Selection and Scheduling Problem in Continuous Timewith Pairwise-interdependencies

Ivan Blecic, Arnaldo Cecchini and Giuseppe A. Trunfio

University of SassariItaly

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?

Modeling interdependencies in continuous time

• Stand-alone performance function

performance at time t of the action iimplemented at the time ti

• Pairwise-interdependency performance function

performance at time t of the action iimplemented at the time ti , given that the

action j is implemented at time tj .

Modeling interdependencies in continuous time

• Example of a stand-alone performance function

t

P

ti

pi – maximum performanceei – time required to reach maximum performance

pi

ei

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

Modeling interdependencies in continuous time

• Example of a pairwise-performance function

Modeling interdependencies in continuous time

• Example of a pairwise-performance function

Modeling interdependencies in continuous time

• Multi-interdependency performance function– Depends only on pairwise interdependenciesHence: given actions {1, 2, …, m}

implemented at times {t1, t2, …, tm}

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

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:

Search heuristics• The selection-and-scheduling problem with

interdependencies know to be NP-hard(Ehrgott&Gandibleux (2000), Robertset al. 2008) )

• We used Covariance Matrix AdaptationEvolution Strategy (CMA-ES)

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

Experiments

Experiments

Experiments

Without interdependecies With interdependencies