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A novel fruit fly framework for multi-objective shape design of tubular linear synchronous motor

Abstract

This paper addresses a novel multi-objective fruit fly optimization algorithm (MOFOA) for solving multi-objective optimization problems. The essence of MOFOA lies in its having two characteristic features. For the first feature, a population of random fruit flies initializes the algorithm. During this initialization phase, the dominated fruit fly is replaced by the nearest non-dominated one. Subsequently, the fruit flies undergo evolution by flying randomly around the non-dominated solution or around the reference point, i.e., the best location of the individual objectives. Afterwards, the fruit flies are updated according to the nearest location whether from the reference point or the previous non-dominated location. For the second feature, the weighted sum method is incorporated to update the previous best locations of fruit flies and the reference point to emphasize the convergence of the non-dominated solutions. To prove the capability of the proposed MOFOA, two standard benchmark problems in addition to the real world application, namely, multi-objective shape design of tubular linear synchronous motor (TLSM) are checked. The corresponding TLSM objective functions aims to maximize operating force and to minimize the flux saturation. The outcomes clearly demonstrate the effectiveness of the proposed algorithm for finding the non-dominated solutions.

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Abbreviations

\(\mathbf{x}\) :

Decision variable vector

\(F(\mathbf{x})\) :

Objective function vector

\(g_i (\mathbf{x})\) :

\(i\hbox {th}\) inequality constraint

\(h_j (\mathbf{x})\) :

\(j\hbox {th}\) equality constraint

\(f_k (\mathbf{x})\) :

\(k\hbox {th}\) element of objective function vector

\(x_i^{L}\) :

Lower bound vector on the \( i\hbox {th}\) decision variable

\(x_i^{U}\) :

Upper bound on the \( i\hbox {th}\) decision variable

\(\Omega ,{\Omega }'\) :

Feasible regions

\({\mathbb R}\) :

Set of real numbers

\({\mathbb R}^{n}\) :

Set of n-dimensional real vectors

K :

Number of objective functions

m :

Number of inequality constraints

N :

Number of equality constraints

n :

Number of decision variables

\(\succ \) :

Worse than

\(k_{\mathrm{fill}}\) :

Fill factor

i :

Current in the conductor

ns :

Number of conductors

ins :

Slot current

\(\tau _p\) :

Pole pitch

p :

Number of poles

B :

Flux density

l :

Length of magnetic pole

\(h_{s} \) :

The height of the slot

\(h_{m} \) :

Height of the magnet

\(t_{s} \) :

Slot thickness

rand():

Random number generator between 0 and 1

PS:

The population size of fruit flies

EA :

Repositories to store the non-dominated solutions

RP :

Repository to store the value of best variable corresponding to the individual objective

R :

Search radius

\(\Delta (t,R)\) :

Step function returns a value in the range [0, R]

T :

Maximum number of iterations

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Correspondence to Gai-Ge Wang.

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Rizk-Allah, R.M., El-Sehiemy, R.A., Deb, S. et al. A novel fruit fly framework for multi-objective shape design of tubular linear synchronous motor. J Supercomput 73, 1235–1256 (2017). https://doi.org/10.1007/s11227-016-1806-8

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Keywords

  • Fruit fly optimization algorithm
  • Multi-objective optimization problem
  • Pareto-optimal solutions
  • Reference point
  • Tubular linear synchronous motor