Using Imperialist Competitive Algorithm in Optimization of Nonlinear Multiple Responses

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1 International Journal of Industrial Engineering & Production Research Septeber 03, Volue 4, Nuber 3 pp ISSN: Using Iperialist Copetitive Algorith in Optiization of Nonlinear Multiple Responses M. Bashiri * & M. Bagheri Mehdi Bashiri, Industrial Engineering Departent, Shahed University, Tehran, Iran, bashiri@shahed.ac.ir Masoud Bagheri, Industrial Engineering Departent, Shahed University, Tehran, Iran,.bagheri@shahed.ac.ir KEYWORDS Multi-response Optiization; Iperialist copetitive Algorith; Genetic Algorith; Experiental Design ABSTRACT The quality of anufactured products is characterized by any controllable quality factors. These factors should be optiized to reach high quality products. In this paper we try to find the controllable factor s levels with iniu deviation fro their target and with a least variation. To solve such probles a siple aggregation function is used to aggregate the ultiple response functions followed by an iperialist copetitive algorith used to find the best level of each controllable variable. Moreover the proble has been better analyzed by Pareto optial solution to release the aggregation function. Then the proposed ultiple response iperialist copetitive algorith (MRICA) has been copared with Multiple objective Genetic Algorith (MOGA). The experiental results show efficiency of the proposed approach in both aggregation and non aggregation ethods for optiization of the nonlinear ultiresponse prograing. 03 IUST Publication, IJIEPR, Vol. 4, No. 3, All Rights Reserved.. Introduction Multi Response Optiization (MRO) ethods can be classified into three general categories: Desirability viewpoints; in this ethod which is proposed by Derringer and Suich [], response variables are aggregated into one function called desirability function. Then this single function should be optiized using an optiization ethod. Also a score between zero and one is assigned to each response. The score value of zero indicates that the predicted response value is copletely undesirable and the value of one indicates that the corresponding response has reached its desired target value. * Corresponding author: Mehdi Bashiri Eail: bashiri@shahed.ac.ir Paper first received Jan, 8, 0, and in accepted for Sep, 4, 0. Priority based techniques; in this category response variables have different iportance according to the decision aker opinion. So these techniques optiize the responses based on their priority factors or iportance weights. Loss function; in this case all response values are aggregated and converted to a single one based on Taguchi function considering the location and dispersion effects. It is obvious that when a single objective function optiization ethod is desired; an aggregation function should be used in ost of MRO probles. Soe recent works in these three areas are reviewed as follows: Saurav et al [] have proposed a hybrid of Taguchi ethod and principal coponent analysis to solve the proble. Also they have copared the proposed approach with Grey-Taguchi ethod. Bashiri and Hejazi [3] have converted the MRO proble to a single response optiization proble in order to

2 M. Bashiri & M. Bagheri Using Iperialist Copetitive Algorith in Optiization 30 analyze the robust experiental design by ipleenting soe Multiple Attribute Decision Making (MADM) techniques such as VIKOR, TOPSIS and PROMETHEE. They showed that their proposed ethod decreases the statistical error. Furtherore, Chang et al [4] have generalized the Taguchi ethod in order to use it in different situations. The proposed odel is presented using a weighted convex loss function. Alvarez et al. [5] have applied Response Surface Methodology (RSM) in designing and optiizing of a capacitive acceleroeter. In their research a facedcentered cube design is ipleented in the experientation. Hsieh [6] has used neural networks to estiate the relation between control variables and responses. Onur koksoy [7] and Lorenz Ihof [8], have proposed exact solution ethods in order to find the optial paraeter settings of ulti-response probles. Also Wu and Yeh [9] have proposed ultiple polynoial regression odels to iniize the total quality loss of dynaic ultiple response systes. Chang [0] has introduced an artificial neural network (ANN) approach to solve a dynaic ulti-response proble. However, these ethods can obtain the best solution aong the specified controllable factor levels. In other words in presence of continuous values of controllable factors, the entioned ethods are unable to achieve the real optial factor cobination. So it sees that introducing new heuristic and eta-heuristics which have abilities to find the real optial solution in a reasonable coputational tie, is interesting. Recently any eta-heuristic solution approaches have been proposed to solve the MRO probles. Correia et al. [] have copared the RSM and GA in optiization of welding process. Ozcelik and Erzurlu [] have proposed an RSM and genetic integrated optiization ethod to iniize the bend of sli layer of plastic. Suresh et al. [3] have proposed second level odel for the expecting roughness degree of steel parts. Their research goal was the optiization of roughness degree of the parts using a genetic algorith. Okte et al. [4] have presented a genetic algorith and RSM. In their proposed procedure first RSM provides a odel to deterine the level of paraeters followed by a GA which optiizes these levels. It can be understood fro the literature that a few researches have been conducted with focus on heuristic and eta-heuristic approaches ipleented to find the best levels of controllable variables. In this paper the relation function between response variables and controllable factors are estiated. Then an aggregated single objective function has been optiized using an adaptive ICA. Also a new ultiresponse Iperialist Copetitive Algorith (MRICA) is proposed in order to find the best settings of controllable factors. The rest of presented paper is organized as follows: In section, the RSM will be discussed in details. The proposed solution ethod will be introduced in section 3. Experiental results coparing to other existing solution approaches have been illustrated in section 4 followed by conclusion in section 5.. Proble Description Relation function between a response and controllable factors can be forulated as equation (). y f ( x, x,... x ) i 0 n y ( x, z ) x x Bx z x z () This equation illustrates the relation function between the controllable factors ( x, x,... x ) and the i-th n process response ( y i ). The first step in the RSM is finding the best approxiated relation function between a response and a set of independent controllable variables. If there is a curvature in the total syste, the second-order or higher polynoial odels should be ipleented. A second-order relation function taken fro [5] can be odeled as equation (). () Where y( x, z) denotes the response variable. x and z denote the vector of controllable and noise factors, respectively. In this odel 0 is the intercept of the regression function. is the vector of linear effects of control variables. B is the atrix which its ain diagonals denote the regression pure factors quadratic coefficients and its off-diagonals denote the coefficient of half quadratic factors interactions. is the vector of coefficients for the linear effects in the noise variables and is a atrix which shows the interactions between the controllable variables and noise factors, followed by which is the rando error with NID (0, ). In ost of RSM related studies, the proble is investigated considering a single response variable. However in practice there are ore than one responses and a practitioner should optiize all of these factors, siultaneously. In this paper we assue that the responses are independent or uncorrelated. During the quality control process, to create a robust design with optial level of factors, the ean and variance of syste should be considered. This procedure is called Dual-Response Syste (DRS). Many effective ethods for DRS have been proposed recently ([6], [7], [8]). One of the popular approaches to the DRS optiization is the MSE criterion proposed by Lin and Tu[5]. They have International Journal of Industrial Engineering & Production Research, Septeber 03, Vol. 4, No. 3

3 3 M. Bashiri & M. Bagheri Using Iperialist Copetitive Algorith in Optiization suggested three basic approaches of MSE for DRS optiization which All of the should be iniized. The first is used when the saller value of a response is preferable. MSE ( [ y ( x )]) [ y ( x )] (3) ˆ ˆ ˆ ˆ z z The second approach iplies that the response value should be axiized: MSE {( [ y ( x )]) [ y ( x )]} (4) ˆ ˆ ˆ ˆ z z And the last approach is useful when the target value of the response variable is desirable: MSE ( [ y ( x )] T ) [ y ( x )] (5) ˆ ˆ ˆ ˆ 3 z z In Eq (5), T is the target value of the response. 3. Solution Approaches Iperialist copetitive algorith is a new etaheuristic evolutionary algorith that starts with a population of answers, any of the called a country [9]. These countries are divided into two basic groups. The best answers or the powerful countries becoe the iperialists and the rest of the solutions will be treated as the colonies of iperialists. In the ain algorith there are two types of inner and outer copetitions which will be discussed in the following parts. 3-. Inner Copetition: In each epire it is preferable that the colonies ove towards the iperialist and by this strategy, we can explore the search space ore efficiently. 3-. Outer Copetition: The basic copetition in an iteration of the algorith is between the epires. Based on this copetition each iperialist wants to take the possession of other iperialists colonies. And after running the algorith in the next iterations the weak iperialists will collapsed and at the end of the algorith only the powerful epire with the best objective function value will be reained which is the optial or at least close to optial solution. In each iteration, the weakest epire is selected and its weakest country is assigned to the ore powerful epires. This strategy will lead to increase in the searching diversity. In this research two ain solution approaches have been considered. In the first one an aggregated single function is optiized by an adaptive ICA and as the second approach the Pareto frontier is constructed and analyzed using the proposed MRICA The First Approach: Aggregated Function One of the basic contributions of this research is proposing of an adaptive ICA in the optiization of an aggregated function of MRO proble. Suppose that we have two response functions based on MSE criteria, so we define the following stateent as the single objective function: R MSE MSE (6) Where. The pseudo code for the proposed ICA ethod in optiization of the aggregated function can be written as follow:. Start. Create the initial countries with special controllable variables values randoly. 3. Select the predefined nuber of best strings as the iperialists and construct the epires. 4. Divide the rest of the strings (countries) between the epires based on their noralized power. 5. If only one epire reains, go to step Start the inner copetition: -Apply soe of heuristic operators to each epire. 7. Start the outer copetition: -Find the weakest epire based on the total noralized power of the epires. -Find the weakest country in that epire and release it fro its epire. -For the vector P consist of the possession probability of each epire -For the vector E with the sae size of P vector and create the eleents of E between zero and one randoly. -For the vector R as R=P-E; -Select the epire which has the axiu value of R. -Assign the released country to the selected epire. -Go to step ; 8. End; 3-4. The Second Approach: MRICA To reach the non doinated solutions in the ultiple responses proble a new algorith which is called Multiple Response Iperialist Copetitive Algorith is proposed in this section. The ain structure of the MRICA has been depicted in figure. International Journal of Industrial Engineering & Production Research, Septeber 03, Vol. 4, No. 3

4 M. Bashiri & M. Bagheri Using Iperialist Copetitive Algorith in Optiization 3 Start Create the initial countries (solutions) and create the epires Apply the inner and outer copetition Save the best countries in the Pareto solutions pool Is reain only one epire? For the best frontier based on the constructed pool End Fig.. Main steps of the proposed MRICA The overall purpose is finding the efficient solutions for MRO proble solving in a situation that the aggregated function can t be considered. However the inner and outer copetitions reain unaltered according to the basic ICA. In the next section the proposed approaches will be discussed in details using a nuerical exaple taken fro the literature. 4. Nuerical Exaple In this paper we use the exaple proposed by Onur koksoy [5]. In entioned industrial case study there are three controllable variables naed x, x, x and also 3 two noise factors naed z, z. Table reports the levels of various factors. Two basic responses y, y have been introduced. In this exaple it is assued that there is no correlation between responses. Also it is assued that the first response should be iniized while the second should reach the target (T=). Table illustrates the Central Coposite Design (CCD) and experiental data for the exaple [5]. Tab.. Levels of factors for the exaple Levels Factors - 0 x x 8 4 x z z Tab.. The experiental results of Central Coposite Design for the nuerical exaple x x x3 z z y y The estiated regression odels for two responses using the Minitab 6 can be stated as follows: yˆ ( x, z ) x.55 x. 077 x.48 x x.08 x x.030x x x.0588 z.06 z.000x z yˆ ( x, z ) x.438x.0950 x.30 x x.43 x x.0 x.0794x z 3 3 (7) (8) The ean and variance of responses can be obtained as following equations: ˆ [ y ] x.55x.077 x.48 x x.08 x x.030 x x.048x ˆ [ y ] x.438x.0950 x.30 x x.43 x x.0x 3 3 (9) (0) International Journal of Industrial Engineering & Production Research, Septeber 03, Vol. 4, No. 3

5 33 M. Bashiri & M. Bagheri Using Iperialist Copetitive Algorith in Optiization ˆ [ y ] ( x ) (.06) s () z z ˆ [ y ] (.0794 x ) s () z Eq(3) and Eq(5) are ipleented to this exaple with respect to the MSE criteria. Firstly we use the aggregated single objective for optiization according to equation (6) with predefined weights given in Table 3. This table also illustrates a solutions obtained by ICA. The results are copared with GRG algorith proposed by Onur koksoy[5]. Response coefficient Tab. 3. The solution coparison between ICA and GRG ethod X X X 3 MSE (ICA) MSE (ICA) MSE (GRG) MSE (GRG) Efficiency of MSE Efficiency of MSE = %00 90% As the GRG ethod is an exact solution approach it can only reach an optial solution by restricting one of the controllable variables into a fixed value. The proposed ICA solves the proble with respect to all variables in a reasonable coputational tie. The reported results in Table 3 show that the proposed ICA can find the results reached by the GRG ethod considering proper weights. It is worth to ention that the proposed solution approach is independent of the restrictions of exact approaches. In the next stage the proposed MRICA was applied for the nuerical exaple to find the Pareto frontier. To show the efficiency of the proposed MRICA, it is copared to the ulti-objective Genetic Algorith. It is clear that the algorith efficiency can be considered by soe easures related to the extracted non doinated solutions (NDS). Nuber of extracted non doinated solutions is a basic easure and its higher value is desirable. Moreover the diversity of solutions in the deterined non doinated solutions can be coputed as another quality easure which is known as crowding distance and the algorith with higher crowding distance value is ore efficient. It counts the distances between two solutions in both sides of a solution. This factor can be calculated using equations 3 and 4. CD CD i T f ( x i ) f ( x i ) f ( x ) f ( x ) n M i ax CD i in (3) (4) Where CD is the crowding distance of i th solution i considering the objective. f ( x i) is the th objective function value of i th solution. CD is the total crowding distance. T In order to reach a robust algorith design the essential paraeters, should be adjusted into their appropriate levels using the experiental design techniques. In this paper a Taguchi ethod has been used for paraeter tuning of both algoriths of MRICA and MOGA. For instance the essential paraeters of genetic algorith are population size and total nuber of iterations. A three-level Taguchi design has been created with respect to the lower and upper bounds for each factor given in table 4. The experiental results of paraeter tuning for ultiple objective Genetic Algorith have been depicted in this table. In the experiental study of paraeters tuning two responses are studied including the coputational tie and nuber of extracted non doinated solutions. The sae experiental study has been done for the MRICA. Tab. 4. The Taguchi design for ipleented MOGA Factor Factor Response Response Population size Levels: Total nuber of algorith iterations Levels: Coputational tie(s) Nuber of obtained non doinated solutions Figure deonstrates the ean of signal to noise (SN) ratio based on two paraeters. According to this figure the values of 300 and 3000 are selected as population size and total iteration nubers, respectively. For the second solution algorith (MRICA) the essential paraeters are population size and nuber of iperialists. By running the Taguchi ethod the values of 300 and 5 are obtained as population size and nuber of iperialists, respectively. International Journal of Industrial Engineering & Production Research, Septeber 03, Vol. 4, No. 3

6 MOGA MRICA Algorith Population size M. Bashiri & M. Bagheri Using Iperialist Copetitive Algorith in Optiization 34 Fig. The ean SN ratios for population size and total iteration nuber based on Taguchi ethod Then the proble was solved by both tuned algoriths. The coparison results of both MRICA and MOGA algoriths are reported in Table 5. These results are obtained after 0 algoriths replications. Table5. The coparison between MRICA and MOGA Mean of Coputation al tie(s) Mean of nuber of non doinated solutions Mean of Total Crowding distance According to the reported results of Table 5, it is clear that the proposed MRICA is ore efficient than MOGA. It has better crowding distance and deterined non doinated solutions in a predefined population and coputation tie. The reason can be described as follow; in iperialist copetitive algorith we have predefined nuber of iperialists in which the inner copetition is running siultaneously and this process yields to obtain better solutions in lower coputational tie than the genetic algorith. Also the outer copetition quarantines the diversity in solution space by aditting an unrelated but new solution to an epire copetition. Also the results can be copared with other exact ethods such as Branch & Bound technique using Lingo optiization package. This software can find only an optial answer with respect to the various MSE criteria. In this case two solutions and.4 are obtained for MSE and MSE criteria, respectively by considering the responses separately. This results show that the optial solution corresponding to the MSE can t be found using branch and bound technique because better MSE values have been found by the proposed MRICA (Fig.3). The coputational tie for the exact solution is 0.04(s). However the proposed eta-heuristic can find 3 non doinated solutions in only 5(s) containing the obtained solution by the exact approach. So the proposed approach can be used widely in ulti response optiization probles. Fig. 3. The Pareto optial frontier obtained by ICA and GA Figure 3 deonstrates the Pareto frontier obtained by each algorith. It is worth to be entioned that the coparison has been done with an equal initial population size. However both of the MRICA and MOGA approaches can reach the optial solutions of GRG and ICA ethods in the reported non doinated solutions. 5. Conclusion Iperialist copetitive algorith is a new etaheuristic algorith which tries to optiize the probles. Literally, any probles especially in industrial fields have any factors which contradict each other. The optiization of these odels is difficult and need to a high value of coputational tie. In this research a new eta-heuristic based approach has been ipleented to optiize a nonlinear ulti-response prograing odel. The experiental results show that the efficiency of the proposed MRICA is coparable with those which obtain to an optiu solution. The proposed approach achieve to the optiu solution or at least near to International Journal of Industrial Engineering & Production Research, Septeber 03, Vol. 4, No. 3

7 35 M. Bashiri & M. Bagheri Using Iperialist Copetitive Algorith in Optiization optiu solution in a reasonable coputational tie. Furtherore the flexibility of this approach will perit the practitioner to choose the factor which is ore iportant to be optiized. However the study of correlated responses will be worth as future studies. References [] Derringer, G., Suich, R., Siultaneous Optiization of Several Response Variables, Journal of Quality Technology, 980, pp [] Saurav, D., Gouta, N., Bandyopadhyay, A., Pradip, K.P., Application of PCA-Based Hybrid Taguchi Method for Correlated ulticriteria Optiization of Suberged Arc Weld: a Case Study, International Journal of Advanced Manufacturing Technology. 45 (3 4), 009, pp [3] Bashiri, M., Hejazi, T.H., An Extension of Multi Response Optiization in MADM view, Journal of Applied Polyer Science. 9 (9), 009, pp [4] Chang, Ch.Y., Ti, L.Ch., Liang, H.W., Optiization of Process Paraeters using Weighted Convex Loss Functions, European Journal of Operations Research, 96, 009, pp [5] Alvarez, M.J., Gil-Negrete, N., Ilzarbe, L., Tanco, M., Viles, E., Asensio, A., A Coputer Experient Application to the Design and Optiization of a Capacitive Acceleroeter, Applied Stochastic Models in Business and Industry, 5 (), 009, pp [6] Hsieh, K.L., Paraeter Optiization of a Multi- Response Process for Lead Frae Manufacturing by Eploying Artificial Neural Networks, International Journal of Advanced Manufacturing Technology, 8, 006, pp Plastic Parts using Integrated Response Surface Method and Genetic Algorith, International Counications in Heat and Mass Transfer 3, 005, pp [3] Suresh, P.V.S., Rao, P.V., Deshukh, S.G., A Genetic Algorith Approach for Optiization of Surface Roughness Prediction Model, International Journal of Machine Tools & Manufacture 4, 00, pp [4] Okte, H., Erzurulu, T., Kurtaran, H., Application of Response Surface Methodology in the Optiization of Cutting Conditions for Surface Roughness, Journal of Materials Processing Technology 70, 005, pp. 6. [5] Lin, D.K.J., Tu, W., Dual Response Surface Optiization, Journal of Quality Technology 7, 995, pp [6] Tang, L.C., Xu, K.A., Unified Approach for Dual Response Surface Optiization, Journal of Quality Technology 34, 00, pp [7] Koksoy, O., Doganaksoy, N., Joint Optiization of Mean and Standard Deviation using Response Surface Methods, Journal of Quality Technology 35, 003, pp [8] Robinson, T.J., Borror, C.M., Myers, R.H., Robust Paraeter Design: a Review, Quality and Reliability Engineering International 0, 004, pp [9] Atashpaz, E., lucas, C., Iperialist Copetitive Algorith: An Algorith for Optiization Inspired by Iperialist Copetition, IEEE 007, /07$5.00_c. [7] Koksoy, O., A Nonlinear Prograing Solution to Robust Multi-Response Quality Proble, Applied atheatics and coputation 96, 008, pp [8] Ihof, L., Optiu Designs for a Multi-Response Regression Model, Journal of Multivariate Analysis 7, 000, pp [9] Wu, F.C., Yeh, C.H., Robust Design of Multiple Dynaic Quality Characteristics, International Journal of Advanced Manufacturing Technology 5, 005, pp [0] Chang, H.H., Dynaic Multi-Response Experients by Back Propagation Networks and Desirability Functions, Journal of the Chinese Institute of Industrial Engineers 3(4), 006, pp [] Correia, D.S., Goncoalves, C.V., Da Cunha, S.S., Ferraresi, V.A., Coparison Between Genetic Algorith and Response Surface Methodology in GMAW Welding Optiization, Journal of Materials Processing Technology 60, 005, pp [] Ozcelik, B., Erzurlu, T., Deterination of Effecting Diensional Paraeters on War Page of Thin Shell International Journal of Industrial Engineering & Production Research, Septeber 03, Vol. 4, No. 3

8 M. Bashiri & M. Bagheri Using Iperialist Copetitive Algorith in Optiization 36 International Journal of Industrial Engineering & Production Research, Septeber 03, Vol. 4, No. 3