P1253297875rnscL

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Resource-Constrained Project Scheduling Problem
(RCPSP)
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Resource-Constrained Project Scheduling Problem
(RCPSP)

• For each activity j (j 1, 2, , n) of the
  project, we have to determine a starting period
  in order to minimize the total duration
  (makespan) of the project while satisfying the
  precedence and resource constraints.
• K resources (k 1, 2, , K) are required to
  complete the activities
• Akt units of resource k are
  available during period t
• Activity j characteristics
• dj duration (number of periods)
• Pj set of predecessors (to be
  completed before j)
• rjk number of units of resource k
  required by j
• during each period of its
  completion
• Hypothesis Activity completed without
  interruption

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Exemple of RCPSP

• One resource required.
• Availability 6 units
• in each period

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First Solution representation (or encoding)

• Activity list (or permutation based)
  representation
• j1, j2,,
  jn
• is a permutation of the activity
  indices where

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Serial SGS to decode the representation into a
schedule

• Activity are scheduled sequentially according to
  their position in the permutation
• Each activity is scheduled to start as early as
  possible according to the precedence and resource
  constraints.

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Activity list representation and Serial SGS
Exemple

• 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

Activity list
Activity list
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Initial population

• The solutions in the initial population are
  generated with the following procedure
• The activities are selected
  sequetially.
• Each time a new activity is selected
  to be the next element of
• the vector, it is selected randomly
  among those having all their
• predecessors already selected.

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Selection Operator

• The parent-solutions are the individuals in the
  population, and they are paired randomly.

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One point crossover
1, 2, 3, 4, 5, 6, 7, 8, 9, 10
1, 3, 2, 6, 4, 8, 5, 7, 9, 10
?4
1, 2, 3, 4, 6, 8, 5, 7, 9, 10
1, 3, 2, 6, 4, 5, 7, 8, 9, 10
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Mutation operator

• Consider each activity
• Select randomly a position where the activity can
  be moved according to the precedence constraints
• Move the activity at this position with a
  probability equal to ßmax
• i.e. Select randomly ß ? 0, 1
• If ß lt ßmax
  them move the activity.

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Second solution representation (encoding)

• Priority based representation
• z z1, z2,, zn
• The smaller is the value of zj , the
  higher is the activity j priority to be scheduled

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Serial SGS to decode the representation into a
schedule

• Activity are scheduled sequentially
• Each time a new activity is scheduled, it is
  selected as one with the highest priority (i.e.,
  with the smallest component in the genotype
  vector) among those having all their predecessors
  already scheduled.
• This activity is scheduled to start as early as
  possible according to the precedence and resource
  constraints.

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Initial Population

• The individuals of the initial population are
  generated randomly by assigning a random integer
  number in 0, 1, , n -1 to each component of
  the genotype vector.
• One of the individual is generated according to
  the LFT rule where the component of the activity
  having the largest latest finishing time is equal
  to (n 1).

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Selection Operator

• Pseudo-elitist selection operator
• Each parent solution is selected randomly in a
  subset of the current population obtained by
  eliminating the 25 less fitted solutions and the
  25 best fitted solutions.
• In each pair, the parent-solutions are selected
  to be different

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One point crossover

• The one point crossover generates two
  offspring-solutions from the two parent-solutions
• z1 z11, z21, ,
  zm1
• z2 z12, z22, ,
  zm2
• as follows
• i) Select randomly a position (index) ?,
  0 ? m.
• ii) Then the offspring-solutions are
  specified as follows
• oz1 z11, z21, , z?1,
  z?12, , zm2
• oz2 z12, z22, , z?2,
  z?11, , zm1
• The first ? components of offspring oz1
  (offspring oz2) are the corresponding ones of
  parent 1 (parent 2), and the rest of the
  components are the corresponding ones of parent 2
  (parent 1)

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Mutation Operator

• For each offspring-solution, the operator is
  applied with a probability of pmut.
• To apply the operator, we first select randomly
  an element
• .
• Then select randomly a component of the
  offspring vector that has a value different from
  , and replace it by .