1. General Purpose Procedures Applied to Scheduling
Contents
Constructive approach
1. Dispatching Rules
Local search
1. Simulated Annealing
2. Tabu-Search
3. Genetic Algorithms

2. Constructive procedures:
1. Dispatching Rules
2. Composite Dispatching Rules
3. Dynamic Programming
4. Integer Programming
5. Branch and Bound
6. Beam Search
Local Search
1. Simulated Annealing
2. Tabu-Search
3. Genetic Algorithms
Heuristic technique is a method which seeks good (i.e. near-optimal solutions) at a reasonable cost without being able to guarantee optimality.

3. Dispatching Rules
A dispatching rule prioritises all the jobs that are waiting for processing on a machine.
Classification
Static: not time-dependent
Dynamic: time dependent
Local: uses information about the queue where the job is waiting or machine where the job is queued
Global: uses information about other machines (e.g. processing time of the jobs on the next machine on its route, or the current queue length

5. Local Search Step. 1. Initialisation k=0
Select a starting solution S0S
Record the current best-known solution by setting Sbest = S0 and best_cost = F(Sbest)
Step 2. Choice and Update
Choose a Solution Sk+1N(Sk) If the choice criteria cannot be satisfied by any member of N(Sk), then the algorithm stops
if F(Sk+1) < best_cost then Sbest = Sk+1 and best_cost = F(Sk+1)
Step 3. Termination
If termination conditions apply
then the algorithm stops
else k = k+1 and go to Step 2.

6. Global Optimum: better than all other solutions
Local Optimum: better than all solutions in a certain neighbourhood

7. 1. Schedule representation
2. Neighbourhood design
3. Search process
4. Acceptance-rejection criterion
1. Schedule representation
Nonpreemptive single machine schedule
permutation of n jobs
Nonpreemptive job shop schedule
m consecutive strings, each representing a permutation of n operations on a machine

8. 2. Neighbourhood design
Single machine:
adjacent pairwise interchange
take an arbitrary job in the schedule and insert it in another positions
Job shop:
interchange a pair of adjacent operations on the critical path of the schedule
one-step look-back interchange

9. current schedule
(h, l)
(h, k)
machine h
(i, j)
(i, k)
machine i
schedule after interchange of (i, j) and (i, k)
(h, l)
(h, k)
machine h
(i, k)
(i, j)
machine i
schedule after interchange of (h, l) and (h, k)
(h, k)
(h, l)
machine h
(i, k)
(i, j)
machine i

10. 3. Search process
select schedules randomly
select first schedules that appear promising for example, swap jobs that affect the objective the most
4. Acceptance-rejection criterion
probabilistic: simulated annealing
deterministic: tabu-search

11. Simulated Annealing Contents 1. Basic Concepts 2. Algorithm
3. Practical considerations

12. Basic Concepts
Allows moves to inferior solutions in order not to get stuck in a poor local optimum.
c = F(Snew) - F(Sold) F has to be minimized
inferior solution (c > 0) still accepted if
U is a random number from (0, 1) interval
t is a cooling parameter: t is initially high - many moves are accepted
t is decreasing - inferior moves are nearly always rejected
As the temperature decreases, the probability of accepting worse moves decreases.
c > 0 inferior solution
-c < 0
t  

13. Algorithm
Step 1.
k=1
Select an initial schedule S1 using some heuristic and set Sbest = S1
Select an initial temperature t0 > 0 Select a temperature reduction function (t)
Step 2.
Select ScN(Sk)
If F(Sbest) < F(Sc)
If F(Sc) < F(Sk) then Sk+1 = Sc
else
generate a random uniform number Uk
If Uk < then Sk+1 = Sc
else Sk+1 = Sk
else Sbest = Sc
Sk+1 = Sc

14. Step 3.
tk = (t)
k = k+1 ;
If stopping condition = true then STOP
else go to Step 2

15. Exercise.
Consider the following scheduling problem 1 | dj | wjTj .
Apply the simulated annealing to the problem starting out with the 3, 1, 4, 2 as an initial sequence.
Neighbourhood: all schedules that can be obtained through adjacent pairwise interchanges.
Select neighbours within the neigbourhood at random. Choose (t) = 0.9 * t
t0 = 0.9
Use the following numbers as random numbers: 0.17, 0.91, ...

16. Sbest = S1 = 3, 1, 4, 2
F(S1) = wjTj = 1·7 + 14· ·0+ 12 ·25 = 461 = F(Sbest)
t0 = 0.9
Sc = 1, 3, 4, 2
F(Sc) = 316 < F(Sbest)
Sbest = 1, 3, 4, 2
F(Sbest) = 316
S2 = 1, 3, 4, 2
t = 0.9 · 0.9 = 0.81
Sc = 1, 3, 2, 4
F(Sc) = 340 > F(Sbest)
U1 = 0.17 > = 1.35*10-13
S3= 1, 3, 4, 2
t = 0.729

17. Sc = 1, 4, 3, 2
F(Sc) = 319 > F(Sbest)
U3 = 0.91 > = 0.016
S4= S4 = 1, 3, 4, 2
t =
...

18. Practical considerations
Initial temperature
must be "high"
acceptance rate: 40%-60% seems to give good results in many situations
Cooling schedule
a number of moves at each temperature
one move at each temperature
t =  ·t  is typically in the interval [0.9, 0.99]
 is typically close to 0
Stopping condition
given number of iterations
no improvement has been obtained for a given number of iteration
Automated Scheduling, School of Computer Science and IT, University of Nottingham

19. Tabu Search Contents 1. Basic Concepts 2. Algorithm
3. Practical considerations
Automated Scheduling, School of Computer Science and IT, University of Nottingham

20. Basic Concepts Algorithm
Tabu-lists contains moves which have been made in the recent past but are forbidden for a certain number of iterations.
Algorithm
Step 1.
k=1
Select an initial schedule S1 using some heuristic and set Sbest = S1
Step 2.
Select ScN(Sk)
If the move Sk  Sc is prohibited by a move on the tabu-list
then go to Step 2
Modern Heuristic Search Methods, (Eds.) V.J. Rayward-Smith, I.H.Osman, C.R. Reeves, G.D. Smith, John Wiley & Sons Ltd. 1996
Automated Scheduling, School of Computer Science and IT, University of Nottingham

21. If the move Sk  Sc is not prohibited by a move on the tabu-list
then Sk+1 = Sc
Enter reverse move at the top of the tabu-list
Push all other entries in the tabu-list one position down
Delete the entry at the bottom of the tabu-list
If F(Sc) < F(Sbest) then Sbest = Sc
Go to Step 3.
Step 3.
k = k+1 ;
If stopping condition = true then STOP
else go to Step 2
Automated Scheduling, School of Computer Science and IT, University of Nottingham

22. F(S1) = wjTj = 12·8 + 14·16 + 12·12 + 1 ·36 = 500 = F(Sbest)
Example.
1 | dj | wjTj
Neighbourhood: all schedules that can be obtained through adjacent pairwise interchanges.
Tabu-list: pairs of jobs (j, k) that were swapped within the last two moves
S1 = 2, 1, 4, 3
F(S1) = wjTj = 12·8 + 14· · ·36 = 500 = F(Sbest)
F(1, 2, 4, 3) = 480
F(2, 4, 1, 3) = 436 = F(Sbest)
F(2, 1, 3, 4) = 652
Tabu-list: { (1, 4) }
Automated Scheduling, School of Computer Science and IT, University of Nottingham

23. S2 = 2, 4, 1, 3, F(S2) = 436
F(4, 2, 1, 3) = 460
F(2, 1, 4, 3) (= 500) tabu!
F(2, 4, 3, 1) = 608
Tabu-list: { (2, 4), (1, 4) }
S3 = 4, 2, 1, 3, F(S3) = 460
F(2, 4, 1, 3) (= 436) tabu!
F(4, 1, 2, 3) = 440
F(4, 2, 3, 1) = 632
Tabu-list: { (2, 1), (2, 4) }
S4 = 4, 1, 2, 3, F(S4) = 440
F(1, 4, 2, 3) = 408 = F(Sbest)
F(4, 2, 1, 3) (= 460) tabu!
F(4, 1, 3, 2) = 586
Tabu-list: { (4, 1), (2, 4) }
F(Sbest)= 408
Automated Scheduling, School of Computer Science and IT, University of Nottingham

24. Practical considerations
Tabu tenure: the length of time t for which a move is forbiden
t too small - risk of cycling
t too large - may restrict the search too much
t=7 has often been found sufficient to prevent cycling
Number of tabu moves: 5 - 9
If a tabu move is smaller than the aspiration level then we accept the move
Modern Heuristic Search Methods, (Ed) V.J. Rayward-Smith, I.H.Osman. C.R.Reeves and g Smith, 1996, John Wiley & Sons Ltd. Chapter 1.
Summary for SA and TS
How to Solve It:Modern Heuristics, Z.Michalewicz, D.Fogel, Springer, 2000, p 134
Automated Scheduling, School of Computer Science and IT, University of Nottingham

25. Genetic Algorithms Contents 1. Basic Concepts 2. Algorithm
3. Practical considerations

26. individuals surviving from the previous generation
Basic Concepts
Simulated Annealing Tabu Search
versus
Genetic Algorithms
a single solution is carried over from one iteration to the next
population based method
Individuals (or members of population or chromosomes)
individuals surviving from the previous generation +
children
generation

27. Fitness of an individual (a schedule) is measured by the value of the associated objective function
Representation
Example.
the order of jobs to be processed can be represented as a permutation: [1, 2, ... ,n]
Initialisation
How to choose initial individuals?
High-quality solutions obtained from another heuristic technique can help a genetic algorithm to find better solutions more quickly than it can from a random start.

28. Reproduction
Crossover: combine the sequence of operations on one machine in one parent schedule with a sequence of operations on another machine in another parent.
Example 1. Ordinary crossover operator is not useful!
Cut Point
P1 = [ ]
P2 = [ ]
O1 = [ ]
O2 = [ ]
Example 2. Partially Mapped Crossover
Cut Point 1
Cut Point 2
31
42
55
P1 = [ ]
P2 = [ ]
O1 = [ ]
O2 = [ ]

29. Example 3. Preserves the absolute positions of the jobs taken from P1 and the relative positions of those from P2
Cut Point 1
P1 = [ ]
P2 = [ ]
O1 = [ ]
O2 = [ ]
Example 4. Similar to Example 3 but with 2 crossover points.
Cut Point 1
Cut Point 2
P1 = [ ]
P2 = [ ]
O1 = [ ]

30. Mutation enables genetic algorithm to explore the search space
Mutation enables genetic algorithm to explore the search space not reachable by the crossover operator.
Adjacent pairwise interchange in the sequence
[1,2, ... ,n]
[2,1, ... ,n]
Exchange mutation: the interchange of two randomly chosen elements of the permutation
Shift mutation: the movement of a randomly chosen element a random number of places to the left or right
Scramble sublist mutation: choose two points on the string in random and randomly permuting the elements between these two positions.

31. Selection
Roulette wheel: the size of each slice corresponds to the fitness of the appropriate individual.
slice for the 1st individual
slice for the 2nd individual
selected individual .
Steps for the roulette wheel
1. Sum the fitnesses of all the population members, TF
2. Generate a random number m, between 0 and TF
3. Return the first population member whose fitness added to the preceding population members is greater than or equal to m

32. Tournament selection
1. Randomly choose a group of T individuals from the population.
2. Select the best one.
How to guarantee that the best member of a population will survive?
Elitist model: the best member of the current population is set to be a member of the next.

33. Algorithm
Step 1.
k=1
Select N initial schedules S1,1 ,... , S1,N using some heuristic
Evaluate each individual of the population
Step 2.
Create new individuals by mating individuals in the current population using crossover and mutation
Delete members of the existing population to make place for the new members
Evaluate the new members and insert them into the population
Sk+1,1 ,... , Sk+1,N
Step 3.
k = k+1
If stopping condition = true then return the best individual as the solution and STOP
else go to Step 2

34. Example
1 || Tj
Population size: 3
Selection: in each generation the single most fit individual reproduces using adjacent pairwise interchange chosen at random there are 4 possible children, each is chosen with probability 1/4 Duplication of children is permitted. Children can duplicate other members of the population.
Initial population: random permutation sequences

35. Generation 1
Individual
Cost
Selected individual: with offspring 13245, cost 20
Generation 2
Individual
Cost
Average fitness is improved, diversity is preserved
Selected individual: with offspring 12354, cost 17
Generation 3
Individual
Cost
Selected individual: with offspring 12435, cost 11

36. Generation 4
Individual
Cost
Selected individual: 12435
This is an optimal solution.
Disadvantages of this algorithm:
Since only the most fit member is allowed to reproduce (or be mutated) the same member will continue to reproduce unless replaced by a superior child.

37. Practical considerations
Population size: small population run the risk of seriously under-covering the solution space, while large populations will require computational resources. Empirical results suggest that population sizes around 30 are adequate in many cases, but are more common.
Mutation is usually employed with a very low probability.

38. Summary Meta-heuristic methods are designed to escape local optima.
They work on complete solutions. However, they introduce parameters (such as temperature, rate of reduction of the temperature, memory, ...) How to choose the parameters?
Other metaheuristics
Ant optimization
GRASP