Multi-objective optimization using Trigonometric mutation multi-objective differential evolution algorithm
Ashish M Gujarathia, Ankita Lohumi, Mansi Mishra, Digvijay Sharma, B. V. Babub* a
Lecturer, Department of Chemical Engineering Birla Institute of Technology and Science (BITS), Pilani (Rajasthan) India; Email:
[email protected] b
Dean- Educational Hardware Division & Professor of Chemical Engineering Birla Institute of Technology and Science (BITS), PILANI – 333 031 (Rajasthan) India Phone: +91-01596-245073 Ext. 259; Fax: +91-01596-244183 Email:
[email protected] Homepage: http://discovery.bits-pilani.ac.in/~bvbabu * Corresponding Author: Abstract: In this study a trigonometric mutation strategy of multi-objective differential algorithm (MODE) is used successfully on selected test problems. The trigonometric mutation strategy consists of working principle of MODE-III algorithm with probability based an enhanced mutation strategy. The developed algorithm is also applied successfully on multi-objective optimization of benchmark test problems the Pareto optimal fronts are obtained. The important metrics such as the convergence and divergence metric are reported. The convergence and divergence history (with respect to number of generations) is reported and is compared with the other strategy of MODE. It is found that when a high value of trigonometric mutation probability (TMP) constant is used, the algorithm converges to a local region. However, a small value of TMP constant ensures that the algorithm converge to the true Pareto front at a faster rate. Such algorithm is useful in solving engineering optimization problems.
Keywords: Multi-objective optimization; Evolutionary algorithms; Multiobjective differential evolution (MODE); Trigonometric mutation
Introduction When more than one objective problem is solved simultaneously it results in a set of solutions rather than a single point solution. Such solution technique is called as the multi-objective optimization. The speciality of these set of solutions is that they are equally good with respect to each other and are called as the non-dominated set. An efficient algorithm may result in a Pareto optimal set of solutions which is the set of non-dominated solutions. Several algorithms such as Non-dominated sorting genetic algorithm (NSGA) and its improved variants, Multi-objective simulated annealing (MOSA), Multi-Objective Differential Evolution (MODE) and its other improved strategies, Strength Pareto Evolutionary Algorithms (SPEA), Pareto Archived Evolutionary Strategy (PAES), etc. [1, 2] are developed to solve multi-objective optimization problems. In this work, trigonometric mutation operation is successfully incorporated in evolutionary multi-objective differential evolution algorithm. Simple mutation operation involves the random selection of first individual vector out of randomly selected three vectors. The scaled difference is added to this randomly selected individual. In case of trigonometric mutation operation, the center point of the hyper geometric point is taken as the vector to be perturbed. The developed algorithm is tested on various test problems and the Pareto optimal fronts are obtained. The important metrics such as the convergence and divergence metric are reported. The convergence and divergence history (with respect to number of generations) is reported and is compared with the other strategy of MODE. It is found that when a high value of trigonometric mutation probability (TMP) constant is used, the algorithm converges to a local region. However, a small value of TMP constant ensures that the algorithm converge to the true Pareto front at a faster rate. Such algorithm is useful in solving engineering optimization problems. The manuscript is organized in the following manner. A brief working principle of trigonometric MODE is give in next section, followed by problem formulation of test problems, the results and discussions and the conclusions. Trigonometric mutation differential Evolution algorithm [3] The mutation operation is carried out in MODE [4] and MODE-III [4] is based on simple differential evolution algorithm [6 - 8], where three distinct vectors are used. These vectors are perturbed by applying a scale factor so that hopefully an efficient vector (noisy random vector) is created [6]. However, the noisy random vector thus created does not always guaranty for direction towards the better function value. In this study, mutation is carried out using trigonometric mutation operation [9-10 concept to the selected vectors. Earlier trigonometric mutation operation was applied to differential evolution algorithm to solve single objective optimization problems [9-10]. In this work we apply trigonometric mutation operation on MODE III algorithm to solve multi-objective optimization problems. The pseudo-code of trigonometric mutation operation carried out in trigonometric MODE algorithm may be represented as below. For detailed discussion on trigonometric mutation, readers may refer literature [9-10]: **Pseudo-code for trigonometric mutation operation for MOO Select three distinct vectors Xa, Xb and Xc other than Xi do {
r1=round (rand*NP) r2=round (rand*NP) r3=round (rand*NP) } while [(r2== i) || (r1==i) ||(r3== i) ||(r1== r2 ) ||(r1== r3) ||(r2== r3)] If (rand(0,1) >Mt) Tempp1=f(Xr1); Tempp2=f(Xr2);; Tempp3=f(Xr3); Sum= |Tempp1|+| Tempp1|+| Tempp1| p1= |Tempp1|/sum; p2= |Tempp2|/sum; p3= |Tempp3|/sum; for j=1:D Xt,i,j = (Xr1(1,j)+X r2(1,j) + X r3(1,j))/3 + abs(p2-p1)*(X r1 (1,j) - X r2 (1,j)) + abs(p3-p2)*(X r2(1,j) - Xr3(j)) + abs(p1-p3)*(X r3 (1,j)-X r1(1,j)); end for else Xt,i=Xr1 + F (Xr2-Xr3) %Trial vector End Simple mutation operation involves the random selection of first individual vector out of the randomly selected three vectors. The weighted difference of the two vectors is added to the third randomly selected individual. In case of trigonometric mutation operation, the center point of the hyper geometric point is taken as the vector to be perturbed. As seen from the pseudo-code of trigonometric mutation operation for MOO, the perturbation in the trigonometric mutation operation is contributed together by three vertices of the triangle defined by three randomly selected vectors. The weights applied to the vectors differentials i.e. (p2-p1), (p3-p2), and (p1-p3) ensures that the new point moves in the direction of improved objective function value. In case of single objective optimization, Tempp1, Tempp2 and Tempp3 (as given in pseudo-code above) are the variables which contain a single objective function value. However, in case of multi-objective optimization Tempp1, Tempp2 and Tempp3 are the vectors which contain the values of evaluated multiple objective functions. The sum is also a vector which contains the sum of individual objective functions. The mean value of all the objectives is calculated as shown by p1, p2 and p3 variables. The noisy random vector is now created by using the p1, p2 and p3 variables as weight as given in above pseudo-code. It is ensured that the noisy random vector moves towards a better direction where there is an improvement in the objective function value. The mutation probability used in this algorithm is 0.5%, i.e., if the random number generated is greater than 0.5 then trigonometric mutation operation is carried out otherwise simple mutation operation as given in MODE III algorithm is used [3].
Problem formulation of test problems Three benchmark test problems are considered namely SCH1, SCH2 and KUR (Table 1) for present study. Two important performance measures namely convergence metric and divergence metric [1] are used to evaluate the performance of trigonometric MODE algorithm. Table 1. MOO test problems used in the present study Sr. Test No. Problem
Problem Type
1
SCH-1 (Schaffer, 1984)
Min-Min Convex
FON (Fonseca and Fleming, 1995)
Min-Min Nonconvex
KUR (Kursawe, 1990)
Min-Min Nonconvex Disconnected
2
3
Objective Functions
Number variables and Bounds n =1
f ( x) = x 2 1
10 − 3 ≤ x ≤ 103
f ( x) = ( x − 2)2 2
f ( x) = 1 − exp − 1
f ( x) = 1 f ( x) = 2
i =1 n i =1
n
1
x − i n i =1
f ( x) = 1 − exp − 2 n −1
n
1 x + i n i =1
2
0.8
n=3 −4≤ x ≤ 4 i
2
− 10 exp 0.2 x 2 + x 2 i i +1 x i
of
n=3 −5≤ x ≤ 5 i
+ 5 sin x3 i
Results and Discussion
Three benchmark test problems are used to judge the performance of algorithm. Convergence and divergence metrics are evaluated and are compared with other well known algorithms from the literature. Table 2 shows the mean (γ) variance ( σ2γ) of convergence and mean and variance of divergence metrics respectively using MODE-III and trigonometric MODE algorithm. The mean and variance values of other algorithms (such as NSGA-II (both real- and binary-coded variants), SPEA, PAES) are also shown in the same table which are taken from the literature.
1.0
4.5 4.0
MODE-III Trigonometric MODE Real Coded NSGA-II True Pareto
3.5 3.0
0.8
0.6
f2
f2
2.5 2.0
0.4
1.5 1.0
MODE III Trigonometric MODE Real coded NSGA III True Pareto
0.2
0.5
0.0
0.0 -0.5 0
1
2
3
0.0
4
0.2
0.4
0.6
0.8
1.0
f1
f1
Fig. (a)
Fig. (b)
0 -2
f2
-4 -6 -8
MODE III Trigonometric MODE Real coded NSGA II True Pareto
-10 -12 -20
-18
-16
-14
f1
Fig. (c) Fig. 1. Pareto fronts obtained using MODE III, Trigonometric MODE, and Real Coded NSGA II and True Pareto fronts for (a) SCH, (b) FON and (c) KUR Test problems
All the simulation runs are taken with same value of control parameters as reported in the literature [1]. The Pareto fronts obtained using MODE III, Trigonometric MODE, Real Coded NSGA II are plotted along with true Pareto fronts for SCH, FON and KUR test problems and are shown in Fig. 1. Results in Table 2 show that results obtained using trigonometric MODE algorithm are comparable in terms of mean and variance of convergence metric, however, divergence metric is higher than NSGA-II strategies.
Table 2. Convergence and divergence metrics obtained using several algorithms Algorithm Real coded NSGA-II~
SCH FON 0.003391 0.001931 γ 2 0 0 σγ 0.477899 0.378065 ∆ 2 0.003471 0.000639 σ∆ ~ 0.002833 0.002571 NSGA-II γ 0.000001 0 Binary σ2γ 0.449265 0.395131 ∆ 2 0.002062 0.001314 σ∆ ~ 0.003465 0.010611 SPEA γ 2 0 0.000005 σγ 0.818346 0.804113 ∆ 2 0.004497 0.002961 σ∆ 0.001313 0.151263 PAES~ γ 0.000003 0.000905 σ2γ 1.063288 1.162528 ∆ 2 0.002868 0.008945 σ∆ 0.0021 0.02554 MODE* γ 2 0 0.00063 σγ 0.67099 0.70069 ∆ 2 0.01332 0.03397 σ∆ 0.002236 0.003381 MODE III* γ 2 0 0 σγ 0.59953 0.620052 ∆ 2 0.00155 0.00095 σ∆ * 0.002035 0.003151 Trigonometric MODE γ 0 0 σ2γ 0.596636 0.612571 ∆ 2 0.000926 0.002878 σ∆ ~ *Results obtained in the present study; Results reported in Ref [1]
KUR 0.028964 0.000018 0.411477 0.000992 0.028951 0.000016 0.442195 0.001498 0.049077 0.000081 0.880424 0.009066 0.057323 0.011989 1.079838 0.013772 0.03837 0.00057 0.82097 0.0053 0.003028 0 0.671036 0.00192 0.00665 0.000009 0.661016 0.001459
Though the algorithm is able to converge to the true Pareto front, diversity of solutions obtained on the Pareto front needs further attention. The Pareto fronts obtained using MODE-III and trigonometric MODE algorithms for SCH1, FON and KUR test problems is shown through Fig. 1a-1c respectively. In the figures, true Pareto front and results obtained using real-coded NSGAII are also plotted. The source code for real-coded NSGA-II is downloaded from KanGAL (http://www.iitk.ac.in/kangal/codes.shtml). A value of TMP is set to 0.5. It is observed that if only trigonometric mutation operation is used, the algorithm converges to a local Pareto front. Therefore it is ensured that simple mutation strategy of MODE-III and trigonometric mutation strategy works with equal probability (i.e. 50%).
Conclusions
The working principle of trigonometric mutation strategy of MODE algorithm is presented in this paper. Three test problems (SCH, FON, and KUR) are considered for MOO. Pareto fronts are obtained fusing MODE III, trigonometric MODE and real-coded NSGA-II algorithms and are compared with true Pareto fronts. Important performance metrics (such as convergence and divergence metrics) are evaluated for both the strategies of MODE and are compared with other algorithms such as NSGA II (real- and binary –coded variants), SPEA and PAES. It is observed that convergence of strategies of MODE are comparable as compared to other algorithms, however, diversity of solutions is relatively poor. References:
1.
Deb, K. (2001) Multi-Objective Optimization using Evolutionary Algorithms, John Wiley & Sons Limited, New York. 2. Tan, K. C., Khor, E. F., Lee, T. H. (2005) Multiobjective evolutionary algorithms and applications, Springer-Verlag: London. 3. Gujarathi, A.M., Babu, B.V. (2009) Improved Strategies of Multi-objective Differential Evolution (MODE) for Multi-objective Optimization. In Proceedings of 4th Indian International Conference on Artificial Intelligence (IICAI-09), Tumkur, Bangalore, December 16-18, 2009. 4. Babu, B.V., Mubeen, J.H.S., Chakole, P.G. (2007) Modeling, simulation, and optimization of wiped film poly ethylene terephthalate (PET) reactor using multi-objective differential evolution (MODE), Materials and Manufacturing Processes: Special Issue on Genetic Algorithms in Materials, Vol. 22 (5), pp. 541-552. 5. Babu, B.V., Gujarathi, A.M., Katla, P., Laxmi, V.B. (2007) Strategies of multi-objective differential evolution (MODE) for optimization of adiabatic Styrene reactor, In Proceedings of International Conference on Emerging Mechanical Technology-Macro to Nano, BITS Pilani, India, Feb 16-18, pp. 243-250. 6. Price, K.V., Storn, R. (1997) Differential evolution a simple evolution strategy for fast optimization, Dr. Dobb s J., Vol. pp. 22, 18—24. 7. Babu, B. V. (2004) Process Plant Simulation, New York: Oxford University Press. 8. Onwubolu G. C. and Babu, B. V. (2004) New Optimization Techniques in Engineering, Springer-Verlag, Heidelberg, Germany. 9. Fan, H. Y., Lampinen, J. (2003). A trigonometric mutation operation to differential evolution. Journal of Global Optimization, vol. 27, 105-129. 10. Angira, R., Alladwar, S. (2007). Optimization of dynamic systems: A trigonometric differential evolution approach, Computers and Chemical Engineering, Vol. 31, 1055–1063.
