution time. I have two types of constraints. first type is the precedence relationships between tasks. The second constraint type is allowing set of tasks to be in execution at the same time. For example : I have a graph G with 6 tasks and the following edges (T1,T2),(T2,T3),(T4,T3),(T4,T5) and (T6,T5). Suppose that T1,T4 are able to execute together and also T1,T6 but not T4,T6. Taking into account the execution time for each task. How to find the schedule which satisfies the precedence relationships between tasks and also minimize the length of the schedule taking into consideration the parallel execution of some tasks.

Answer 1

If the exclusion constraints ("T1,T4 are able to execute together") wouldn't be there (and no other constraints added), you could just start every task by taking the maximum finish time of all of its preceding tasks. That would be optimal and scale well. You would automatically get the shortest makespan. It would not be NP-complete/hard, and not job shop scheduling.

Unfortunately, the exclusion constraint (and potentially any other you add in the future) turn it into job shop scheduling (as mentioned by Lars), which is NP-complete/hard. See this video of job shop scheduling variant of an open source java implementation, that demo's why some tasks start later than their preceding tasks are finished. To solve that, look into heuristics, Metaheuristics (Tabu Search, ...) or other related techniques.

Answer 2

To remain simple, you can use a constructive heuristic based on priority rules along with a schedule generatiin scheme or also called SGS, see this for further reference. The heuristic will generate a ordered list of activities according to some criteria and the SGS will take this list as input and will generate the schedule. In your implementation of the SGS, you will tell if two tasks may or may not be executed in parallel basing in your second constraint.

If you want something more robust, you can use a Metaheuristic, where basically you will generate a solution (list of tasks) and modify this solution using local search techniques, exploring your solutions search space. You could generate solutions based on priority rules and evaluate them with the SGS implementation. This is just a simplified explanation of how a Metaheuristic will works, there are several variances. A good example of Metaheuristic is the Simulated Annealing, applied to the RCPSP problem: http://www.sciencedirect.com/science/article/pii/S0377221702007610.