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# Selection and Scheduling Problem in Continuous Time with Pairwise-interdependencies Ivan Blečić, Arnaldo Cecchini, Giuseppe A. Trunfio - Department of Architecture, Planning and Design, University of Sassari, Alghero

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### Selection and Scheduling Problem in Continuous Time with Pairwise-interdependencies Ivan Blečić, Arnaldo Cecchini, Giuseppe A. Trunfio - Department of Architecture, Planning and Design, University of Sassari, Alghero

1. 1. Selection and Scheduling Problem in Continuous Time with Pairwise-interdependenciesIvan Blecic, Arnaldo Cecchini and Giuseppe A. Trunfio University of Sassari Italy
2. 2. Selection and scheduling of Projects/Actions• Portfolio selection problem: what to do? – constraints, objective criteria – interdependencies among actions (combinatorial aspects)• Scheduling problem: when to do what? – How to model time? – How to model interdependencies?
3. 3. Modeling interdependencies in continuous time • Stand-alone performance function performance at time t of the action i implemented at the time ti • Pairwise-interdependency performance function performance at time t of the action i implemented at the time ti , given that the action j is implemented at time tj .
4. 4. Modeling interdependencies in continuous time • Example of a stand-alone performance function P pi ti ei t pi – maximum performance ei – time required to reach maximum performance
5. 5. Modeling interdependencies in continuous time • Pairwise-performance function – Another assumption: influence at the time t of the action j on the performance of the action i is proportional to the fraction of the full performance reached by the action j at the time t. Hence: - the marginal performance of the action i due to the interdependency from the action j with respect to time
6. 6. Modeling interdependencies in continuous time • Example of a pairwise-performance function
7. 7. Modeling interdependencies in continuous time • Example of a pairwise-performance function
8. 8. Modeling interdependencies in continuous time • Multi-interdependency performance function – Depends only on pairwise interdependencies Hence: given actions {1, 2, …, m} implemented at times {t1, t2, …, tm}
9. 9. Modeling interdependencies in continuous time • Total performance of a subset of actions {1, 2, …, s) is the sum of the multi-interdependency performance functions for all the actions in the subset. (yields the instantaneous performance of all actions in the subset at any particular time t) • The overall performance in a given time interval is it’s defined integral over that interval. That is our objective function
10. 10. Budget constraint• Each action has a cost (has to be paid upfront)• There is an initial endowment of budget resources and an inflow at costant rate of• Thus, given a time-ordered bundle of actions {1, 2, …, m} implemented respectively at times {t1, t2, …, tm} , we have the following set of m constraints:
11. 11. Search heuristics• The selection-and-scheduling problem with interdependencies know to be NP-hard (Ehrgott&Gandibleux (2000), Robertset al. 2008) )• We used Covariance Matrix Adaptation Evolution Strategy (CMA-ES)
12. 12. Experiments• 10 projects, with respective values for e and p, and all the pairwise bs (90 values),• Ran for time horizon of 20 under 4 configurations: v = 0 and v = 20, with and without interdependencies
13. 13. Experiments
14. 14. Experiments
15. 15. ExperimentsWithout interdependecies With interdependencies