ResourceConstrained Project Scheduling Problem RCPSP ResourceConstrained Project Scheduling

Resource-Constrained Project Scheduling Problem (RCPSP)

Resource-Constrained Project Scheduling Problem (RCPSP) o o 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

Exemple of RCPSP One resource required. Availability: 6 units in each period

First Solution representation (or encoding) o Activity list (or permutation based) representation [j 1, j 2, …, jn] is a permutation of the activity indices where

Serial SGS to decode the representation into a schedule n n 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.

Activity list representation and Serial SGS : Exemple [1, 2, 5, 3, 4, 6, 8, 7, 9, 10] Different activty lists Same schedule [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] Activity list

Initial population o 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.

Selection Operator o The parent-solutions are the individuals in the population, and they are paired randomly.

One point crossover [1, 2, 3, 4, 5, 6, 7, 8, 9, 10] [1, 3, 2, 6, 4, 8, 5, 7, 9, 10] ρ=4 [1, 2, 3, 4, 6, 8, 5, 7, 9, 10] [1, 3, 2, 6, 4, 5, 7, 8, 9, 10]

Mutation operator o o o 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 βmax : i. e. Select randomly β є [0, 1] If β < βmax them move the activity.

Second solution representation (encoding) o Priority based representation z =[z 1, z 2, …, zn] The smaller is the value of zj , the higher is the activity j priority to be scheduled

Serial SGS to decode the representation into a schedule n n n 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.

Initial Population o o 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).

Selection Operator o o o 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

One point crossover o The one point crossover generates two offspring-solutions from the two parent-solutions z 1 = [ z 11, z 21, …, zm 1] z 2 = [ z 12, z 22, …, zm 2] as follows: i) Select randomly a position (index) ρ, 0 ≤ ρ ≤ m. ii) Then the offspring-solutions are specified as follows: oz 1 = [ z 11, z 21, …, zρ1, zρ+12, …, zm 2] oz 2 = [ z 12, z 22, …, zρ2, zρ+11, …, zm 1] The first ρ components of offspring oz 1 (offspring oz 2) are the corresponding ones of parent 1 (parent 2), and the rest of the components are the corresponding ones of parent 2 (parent 1)

Mutation Operator o o 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.