970 IEEE TRANSACTIONS ON MAGNETICS, VOL. 44, NO. 6, JUNE 2008

Multiobjective Tabu Search Algorithms for OptimalDesign of Electromagnetic Devices

Sara Carcangiu, Alessandra Fanni, and Augusto Montisci

Electrical and Electronic Engineering Department, University of Cagliari, 09123 Cagliari, Italy

In this paper, an original algorithm to solve multiobjective optimization problems, which makes use of the tabu search meta-heuristic,is presented. Scalarization of the vector problem is performed by introducing fitness functions that take under control both the Paretooptimality of the solutions, and the uniformity in the Pareto front sampling. The performance of the proposed algorithm is comparedwith that of a scalar tabu search method, coupled with the -constraint strategy. The results on analytical and electromagnetic problemsdemonstrate the effectiveness of the method.

Index Terms—Multiobjective optimization, Pareto front, tabu search.

I. INTRODUCTION

I N the last decades, stochastic techniques to solve optimiza-tion problems with multiple conflicting objectives have been

widely investigated [1], most of which focused on evolutionaryalgorithms. These problems are called multiobjective optimiza-tion problems (MOPs), or vector optimization problems. In thefield of optimal design of electromagnetic devices, a wealth ofmultiobjective stochastic optimizers has been developed, basedon local or global search methods. In [2], the authors claimedthat evolutionary algorithms are often overshadowed by tabusearch (TS) methods. However, using these last methods, somescalarization techniques must be used to adapt the scalar schemaof the tabu search meta-heuristic to the vector MOP. Differentapproaches have been presented in the literature to solve MOPin the field of electromagnetic design, which make use of TSmeta-heuristics. In [3], a tabu search meta-heuristic [4], cou-pled with -constraint to handle the vector optimization, hasbeen compared with a multi-objective multi-individual evolu-tion strategy. The case study was the automated shape design ofa magnetic pole. In [2], a fitness sharing function, which pre-serves the diversity of the Pareto solutions, is introduced forscalarization of the TS schema.

In the present paper, the concept of fitness sharing functionintroduced in [2] is used to develop an innovative multiobjectiveTS algorithm.

The performance of the proposed Multiobjective-TS algo-rithm (MO-TS) has been compared with that of the -con-straint TS algorithm proposed in [3] on some analytical andelectromagnetic MOPs, whose Pareto front is available from theliterature.

II. MULTIOBJECTIVE OPTIMIZATION PROBLEM

A MOP is characterized by a vector of mutually conflictingobjective functions. This vector problem can be also seen asa constrained optimization problem where one of the quanti-ties is assumed as the function to be optimized, whereas theother quantities are defined as constraints of the optimization

Digital Object Identifier 10.1109/TMAG.2007.916336

problem. The two definitions are similar and all the quantitiesinvolved can be treated as objective functions of the MOP, ne-glecting their division into goals and constraints.

A MOP can be formalized as follows:

(1)

where is the vector of the input variables (design variables),and is the vector of objective functions . Here

represent the problem constraints. In a MOP, a solu-tion is dominated by another solution if is better than

on all objectives, and it will be denoted, here, by . Asolution is a Pareto optimal solution if no objective functioncan be improved without worsening at least one other objectivefunction. Such solution is not unique, and the set of the Paretooptimal solutions are know as the Pareto front. The performanceof a MOP solver can be evaluated on the basis of different cri-teria: capability of finding Pareto optimal solutions, capabilityof uniformly sampling the Pareto front, and limited computa-tional cost.

III. MULTIOBJECTIVE TABU SEARCH METHODS

In this paper, a fully vectorial multiobjective TS (MO-TS) ispresented.

Tabu search is a family of meta heuristic procedures, whichperform the search for the optimal solution by exploring thevariable space and storing the features that correspond to badprevious moves. Such features are labeled as tabu and they areavoided during the search for the optimum. The core blockof MOP tabu search algorithms follow a reactive scheme: thetabu list dimension (called tabu tenure TT) can be fixed orvariable. The algorithm has to be able to deeply explore aregion that looks promising, and to leave a region that does notlook promising. This is obtained by a dynamic management ofthe tabu tenure. That schema is inherently scalar, and differentscalarization strategies have been implemented and comparedin the present paper. In particular, the MO-TS performanceis compared with that of a single objective TS algorithmcoupled with the -constraint formulation of the multiobjectiveproblem [3].

0018-9464/$25.00 © 2008 IEEE

CARCANGIU et al.: MULTIOBJECTIVE TABU SEARCH ALGORITHMS FOR OPTIMAL DESIGN OF ELECTROMAGNETIC DEVICES 971

A. -constraint Tabu Search

In the -constraint method, the multiobjective optimization isreduced to a single-objective one by formalizing the MOP as inthe following:

subject to

(2)

where is the level of function prescribed to individual .In other words, for each function values are assigned toeach of the other functions, and single-objective con-strained optimizations are successively run.

B. MO-TS

In the proposed MO-TS, the scalarization of the vector op-timization problem is performed by assigning to each visitedsolution two fitness values as follows:

(3)

where, referring to the list of the Pareto optimal solutionspresently founded, is the number of such solutionsthat are dominated by the current solution isthe number of solutions dominating is the smallest distancebetween and the points of and are parameters thatdetermine the sampling step of the Pareto front. The first fitnessfunction allows to store a set of solutions that are not dominatedby those already present in the list. At each iteration, some so-lutions in the Pareto optimal list could become dominated bythe new solution, and then they have to be removed. The secondfitness function has two terms: the first one that favors the newpoints at distance from the nearest point of , and the secondone that penalizes the points that are very close to a point of .

IV. RESULTS

In order to show the performance of the proposed search algo-rithm, an analytical benchmark, constructed with the problem-generation technique suggested in [5], has been used. Moreover,the method has been used to find the optimal configuration oftwo electromagnetic devices test beds [6], [7].

All the tests in this section have been performed using anAMD XP 3000+ CPU with 512 MB of RAM running underWindows XP Professional.

A. Analytical Test

The first benchmark is an analytical two parameters-two ob-jectives problem [5], where the two cost functions to be mini-mized are

(4)where

(5)

Fig. 1. Analytical test problem: circles: Pareto front obtained with MO-TSmethod; squares: approximated Pareto front obtained with e-constraint method;line: analytically calculated Pareto front.

Fig. 2. Model of the magnetic pole.

The Pareto-optimal front is analytically available, and it isdescribed by , where is the globalminimum value of . The Pareto front obtained with the-constraint and MO-TS are shown in Fig. 1, together with the

analytically calculated Pareto front. As can be noted, both al-gorithms well fit the analytical Pareto front and the respectivesamples are uniformly distributed. Note that in Fig. 1, the differ-ence between the sampling steps in the right and left branchesof the Pareto front is due to the different ranges of the two ob-jective functions.

B. Optimal Electromagnetic Devices Design

1) Magnetic Pole: In this section, the optimal shape designof a magnetic pole is first considered [6]. In Fig. 2, the modelof the device is shown. Because of the symmetry with respectto the y-axis, only a half of the magnetic pole rectilinear sectionhas been modeled. The current density is uniform in the windingand is zero elsewhere. The nonlinear permeability of the ferro-magnetic material is taken into account. As far as the inverseproblem is concerned, four design variables(see Fig. 2) are selected. The feasible region of the design vari-ables is defined by the conditions of both geometric congruencyand nonsaturation of the material.

Two objective functions are defined:— is the maximum component of magnetic induction

in the y-axis direction along the air gap midline, to bemaximized;

— is the average component of magnetic induction inthe x-axis direction in the winding, to be minimized.

Both the proposed iterative procedures involve the solutionof a large number of time-consuming direct problems. For thisreason, a multilayer perceptron neural network (MLP) is trained

972 IEEE TRANSACTIONS ON MAGNETICS, VOL. 44, NO. 6, JUNE 2008

Fig. 3. Magnetic pole: approximated Pareto front.

to solve the direct problem and to calculate the objective func-tions. The MLP associates the objective function values to eachfeasible vector , by means of a set of examples. These trainingexamples are calculated by means of FEM simulations.

The optimization procedures described in the previous sec-tion have been compared in approximating the Pareto front ofthat case study. In Fig. 3, an approximation of the Pareto frontis shown with the same number of sample points (28 in the ex-ample) for each method.

In order to obtain the list of 28 Pareto optimal solutions,70 000 000 iterations are performed by the MO-TS with acomputation time of 15 min, whereas 60 000 000 iterationswith a computation time of 11 min are necessary to the -con-straint TS method. Note that, using -constraint TS method, themultiobjective optimization is reduced to a 60 single-objectiveproblems.

As can be noted, both algorithms are able to find uniformlysampled Pareto optimal solutions.

2) TEAM Workshop Problem 25: The problem consists inthe shape molds optimization of a die press with electromagnet[7] in order to obtain a radial magnetic flux distribution in thecavity region.

The design variables of the optimization problem are reportedin Fig. 4: the radius of the inner die, the length of theouter die, and the axes and of the ellipse. The desiredcomponents of the flux density and along the e-f line (acircular arc of 45 having radius of 11.75 mm) have to vary as

(6)

where is the angle from the x-axis.The objective function to be minimized (global error) is

(7)

where is the number of points along the e-f line, andare the prefixed and obtained values. The objective function

measures the homogeneity of the magnetic distribution.On the standard TEAM Problem 25, together with the value

of the global error (7), the obtained solutions are compared onthe basis of the following two values, which measure the localquality of the induction magnetic distribution:

Fig. 4. TEAM Workshop Problem 25: die press molds.

the maximum deviation on the amplitude (amplitude localerror), defined as

% (8)

and the maximum error on the angle of flux density vector(angle local error), defined as

(9)

In [8], it was proposed to solve the TEAM 25 as a multiob-jective problem, by assuming (7), (8), and (9) as conflicting ob-jectives. In that formulation, a solution is searched such that themagnetic induction distribution is as homogeneous as possible,the local deviation on the magnetic induction amplitude is min-imum, and the local deviation on the magnetic induction vectorangle is minimum.

Note that the design problem has not changed, but the op-timization problem has changed. In this paper, this MOP for-mulation has been used to test the proposed algorithm. Finiteelement analyses have been performed in order to have datasets to train, validate, and test the neural network used to solvethe direct problem and to calculate the three objective func-tions. As suggested in the benchmark problem, a bidimensionalmodel has been built and a magnetostatic nonlinear analysis hasbeen performed. The network model has four input nodes, corre-sponding to the four design parameters of the problem, and 20output nodes, which correspond to the difference between thecalculated values of and and the specified values in the10 distinct points in (7). Fig. 5 shows the approximated Paretofront obtained by means of MO-TS method. The Pareto front isthe same as reported in [8]. The same Pareto front has been ob-tained with -constraint TS method too, but the computationalcost is higher. To obtain a list P of 90 Pareto optimal solutions,400 000 000 iterations are performed using MO-TS with a com-putation time of 2.7 h, whereas, using -constraint TS method,the multiobjective optimization is reduced to a 1050 single-ob-jective problems, therefore 1 050 000 000 iterations with a com-putation time of 7.3 h are needed.

Note that, for -constraint TS the number of iterations de-pends on the number of samples in the Pareto front, and expo-

CARCANGIU et al.: MULTIOBJECTIVE TABU SEARCH ALGORITHMS FOR OPTIMAL DESIGN OF ELECTROMAGNETIC DEVICES 973

Fig. 5. TEAM Workshop Problem 25: circles: approximated Pareto front ob-tained with MO-TS method.

TABLE ITEAM WORKSHOP PROBLEM 25: COMPARISON BETWEEN GLOBAL ERROR

VALUES EVALUATED BY NN AND BY FEM MODELS

TABLE IITEAM WORKSHOP PROBLEM 25: RESULTS FROM OTHER AUTHORS

nentially increases with the number of objectives in the MOP,whereas for MO-TS that number only depends on the desiredsampling of the Pareto front.

Table I shows the global error calculated by the neuralnetwork model in ten points of the Pareto front obtained withMO-TS. The same Table I reports also the actual correspondingvalues, obtained by finite element analyses. In Table II, other

optimal results taken from literature are reported. As can benoted comparing Tables I and II, our values show a goodagreement with the configurations obtained by other authors.

V. CONCLUSION

Two different implementations of the tabu searchmeta-heuristic to solve MOPs have been presented. In order toreduce the computational cost of objective function evaluation,the FEM model is substituted with a neural model. The obtainedresults show that both the algorithms are able to sample theoptimal Pareto front with a negligible error. The uniformity ofthe sampled Pareto front is good for both algorithms, whereasMO-TS over performs -constraint in terms of computationalcost, as much as the number of objective functions of the MOPgrows.

REFERENCES

[1] K. Deb, A. Pratap, S. Agarwal, and T. Meyarivan, “A fast and elitistmultiobjective genetic algorithm: NSGA II,” IEEE Trans. Evol.Comput., vol. 6, no. 2, pp. 182–197, Apr. 2002.

[2] S. Y. Yang et al., “An improved tabu based vector optimal algorithm fordesign optimizations of electromagnetic devices,” IEEE Trans. Magn.,vol. 40, no. 2, pp. 1140–1143, Mar. 2004.

[3] S. Carcangiu, P. Di Barba, A. Fanni, M. E. Mognaschi, and A. Mon-tisci, “Comparison of multi-objective optimisation approaches for in-verse magnestostatic problems,” COMPEL, vol. 26, no. 2, pp. 293–305,2007.

[4] E. Cogotti, A. Fanni, and F. Pilo, “A comparison of optimization tech-niques for Loney’s solenoids design: An alternative tabu search ap-proach,” IEEE Trans. Magn., vol. 36, no. 4, pp. 1153–1157, Jul. 2000.

[5] K. Deb, “Multi-objective genetic algorithms: Problem difficultiesand construction of test problems,” Evol. Comput., vol. 7, no. 3, pp.205–230, Fall 1999.

[6] P. Di Barba and M. E. Mognaschi, “Recent experiences of multiob-jective optimisation in electromagnetics: A comparison of methods,”COMPEL, vol. 24, no. 3, pp. 921–930, 2005.

[7] N. Takahashi, “Optimization of die press model,” in Proc. TEAM Work-shop, Okayama, Japan, Mar. 20–21, 1996, pp. 61–69.

[8] L. Lebensztajn and J. L. Coulomb, “TEAM workshop problem 25:A multiobjective analysis,” IEEE Trans. Magn., vol. 40, no. 2, pp.1402–1405, Mar. 2004.

[9] P. Alotto and M. A. Nervi, “An efficient algorithm for the optimizationof problems with several local minima,” Int. J. Numer. Methods Eng.,no. 50, pp. 847–868, 2001.

[10] A. Canova, G. Gruosso, and M. Repetto, “Magnetic design optimiza-tion and objective function approximation,” IEEE Trans. Magn., vol.39, no. 5, pp. 2154–2162, Sep. 2003.

[11] N. Takahashi, K. Muramatsu, M. Natsumeda, K. Ohashi, K. Miyata,and K. Sayama, “Solution of problem 25 (Optimization of die pressmodel),” in Proc. ICEF’96, Hubei, China, Oct. 9–11, 1996, pp.383–386.

[12] D. Cherubini, A. Fanni, A. Montisci, and P. Testoni, “Inversion of MLPneural network for direct solution of inverse problems,” IEEE Trans.Magn., vol. 41, no. 5, pp. 1784–1787, May 2005.

Manuscript received June 24, 2007. Corresponding author: S. Carcangiu(e-mail: s.carcangiu@diee.unica.it).