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OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
Robust optimization for resource-constrained projectscheduling with uncertain activity durations
Christian Artigues1, Roel Leus2 and Fabrice Talla Nobibon2
1LAAS-CNRS, Université de Toulouse, France
2Research group ORSTAT, Faculty of Business and Economics,
K.U.Leuven, Leuven, Belgium
partly funded by ANR �Blanc� program
ROBOCOOP project - ANR-08-BLAN-0331-01
Artigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
1 Introduction
Robust optimization
Resource-constrained project scheduling
Robust project scheduling
Problem complexity and issues
2 Evaluation of the maximal regret of a given selection
Restriction to extreme scenarios
Lower and upper bounds for maximal
Integer linear programming for maximal regret evaluation
3 Solving the AR-RCPSP
Artigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
Robust optimizationResource-constrained project schedulingRobust project schedulingProblem complexity and issues
Robust combinatorial optimization
Combinatorial optimization and uncertainty scenarios
Combinatorial Optimization Problem : minx∈X⊆{0,1}n cx .
Suppose c is uncertain with c ∈ C, set of uncertainty scenarios.
Minimax cost or minimax regret
Robust combinatorial optimization consists in
Minimize the worst cost over all scenarios minx∈X maxc∈C cx
Minimize the worst absolute regret over all scenarios
minx∈X maxc∈C(cx −miny∈X cy)
Minimize the worst relative regret over all scenarios
minx∈X maxc∈C(cx−miny∈X cy)
miny∈X cy
Artigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
Robust optimizationResource-constrained project schedulingRobust project schedulingProblem complexity and issues
Resource-constrained project scheduling
De�nition
V = {0, 1, . . . , n, n + 1}, set of activities (project) with 0
dummmy start activity and n + 1 dummy end activity,
pi duration of activity i ∈ V ,
R set of resources,
bk , availability of resource k ∈ R ,
bik demand of activity i for resource k ∈ R ,
E precedence constraints,
Si start time of i (to be determined)
RCPSP (Resource-Constrained Project Scheduling Problem) :
Minimize total project duration subject to precedence and resource
constraints.
Artigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
Robust optimizationResource-constrained project schedulingRobust project schedulingProblem complexity and issues
�conceptual� RCPSP formulation
Decision : set activity start times (S ∈ Rn+2)
S : (in�nite) set of feasible schedules = set of vectors
S ∈ Rn+2 verifying
Sj ≥ Si + pi ∀(i , j) ∈ E (1)∑sj≤t<sj+pi
bik ≤ Bk ∀t ≥ 0,∀k ∈ R (2)
Sj ≥ 0 i ∈ V (3)
Formulation 1 (direct)
(RCPSP)minS∈S Sn+1
Artigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
Robust optimizationResource-constrained project schedulingRobust project schedulingProblem complexity and issues
RCPSP as a combinatorial optimization problem
Decision variables : select a feasible selection X ⊆ V 2 (set of
activity pairs representing additional precedence constraints)
A selection X ⊆ V 2 is feasible if S(X ) ⊆ S with
S(X ) = {S ≥ 0|Sj ≥ Si + pi , ∀(i , j) ∈ E ∪ X}, the set of start
times verifying the precedence constraints.
X : set of feasible selection.
Cmax(X ) : length of the longest path in G (V ,E ∪X ) (each arc
having a length equal to origin activity duration).
Formulation 2 (Combinatorial optimization)
(RCPSP)minX∈X Cmax(X )
Artigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
Robust optimizationResource-constrained project schedulingRobust project schedulingProblem complexity and issues
Uncertain duration and selections
For each activity i , uncertain duration pi ∈ Pi with Pi �nite
continuous (interval) or discrete set.
Cmax(X , p) : length of the longest path in G (V ,X ∪ E ) (each
arc having a length equal to origin activity duration in scenario
p).
A selection is feasible for any duration scenario.
A selection de�nes a policy for scheduling under uncertain durations
(Earliest-Start policy).
Artigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
Robust optimizationResource-constrained project schedulingRobust project schedulingProblem complexity and issues
Robust resource-constrained project scheduling problem
Absolute regret of a selection X for a duration scenario p :
RA(X , p) = (Cmax(X , p)−minY∈X Cmax(Y , p))
Minimax absolute regret resource-constrained project scheduling
problem
(AR − RCPSP)minX∈X maxp∈P RA(X , p)
Relative regret of a selection X for a duration scenario p :
RR(X , p) = (Cmax(X ,p)−minY∈X Cmax(Y ,p))minY∈X Cmax(Y ,p)
Minimax relative regret resource-constrained project scheduling
problem
(RR − RCPSP)minX∈X maxp∈P RR(X , p)
Artigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
Robust optimizationResource-constrained project schedulingRobust project schedulingProblem complexity and issues
Problem complexity and issues
Complexity
Given a selection X and a durations scenario p, computing the
absolute regret RA(X , p) or the relative regret RR(X , p) is NP-hard(RCPSP).
Issues :
Compute lower bounds and upper bound of the minimax
regret ?
Propose a solution method that can be used in practice ?
Artigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
Restriction to extreme scenariosLower and upper bounds for maximalInteger linear programming for maximal regret evaluation
absolute maximal regret : restriction to extreme scenarios
Let pmini (pmax
i ) denote the minimum (maximum) element of
Pi .
A scenario p is extreme if pi = pmini or pi = pmax
i for any
activity i ∈ V .
Theorem
Given a selection X , absolute maximal regret is reached on an
extreme scenario.
Artigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
Restriction to extreme scenariosLower and upper bounds for maximalInteger linear programming for maximal regret evaluation
Relative maximal regret and extreme scenarios
Counterexample
Let n = 2, P1 = {2, 3, 6} and P2 = {1, 3, 5}, no precedence
constraints, no resource constraints.
For any scenario, optimal makespan is C ∗max(p) = max(p1, p2).
Let X = {(1, 2)}. Cmax(X , p) = p1 + p2.
Absolute regret of X for a scenario p is
RA(X , p) = p1 + p2 −max(p1, p2) of maximum set by p1 = 6
and p2 = 5.
Relative regret if X for a scenario p is
RR(X , p) = p1+p2max(p1,p2) − 1 of maximum set by unique scenario
p1 = p2 = 3, which is not extreme.
Artigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
Restriction to extreme scenariosLower and upper bounds for maximalInteger linear programming for maximal regret evaluation
Maximal regret lower bound
Let Y denote a feasible selection.
ra(X ,Y ) = maxp∈P(Cmax(X , p)− Cmax(Y , p))
rr(X ,Y ) = maxp∈PCmax(X ,p)−Cmax(Y ,p)
Cmax(Y ,p)
ra(X ,Y ) (rr(X ,Y )) is the largest absolute (relative) di�erence
between the longest path lengths in G (V ,E ∪ X ) and
G (V ,E ∪ Y ).
Lower bounds for absolute and relative maximal regrets
If Y is a feasible selection, maxp∈P RA(X , p) ≥ ra(X ,Y ) and
maxp∈P RR(X , p) ≥ rr(X ,Y )
Artigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
Restriction to extreme scenariosLower and upper bounds for maximalInteger linear programming for maximal regret evaluation
Maximal regret upper bounds
A �necessary� selection Y is a (non-necessarily feasible)
selection such that ∀p ∈ P , Cmax(Y , p) is a lower bound of
C ∗max(p).
A trivial necessary selection is obtained by setting Y = ∅.
Absolute and relative maximal regret upper bounds
If Y is a necessary selection, maxp∈P RA(X , p) ≤ ra(X ,Y ) et
maxp∈P RR(X , p) ≤ rr(X ,Y )
Open problems : complexity of ra(X ,Y ) and rr(X ,Y )computations.
Artigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
Restriction to extreme scenariosLower and upper bounds for maximalInteger linear programming for maximal regret evaluation
Integer linear programming for absolute maximal regretevaluation
Simultaneous computation of the maximal regret and of the
optimal RCPSP solution
Variable ai ∈ {0, 1} for selection of minimun or maximum
duration.
Continuous �ow variables φminij ∈ [0, ai ] and φ
maxij ∈ [0, 1− ai ]
for longest path length computation in G (V ,E ∪ X ).
Continuous start time variables Si for the optimal RCPSP
solution under scenario p set by ai variables.
Variables yij ∈ {0, 1} for the optimal selection corresponding
to the optimal solution given by SiContinuous resource �ow variables fijk for feasibility conditions
of the selection Y .
Multi-mode RCPSP with a composite linear objective functionArtigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
Restriction to extreme scenariosLower and upper bounds for maximalInteger linear programming for maximal regret evaluation
ILP for maximal regret computation (extract)
RA∗(X ) = max
∑
(i,j)∈E∪Xpmin
i φmin
ij + pmax
i φmax
ij
− Sn+1
s.c. :∑
(i,j)∈E∪Xφmin
ij + φmax
ij =∑
(j,i)∈E∪Xφmin
ji + φmax
ji ∀i ∈ V \ {0, n + 1}
∑(0,j)∈E∪X
φmin
0j + φmax
0j =∑
(j,n+1)∈E∪Xφmin
j,n+1+ φmax
j,n+1= 1
∑(i,j)∈E∪X
φmax
ij ≤ ai ,∑
(i,j)∈E∪Xφmin
ij ≤ 1− ai ∀i ∈ V \ {0, n + 1}
φmin
ij , φmax
ij ≥ 0 ∀(i , j) ∈ E ∪ X
Sj ≥Si + (1− ai )pmin
i + aipmax
i −M(1− yij ) ∀(i , j) ∈ E
S0 = 0
ai ∈ {0, 1} ∀i ∈ V
a0 = an+1 = 0
Y ∈ XArtigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
AR-RCPSP formulation by explicit consideration of allscenarios
Let p1, . . . , ph, . . . , p|P| be the list of all scenarios :
ρ∗ = min ρ
ρ ≥ Shn+1 − C∗max(ph) ∀ph ∈ P
Shj ≥ Shi + phi −M(1− xij ) ∀(i , j) ∈ V × V , i 6= j , ∀ph ∈ P
Shi ≥ 0 ∀i ∈ V , ∀ph ∈ PX ∈ X
considering a subset of scenarios P̃ ⊆ P lead to a lower bound
of the minimax absolute regret.
We propose an iterative scenario relaxation-based method to solve
the AR-RCPSP, progressively increasing the lower bound.
Artigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
First scenario relaxation-based algorithm
1: set UB =∞ and LB = 0.2: select a scenario p1 (e.g. pmin) and solve the RCPSP. P̃ ← {p1}.
h← 1.3: Solve the AR-RCPSP with P̃ and obtain a lower bound LB and a
selection X .4: If LB = UB, Stop.5: Else compute the maximal regret of X solving the multi-mode
RCPSP, update upper bound UB, get scenario ph+1 and optimalmakespan C∗
max(ph+1). h← h + 1.
6: If LB = UB, Stop.7: Else insert ph+1 in P̃ and return to step 2.
Converges in at most 2n iterations.
(see also Assavapokee et al. (COR 35(6), 2093-2102, 2008)) for a general robust
optimization method)
Artigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
Variant of the scenario relaxation-based algorithm
1: select a scenario p1 (e.g. pmin) and solve the RCPSP. P̃ ← {p1}.h← 1.
2: Solve the AR-RCPSP with P̃ and obtain a lower bound LB and aselection X .
3: If LB = UB, Stop.4: Else �nd a solution of the multimode RCPSP with an objective larger
than LB, and get corresponding scenario ph+1. h← h + 1.5: If a solution was found, solve the RCPSP to obtain C∗
max(ph+1),
insert ph+1 in P̃ and return to step 2.6: Else Stop.
Advantages : The multi-mode RCPSP has not to be solved
optimally
Drawback : A RCPSP has to be solved at each iteration.
Artigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
Preliminary numerical experiments
Two examples
i pmin
ipmax
ibi1 bi2 Γi
1 4 8 2 1 102 0 2 1 0 5, 63 0 2 3 1 74 1 3 2 0 85 2 4 1 1 96 4 6 2 1 107 4 8 3 0 −8 2 4 1 2 −9 1 3 1 2 1010 3 5 1 1 −bk 7 4
i pmin
ipmax
ibi1 bi2 bi3 bi4 Γi
1 1 3 10 10 5 5 7, 8, 92 1 3 10 2 3 8 5, 6, 73 1 4 5 9 2 8 4, 5, 64 6 8 3 2 10 10 85 1 3 4 6 10 8 66 1 3 1 7 2 1 87 8 10 10 8 8 2 −8 6 8 8 4 4 3 −9 6 8 4 2 3 2 −10 6 8 2 9 2 5 −bk 19 18 19 17
Artigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
Results
Algorithm 1 solves example 1 in 9 iterations and 9.3 seconds
and example 2 in 7 iterations and 2801 seconds.
Algorithm 2 solves example 1 in 9 iterations and 4.8 seconds
and example 2 in 13 iterations and 1843 seconds.
The proposed variant seems faster than the algorithm inpired by
Assavapokee et al.
Artigues, Leus and Talla Nobibon Robust Optimization for the RCPSP
OutlineIntroduction
Evaluation of the maximal regret of a given selectionSolving the AR-RCPSP
Conclusion
De�nition of the robust RCPSP based on selection
representation.
Results for the minimax regret RCPSP with uncertain
duration : structural properties and improvement of
general-purpose scenario relaxation-based robust optimization
methods.
Towards a practical robust optimization algorithm. Improve
lower bound computations and design of a scenario
relaxation-based heuristic.
Necessity of considering less conservative approaches
(Bertsimas et Sim, Math Prog, 2004, ...).
Artigues, Leus and Talla Nobibon Robust Optimization for the RCPSP