Beta-Robust Solutions for the Fuzzy Open Shop Scheduling

-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


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.


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.


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



p1 p2 p3

p4 p5 p6

p7 p8 p9



Job 1:

Job 2:

Job 3:

p1 p2 p3

p4 p5 p6

p7 p8 p9



Machine 1 Machine 2 Machine 3

p1 p2 p3

p4 p5 p6

p7 p8 p9



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).


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


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.


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)


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


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 .


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.


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 .


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.


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


Results: Evolution of Adaptive GA

0

0.2

0.4

0.6

0.8

1

1020

1040

1060

1080

1100

1120

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).


Results: a-priori versus a-posteriori (7 7)


Results: a-priori versus a-posteriori (8 8)


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.


Thank you!

Questions?


IntroductionThe Fuzzy Open Shop ProblemOpen ShopFuzzy Durations

*-Robust SchedulesAre *-Robust Schedules Actually Robust?Experimental Results

Beta-Robust Solutions for the Fuzzy Open Shop Scheduling

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