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A survey of simulated annealing as a tool for single and multiobjective optimization

  • Review Paper
  • Published:
Journal of the Operational Research Society

Abstract

This paper presents a comprehensive review of simulated annealing (SA)-based optimization algorithms. SA-based algorithms solve single and multiobjective optimization problems, where a desired global minimum/maximum is hidden among many local minima/maxima. Three single objective optimization algorithms (SA, SA with tabu search and CSA) and five multiobjective optimization algorithms (SMOSA, UMOSA, PSA, WDMOSA and PDMOSA) based on SA have been presented. The algorithms are briefly discussed and are compared. The key step of SA is probability calculation, which involves building the annealing schedule. Annealing schedule is discussed briefly. Computational results and suggestions to improve the performance of SA-based multiobjective algorithms are presented. Finally, future research in the area of SA is suggested.

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Acknowledgements

The correspondence of Balram Suman with Kiran Mishra, graduate student, The State University of New Jersey, Rutgers and her valuable suggestions are gratefully acknowledged.

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Appendix

Appendix

Acronyms

ASA:

Adaptive Simulated Annealing

GA:

Genetic Algorithm

PDMOSA:

Pareto Dominant based Multiobjective Simulated Annealing

PSA:

Pareto Simulated Annealing

SA:

Simulated Annealing

CSA:

Chaotic Simulated Annealing

SMOSA:

Suppapitnarm Multiobjective Simulated Annealing

UMOSA:

Ulungu Multiobjective Simulated Annealing

WMOSA:

Weight based Multiobjective Simulated Annealing

Notation

a, b :

decision vector

B, C, C 1, C 2 :

constant

ΔE :

change in energy

F :

objective function vector

f :

objective function value

f max :

estimated maximum value of the objective function

Δf :

multidimensional criteria space

Δf 0 :

range of change in the value of the objective function

g :

inequality constraint violation

h :

equality constraint violation

I :

a column vector with all elements equal to 1.

i, j :

index

J :

number of constraints on the solution vector

K :

a constant

K B :

Boltzmann's constant

L :

set of uniform random weight vectors

L :

Chebysev norm

l :

number of equality constraint

m :

no of inequality constraint

M k :

objective function at eqilibrium

N :

number of objective functions

n :

number of decision variables

Prob :

probability

S(A):

spacing of a Pareto set A

ΔS′:

change in fitness value

Δs :

one-dimensional space

T :

annealing temperature (controlling parameter in the algorithm)

W :

weight vector

X :

current solution vector

Y :

generated solution vector

Z :

vector whose elements are objective function value ie z i after applying the penalty function approach

Z k :

Chaotic variable

Greek symbols

α :

a constant

β :

a constant

δ :

parameter for probability calculation

λ :

weight vector

σ 2 :

variance of the objective function at equilibrium

μ :

bifurcation parameter of the system

χ 0 :

defined as the number of accepted bad moves divided by the number of attempted bad moves

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Suman, B., Kumar, P. A survey of simulated annealing as a tool for single and multiobjective optimization. J Oper Res Soc 57, 1143–1160 (2006). https://doi.org/10.1057/palgrave.jors.2602068

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