Gene Adaptations
Step 4) RETURN TO BASE
Return to an uncrowded base point periodically (so as to generate a more continuous Pareto set, by locally exploring around uncrowded points in the non-dominated set)
First return to base (selected an uncrowded point; see later) is done after NB,1 [= NT,1] iterations. Thereafter, the return to base is after NB,j
iterations, j = 2, 3, . . . :
NB,2 = 2NT,2; NB,j = rB NB,j - 1; j = 3, 4, . . .
0 ≤ rB ≤ 1 [rB = 0.9] s.t.: NB,j ≥ 10
Generate File 3 (of uncrowded members in File 2)
Normalize Ii (using max and min values) for all AS non-dominated
archived points in File 2, so that 0 ≤ Ii ≤ 1
Find the crowding distances (Idist,i) of all AS points in File 2 (as in
NSGA-II)
Copy Aj of the most uncrowded points (including the boundary points) into File 3, where
Aj = Φj AS; Φ1 = 1; Φj+1 = rΦj; r = 0.9; Aj ≥ AMIN = 6
Use a random number (digitized appropriately) to select the uncrowded base point to be returned to, from File 3
Go to Step 2 (giving a perturbation to this selected uncrowded point) ---
Comments
Since File 2 contains all the non-dominated solutions till the current iteration, there is no concept of elitism in MOSA. Also, Step A in Step 2 is omitted in MOSA. For SSA, Steps A, B (in Step 2) and 4 are omitted.
MOGA and MOSA with JG Adaptations 125
Fig. 4.A 1 Flowchart for MOSA-JG/aJG START
Generate a feasible point, x(0), initialize S(0), TSTART (large), k, l = 0
Randomly perturb x(k) to give a feasible x(k + 1)
Evaluate Ij (x(k + 1)); j = 1, 2, . . . . , m Check for archiving (non-dominance)
Calculate the probability of acceptance
Is x (k + 1) to be accepted? Accept x(k + 1)
Periodically, reduce T/return to base NO Is x(k + 1) to be archived? YES YES NO Check for stopping YES Output archive contents
STOP NO
k = k + 1
Do jumping gene operation to give a feasible x(k+1)
Check annealing schedule and return to base
YES NO
Nomenclature
AMIN lower limit on the number of archived solutions taken in the return-to-base operation
B parameter for temperature in the annealing schedule C parameter used to call the annealing schedule fb fixed length of the JG
Ii i
th
objective function
lchr length of chromosome
lstring,i number of binaries used to represent the i
th
decision variable
m number of objective functions
N number of accepted solutions after initial return-to-base NB,i number of iterations to be performed for call to return-to-base
Nd number of decision variables in SA
Ngen number of generations
Ngen,max maximum number of generations
Np population size
n number of decision variables in GA
Nseed random seed
NT,i number of iterations to be performed before reducing the temperature
PaJG probability of carrying out the aJG operation
Pcross probability of carrying out the crossover operation
P11…1 probability for changing all binaries of a selected decision
variable to zero
PJG probability of carrying out the JG operation PmJG probability of carrying out the mJG operation Pmut probability of carrying out the mutation operation PsJG probability of carrying out the sJG operation
PsaJG probability of carrying out the saJG operation
rB return-to-base parameter
rI parameter to update Φi
R random number
S(k) Nd-dimensional vector of step sizes in the k th
acceptance
T m-dimensional vector of computational temperatures in MOSA
U Nd-dimensional vector of normalized decision variables, Ui x vector of decision variables, xi
MOGA and MOSA with JG Adaptations 127
Greek symbols
αi Temperature reducing parameter
σi Standard deviation of accepted solutions
Φ1 Fraction of archived solutions taken for the return-to-base operation in the first call
∆f Difference in new and old values of the objectivefunction
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Exercises
1. Solve the ZDT4 Problem 1 using NSGA-II-aJG and MOSA-aJG (the two codes in the CD give the solutions).
2. Modify the codes in the CD to solve the ZDT2 Problem 2 and the ZDT3 Problem 3, using NSGA-II-aJG and MOSA-aJG.
August 25, 2008 19:30 World Scientific Book - 9in x 6in book
Chapter 5