1. Evolution Programs for Various Discrete Problems 報告者：王 敬 育

2. Scheduling(1/13) Example ： Assume there are three orders ， o 1 ， o 2 and o 3 。 For each order ， the parts and the number of units to be produced are ： o 1 ： 30 * part a o 2 ： 45 * part b o 3 ： 50 * part c

3. Scheduling(2/13) Each part has one or more alternative process plans ： a ： plan # 1 a (opr 2 ， opr 2 ， opr 9 ) a ： plan # 2 a (opr 1 ， opr 3 ， opr 7 ， opr 8 ) a ： plan # 3 a (opr 5 ， opr 6 ) b ： plan # 1 b (opr 2 ， opr 6 ， opr 7 ) b ： plan # 2 b (opr 1 ， opr 9 )

4. Scheduling(3/13) Each operation requires some times on one or more machines ， these are ： opr 1 ： (m 1 10) (m 3 20) opr 2 ： (m 2 20) opr 3 ： (m 2 20) (m 3 30) opr 4 ： (m 1 10) (m 2 30) (m 3 20) opr 5 ： (m 1 10) (m 3 230)

5. Scheduling(4/13) opr 6 ： (m 1 40) opr 7 ： (m 3 20) opr 8 ： (m 1 50) (m 2 30) (m 3 10) opr 9 ： (m 2 20) (m 3 40) finally ， each machine has its setup time necessary for changes in operation ： m 1 ： 3 m 2 ： 5 m 3 ： 7

6. Scheduling(5/13) However ， most operators (mutations ， crossovers) applied to such a message would result in illegal schedules 。 Davis ： encoding/decoding strategy m 1 ： (40 o 3 o 1 o 2 ‘ wait ’ ‘ idle ’ ) This representation guarantees a legal schedule 。

7. Scheduling(6/13) The operators wee problem specific (they were derived from deterministic methods) ： run-idle ： this operator is applied to preference lists of the machines that have been waiting for more than an hour 。 It inserts the ‘idle’ as the second member of the preference list and reset the first member (time) of the preference list to 60(minutes)

8. Scheduling(7/13) scramble ： the operator “scramble” the members of a preference list crossover ： the operator exchanges preference list for selected machines The probability of run-idle was set to the percentage of the time the machine spent waiting ， divided by the total time of the simulation

9. Scheduling(8/13) TSP ： Davis represented an ordering as a list of nodes and use two operators ： the order- based crossover and scramble sublist mutation 。 Even mutation ， which should carry out a local modification of a chromosome ， seems to be problem dependent

10. Scheduling(9/13) The scramble sublist mutation selects a sublist of nodes from a parent and scrambles it in the offspring ： P = (2 4 | 7 1 4 8 | 3 5 9) may produce the offspring O = (2 4 | 4 8 1 7 | 3 5 9)

11. Scheduling(10/13) In general ， and for the scheduling problems in particular ， this is the direction to follow ： to incorporate the problem –specific knowledge not only in operators but in the chromosome structures as well

12. Scheduling(11/13) Compare three representations ， from the simplest ： (o 1 ) (o 2 ) (o 3 )… The intermediate ： (o 1 plan # 1 a ) (o 2 plan # 2 b ) (o 3 plan # 3 a ) The most complex ： (o 1 ) (o 2 ) (o 3 )

13. Scheduling(12/13) The operators themselves must be adjusted to suit the domain requirements 。 The chromosome representation should contain all the information that pertain to the optimization problem In summary ， it is possible to classify all GA- based approaches as various scheduling problems on the basis of chromosome representations 。 These fall into two categories ：

14. Scheduling(13/13) indirect representations ： 1. performed by a special decoder (schedule builder) 2. these representations can be divided into domain independent and problem-specific representations direct representation ： usually this representation requires some problem-specific operators

15. The timetable problem(1/4) example ： ● a list of teachers {T 1 ， … ， T m } ● a list of time intervals (hours) {H 1 ， … ， H n } ● a list of classes {C 1 ， … ， C k }

16. The timetable problem(2/4) The most nature chromosome representation of a potential solution of the timetable problem is a matrix representation ： a matrix (R) ij (1 ≤ i ≤ m ， and 1 ≤ j ≤ n) The following genetic operators were used ： mutation of order k ： select two contiguous sequences of k elements from the same row in the matrix R and swaps them

17. The timetable problem(3/4) day mutation ： a special case of the previous one ： is selects two groups of columns (hours) of the matrix R which correspond to different days and swap them

18. The timetable problem(4/4) crossover ： two matrices R 1 and R 2 ， the operator sorts the rows of decreasing values of a so-called local fitness function and the best b rows are taken as a building block; the remaining m – b rows are taken from the matrix R 2

19. Partitioning objects and graphs partitioning n objects into k categories ： ex. bin packing problem ， graph coloring problem ， etc… An interesting approach to the partitioning problem is presented by Bon Laszewski ， he encodes partitions using group-number encoding ， i.e.. partitions are represented as n-strings of integer numbers ， (i 1 …i n )

20. Partitioning objects and graphs Where the j-th integer i j {1…k} indicates the group number assigned to object j 。 This representation is supported by “intelligent structural operators” ： structural crossover and structural mutation 。 structural crossover ： Ex. p 1 = (112311232233) ， p 2 = (112123122333)

21. Partitioning objects and graphs These string decode into the following partitions ： p1 ： {1 ， 2 ， 5 ， 6} ， {3 ， 7 ， 9 ， 10} ， {4 ， 8 ， 11 ， 12} p2 ： {1 ， 2 ， 4 ， 7} ， {3 ， 5 ， 8 ， 9} ， {6 ， 10 ， 11 ， 12} From the view of “element” ， that means ： p1 ： {1 ， 1 ， 1 ， 1} ， {2 ， 2 ， 2 ， 2} ， {3 ， 3 ， 3 ， 3}

22. Partitioning objects and graphs First ， a random partition is chosen ： say ， partition #2 。 This partition is copied from p 1 into p 2 ： P 2 = (112123222233) The copying process usually destroys the requirement of equal partition sizes ， hence we apply a repair algorithm 。 P 2 = (1121*32*2233)

23. Partitioning objects and graphs and replaced (randomly) by numbers of other partition ， which were overwritten in the copying step 。 Thus the final offspring might be P 2 = (112133212233)

24. Partitioning objects and graphs structural mutation ： a parent ： P = (112133212233) may produce the following offspring (the number on positions 4 and 6 are swapped) ： P’ = (112331212233)

25. Partitioning objects and graphs Falkenauer proposed so-called Grouping Genetic Algorithm (GGA) to deal with a variety of grouping (partitioning ) problems consists of two parts ： object part and group part The object part uses the group-number encoding ： it consistof n-strings of integer numbers where the j-th integer i j {1…k} indicates the group number assigned to object j

26. Partitioning objects and graphs The group part is of variable length and represents only groups 。 Ex. (1 2 1 3 3 3 1 1 2 ： 1 2 3) group number 1 ： {1 ， 3 ， 7 ， 8} group number 2 ： {2 ， 9} group number 3 ： {4 ， 5 ， 6} Follow above ， for the bin packing problem ， the proposed Bin Packing Crossover Operator BPCX

27. Partitioning objects and graphs works as follows ： (121333112 ： 1 2 3) and (233514226 ： 2 3 4 5 6) two crossing sites are selected (at random) for each group part of the chromosomes ： (121333112 ： 1 |2 3|) and (233514226 ： 2|3 4 |5 6) Then it becomes (… ： 2 2 2 1 3 1 3 2 4 2 5 2 6 2 ) After eliminating conflicts ， the offspring chromosome has the following shape ：

28. Partitioning objects and graphs (2 2 2 1 ? 3 1 3 1 3 1 2 2 2 2 2 1 ： 2 2 2 1 3 1 ) After translate ， (1 2 ? 3 3 3 1 1 2 ： 1 2 3) Falkenauer used the first fit descending heuristic to insert missing objects 。 If there is no room for the object number 3 in any of the three bins in the above chromosome ， it necessary to create an additional bin ：

29. Partitioning objects and graphs (1 2 4 3 3 3 1 1 2 ： 1 2 3 4)

30. Path planning in a mobile robot environment The Evolutionary Navigator (EN) ， unifies off-line and on-line planning with a simple map of high fidelity and an efficient planning algorithm 。 off-line planner ： search for the optimal global path from the start to the destination on-line planner ： handling possible collisions or previously unknown objects by replacing a part of the original path by the optimal subtour

31. Path planning in a mobile robot environment The EN procedure ： see p.255 Fig.11.1 FEG ： oFf-line Evolutionary alGorithm ， generates a near-optimal global path 。 See Fig.11.2 NEG ： oN-line Evolutionary alGorithm ， generate local paths to deal with unexpected collisions and objects

32. Path planning in a mobile robot environment For NEG and FEG ， the difference is only the parameters they use ， ex. population_size ， number of generation ， maximum length of a chromosome Note that both EFG and NEG do global planning ， even if NEG usually generates a local path ， it operates on the updated global map

33. Path planning in a mobile robot environment Chromosome are ordered lists of path nodes 。 Each of the path nodes ， apart from the pointer to the next node ， consists of x and y coordinates of an intermediate knot point along the path and a Boolean variable b ， which indicates whether the given node is feasible or not → … → x1x1 y1y1 b1b1 xnxn ynyn bnbn

34. Path planning in a mobile robot environment If the node is feasible ， b is set to TRUE; otherwise ， FALSE The fitness (the total path cost) of a chromosome p = is determined by two separate evaluation functions (feasible and infeasible individuals)

35. Path planning in a mobile robot environment For a feasible path p ： Path_Cost(p) = w d · dist(p) + w s · smooth(p) + w c · clear(p) w d ， w s ， w c normalize the total cost of a path dist(p) = d( m i ， m i+1 ) is the distance between knot points m i and m i+1 ， the function dist(p) return total length of the path p

36. Path planning in a mobile robot environment smooth(p) = max s(m i ) ， where s(m i ) = Ө i min{d( m i-1 ， m i ) ， d( m i ， m i+1 } The function return the largest curvature of p at a knot point

37. Path planning in a mobile robot environment clear(p) = c i ， where c i = d i – г if d i ≥ г ， otherwise ： a(г- d i ) d i is the minimum distance between the segment (m i ， m i+1 ) of the path and all known objects г define a safe distance and a is a coefficient The function returns the largest number which measures clearance between all segments of p and the objects

38. Path planning in a mobile robot environment For an infeasible path p ： Path_Cost(p) = α+β+γ α ： the number of intersections of the path p with all walls of the objects β ： average number of intersections per infeasible segment γ ： provides the cost of the worst feasible path in the current population

39. Path planning in a mobile robot environment crossover ： similar to the classical one-point crossover widely used in GA Mutation_1 ： x’= x - δ(t ， x-a) ， if r = 0 x +δ(t ， b-x) ， if r = 1 Mutation_2 ： x’= x - δ(t ， x-a) ， if r = 0 x +δ(t ， b-x) ， if r = 1

40. Path planning in a mobile robot environment insertion ： inserts a new node into the existing path deletion ： delete a node from the path smooth ： smooth a part of the path by cutting sharp turns swap ： select chromosome into two parts (the splitting point is determined at random) and swaps these parts