1. Optimization with Meta-Heuristics
Question: Can you ever prove that a solution generated using a meta-heuristic is optimal?
Answer: Yes, for a minimization problem, if the value of the solution equals a lower bound.
Question: If the solution of a meta-heuristic for a minimization problem does not equal the lower bound, does that mean the solution is not optimal?
Answer: Not necessarily, you just don’t know.
Observation: Developing a good lower bound just as important as developing a good meta-heuristic algorithm.

2. DOE for Meta-Heuristics
Question: With all the seeming randomness, and choices of neighborhoods and algorithm parameters, how do you know you have developed a good approach or not?
Answer: Design of Experiments

3. DOE for Meta-Heuristics
Recall the classic Johnson, et al. simulated annealing algorithm:
1. Get an initial solution S.
2. Get an initial temperature T > 0.
3. While not yet frozen do the following:
3.1 Perform the following loop l time.
3.1.1 Pick a random neighbor S’ of S.
3.1.2 Let D = cost(S’) – cost(S)
3.1.3 If D <= 0 (downhill move), Set S = S’.
3.1.4 If D > 0 (uphill move), Set S = S’ with probability e-D/T.
3.2 Set T = rT (reduce temperature).
4. Return S.
What are potential experimental parameters?

4. DOE for Meta-Heuristics
Design parameters for simulated annealing algorithm include:
Problem instances
Cooling approach
Starting temperature
Number of iterations at temperature
Temperature reduction rate
Termination condition
Variance from Johnson’s classic algorithm
Neighborhoods
Acceptance probability function

5. DOE for Meta-Heuristics
Obtaining Problem Instances:
Benchmark problems
many others
Problem generator
How many problems
Size of problem
Problem characteristics (unique for different problem types)

6.
Shop Scheduling Benchmark Problems
General Information
C Programs For Problem Generation
Parameter Values For Problem Generation
J//Cmax Problems
J//Lmax Problems
J/2SETS/Cmax Problems
J/2SETS/Lmax Problems
F//Cmax Problems
F//Lmax Problems

7. DOE for Meta-Heuristics
How to report results:
Must evaluate to something (solution value – lower bound)
Compare solution versus run time
Compare over some problem generation parameter (due date range)

8. DOE for Meta-Heuristics
How to report results: different sized problem instances

9. DOE for Meta-Heuristics
How to report results: Comparison to benchmark problems
VFSA: K=4,S=1.5,B=0.8
Problem Name LB
Previous UB (from Uzsoy)
UB (best known from Balas)
Balas
CPU sec.
After
5 min
30 min
60 min
120 min
Best VFSA
r_20_15_1_1_2
1140
1464
1263
175
1299
1275
1244
r_20_15_1_1_3
1182
1501
1304 98
1271
1268
1266
1258
r_20_15_1_1_4
1160
1492
1396
127
1367
1332
1326
r_20_15_1_1_6
1027
1448
145
1229
1222
1208
r_20_15_1_1_8
1127
1552
1459
135
1449
1417
1388
r_20_15_1_2_1
1721
2090
1817 85
1818
r_20_15_1_2_10
1775
2092
1873 39
r_20_15_1_2_5
1925
2181
1949 37
1930
r_20_15_1_2_8
1599
1785
1636 15
r_20_15_1_2_9
1956
2246
2020
r_20_15_2_1_1
2165
2000 11
2014
1972
1970
1963
1960
r_20_15_2_1_3
1727
2100
1976
131
1971
1918
1901
1900
1898
r_20_15_2_1_5
1521
1839
1726
121
1697
1690
1684
1683
r_20_15_2_1_7
1575
1957
1908
118
1846
1830
1824
r_20_15_2_1_9
1858
2143
1968
110
1955
1929
1914
1913

10. DOE for Meta-Heuristics
How to report results:

11. DOE for Meta-Heuristics
How to report results:

12. DOE for Meta-Heuristics
In class assignment: Develop a DOE for the traveling salesman problem.