Robust optimization for resource-constrained project scheduling with uncertain activity durations

Transcription

1 Robust optimization for resource-constrained project scheduling with uncertain activity durations Christian Artigues 1, Roel Leus 2 and Fabrice Talla Nobibon 2 1 LAAS-CNRS, Université de Toulouse, France 2 Research group ORSTAT, Faculty of Business and Economics, K.U.Leuven, Leuven, Belgium partly funded by ANR Blanc program ROBOCOOP project - ANR-08-BLAN

2 1 Robust optimization Resource-constrained project scheduling Robust project scheduling Problem complexity and issues 2 Restriction to extreme scenarios Lower and upper bounds for maximal Integer linear programming for maximal regret evaluation 3

3 Robust combinatorial optimization Robust optimization Resource-constrained project scheduling Robust project scheduling Problem complexity and issues Combinatorial optimization and uncertainty scenarios Combinatorial Optimization Problem : min x 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 min x X max c C cx Minimize the worst absolute regret over all scenarios min x X max c C (cx min y X cy) Minimize the worst relative regret over all scenarios min x X max c C (cx min y X cy) min y X cy

4 Resource-constrained project scheduling Robust optimization Resource-constrained project scheduling Robust project scheduling Problem complexity and issues Denition V = {0, 1,..., n, n + 1}, set of activities (project) with 0 dummmy start activity and n + 1 dummy end activity, p i duration of activity i V, R set of resources, b k, availability of resource k R, b ik demand of activity i for resource k R, E precedence constraints, S i start time of i (to be determined) RCPSP (Resource-Constrained Project Scheduling Problem) : Minimize total project duration subject to precedence and resource constraints.

5 conceptual RCPSP formulation Robust optimization Resource-constrained project scheduling Robust project scheduling Problem complexity and issues Decision : set activity start times (S R n+2 ) S : (innite) set of feasible schedules = set of vectors S R n+2 verifying S j S i + p i (i, j) E (1) s j t<s j +p i b ik B k t 0, k R (2) S j 0 i V (3) Formulation 1 (direct) (RCPSP) min S S S n+1

6 Robust optimization Resource-constrained project scheduling Robust project scheduling Problem 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 S j S i + p i, (i, j) E X }, the set of start times verifying the precedence constraints. X : set of feasible selection. C max (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) min X X C max (X )

7 Uncertain duration and selections Robust optimization Resource-constrained project scheduling Robust project scheduling Problem complexity and issues For each activity i, uncertain duration p i P i with P i nite continuous (interval) or discrete set. C max (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 denes a policy for scheduling under uncertain durations (Earliest-Start policy).

8 Robust optimization Resource-constrained project scheduling Robust project scheduling Problem complexity and issues Robust resource-constrained project scheduling problem Absolute regret of a selection X for a duration scenario p : RA(X, p) = (C max (X, p) min Y X C max (Y, p)) Minimax absolute regret resource-constrained project scheduling problem (AR RCPSP) min X X max p P RA(X, p) Relative regret of a selection X for a duration scenario p : RR(X, p) = (Cmax(X,p) min Y X C max(y,p)) min Y X C max(y,p) Minimax relative regret resource-constrained project scheduling problem (RR RCPSP) min X X max p P RR(X, p)

9 Problem complexity and issues Robust optimization Resource-constrained project scheduling Robust project scheduling 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?

10 Restriction to extreme scenarios Lower and upper bounds for maximal Integer linear programming for maximal regret evaluation absolute maximal regret : restriction to extreme scenarios Let pi min P i. (pi max ) denote the minimum (maximum) element of A scenario p is extreme if p i = pi min activity i V. Theorem or p i = p max i for any Given a selection X, absolute maximal regret is reached on an extreme scenario.

11 Restriction to extreme scenarios Lower and upper bounds for maximal Integer linear programming for maximal regret evaluation Relative maximal regret and extreme scenarios Counterexample Let n = 2, P 1 = {2, 3, 6} and P 2 = {1, 3, 5}, no precedence constraints, no resource constraints. For any scenario, optimal makespan is C max(p) = max(p 1, p 2 ). Let X = {(1, 2)}. C max (X, p) = p 1 + p 2. Absolute regret of X for a scenario p is RA(X, p) = p 1 + p 2 max(p 1, p 2 ) of maximum set by p 1 = 6 and p 2 = 5. Relative regret if X for a scenario p is RR(X, p) = p 1+p 2 max(p 1,p 2 ) 1 of maximum set by unique scenario p 1 = p 2 = 3, which is not extreme.

12 Restriction to extreme scenarios Lower and upper bounds for maximal Integer linear programming for maximal regret evaluation Maximal regret lower bound Let Y denote a feasible selection. ra(x, Y ) = max p P (C max (X, p) C max (Y, p)) rr(x, Y ) = max p P C max(x,p) C max(y,p) C max(y,p) ra(x, Y ) (rr(x, Y )) is the largest absolute (relative) dierence 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, max p P RA(X, p) ra(x, Y ) and max p P RR(X, p) rr(x, Y )

13 Maximal regret upper bounds Restriction to extreme scenarios Lower and upper bounds for maximal Integer linear programming for maximal regret evaluation A necessary selection Y is a (non-necessarily feasible) selection such that p P, C max (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, max p P RA(X, p) ra(x, Y ) et max p P RR(X, p) rr(x, Y ) Open problems : complexity of ra(x, Y ) and rr(x, Y ) computations.

14 Restriction to extreme scenarios Lower and upper bounds for maximal Integer linear programming for maximal regret evaluation Integer linear programming for absolute maximal regret evaluation Simultaneous computation of the maximal regret and of the optimal RCPSP solution Variable a i {0, 1} for selection of minimun or maximum duration. Continuous ow variables φ min ij [0, a i ] and φ max ij [0, 1 a i ] for longest path length computation in G(V, E X ). Continuous start time variables S i for the optimal RCPSP solution under scenario p set by a i variables. Variables y ij {0, 1} for the optimal selection corresponding to the optimal solution given by S i Continuous resource ow variables f ijk for feasibility conditions of the selection Y. Multi-mode RCPSP with a composite linear objective function

15 Restriction to extreme scenarios Lower and upper bounds for maximal Integer linear programming for maximal regret evaluation ILP for maximal regret computation (extract) RA (X ) = max (i,j) E X p min i φ min ij + p max i φ max ij S n+1 s.c. : φ min ij + φ max ij = φ min ji + φ max ji i V \ {0, n + 1} (i,j) E X (j,i) E X φ min 0j + φ max 0j = φ min j,n+1 + φmax j,n+1 = 1 (0,j) E X (j,n+1) E X φ max ij a i, φ min ij 1 a i i V \ {0, n + 1} (i,j) E X (i,j) E X φ min ij, φ max ij 0 (i, j) E X S j S i + (1 a i )p min i + a i p max i M(1 y ij ) (i, j) E S 0 = 0 a i {0, 1} a 0 = a n+1 = 0 Y X i V

16 AR-RCPSP formulation by explicit consideration of all scenarios Let p 1,..., p h,..., p P be the list of all scenarios : ρ = min ρ ρ S h n+1 C max (ph ) p h P S h j S h i + p h i M(1 x ij ) (i, j) V V, i j, p h P S h i 0 i V, p h P X X considering a subset of scenarios P 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.

17 First scenario relaxation-based algorithm 1: set UB = and LB = 0. 2: select a scenario p 1 (e.g. p min ) and solve the RCPSP. P {p 1 }. h 1. 3: Solve the AR-RCPSP with P 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 p h+1 and optimal makespan C max(p h+1 ). h h : If LB = UB, Stop. 7: Else insert p h+1 in P and return to step 2. Converges in at most 2 n iterations. (see also Assavapokee et al. (COR 35(6), , 2008)) for a general robust optimization method)

18 Variant of the scenario relaxation-based algorithm 1: select a scenario p 1 (e.g. p min ) and solve the RCPSP. P {p 1 }. h 1. 2: Solve the AR-RCPSP with P and obtain a lower bound LB and a selection 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 p h+1. h h : If a solution was found, solve the RCPSP to obtain C max(p h+1 ), insert p h+1 in P 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.

19 Preliminary numerical experiments Two examples i p min i p max i b i1 b i2 Γ i , b k 7 4 i p min p max b i i i1 b i2 b i3 b i4 Γ i , 8, , 6, , 5, b k

20 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.

21 Conclusion Denition 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,...).