Applications%20of%20Genetic%20Algorithms%20to%20Resource-Constrained%20Scheduling%20Tasks

1
Applications of Genetic Algorithms to
Resource-Constrained Scheduling Tasks

• Keith Downing
• Dept. of Computer and Information Sciences
• The Norwegian University of Science Technology
• Trondheim, Norway
• keithd_at_idi.ntnu.no

2
Outline

• Introduction to Evolutionary Algorithms
• Basic Concepts
• Simple Scheduling Example
• Applying EAs to Scheduling Problems
• Travelling Salesman
• Job Sequencing
• Classic Job-Shop Scheduling
• Main Focus Representational Issues

3
Darwinian Evolution
Physiological, Behavioral
Phenotypes
Natural Selection
Ptypes
Reproduction
Gtypes
Genotypes
Genetic
4
Evolutionary Algorithms
Semantic
Parameters, Code, Neural Nets, Rules
Performance Test
P,C,N,R
Generate
Bits
Bit Strings
Syntactic
5
Evolutionary Computation Parallel Stochastic
Search
Biased Roulette Wheel
6
1
5
Selection Biasing
4
2
Translation Performance Test
3
Selection
Crossover
Mutation
6
Types of Evolutionary Algorithms

• Genetic Algorithms (Holland, 1975)
• Representation Bit Strings gt Integer or real
  feature vectors
• Syntactic crossover (main) mutation
  (secondary)
• Evolutionary Strategies (Recehenberg, 1972
  Schwefel, 1995)
• Representation Real-valued feature vectors
• Semantic mutation (main) crossover (secondary)
• Evolutionary Programs (Fogel, Owens Walsh,
  1966 Fogel, 1995)
• Representation Real-valued feature vectors or
  Finite State Machines
• Semantic mutation (only)
• View each individual as a whole species, hence
  no crossover
• Genetic Programs (Koza, 1992)
• Representation Computer programs (typically in
  LISP)
• Syntactic crossover (main) mutation (secondary)

7
Evolutionary Computation Requirements

• Domain that supports quantitative fitness
  assignment
• Fitness function that accurately evaluates
  performance
• Representation for solutions that tolerates
  mutation crossover

Classic Genetic Algorithm
8
Using Evolutionary Algorithms

• When
• Large, rough search spaces
• Satisficing or Optimization problems
• Entire solutions are easily generated and tested
• Exhaustive search methods are too slow
• Heuristic search methods cannot find good
  solutions (e.g. Stuck at local max)
• How
• Determine EA-amenable representation of solutions
• Define fitness function
• Define selection function roulette-wheel
  biasing function (f fitness -gt area)
• Set key EA parameters population size, mutation
  rate, crossover rate, generations, etc.
• EAs are easy to write, and theres lots of
  freeware!
• Specific problems often require specific
  representations genetic operators

9
Application Areas for Evolutionary Algorithms

• Optimization Controllers, Job Schedules,
  Networks(TSP)
• Electronics Circuit Design (GP)
• Finance Stock time-series analysis prediction
• Economics Emergence of Markets, Pricing
  Purchasing Strategies
• Sociology cooperation, communication, ANTS!
• Computer Science
• Machine Learning Classification, Prediction
• Algorithm design Sorting networks
• Biology
• Immunology natural virtual (computer immune
  system)
• Ecology arms races, coevolution
• Population genetics roles of mutation, crossover
  inversion
• Evolution Learning Baldwin Effect, Lamarckism

10
Process Scheduling (Kidwell, 1993)

• Task pairs (run communicate result) to be run
  on a set of processors
• ((7, 16) (11, 22) (12, 40), (15, 22).)
• A tasks run must finish before result sending
  begins
• All processors share a central communication line
  (bus)
• Each processor can handle only one task at a
  time.
• Each processor is capable of running any of the
  tasks
• Only one processor at a time can send its message
• A task cannot be removed from a processor until
  both run send are finished
• Tasks run on the main processor, P0, require no
  communication time, whereas tasks run on all
  other processors must send their message to P0
• Goal Schedule the tasks on processors so as to
  minimize the total timespan

11
Process Schedule Optimization using the Genetic
Algorithm

• Use GA to search the space of possible schedules
  (solutions)
• 1. Represent schedule in a GA-amenable form (i.e.
  linearize it)

2. Define a fitness function Fitness
MaxTimespan - Timespan Lower timespan gt
Higher fitness Other possibilities 1/(1
(Timespan - MinTimespan))
12
GA-based Schedule Optimization

• Compute schedules timespan by running on a
  process-network simulator.
• 1. Remove next task (whose assigned
  processor is open) from task-list and
• start simulating it on that
  processor.
• 2. Remove tasks from processors as
  soon as they finish running sending
• 3. Define a biasing function to convert fitness
  to a proportion of the roulette wheel
• Sigma Scaling (One of many standard biasing
  functions)
• ExpVal(x) Max ( 0, 1 (Fitness (x) -
  AvgFitness) / (2 StDevFitness))
• Normalizing
• Roulette-Wheel(x) ExpVal(x) /
  SumofAllExpVals
• 4. Select Mutation and Crossover Rates pmut
  0 .01 pcross 0.75
• 5. Select a population size popsize 10
• 6. Select of generations numgen 10
• 7. Run the Genetic Algorithm
• Generates a random initial population (of
  schedules) and evolves them via sigma-scaling
  selection, crossover and mutation for numgen
  generations

13
Kidwells (1993) Task List

• ((7 16) (11 22) (12 40) (15 22) (17 23)
• (17 23) (19 23) (20 28) (20 27) (26 27)
• (28 31) (36 37) (31 29) (28 22) (23 19)
• (22 18) (22 17) (29 16) (27 16) (35 15))
• MaxTime Sum of all run send times 916 time
  units
• Use 8 processors P0 - P7, with P0 being the
  master processor

14
Population of Schedules (Generation 0)

• These are randomly-generated
• Processor List
• 1 Span 401 Fitn 515 (11.0) (1 2 1 3 6 7 2 0
  0 4 5 1 6 0 5 5 4 7 4 6)
• 2 Span 383 Fitn 533 (13.7) (1 0 1 0 0 4 0 1
  0 5 3 6 2 5 5 5 1 3 5 7)
• 3 Span 426 Fitn 490 ( 7.1) (6 2 2 7 6 5 2 6
  7 6 6 5 3 4 0 1 0 2 0 7)
• 4 Span 415 Fitn 501 ( 8.8) (5 0 5 7 1 2 5 4
  2 6 3 6 2 0 0 4 1 7 3 1)
• 5 Span 377 Fitn 539 (14.7) (4 2 0 6 2 1 2 0
  1 6 3 2 2 3 3 4 0 3 0 1)
• 6 Span 435 Fitn 481 ( 5.7) (3 2 7 7 6 0 3 1
  4 7 7 5 1 0 4 6 5 5 5 6)
• 7 Span 439 Fitn 477 ( 5.1) (0 3 2 3 2 2 7 0
  4 5 2 1 6 3 1 7 1 0 3 2)
• 8 Span 410 Fitn 506 ( 9.6) (2 5 2 1 0 0 2 2
  5 0 1 3 6 1 3 3 4 6 2 0)
• 9 Span 337 Fitn 579 (20.9) (1 0 0 7 0 4 3 1
  1 0 0 2 4 1 2 4 6 4 7 6)
• 10 Span 449 Fitn 467 ( 3.5) (2 2 2 4 1 4 4 6
  6 4 7 4 0 6 2 5 2 7 7 7)
• Avg Fitness 508.80 StDev Fitness 32.28

15
Roulette Wheel (Generation 0)
16
Population of Schedules (Generation 1)

• 1 Span 366 Fitn 550 (12.1) (1 0 0 3 0 4 2 1
  0 1 3 6 4 5 7 4 5 7 5 6)
• 2 Span 406 Fitn 510 ( 7.9) (1 0 1 4 0 4 1 1
  1 4 0 2 2 1 0 5 2 0 7 7)
• 3 Span 377 Fitn 539 (11.0) (4 2 0 6 2 1 2 0
  1 6 3 2 2 3 3 4 0 3 0 1)
• 4 Span 464 Fitn 452 ( 1.8) (1 0 1 3 0 4 2 0
  1 4 4 2 4 1 5 5 4 6 4 6)
• 5 Span 366 Fitn 550 (12.1) (1 2 0 7 6 7 3 1
  0 0 1 1 6 0 2 4 6 5 7 6)
• 6 Span 337 Fitn 579 (15.2) (1 0 0 7 0 4 3 1
  1 0 0 2 4 1 2 4 6 4 7 6)
• 7 Span 337 Fitn 579 (15.2) (1 0 0 7 0 4 3 1
  1 0 0 2 4 1 2 4 6 4 7 6)
• 8 Span 415 Fitn 501 ( 7.0) (5 0 5 7 1 2 5 4
  2 6 3 6 2 0 0 4 1 7 3 1)
• 9 Span 465 Fitn 451 ( 1.7) (5 5 7 7 1 2 5 2
  3 6 1 6 2 1 0 5 1 7 2 1)
• 10 Span 329 Fitn 587 (16.0) (2 0 0 1 0 0 2 4
  4 0 3 3 6 0 3 2 4 6 3 0)
• Avg Fitness 529.80 StDev Fitness 47.43

17
Roulette Wheel (Generation 1)
18
Population of Schedules (Generation 2)

• 1 Span 337 Fitn 579 (11.7) (1 0 0 7 0 0 6 0
  2 6 3 6 0 4 1 4 1 7 1 1)
• 2 Span 465 Fitn 451 ( 0.0) (5 0 5 3 1 6 1 5
  0 1 3 6 6 1 6 4 5 7 7 6)
• 3 Span 366 Fitn 550 ( 8.8) (1 2 0 7 6 7 3 1
  0 0 1 1 6 0 2 4 6 5 7 6)
• 4 Span 329 Fitn 587 (12.5) (2 0 0 1 0 0 2 4
  4 0 3 3 6 0 3 2 4 6 3 0)
• 5 Span 329 Fitn 587 (12.5) (2 0 0 1 0 0 2 4
  4 0 3 3 6 0 3 2 4 6 3 0)
• 6 Span 270 Fitn 646 (18.4) (1 0 0 5 0 0 2 0
  0 0 3 3 4 0 2 2 4 6 3 2)
• 7 Span 364 Fitn 552 ( 9.0) (2 0 0 3 0 4 3 5
  5 0 0 2 6 1 3 4 6 4 7 4)
• 8 Span 337 Fitn 579 (11.7) (1 0 0 7 0 4 3 1
  1 0 0 2 4 1 2 4 6 4 7 6)
• 9 Span 347 Fitn 569 (10.7) (5 0 1 7 1 6 1 4
  0 0 1 2 2 0 0 4 5 7 7 0)
• 10 Span 410 Fitn 506 ( 4.5) (1 0 4 7 0 0 7 1
  3 6 2 6 4 1 2 4 2 4 3 7)
• Avg Fitness 560.60 StDev Fitness 49.61

19
Roulette Wheel (Generation 2)
20
Population of Schedules (Generation 6)

• 1 Span 263 Fitn 653 (10.6) (1 0 0 7 0 4 6 0
  0 4 3 7 0 0 0 0 5 6 3 6)
• 2 Span 232 Fitn 684 (16.7) (1 0 0 5 0 0 2 0
  0 6 3 3 0 0 2 2 0 7 1 0)
• 3 Span 325 Fitn 591 ( 0.0) (1 0 0 5 0 0 6 0
  0 2 3 3 4 0 3 4 5 7 3 2)
• 4 Span 261 Fitn 655 (11.0) (1 0 0 7 0 0 2 0
  0 0 3 2 0 0 2 0 1 6 3 2)
• 5 Span 249 Fitn 667 (13.3) (1 0 0 5 0 0 2 0
  0 2 3 6 0 0 3 0 5 6 3 2)
• 6 Span 255 Fitn 661 (12.1) (1 0 0 5 0 0 6 0
  0 0 3 3 0 0 1 0 5 7 3 2)
• 7 Span 292 Fitn 624 ( 4.8) (1 0 0 7 0 0 3 0
  0 0 3 3 4 0 0 4 1 6 3 2)
• 8 Span 269 Fitn 647 ( 9.4) (1 0 0 7 0 4 6 0
  0 4 3 7 0 0 2 0 4 6 3 6)
• 9 Span 247 Fitn 669 (13.7) (1 0 0 5 0 0 2 0
  0 0 3 7 4 0 0 4 1 6 3 2)
• 10 Span 274 Fitn 642 ( 8.4) (1 0 0 7 0 0 2 0
  0 0 3 3 0 0 3 0 5 6 3 2)
• Avg Fitness 649.30 StDev Fitness 24.87

21
Roulette Wheel (Generation 6)
22
Population of Schedules (Generation 9)

• 1 Span 265 Fitn 651 ( 0.4) (1 0 0 5 0 0 2 0
  0 2 3 6 0 0 1 0 5 7 3 2)
• 2 Span 241 Fitn 675 (12.3) (1 0 0 7 0 0 2 0
  0 4 3 3 0 0 3 0 0 6 1 0)
• 3 Span 238 Fitn 678 (13.8) (1 0 0 5 0 0 6 0
  0 6 3 3 0 0 1 0 1 6 1 0)
• 4 Span 258 Fitn 658 ( 3.8) (1 0 0 5 0 0 6 0
  0 6 3 2 0 0 0 0 0 7 1 0)
• 5 Span 248 Fitn 668 ( 8.8) (1 0 0 5 0 0 2 0
  0 2 3 3 0 0 1 0 1 6 3 0)
• 6 Span 246 Fitn 670 ( 9.8) (1 0 0 5 0 0 6 0
  0 6 3 7 0 0 1 0 5 7 1 2)
• 7 Span 246 Fitn 670 ( 9.8) (1 0 0 5 0 0 2 0
  0 4 3 3 0 0 1 0 5 7 1 2)
• 8 Span 242 Fitn 674 (11.8) (1 0 0 5 0 0 6 0
  0 2 3 6 0 0 3 0 1 6 3 0)
• 9 Span 226 Fitn 690 (19.7) (1 0 0 5 0 0 2 0
  0 0 3 6 0 0 3 0 5 6 3 2)
• 10 Span 246 Fitn 670 ( 9.8) (5 0 0 5 0 0 2 0
  0 2 3 6 0 0 3 0 5 7 3 2)
• Avg Fitness 670.40 StDev Fitness 10.06
  (Convergence)

23
Roulette Wheel (Generation 9)
24
Process-Schedule Evolution
GA falls off a peak
Convergence
25
Travelling Salesman Problem (TSP)

• Given N cities matrix of distances between
  them.
• Find Shortest cyclic tour that visits all
  cities.

• NP-Hard Exponential to both find solutions to
  verify solutions
• Heuristic Methods Find optimal solutions when Nlt
  1000.
• Genetic Algorithm Find good solutions for any N
• Applications Network building, Delivery routing,
  Sequence scheduling...

26
Applying GAs to TSP

• Fitness Function 1/tour-length or
  optimal-tour-length/tour-length
• Chromosome
• Direct Representation List of cities (standard
  approach)
• 7 3 25 31 12 1 5 14.
• Indirect Representation List of next city to
  pull from ordered list and insert into the
  solution sequence
• 1 1 2 3 5. gt 1 2 4 6 9
• Crossover
• Standard Bit or Integer Cross Only works for
  indirect representations
• Location Preserving Children inherit, as much
  as possible, cities in same gene location as
  parents
• Edge Preserving Children inherit, as much as
  possible, city-city edges from parents (actual
  edge locations in the chromosome may vary from
  parent to kid)
• Crossover is the key element to TSP GAs - and
  where most research is done.

27
Representational Issues for GA-based TSP

• Standard Crossover Mutation are purely
  syntactic gt pay no attention to semantics (i.e.
  The meanings of the bits or integers that they
  manipulate).
• Direct representations often embody constraints
  that simple crossover mutation cannot enforce.
• Indirect representations usually involve fewer
  constraints, so simple crossover mutation are
  often sufficient.
• Compare to Process Scheduling (Kidwell), where
  constraints (i.e. All alleles between 0 and N)
  were easy to enforce.

28
Partially-Mapped Crossover (PMX)

• Goldberg (1985)
• 1. Choose equal-length segments to swap
• 2. Create a mapping between corresponding elems
  of the segments
• 3. Swap the segments
• 4. Apply the map to each child to restore
  completeness
• Select Segments
• P1 1 6 3 4 5 7 2 8
• P2 3 7 6 5 8 1 2 4
• Create Map
• (4 ltgt 5 5 ltgt 8 7ltgt 1)
• Swap
• K1 1 6 3 5 8 1 2 8 Blue positions must be
  changed via mapping
• K2 3 7 6 4 5 7 2 4
• Apply Map (twice for last position in this eg.)
• K1 7 6 3 5 8 1 2 4
• K2 3 1 6 4 5 7 2 8

29
Subtour Operations (Cleveland Smith, 89)

• Subtour Chunking
• Select random chunks from the parents
• Prune chunk of cities that already exist in the
  child
• Insert chunk into child position as close as
  possible to its position in parent
• P1 (9 4 3)3 2 (6 8 7 1)1 (5 10)5
• P2 3 5 8 (7 10 4 6)2 9 (2 1)4
• K1 9 3 10 4 6 8 7 1 2 5
• Subtour Replacement
• Find chunks with same elems in both parents and
  swap them
• P1 1 5 6 3 2 4 8 7
• P2 2 7 4 8 1 6 5 3
• K1 1 5 6 3 2 7 4 8
• K2 2 4 8 7 1 6 5 3

30
Inheritance

• For GAs to make progress, they must pass on many
  of the good features from generation G to
  generation G1.
• Hence, when parents crossover, the good features
  of each should be preserved in at least one of
  the children.
• Biology Heritability degree to which children
  resemble their parents
• (Xkid-avg - Xpop-avg) heritability(Xparent-avg
  -Xpop-avg)
• Location Preservation (1985 - 1988)
• With PMX Subtour exchanges, kids inherit many
  cities in the same position as in one of the
  parents.
• Results not that promising
• gt GA viewed as inappropriate for TSP.
• Edge Preservation (1989 - present)
• Edges are the key contributors to TSP costs
  (fitness), so what we really need to preserve are
  city pairs (i.e. Edges) of TSP tours.
• GAs with edge focus perform much better, near
  optimal
• gt GA useful for TSP!

31
Edge Recombination Operator (ERO)

• Whitley, Starkweather Fuquay (1989)
• Maximizes edge inheritance
• 1. Create Edge Map
• 2. Init Kid with one of the parents first city
• 3. Remove new kid city, NKC, from edge map
• 4. NKC neighbor of NKC with the smallest edge
  list. If no remaining neighbors, randomly pick
  an unvisited city (gt violate edge inheritance)
• 5. If tour not complete goto 3
• Edge Map
• A C B E B A C F
• E.g. ACDEFB x EABCFD C A B D F D C E
  F
• E A D F F B C D E
• A gtC(3), B(2), E(2) gt AE gt D(2), F(3) gt
  AED gt C(2), F(2)
• gt AEDC gt B(1), F(1) gt AEDCB gt F(0) gt
  AEDCBF
• All edges in kid except FA come from one or
  another parent

32
Binary Matrix Crossover

• Homaifar, Guan Liepins (1993)
• During crossover convert TSP tours to
  unidirectional connection matrices
• (c a b e f d) x (e b f
  a d c) gt illegal tour gt (a d c e
  b f)
• a b c d e f a b
  c d e f a b c d e f a
  b c d e f
• a 0 1 0 0 0 0 0 0 0
  1 0 0 0 1 0 1 0 0 0 0 0 1 0 0
• b 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0
  0 0 1 0 0 0 0 0 1
• c 1 0 0 0 0 0 X 0 0 0 0 1 0
  gt 1 0 0 0 1 0 gt 0 0 0 0 1 0
• d 0 0 1 0 0 0 (2nd bit) 0 0 1 0 0
  0 0 0 1 0 0 0 0 0 1 0 0 0
• e 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0
  0 0 0 1 0 0 0 0
• f 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0
  0 0 1 0 0 0 0 0
• Problem Redundancies (rows 1 3) and omissions
  (rows 5 6) in child
• Solution Move 1s from redundant rows to
  all-zero rows in a manner that preserves as many
  parent edges as possible.
• 100 edge preservation in example above.

33
Binary Matrix Crossover (2)

• Problem Cycles of length lt N
• Solution Cut cycles and connect to one another

Whenever possible, the cycle-connecting edges
should come from a parent
Inversion Choose a segment of the tour and
reverse it. Hill Climbing Only keep inversion
products that yield higher fitness (shorter
tours) Evolution (GA) Learning (Hill Climbing)
Lamarckism (Reverse-encoding of learned
improvement into the genome/tour)
34
TSP Benchmark Comparison

• Edge Inheritance beats position inheritance.
• Evolution Learning beats evolution

35
Indirect Representations (Grefenstette, 1985)
2 3 1 2 2 4...
Pull 2nd element from the sorted list of unused
cities and insert it into the next spot in the
solution path

• Advantages
• Simple Representation
• Crossover Mutation create legal tours gt no
  (time consuming) special ops required
• Disadvantages
• Low Heritability (of positions and edges)
• Crossover can disrupt position sequences on right
  side of the crossover point

36
Inheritance Problems with Indirect
Representations

• 6-city TSP
• P1 1 1 1 1 1 1 gt 1 2 3 4 5 6
• P2 2 3 2 3 2 3 gt 2 4 3 6 5 1
• Crossover after 2nd gene
• K1 1 1 2 3 2 3 gt 1 2 4 6 5 3
• Inheritance 66.6 position, 50 edge
• K2 2 3 1 1 1 1 gt 2 4 1 3 5 6
• Inheritance 66.6 position, 33.3 edge

37
Job Sequencing Problems

• Flow-Shop Scheduling Problem
• J items to be processed by a system machine or
  sequence of machines.
• Determine proper order to introduce items into
  the system. Once in the system, its out of the
  schedulers control, so the assembly line is
  equivalent to a single machine from schedulers
  point of view.
• Different items may require different operations,
  with each op type demanding a setup time on a
  machine gt often useful to group similar-op items
  in the scheduled sequence to reduce number of
  setups (i.e. Retooling time).
• Time constraints may make it important that X
  items are completely processed per day. Hence,
  some retooling may be necessary.
• Each work area on the assembly line may be a set
  of machines, each capable of similar operations.
  buffer for waiting items.
• Similar to TSP
• sequence of cities -vs- sequence of items
• edges important city distances -vs- retooling
  times

38
Computer Board Assembly
Cleveland Smith (1989)
15 fail
15 fail
Solution Sequence B3,B3,B1,C3,A1,A1,A2, A2,B1,B1,
C1,C1,C1,C1, C2, D1, A2, A2, A2
Example Problem Contract 1 2xA, 3xB, 4xC,
1xD (due date 72 hours) Contract 2 5xA,
1xC, 2xD (due date 120 hours) Contract 3
2xB, 1xC (due date 24 hours)
39
Weighted Subtour Chunking

• Cleveland Smith (1989)
• Special Subtour operation for job-sequencing
  problems
• Same procedure as standard subtour chunking, but
• Decision of where to place chunks relative to one
  another is based on average due dates of jobs in
  the chunk
• Assume (10 4) has later average due date than (6
  8 7 1)
• P1 (9 4 3)3 2 (6 8 7 1)1 (5 10)5
• P2 3 5 8 (7 10 4 6)2 9 (2 1)4
• K1 9 3 5 2 6 8 7 1 10 4 (10 4) placed after
  (6 8 7 1)
• Knowledge-Intensive genetic operators
• Semantics (meaning) of the chromosomal
  bits/integers influence result of crossover.
• PMX, simple subtour chunking, subtour replacement
  and most genetic ops on indirect representations
  are blind, purely syntactic.
• In nature, genetic operations are blind, but GAs
  can exploit semantics during crossover mutation
  to speed convergence to optimal solutions.

40
GA -vs- Heuristic Methods Job Sequencing

• Cleveland Smith (1989)
• Heuristics
• EDD Earliest Due Date jobs released first
• SPT Shortest Processing Time jobs released
  first
• LST Least Slack Time jobs released first
• Slack Time Due Date - Total Processing
  Time

• Table values costs (earliness and lateness are
  penalized)
• Weighted Chunking always among the best
  performing crossover ops
• Subtour Replacement among the worst (since its
  often hard to find matching chunks in
• the two parents)

41
JSSP Job-Shop Scheduling Problem

• J Jobs to be performed on M machines. Each job
  consists of M operations, one for each of the
  machines.
• The operations must be done in the proper
  sequence gt at time t, no more than one machine
  can be working on an operation for job j.
• Operations may require different amounts of time.
  
• NP-Hard Problem gt Exponential time required both
  to find optimal solutions and to verify them.
• Search Space Size (J!)M Choose appropriate
  job sequence for each machine.
• Common Solution Methods
• Branch Bound techniques exhaustive optimizing
  search with deterministic node pruning. Too
  expensive for large J M (gt 20)
• Heuristic search priority rules resolve choices
  (of next ops to schedule).
• Satisficing that often finds optimal
  solutions, but no guarantee.
• More feasible approach to large JSSP problems.

42
JSSP Matrices

• Machine-Sequence Processing-Time Matrix
• Machine (Processing Time)
• Job1 3(1) 1(3) 2(6) 4(7) 6(3) 5(6)
• Job2 2(8) 3(5) 5(10) 6(10) 1(10) 4(4)
• Job3 3(5) 4(4) 6(8) 1(9) 2(1) 5(7)
• Job4 2(5) 1(5) 3(5) 4(3) 5(8) 6(9)
• Job5 3(9) 2(3) 5(5) 6(4) 1(3) 4(1)
• Job6 2(3) 4(3) 6(9) 1(10) 5(4) 3(1)
• 6x6 Benchmark from Muth Thompson, Industrial
  Scheduling (1963)
• Job-Sequence Matrix
• Job
• M1 1 4 3 6 2 5
• M2 2 4 6 1 5 3
• M3 3 1 2 5 4 6
• M4 3 6 4 1 2 5
• M5 2 5 3 4 6 1
• M6 3 6 2 5 1 4

43
JSSP Matrices (2)

• Machine Sequence Matrix (Mseq) sequences of
  operations (denoted by the of the machine on
  which they must be performed) for each job
• Processing Time Matrix (Tseq) durations for
  each operation of each job
• Job Sequence Matrix (Jseq) sequences of job
  operations to be performed on each machine.
• Schedule assignment of starting times (on the
  required machines) to all operations. Often
  drawn as a Gantt chart.
• M1 111 44444333333333
  66666666662222222222555
• M2 2222222244444666111111555 3
• M3 3333331 2222255555555544444
• M4 3333 666
  4441111111 22225
• M5 2222222222
  555553333333444444446666111111
• M6 33333333
  66666666622222222225555111444
• Total Time 55 minimum makespan

44
Schedule Classifications

• Complete All job operations scheduled exactly
  once.
• Feasible Complete Satisfies Mseq and Tseq
  Some ops may be started earlier without changing
  the machines order of operations

Machine X
Semi-Active Feasible Some ops may be started
earlier, but theyll necessarily alter the
operation order.
Backward Filling
Machine X
Active Semi-Active Some ops may be started
earlier, but theyll necessarily delay other
operations.
A
B
C
Machine X
Moving C delays B
45
Generating Schedules Simple Approaches

• MT Directly from an Mseq Tseq (Left-to-Right
  packing)
• 1. Randomly pick a job, j, and select its
  leftmost unschedule operation, op.
• 2. Schedule it on its machine,m, for the
  earliest time point, t, where
• a. All previously-scheduled operations on m
  have finished by t
• b. All previously-scheduled operations for j
  have finished by t.
• 3. Repeat until all job operations have been
  scheduled.
• Guaranteed semi-active schedule, but rarely
  optimal.
• MTBF MT with backward filling gt active
  schedules.
• MJT From Mseg, Jseq Tseq
• 1. Randomly pick any leftmost unscheduled
  operation, op, on any machine, m, in Jseq which
  is also the leftmost unscheduled operation for
  its job in Mseq.
• 2. Schedule op on m at earliest time t, where
  same as step 2 above
• 3. Repeat until all job operations have been
  scheduled.
• No guarantees of feasibility, since deadlocks
  may occur.

46
Scheduling Deadlocks

• Conflicts between Jseq and Mseq
• Job Sequences
• Machine1 Job5 Job4
• Machine2 Job4 Job5
• Machine Sequences
• Job4 M1 M2
• Job5 M2 M1
• Deadlock
• M1 wants Job5, but cant have it until M2 has
  taken Job5 (according to Mseq). But M2 wont do
  Job5 until it does Job4, and Job4 cant be
  performed on M2 until its performed on M1
  (according to Mseq). But Job4 cant be performed
  on M1 until Job5 is performed on M1!

47
Giffler Thompson Algorithm (1960)

• GT Mseq Tseq gt Active Schedules
• 1. C set of next schedulable operations for
  each job
• early-start-time 0 for all ops in C
• 2. o earliest completed task in C
• T(C) completion time of o
• 3. G conflict set all ops in C on
  machine(o) that overlap o (including o)
• 4. Randomly choose o from G schedule it.
• 5. Add successor(o) to C
• 6. Update early-start-time for all ops in C
• 7. If not empty(C) then goto 2
• Choice points
• step 2 Ties for earliest completed operation
• step 4 Multiple conflicts
• All combos of choices gt all active schedules.
• Also used to convert complete schedules to
  active schedules

48
Genetic Algorithm Fitness Functions for JSSP

• Lin, Goodman Punch (1997)
• wj Weight for job j dj Due date for
  job j
• Cj Completion time for job j rj Release
  time for job j
• Lj Lateness of job j Cj - dj Tj
  Tardiness of job j max(Lj, 0)
• Uj penalty for job j 1 if late, 0
  otherwise
• Ej Earliness of job j max (-Lj, 0)

Makespan
Weighted Flow Time
Weighted Tardiness
Maximum Tardiness
Weighted Lateness
Weighted Tardy Jobs
Weighted Earliness Weighted Tardiness
Fitness functions are usually simple inverses of
the above metrics
49
Genetic Algorithm Encoding Problems

• Assume that we want to represent job sequences
  directly in the GA chromosome
• Task
• Machine1 1 4 3 6 2 5
• Machine2 2 4 6 1 5 3
• MachineM 3 6 2 1 3 4
• We could easily convert Jseq into a chromosome
• 143625 246153 362134 gt 001100011110010101 .
• But then both randomly-generated chromosomes and
  the results of crossover may represent illegal
  schedules
• 143625 246153. X 152634 162534 (cross
  point after 3rd entry)
• gt
• 143634 162534. And 152625 246153
• Both children are illegal, since they a) advise
  doing same jobs twice on the same machine, and b)
  leave out certain jobs.

50
Nakano Yamada (1991)
Pairwise job priorities for all machines
GA Chromosome
Local Harmonization
Jseq
Jseq
Mseq
Global Harmonization
MJT
Tseq
Deadlock
Feasible Schedule

• Local Harmonization Removes single-machine
  priority conflicts
• Crossover Mutation Standard bitwise
• Lamarckism Changes to Jseq in Global
  Harmonization are back-coded into chromosome

51
Priority-Based Schedule Representation (1)

• A job-priority schedule establishes pairwise
  priorities of jobA over JobB on each of the M
  machines
• Job1 -vs- Job2 1 1 0 1 0 0
• Job1 -vs- Job3 0 1 1 0 0 0
• N(N-1)/2 pairs

Job2 has priority over Job1 on Machine M(1,5) 6
Job1 has priority over Job3 on Machine M(1,2) 1
Combine all rows to create a GA chromosome of
MN(N-1)/2 bits 110100 011000 110101 001100.
52
Priority-Based Schedule Representation (2)

• Now, neither random schedule generation nor
  crossover creates illegal pairs of priority.
  I.e. you never get situations in which Job1 is
  prioritized over Job2 on machine 4 and Job2 over
  Job1 on machine 4.
• But, triples, quadruples, etc. may be
  inconsistent
• On Machine 4 job1 gt job2 job2 gt job3
  job3 gt job1
• This set of priorities makes it impossible to
  establish a unique priority order for machine 4.
  Hence, we cannot generate a machine schedule
  (I.e. job sequence) for machine 4.

53
Local Harmonization

• Process for converting illegal genotypes
  (schedules) into legal job sequences.
• 1. Convert sequence of pairwise priorities into
  priority tables for each machine.
• 2. Establish a strict priority order for each
  table based on the original priorities
• Jobs Sum
• Job1 0 0 1 1 0 2
• Job2 1 0 0 1 1 3
• Job3 1 1 1 1 0 4
• Job4 0 1 0 0 0 1
• Job5 0 0 0 1 1 2
• Job6 1 0 1 1 0 3
• Job3 is most dominant, so make it dominant EVERY
  job. This entails changing entry (6,3) to 0 and
  (3,6) to 1. This reduces job6s sum to 2.
  Hence, job2 is now the only one with sum 3, so
  its the 2nd most dominant. Change the table so
  that job2 dominates every job but 3and so
  on.until a strict priority order is achieved. Do
  the same for each machines priority table.

54
Global Harmonization

• Local harmonization creates a job sequence for
  each machine.
• Global harmonization is used to remove deadlocks.
• Job Sequences
• Machine1 Job2 Job6 Job1 Job4 ... Blue gt
  already scheduled
• Machine2 Job1 Job 3 Job6 Job5
• Machine Sequences
• Job4 M1 M2
• Job5 M2 M1
• Deadlock Job4 wants to run on M1, but J1 is next
  on M1. Similarly, J5 wants to run on M2, but J3
  and J6 are ahead of it. Assume all other jobs
  are either finished or deadlocked with even
  longer waits.
• Global Harmonization
• Swap Jseq jobs to accomodate J4, since its
  closest to being run.
• Machine1 Job2 Job6 Job4 Job1
• Forcing (Lamarckism) Change gtype so that J4 has
  priority over J1 on M1

55
Shi (1997)
Schedule next task of Job 3

• Lamarckism When MTBF uses backward filling, it
  changes job order in the chromosome
• Job Partition Crossover - Unique subsets of the J
  jobs taken from each parent.
• Necessary since kids must have M copies
  of all J jobs.
• Mutation Job swaps or moves within chromosome

56
Job-Partitioned Crossover (Shi, 1997)

• Select non-overlapping subsets S1, S2 of 1,2J
  S, such that S1 U S2 S
• Choose all and only S1 (S2) genes from parent 1
  (2) and place in child

Kid chromosome is guaranteed to have M copies
of all J jobs, when parents do
57
Kobayashi, Ono Yamamura (1995)
Job Sequence Matrix (Jseq)
GA Chromosome
Mseq
No harmonization needed, since a) init Jseq
pop gend by GT b) All kids run through GT
MJT
Tseq
Active Schedule
Subsequence Exchange Crossover (SXX) Only Jseq
segments with same jobs exchanged
between parents gt kid schedules are complete
Parent Jseq
Kid Jseq
GT
Active Schedule
SXX
Parent Jseq
GT
Kid Jseq
Active Schedule
58
Subsequence Exchange Crossover (SXX)

• Swap similar (but permuted) subsets of parent
  job sequences
• GT applied to kid Jseqs may result in job swaps
  within rows

J2 J3 J5 J1 J4 J6 J4 J1 J2 J3 J6 J5 J3 J2 J1 J6
J5 J4
J1 J4 J5 J6 J3 J2 J5 J6 J3 J4 J2 J1 J1 J3 J5 J2
J4 J6
Parent Jseq2
Parent Jseq1
SXX
J2 J3 J1 J4 J5 J6 J4 J2 J1 J3 J5 J6 J3 J2 J1 J6
J5 J4
J5 J1 J4 J6 J3 J2 J6 J5 J3 J4 J1 J2 J1 J3 J5 J2
J4 J6
Kid Jseq2
Kid Jseq1
To GT
To GT
59
Lin, Goodman Punch (1997)
Active Schedule
GA Chromosome

• Time Horizon Exchange Crossover (THX) Prior to
  scheduling time T, 1st parents scheduled ops
  are used to resolve conflicts. After T, use 2nd
  parent.

Yamada Nakano (1992) use similar method (GA
GT) but choose parent schedule randomly for each
conflict resolution (i.e. O selection)
60
Fang, Ross Corne (1993)
List of MxJ Job choices for MTBF (2 1 3 3 3 4 5 5
1 6 )
GA Chromosome
Mseq
Schedule next task of 5th unfinished job in the
circular job list
MTBF
Tseq
Active Schedule

• Gene Variance Based Operator Targeting (GVOT) -
  choose crossover and mutation
• points based on population variances of genes.
• GVOT biases X-over pt (high variance genes)
• Mutation (low variance genes)
• Standard Bitwise Crossover - Possible due to
  indirect representation

61
Indirect JSSP Representation (Fang, Ross Corne,
1993)

• Genotype for an J ( jobs) x M ( machines) JSSP
• JM chunks, where each chunk has log2(J) bits.
• 12321432443212.
• Take the next operation for the 2nd uncompleted
  job (so this isnt necessarily job 2) and put it
  into the earliest place that it will fit in the
  currently developing schedule.
• Indirect representation since the genes dont
  refer to absolute jobs
• 2232 -vs- 1142
• Assuming each job has more than 3 tasks, then
  these are interpreted as
• Schedule 3rd task of job 2 -vs- Schedule
  the 1st task of job 2
• The same position (4) has the same value (2), but
  it has a different meaning in the two genomes.
• Context sensitivity, epistasis genes interact in
  determining phenotypes

62
JSSP Benchmark Comparison

• All results are makespans from the best runs
  (red values are optimal)
• GA with specialized crossover approaches Branch
  Bound optimality
• (with much less searching)
• Indirect representations (Fang et. Al.) perform
  well, but slightly worse
• than most other direct representations

63
GA Search Efficiency

• Search Space Size (J!)M 3.96 x 1065 for the
  10 x 10 JSSP
• Can be larger depending upon the GA
  representation
• Typical GAs above evaluate around 150,000
  individuals
• E.g. Population size 1000 Generations
  150
• 1.5 x 105 / 3.96 x 1065 3.79 x 10-61
• fraction of search space that we need to
  explore to find a good solution when using GAs.
• Branch Bound techniques explore a significantly
  larger portion of the search space.

64
Search Spaces
Representation Space (Syntax)
Solution Space (Semantics)
Completeness Every point in SS is covered by a
point in RS Soundness Every point in RS maps to
a point in SS gt Anything generated during
RS search is a valid solution Uniqueness No 2
points in RS map to the same point in
SS. Translational Determinism No point in RS
maps to more than 1 point in SS gt The
translation of a representation gives the same
solution every time Heritability Children
solutions resemble their parents
65
Direct Representations

• Reps wherein genes encode absolute values such
  that the same spot on the genome encodes the same
  information, regardless of the values of other
  genes.
• If Position P A in both chromosomes C1 and C2,
  then, when converted into phenotypes, C1 and C2
  will derive the same characteristic from
  position P.
• Phenotypes can be independently read off the
  genome
• Genes that converge to particular alleles can
  only be changed via mutation
• - When the genome encodes a sequence of unique
  values, then crossover can create genomes with
  illegal interpretations (as phenotypes). Hence,
  an extra, often time-consuming, step is required
  to convert the genotypes to legal varieties, e.g.
  harmonization.

66
Indirect Representations

• Reps wherein genes encode relative values such
  that the absolute values of genes are dependent
  upon the values of other (e.g. Earlier) genes.
• All genotypes represent legal phenotypes.
• Easier to manipulate. Can perform syntactic
  operations (mutation ,crossover, etc.) without
  paying attention to the semantics gt crossover is
  cheap!
• - High epistatis (gene interactions) makes
  GA-based search more difficult. It works against
  the Building Block Hypothesis.
• - False Competition
• Two different gtypes may rep the same (or
  similar) ptypes, which will compete with one
  another for repro success, thus reducing the gain
  of each.
• - Low Inheritance via Crossover
• Two different genotypes produce similar
  phenotypes, but crossover produces children with
  vastly different phenotypes from the parents.
• 11122212 x 22211112 (cross after 3rd gene)
• Do 4th task of 1, then 4th task of 2
• 11111112 x 22222212
• Do 7th task of 1, then 1st task of 2 -vs- Do
  1st task of 1, then 7th task of 2

67
Summary

• When search spaces become too large for
  exhaustive (Branch Bound) or too complex for
  heuristic techniques, GAs can help.
• GA cannot guarantee optimality, but good
  solutions normally found, and search is very
  efficient.
• Choice of fitness function is straightforward for
  most GA scheduling applications
• Choice of representation is more difficult.
• One of reps presented above might be useful, but
• For special problems, a new rep may be needed
• If its a direct representation, then special
  mutation crossover ops also needed