Robust optimization for resource-constrained project ...€¦ · Resource-constrained project...

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


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


OutlineIntroduction


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


OutlineIntroduction



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.


OutlineIntroduction



�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


OutlineIntroduction



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 )


OutlineIntroduction



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).


OutlineIntroduction



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)


OutlineIntroduction



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 ?


OutlineIntroduction


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.


OutlineIntroduction



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.


OutlineIntroduction



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 )


OutlineIntroduction



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.


OutlineIntroduction



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



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


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.


OutlineIntroduction


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)


OutlineIntroduction


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.


OutlineIntroduction


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


OutlineIntroduction


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.


OutlineIntroduction


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, ...).


Date post:	03-Aug-2020
Category:	Documents
Upload:	others
View:	7 times
Download:	0 times

Robust optimization for resource-constrained project ...€¦ · Resource-constrained project...

Documents