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-Robust Solutions for the Fuzzy Open ShopScheduling
Juan Jos Palacios1 Ins Gonzlez-Rodrguez2
Camino R. Vela1 Jorge Puente1
1Dept. of Computer Science,University of Oviedo (Spain)
2Dept. of Mathematics, Statistics and Computing,University of Cantabria (Spain)
IPMU 2014, July 1519
Palacios et al. (Uniovi-Unican) -Robustness for FOSP IPMU 2014 1 / 20
Outline
1 Introduction
2 The Fuzzy Open Shop Problem
3 -Robust Schedules
4 Are -Robust Schedules Actually Robust?
5 Experimental Results
Palacios et al. (Uniovi-Unican) -Robustness for FOSP IPMU 2014 2 / 20
Motivation: Scheduling
Scheduling: Plan de execution of tasks in resources (machines)under certain constraints in an optimal way.Traditional approach: Assume well-known deterministic input dataand optimise some performance measure.Drawback: Input variables are not well-known, may change,causing the optimal solution to be of little or no use at themoment of its execution.
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Motivation: Robustness
RobustnessA robust solution should still work satisfactorily when design variableschange slightly.
Different approaches:Practical: Add slack times to task durations so solutionsperformance is not affected by delays.Minimax (regret): Construct solutions with best possibleperformance in worst case.-robustness Maximise the likelihood that a solutionsperformance is not worse than a given target in all cases.
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Open Shop Scheduling Problem (OSP)
Open shop scheduling problem:applications in testing components of electronic systems, medicaldiagnosis. . . :)increasing presence in the literature :)NP-complete in the general case!traditionally assumes perfect-deterministic information :(
We would like to. . .incorporate uncertainty into data: fuzzy durationsfind robust solutions
Palacios et al. (Uniovi-Unican) -Robustness for FOSP IPMU 2014 5 / 20
Open Shop Scheduling Problem (OSP)
p1 p2 p3
p4 p5 p6
p7 p8 p9
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Open Shop Scheduling Problem (OSP)
Job 1:
Job 2:
Job 3:
p1 p2 p3
p4 p5 p6
p7 p8 p9
Palacios et al. (Uniovi-Unican) -Robustness for FOSP IPMU 2014 6 / 20
Open Shop Scheduling Problem (OSP)
Machine 1 Machine 2 Machine 3
p1 p2 p3
p4 p5 p6
p7 p8 p9
Palacios et al. (Uniovi-Unican) -Robustness for FOSP IPMU 2014 6 / 20
Open Shop Scheduling Problem (OSP)
O||Cmaxn jobs: J1, . . . , Jn,m machines: M1, . . . ,Mm,mn tasks or operations: Ok : 1 k mn:
I each belonging to one job and requiring one machine;I cannot overlap with other tasks in the same job or machine.
objective: minimise the makespan Cmax (completion time of lasttask).
Palacios et al. (Uniovi-Unican) -Robustness for FOSP IPMU 2014 6 / 20
Uncertain Task Durations: Triangular Fuzzy Numbers
Assume we only know:an interval of possible values for the duration [a1,a3],the most likely duration a2 in this interval.
This knowledge is represented by a triangular fuzzy number, TFN,A = (a1,a2,a3):
a1 a3a2
Palacios et al. (Uniovi-Unican) -Robustness for FOSP IPMU 2014 7 / 20
Uncertain Task Durations: Triangular Fuzzy Numbers
Assume we only know:an interval of possible values for the duration [a1,a3],the most likely duration a2 in this interval.
This knowledge is represented by a triangular fuzzy number, TFN,A = (a1,a2,a3):
a1 a3a2
Palacios et al. (Uniovi-Unican) -Robustness for FOSP IPMU 2014 7 / 20
Uncertain Durations: Working With TFNs
Arithmetic:addition:
A + B = (a1 + b1,a2 + b2,a3 + b3)
maximum:
max(A,B) (max{a1,b1},max{a2,b2},max{a3,b3})
We obtain a fuzzy schedule:starting and completion times of all tasks are TFNs;but there is no uncertainty re. the task processing order.
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Necessity and Possibility
A TFN can be seen as a possibility distribution on the values aduration may take.We can measure the necessity and possibility that the makespan Cmaxis less than a given number r :
Cmax
r
N(Cmax r)(Cmax r)
Palacios et al. (Uniovi-Unican) -Robustness for FOSP IPMU 2014 9 / 20
and RobustnessAssume we have a target C? for the makespan: A schedule withmakespan Cmax is
necessarily -robust w.r.t. C? if and only if = N(Cmax C?);possibly -robust w.r.t. C? iff = (Cmax C?).
and are the degrees of necessary and possible robustness w.r.t.the threshold C?.
Cmax
N(Cmax r)(Cmax r)
C
Palacios et al. (Uniovi-Unican) -Robustness for FOSP IPMU 2014 10 / 20
and Robustness: Remarks
is the degree of confidence that Cmax will be for sure belowthe threshold C?.If a schedule is -robust and -robust w.r.t. C?,
Pr(Cmax C?) .
Build a robust schedule w.r.t. C? = build a schedule maximisingthe confidence level .
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Robustness Assessment: Semantics of FuzzySchedules
A -robust fuzzy schedule:is a predictive schedule;provides an ordering pi of tasks;provides an estimate (possibility distribution) of the value of Cmax .
When tasks are executed following pi they have exact durations pij , sothe a-posteriori makespan Cmax(pi, pij) is already exact.
Idea for assessmentIn most cases, for the a-posteriori makespan it should hold that
Cmax(pi, pij) C.
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Robustness Assessment: Simulation
Simulate K possible realisations of task durations (coherent withpossibility distributions given by TFNs);calculate a-posteriori makespan for each simulation;measure the proportion of a-posteriori schedules withmakespan below C
ConjenctureA high a-priori confidence level should correspond to a higha-posteriori performance level .
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Experimental Results: Instances
Problem instances are fuzzified versions of well-known OSPbenchmark:
j7 : 9 instances of size 7 7;j8: 9 instances of size 8 8.
Number of a-posteriori simulations: K=1000.
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Obtaining -Robust Schedules: Adaptive GA
Initial Idea: Re-use GA from literature on FOSP with objectivefunction ;Drawback: Schedules in initial population will have = 0 for anyreasonable C, so the GA cannot evolve!!!Improved Idea: Use adaptive GA minimising with successivethreshold approximations:
C0 > C1 > > Cn = C
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Results: Evolution of Adaptive GA
0
0.2
0.4
0.6
0.8
1
1020
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1060
1080
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1140
1160
0 200 400 600 800 1000 1200 1400 1600 1800 2000
-robustness C*
Number of generations
C* Best individual Population Average
Evolution of adaptive threshold Ci and necessary robustness level on oneproblem instance (average of 10 runs).
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Results: a-priori versus a-posteriori (7 7)
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Results: a-priori versus a-posteriori (8 8)
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Conclusions and Future Work
For the FOSP problem:robustness as good averagebehaviour of predictiveschedules;new robustness measure w.r.t.performance threshold basedon possibility/necessity;adaptive GA to obtain robustschedules;empirical assessment basedon simulations.
Future work:Next step: --robustness;Multiobjective approach:minimise makespan +maximise robustness.
Palacios et al. (Uniovi-Unican) -Robustness for FOSP IPMU 2014 19 / 20
Thank you!
Questions?
Palacios et al. (Uniovi-Unican) -Robustness for FOSP IPMU 2014 20 / 20
IntroductionThe Fuzzy Open Shop ProblemOpen ShopFuzzy Durations
*-Robust SchedulesAre *-Robust Schedules Actually Robust?Experimental Results