1. Selection and Scheduling Problem
in Continuous Time
with Pairwise-interdependencies
Ivan Blecic, Arnaldo Cecchini and Giuseppe A. Trunfio
University of Sassari
Italy

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. 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. 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. 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. Modeling interdependencies in continuous time
• Example of a pairwise-performance function

7. Modeling interdependencies in continuous time
• Example of a pairwise-performance function

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. 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. 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. 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. 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. Experiments

14. Experiments

15. Experiments
Without interdependecies With interdependencies