1. 1 Fuzzy Scheduling Contents 1. Introduction to Fuzzy Sets 2. Application of Fuzzy Sets to Scheduling Problems 3. A Genetic Algorithm for Fuzzy Flowshop Scheduling Problem

2. 2 Literature 1. Fuzzy sets, uncertainty, and information, G. J. Klir and T.A. Folger. Englewood Cliffs, N.J : Prentice Hall, 1988. 2. "Flowshop Scheduling with Fuzzy Duedate and Fuzzy Processing Time", by H. Ishibuchi and T. Nurata, in: Scheduling Under Fuzziness, R. Slowinski, and M. Hapke, (eds), Physica-Verlag, A Springer-Verlag Company, 2000, pages 113-143 3. "Genetic algorithms and neighborhood search algorithms for fuzzy flowshop scheduling problems", Ishibuchi, H., Yamamoto, N., Murata, T., Tanaka., H, Fuzzy Sets and Systems Vol. 67, No.1, 1994, pages 81-100.

3. 3 Introduction to Fuzzy Sets crispversusfuzzy conventional logic: statement is true "classical" set theory: an element belongs to a set or not optimisation: a solution is feasible or not Basic foundations of fuzzy sets: Lotfi Zadeh, Fuzzy Sets, Information and Control, Vol. 8, 1965 dichotomous yes-or-no-type more-or-less type

4. 4 Classical set: x  A or x  A Example. A - set of even natural numbers less than 10 A = {2, 4, 6, 8}  A (x) degree of membership of x in A  A (1) = 0  A (2) = 1... or Fuzzy sets takes on values in the range [0,  ],  is usually 1 :

5. 5 Examples. = "real number close to 10" 0 1.0 15 x 510 = ”integers close to 10” = { (7, 0.1), (8, 0.5), (9, 0.8), (10, 1), (11, 0.8), (12, 0.5), (13, 0.1)

6. 6 Fuzzy sets and probability measure two bottles of liquid first bottle has membership 0.91 in the fuzzy set of all drinkable liquids second bottle probability that it contains drinkable liquids is 0.91 Which bottle would you choose?

7. 7 Application of Fuzzy Sets to Scheduling Problems 1. Approximate reasoning Approximate Reasoning is the process by which a possible imprecise conclusion is deduced from a collection of imprecise premises. Fuzzy IF-THEN Production Rules 2. Fuzzy constraints How to calculate satisfaction of the flexible constraints? 3. Fuzzy parameters Most often imprecise or incomplete are: fuzzy due dates fuzzy processing times

8. 8 Example of a rule-based system for dispatching Release job j next Time characteristicsMachine utilisationExternal priority Slack timeWaiting time Approximate Reasoning

9. 9 Waiting time: {long, medium, short} Slack time: {critically short, short, sufficient} Time characteristics: {urgent, not urgent} IF Waiting time is long AND Slack time is critically short THEN Time characteristics is urgent IF Waiting time is medium AND Slack time is critically short THEN Time characteristics is urgent IF Waiting time is short AND Slack time is critically short THEN Time characteristics is urgent IF Waiting time is medium AND Slack time is short THEN Time characteristics is not urgent... 0 1.0 8 T 634.5 j(T)j(T) critically short shortsufficient

10. 10 Waiting time is A - crisp Slack time is B - crisp Fuzzy RulesObserved System StateFuzzy Advice Time criterion is C - fuzzy + DEFUZZIFICATION

11. 11 Fuzzy Constraints  C (x) satisfaction grade of the solution x crisp constraint fuzzy constraint

12. 12 Fuzzy Due Date d j duedate of job j C j completion time of job j satisfaction grade 0 1.0 djdj CjCj j(Cj)j(Cj)

13. 13 0 1.0 djUdjU CjCj j(Cj)j(Cj) djLdjL 0 djUdjU CjCj djLdjL ejLejL ejUejU j(Cj)j(Cj)

14. 14 A Genetic Algorithm for Fuzzy Flowshop Scheduling Problems Problem Statement flow shop scheduling n jobs which have fuzzy duedates m machines sequence of n jobs is represented by a vector x=(x 1, x 2,...,x n ) p i,j - processing time of job j on machine i C i,j - completion time of job j on machine i C 1,x 1 = p 1,x 1 C i,x 1 = C i-1,x 1 + p i,x 1 for machines i=2,3,...,m C 1,x k = C 1,x k-1 + p 1,x k for jobs k=2,3,...,n C i,x k = max {C i-1,x k, C i,x k-1 } + p i,x k for machines i=2,3,...,m and for jobsk=2,3,...,n

15. 15 Objective 1: Maximisation of minimum satisfaction grade Maximise f min = min{  j (C j ) : j=1,2,...,n} Objective 2: Maximisation of total satisfaction grade

16. 16 total number of possible solutions: n! * optimal solution can be found only for small-size problems * for large-size problems search techniques could be used Genetic algorithm Notation N number of individuals in each population x p i i-th individual in the p-th population  p = {x p i : i=1,2,..., N}individuals in the p-th population Characteristics of the algorithm selection based on the roulette wheel two-point order crossover shift mutation elite strategy

17. 17 Step 1. Initialisation Randomly generate N individuals to construct an initial population  1 Step 2. Selection Select N-1 individuals from the current population using the roulette wheel selection method with linear scaling Selection probability g(x)objective function f min or f sum g min (  p ) = min { g(x p j ) : x p j  p } objective function of the individual minimum objective function of the whole population 

18. 18 Step 3. Crossover Two-point order crossover is applied to each of the selected pairs with a pre-specified crossover probability. When the crossover operator is not applied one parent becomes an offspring.

19. 19 Step 4. Mutation Applied to each of the individuals generated by the crossover with a pre-specified mutation probability.

20. 20 Step 5. Elitist strategy Add the best individual from the previous population to the new N-1 individuals generated by the crossover and mutation. Step 6. Termination test If a pre-specified stopping condition is satisfied stop.

21. 21 * The value of the Objective function 1 (maximisation of the minimum satisfaction grade) is 0 if the satisfaction grade of at least one job is 0. 0 1.0 djUdjU CjCj djLdjL ejLejL ejUejU j(Cj)j(Cj)  is a positive constant Objective 1’: Maximisation of minimum satisfaction grade Maximise f’ min = min{ j (C j ) : j=1,2,...,n}

22. 22 Summary * Fuzzy sets and logic can be successfully used to treat various types of uncertainty that exist in scheduling problems. Fuzzy duedate and fuzzy processing time are typical examples of such uncertainty. * Many conventional scheduling problems could be reformulated as fuzzy scheduling problems.

23. 23

24. 24 Experiments Population size N = 50 Crossover probability = 0.9 Mutation probability = 0.3 Stopping condition = 2000 generations Number of machines m = 10 100 problems with n = 10 jobs 100 problems with n = 20 jobs 100 problems with n = 50 jobs Processing time p i,j is a random integer from the interval [1,99]

25. 25 Each job has a fuzzy duedate How to generate d j L and d j U ? 1. Randomly generate a sequence x=(x 1, x 2,...,x n ), n=10, 20, 50 for each test problem 2. Calculate the completion time of each job C j (x) 3. Calculate for each job j d j L = C j (x) -  d j U = C j (x) +   is a random integer from the interval [100, 200] 0 1.0 djUdjU CjCj j(Cj)j(Cj) djLdjL

26. 26 0 1.0 djUdjU CjCj j(Cj)j(Cj) djLdjL  j (C j )=0.5 if the jobs are processed in the order x x is the optimal solution for Objective 1 (maximisation of the minimum satisfaction grade)

27. 27 * The value of the Objective function 1 (maximisation of the minimum satisfaction grade) is 0 if the satisfaction grade of at least one job is 0.

28. 28 0 1.0 djUdjU CjCj djLdjL ejLejL ejUejU j(Cj)j(Cj)  is a positive constant Objective 1’: Maximisation of minimum satisfaction grade Maximise f’ min = min{ j (C j ) : j=1,2,...,n}

29. 29 * Heuristic method for initial solution: Process the given jobs in an increasing order of their duedates. With fuzzy duedates we can use values of d j U to order the jobs Initialisation: one heuristic schedule and N-1 randomly generated * Genetic algorithm can effectively utilise heuristic initial solutions!