Mech090414.qxd The Journal of Engineering Research Vol. 7, No. 1, (2010) 42-52 1. Introduction Submerged arc welding (SAW) is a multi-factor, multi- objective metal joining technology in which several __________________________________________ *Corresponding author’s e-mail: sdatta@nitrkl.ac.in process control parameters interact in a complicated man- ner and influence differently on quality of the prepared Multi-Objective Optimization of Submerged Arc Welding Process Saurav Datta* and Siba Sankar Mahapatra Department of Mechanical Engineering, National Institute of Technology, Rourkela, Orissa-769008, India Received 14 April 2009; accepted 5 September 2009 Abstract: Submerged arc welding (SAW) is an important metal fabrication technology specially applied to join metals of large thickness in a single pass. In order to obtain an efficient joint, several process parameters of SAW need to be studied and precisely selected to improve weld quality. Many methodologies were proposed in the past research to address this issue. However, a good number of past work seeks to optimize SAW process param- eters with a single response only. In practical situations, not only is the influence of process parameters and their interactive effects on output responses are to be critically examined but also an attempt is to be made to optimize more than one response, simultaneously. To this end, the present study considers four process control parameters viz. voltage (OCV), wire feed rate, traverse speed and electrode stick-out. The selected weld quali- ty characteristics related to features of bead geometry are depth of penetration, reinforcement and bead width. In the present reporting, an integrated approach capable of solving the simultaneous optimization of multi-qual- ity responses in SAW was suggested. In the proposed approach, the responses were transformed into their indi- vidual desirability values by selecting appropriate desirability function. Assuming equal importance for all responses, these individual desirability values were aggregated to calculate the overall desirability values. Quadratic Response Surface Methodology (RSM) was applied to establish a mathematical model representing overall desirability as a function involving linear, quadratic and interaction effect of process control parameters. This model was optimized finally within the experimental domain using PSO (Particle Swarm Optimization) algorithm. A confirmatory test showed a satisfactory result. A detailed methodology of RSM, desirability func- tion (DF) and a PSO-based optimization approach was illustrated in the paper. Keywords: Submerged arc welding (SAW), Multi-objective optimization, Response surface methodology (RSM), Desirability function (DF), Particle swarm optimization (PSO) ¢ùWɨdG ΩÉë∏dG á≤jô£d á«Yƒ°VƒŸG √O~©àŸG á«∏YÉØdG GôJÉÑgÉe QÉμfÉ°S ÉÑ«°S , ÉJGO ∞jQGƒ°S ::áá°°UUÓÓÿÿGGΩÉë∏d πeGƒY I~Y ,AƒØc π°üØe ≈∏Y ∫ƒ°ü◊G πLG øe .I~MGh Iôe ‘ IÒÑc ∑ɪ°SG äGP ¿OÉ©e §Hôd â∏ª©à°SG GPEG á°UÉNh ᪡e ™«æ°üJ É«Lƒ∏æμJ ƒg ¢ùWɨdG ΩÉë∏dG âfÉc á≤HÉ°ùdG ∫ɪY’G øe º¡e O~Y ∂dP ™eh .´ƒ°VƒŸG Gòg ™e πeÉ©à∏d á≤HÉ°S çƒëH ‘ É¡MGÎbEG ” ~b ¿OÉ©e º¶f øe ~j~©dG .á«YƒædG Ú°ùëàd áb~H ¢SQ~J ¿G Öéj ¢ùWɨdG ∑Éæg øμdh áb~H É¡°üëa ” ~b äÉLôıG ≈∏Y á∏YÉØàŸG É¡JGÒKÉJ h á«∏ª©dG äGOôØe ÒKÉJ §≤a ¢ù«d ,á«∏ª©dG ´É°Vh’G ‘ .I~MGh áHÉéà°SÉH áJGOôØe h ¢ùWɨdG ΩÉë∏dG π«©Øàd πeCÉJ áYô°S ,á«àdƒØdG :»gh Iô£«°ùdG äGOôØŸ äÉ«∏ªY á©HQG QÉÑàY’G Ú©H òNÉJ á«dÉ◊G á°SGQ~dG ¿Éa ∂dP πLG øeh .áæeGõàe IQƒ°üH √~MGh áHÉéà°SG øe ÌcÉH á«∏YÉa äGP É¡∏©÷ ádhÉfi ܃∏°SG ìGÎbG ” ‹É◊G ôjô≤àdG ‘ .RôÿG ¢VôYh í«∏°ùàdG h ájPÉØædG ≥ªY »g RôÿG á°S~æg ÉjGõà §ÑJôJ QÉàıG ∂∏°ùdG á«Yƒf .OhÎc’G RhôH ,á«ÑfÉ÷G áYô°ùdG ,∂∏°ùdG ájò¨J áaÉμd ájhÉ°ùàe ᫪gG ¢VGÎaÉH .áÑ°SÉæŸG áHƒZôŸG ádG~dG QÉ«àNEG ᣰSGƒH ájOôØdG á≤jô£dG ‘ .¢ùWɨdG ΩÉë∏d äÉ«YƒædG √O~©àŸG áHÉéà°S’G ™e áæeGõàe á«∏YÉa hP πc πHÉb èe~e áHÉéà°S’G »YÉHôdG í£°ùdG á≤jôW .á∏eÉ°ûdG áHƒZôŸG º«≤dG ÜÉ°ù◊ ⩪L ~b áHƒZôŸG ájOôØdG º«≤dG √òg ¿Éa ,äÉHÉéà°S’G RSMπãÁ »°VÉjQ Ωɶf ¢ù«°SÉàd É¡≤«Ñ£J ” ~b äÉ«eRQGƒN ∫ɪ©à°SÉH »ÑjôéàdG π≤◊G ∫ÓN GÒNG á∏«©ØJ ” ~b êPƒªædG Gòg .äGOôØŸG h á«∏ª©∏d »∏YÉØJ »YÉHQ »£N ÒKÉJ ≈∏Y πª°ûJ ¬dG~c á∏eÉ°ûdG áÑZôdGPSOäÉ°Uƒëa . ¤G á∏°üØe á≤jôW .á«°Vôe èFÉàf âæ«H áj~«cCÉJRSM áÑZôdG ádG~dGDF á«∏YÉØdG ܃∏°SG ≈∏Y I~ªà©ŸG á«∏YÉØdG ∂dòchPSO.åëÑdG ‘ áfÉ«ÑJ ” ~b ««MMÉÉààØØŸŸGG ääGGOOôôØØŸŸGG ¢ùWÉZ ΩÉ◊ : áWSA Ö«éà°ùŸG í£°ùdG á≤jôW ,±G~g’G √O~©àe á«∏YÉa ,RSM ¬ÑZôdG ádGO ,DF äÉÄjõ÷G ~°ûM ¬«∏YÉa ,PSO . 43 The Journal of Engineering Research Vol. 7, No. 1, (2010) 42-52 weld. Weld quality depends on various features of bead geometry, mechanical-metallurgical characteristics of the weld as well as on weld chemistry. Moreover, the cumula- tive effect of combined aforesaid quality features deter- mines the extent of joint strength that determines func- tional aspects if the weld is subjected to practical field of application. Therefore, preparation of a satisfactory good quality weld seems to be a challenging job. Complete knowledge regarding the mode of influence of the process control parameters and their interactions are to be exactly known prior to select an optimal process environment capable of producing desired quality weld. However, SAW optimization is a difficult task due to simultaneous fulfillment of multi-quality features which should be close to the desired target value at the optimal setting. In prac- tice, it may happen that an improvement of one response may cause severe loss to another quality feature for a par- ticular parametric combination. Tay and Butler (1996) proposed an application of an integrated method using experimental designs and neural network technologies for modeling and optimizing a metal inert gas (MIG) welding process. Correia et al. (2004) used Genetic Algorithm (GA) to decide near-opti- mal settings of a GMAW welding process. Dongcheol et al. (2002) suggested the use of Genetic Algorithm and Response Surface Methodology (RSM) for determining optimal welding conditions. Hsien-Yu Tseng (2006) pro- posed an integrated approach to address the welding eco- nomic design problem. The integrated approach applied general regression neural network (NN) to approximate the relationship between welding parameters (welding current, electrode force, welding time, and sheet thick- ness) and the failure load. An analytical formula was gen- erated from the trained general regression neural network, and the mathematical model for the economic welding design was constructed. GA was then applied to resolve the mathematical model and to select the optimum weld- ing parameters. These parameters were recommended for use to obtain the preferred welding quality at the least pos- sible cost. Zhao et al. (2006) focused on the performance -predict- ing problems in the spot welding of the body-galvanized steel sheets. Artificial Neural Networks (ANNs) were used to describe the mapping relationship between weld- ing parameters and welding quality. After analyzing the limitation that existed in standard back propagation (BP) networks, the original model was optimized based on a lots of experiments. A lot of experimental data about welding parameters and corresponding spot-weld quality were provided to the ANN for study. The results showed that the improved BP model can predict the influence of welding currents on nugget diameters, weld indentation and the shear loads ratio of spot welds. The forecasting precision was quite high satisfying the practical applica- tion value. Pasandideh and Niaki (2006) presented a new methodology for solving multi-response statistical opti- mization problems. This methodology integrates desir- ability function and simulation approach with a genetic algorithm. The desirability function was used for model- ing the multi-response statistical problem whereas the simulation approach generated required input data and finally the genetic algorithm was implemented to optimize the model. Praga-Alejo et al. (2008) highlighted that the Neural Network (NN) with GA as a complement are good opti- mization tools. The authors compared its performance with the RSM that is generally used in the optimization of the process, particularly in welding. Many designed experiments require the simultaneous optimization of multiple responses. The common trend to tackle such an optimization problem is to develop mathe- matical models of the responses. These indicate the entire process behavior. The effect of process parameters on dif- ferent responses can be analyzed from the developed models. Multiple linear regression and Response Surface Methodology are two common tools available for devel- oping the mathematical models of the responses as a func- tion of process parameters. Depending on the require- ment, each quality features/responses are optimized (max- imized or minimized) to determine the optimal setting of the parameters. However, this method is applicable for the optimization of a single objective function. In a multi- objective case, it is essential to convert these multiple objectives to an equivalent single objective function which has to be optimized finally. A common approach is to use a desirability function combined with an optimization algorithm to find the most desirable settings. In the desirability function approach, individual response desirability values are calculated depending on the target as well as prescribed tolerance limit of the response variables. Individual desirability val- ues are then aggregated to calculate the overall desirabili- ty value. The optimal setting is one which can maximize the overall desirability. In doing so, a mathematical model is required for overall desirability. The model is then opti- mized finally. However, as the number of factors that affect the complexity of a multiple response problem increases, conventional optimization algorithms can fail to find the global optimum. For these situations, a common approach is to implement a heuristic search procedure like the GA and ANN or other optimization algorithms like Controlled Random Search (CRS) Price, W. L. (1977). However, it has been found that GA was adapted many times by previous researchers; less effort was made on application of CRS and even PSO in optimizing features of submerged arc weld. In consideration of the above, the present study aims at evaluating a near optimal parameter setting for the optimization of bead geometry parameters of a submerged arc weld. The study proposes integrating RSM-based desirability function approach and a PSO algorithm for multi-response optimization of SAW. Bead geometry parameters of submerged arc weld on mild steel were selected as multi-objective responses and they were optimized to select the optimal process environment. Finally, the study concludes the effectiveness and applica- tion feasibility of the proposed integrated approach. 44 The Journal of Engineering Research Vol. 7, No. 1, (2010) 42-52 2. Desirability Function (DF) Approach Individual desirability values related to each of the quality parameters are calculated using the formula pro- posed by Derringer and Suich in1980. There are three types of desirability function: Lower- the-Better (LB), Higher-the-Better (HB) and Nominal- the-Best (NB). In the present investigation, for reinforce- ment and bead width LB criteria; and for penetration depth HB criteria have been selected. This is because, the objective of the work was to minimize reinforcement and bead width (to reduce weld metal consumption) and to maximize penetration depth as strength of the welded joint directly depends on penetration depth. The NB crite- rion is generally selected in cases where responses have their fixed target value. (1) (2) (3) (4) (5) (6) The individual response desirability values were accu- mulated to calculate the overall desirability, using the fol- lowing Eq. (7). Here, D is the overall desirability value, di is the individual desirability value of ith quality char- acteristic and n is the total number of responses. wi, is the individual response weightage. (7) 3. Response Surface Methodology (RSM) The response function that represents any of the output features of the weldment can be expressed as An individual desirability value using the Lower-the- better (LB) criterion is shown in Fig. 1. The value of ŷ is expected to be the lower the better. When ŷ is less than a particular criteria value, a desirability value id equal to 1; if ŷ exceeds a particular criteria value, the desirability value equals to 0. id varies within the range 0 to 1. The desirability function of the Lower-the- better (LB) criterion can be written as below (Eqs. 1 to 3). Here, miny denotes the lower tolerance limit of ŷ , the maxy represents the upper tolerance limit of ŷ and r represents the desirability function index, which is to be assigned previously according to the consideration of the optimization solver. If the corresponding response is expected to be closer to the target, the index can be set to the larger value, otherwise a smaller value. Figure 1. Desirability Function (LB) If ,ˆ minyy  1id If ,ˆ maxmin yyy  r i yy yy d          maxmin maxˆ If maxˆ yy  , 0id An individual desirability value using the Higher- the-better (HB) criterion is shown in Fig. 2. The value of ŷ is expected to be the higher the better. When ŷ is exceeds a particular criterion value, according to the requirement, the desirability value id is equals to 1; if ŷ is less than a particular criteria value, ie. less than the acceptable limit, the desirability value is equals to 0. The desirability function of the Higher-the-better (HB) criterion can be written as below (Eqs. 4 to 5). Here, miny denotes the lower tolerance limit of ŷ , the maxy represents the upper tolerance limit of ŷ and r represents the desirability function index, which must have been previously according to the consideration of the optimization solver. If the corresponding response is expected to be closer to the target, the index can be set to the larger value, otherwise to a smaller value. Figure 2. Desirability Function (HB) If ,ˆ minyy  0id If ,ˆ maxmin yyy  r i yy yy d          minmax minˆ If maxˆ yy  , 1id 1 2 1 1 2( ............ )n www w nD d d d  45 The Journal of Engineering Research Vol. 7, No. 1, (2010) 42-52 (8) Here, Y is the response. V = voltage (OCV), Wf = Wire feed rate, Tr = Traverse Speed and N = electrode stick- out. The selected relationship is a second-degree response surface, which is expressed as follows: - (9) The Response Surface Methodology (RSM) is an effi- cient tool, which is widely applied for modeling the out- put response(s) of a process in terms of the important con- trollable variables and then finding the operating condi- tions that optimize the response. The Eq. (9) can be writ- ten as a multiple linear regression model as follows: - (10) The method of least squares can be used to estimate the regression coefficients in Eq. (10). In this study regression coefficients were computed by statistical software pack- age MINITAB (Release 14). 4. Particle Swarm Optimization (PSO) Algorithm Particle swarm optimization (PSO) is a population- based stochastic optimization technique developed by Dr. Eberhart and Dr. Kennedy in 1995, inspired by the social behavior of bird-flocking or fish-schooling. PSO shares many similarities with evolutionary com- putation techniques such as Genetic Algorithms (GA). The system is initialized with a population of random solutions and searches for optima by updating genera- tions. However, unlike GA, PSO has no evolution opera- tors such as crossover and mutation. In PSO, the potential solutions, called particles, fly through the problem space by following the current optimum particles. Each particle keeps track of its coordinates in the prob- lem space which is associated with the best solution (fit- ness) it has achieved so far. (The fitness value is also stored.) This value is called pbest. Another "best" value that is tracked by the particle swarm optimizer is the best value, obtained so far by any particle in the neighbors of the particle. This location is called lbest. When a particle takes all the population as its topological neighbors, the best value is a global best and is called gbest. The particle swarm optimization concept consists of, at each time step, changing the velocity of (accelerating) each particle toward its pbest and lbest locations (local version of PSO). Acceleration is weighted by a random term, with separate random numbers being generated for acceleration toward pbest and lbest locations. In the past several years, PSO has been successfully applied in many research and application areas. It has been demonstrated that PSO gets better results in a faster, cheaper way than other methods. Another reason that PSO is attractive is that there are few parameters to adjust. One version, with slight varia- tions, works well in a wide variety of applications. Particle swarm optimization was used for approaches that can be used across a wide range of applications, as well as for specific applications focused on a specific requirement. 5. Experimentation Bead-on-plate submerged arc welding (on mild steel plates of thickness 10 mm) was carried out following 34 full factorial design which consists of 81 combinations of voltage (OCV), wire-feed rate, traverse speed and elec- trode stick-out. Each process control parameters was var- ied in three different levels during experiments. Interaction effects of process parameters were assumed negligible in the present study. Three responses related to features of bead geometry viz. bead width, reinforcement and depth of penetration were selected in the present study. A copper coated electrode wire of diameter 3.16 mm (AWS A/S 5.17:EH14) was used during the experi- ments. Welding was performed with flux (AWS A5.17/SFA 5.17) with grain size 0.2 to 1.6 mm with basic- ity index 1.6 (Al2O3+MnO2 35%, CaO+MgO 25% and SiO2+TiO2 20% and CaF2 15%). The experiments were performed on a Submerged Arc Welding Machine- INDARC AUTOWELD MAJOR (Maker: IOL Ltd., India). While the weld was being made, the specimens were prepared for metallographic test. Features of bead geometry (macrostructure) were observed in Optical Trinocular Metallurgical Microscope (Make: Leica, GER- ( , , , )Y f V Wf Tr N 2 0 1 2 3 4 11 2 2 2 22 33 44 12 13 14 23 24 34 . . . . . . Y b V b Wf b Tr b N b V b Wf b Tr b N b V Wf b V Tr b V N b Wf Tr b Wf N b Tr N                0 , 4 1i i b  , 4 1ii i b  and all ijb (interaction term coefficients) are constant. The dimensions (units) of the constants should be such that they must take care of the dimensional similarity on both sides of Eq. (9). 0 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 Y x x x x x x x x x x x x x x                               Here, 14 1 ( )i iY f x  and 1 1 2 2 3 3 4 4 5 11 6 22 7 33 8 44 9 12 10 13 11 14 12 23 13 24 14 34 , , , , , , , , , , , , , b b b b b b b b b b b b b b                             1 2 3 4 5 2 2 2 6 7 2 8 9 10 11 12 13 14 , , , , , , , , . , . , . , . , . , . x V x Wf x Tr x N x V x Wf x Tr x N x V Wf x V Tr x V N x Wf Tr x Wf N x Tr N               46 The Journal of Engineering Research Vol. 7, No. 1, (2010) 42-52 Parameters Units Notation Level -1 Level 0 Level +1 Voltage (OCV) Volts V 27 28 29 Wire feed rate cm/min Wf 655 970 1285 Traverse speed cm/min Tr 72 98 124 Stick-out mm N 27 29 31 Table 1. Process control parameters and their limits Design of experiment (Factorial combination) Response data (Bead geometry) Sl. No. V Wf Tr N P R W 1 -1 -1 -1 -1 3.849 1.761 10.061 2 -1 -1 -1 0 3.748 1.725 10.520 3 -1 -1 -1 1 3.627 1.709 11.219 4 -1 -1 0 -1 3.472 1.374 9.151 5 -1 -1 0 0 3.451 1.368 9.320 6 -1 -1 0 1 3.410 1.382 9.729 7 -1 -1 1 -1 3.155 1.287 8.821 8 -1 -1 1 0 3.214 1.311 8.700 9 -1 -1 1 1 3.253 1.355 8.819 10 -1 0 -1 -1 4.149 1.836 10.980 11 -1 0 -1 0 4.038 1.780 11.530 12 -1 0 -1 1 3.907 1.744 12.320 13 -1 0 0 -1 3.762 1.446 9.720 14 -1 0 0 0 3.731 1.420 9.980 15 -1 0 0 1 3.680 1.414 10.480 16 -1 0 1 -1 3.435 1.356 9.040 17 -1 0 1 0 3.484 1.360 9.010 18 -1 0 1 1 3.513 1.384 9.220 19 -1 1 -1 -1 4.649 2.067 11.559 20 -1 1 -1 0 4.528 1.991 12.200 21 -1 1 -1 1 4.387 1.935 13.081 22 -1 1 0 -1 4.252 1.674 9.949 23 -1 1 0 0 4.211 1.628 10.300 24 -1 1 0 1 4.150 1.602 10.891 25 -1 1 1 -1 3.915 1.581 8.919 26 -1 1 1 0 3.954 1.565 8.980 27 -1 1 1 1 3.973 1.569 9.281 28 0 -1 -1 -1 3.638 1.565 11.671 29 0 -1 -1 0 3.577 1.515 11.980 30 0 -1 -1 1 3.496 1.485 12.529 31 0 -1 0 -1 3.321 1.208 10.121 32 0 -1 0 0 3.340 1.188 10.140 33 0 -1 0 1 3.339 1.188 10.399 34 0 -1 1 -1 3.064 1.151 9.151 35 0 -1 1 0 3.163 1.161 8.880 36 0 -1 1 1 3.242 1.191 8.849 37 0 0 -1 -1 3.888 1.670 12.550 38 0 0 -1 0 3.817 1.600 12.950 39 0 0 -1 1 3.726 1.550 13.590 40 0 0 0 -1 3.561 1.310 10.650 41 0 0 0 0 3.570 1.270 10.760 42 0 0 0 1 3.559 1.250 11.110 43 0 0 1 -1 3.294 1.250 9.330 44 0 0 1 0 3.383 1.240 9.150 45 0 0 1 1 3.452 1.250 9.210 Table 2. Design of experiment and data related to bead geometry parameters 47 The Journal of Engineering Research Vol. 7, No. 1, (2010) 42-52 MANY, Model No. DMLM, S6D & DFC320 and Q win Software). The domain of the experiment is shown in Appendix (Table 1). The design of experiment (DOE) and collected experimental data, related to individual quality indicators of bead geometry are listed in Appendix (Table 2). These data were utilized in proposed integrated opti- mization approach, to be discussed later. 6. Results and Discussions of Proposed Opti- mization Approach 6.1 Calculation of Individual Desirability Values and Overall Desirability Function The flow chart of the approach is furnished below in Appendix (Fig. 3). Response data were transformed to their individual desirability values using a desirability function approach (Fuller, D. and Scherer, W., 1998). These are shown in Table 3. For depth of penetration HB (Higher-the-better) and for reinforcement as well as bead width LB (Lower-the-better) criteria were selected. The index of desirability function was the selected one. In this computation the minimum and maximum values of each response (Table 2) were denoted as ymin and ymax respec- tively. Individual desirability values of the responses were clustered to calculate the overall desirability value (Table 3). It was assumed that all responses are equally impor- tant. The same weight was assigned to all responses. 6.2 Development of Response Surface Model of Overall Desirability RSM was applied to derive a mathematical model of overall desirability. Overall desirability was expressed as a function of four process control parameters. The model consists of linear, square (quadratic) and interaction terms of the process parameters affecting the overall desirability value. The constant term and coefficients of the 46 0 1 -1 -1 4.338 1.931 13.089 47 0 1 -1 0 4.257 1.841 13.580 48 0 1 -1 1 4.156 1.771 14.311 49 0 1 0 -1 4.001 1.568 10.839 50 0 1 0 0 4.000 1.508 11.040 51 0 1 0 1 3.979 1.468 11.481 52 0 1 1 -1 3.724 1.505 9.169 53 0 1 1 0 3.803 1.475 9.080 54 0 1 1 1 3.862 1.465 9.231 55 1 -1 -1 -1 3.523 1.509 14.101 56 1 -1 -1 0 3.502 1.445 14.260 57 1 -1 -1 1 3.461 1.401 14.659 58 1 -1 0 -1 3.266 1.182 11.911 59 1 -1 0 0 3.325 1.148 11.780 60 1 -1 0 1 3.364 1.134 11.889 61 1 -1 1 -1 3.069 1.155 10.301 62 1 -1 1 0 3.208 1.151 9.880 63 1 -1 1 1 3.327 1.167 9.699 64 1 0 -1 -1 3.723 1.644 14.940 65 1 0 -1 0 3.692 1.560 15.190 66 1 0 -1 1 3.641 1.496 15.680 67 1 0 0 -1 3.456 1.314 12.400 68 1 0 0 0 3.505 1.260 12.360 69 1 0 0 1 3.534 1.226 12.560 70 1 0 1 -1 3.249 1.284 10.440 71 1 0 1 0 3.378 1.260 10.110 72 1 0 1 1 3.487 1.256 10.020 73 1 1 -1 -1 4.123 1.935 15.439 74 1 1 -1 0 4.082 1.831 15.780 75 1 1 -1 1 4.021 1.747 16.361 76 1 1 0 -1 3.846 1.602 12.549 77 1 1 0 0 3.885 1.528 12.600 78 1 1 0 1 3.904 1.474 12.891 79 1 1 1 -1 3.629 1.569 10.239 80 1 1 1 0 3.748 1.525 10.000 81 1 1 1 1 3.847 1.501 10.001 P (Penetration), R (Reinforacement) and W (bead width) 48 The Journal of Engineering Research Vol. 7, No. 1, (2010) 42-52 factors/interaction of factors were evaluated with the sta- tistical-software package Minitab (Release 14). Minitab's multiple linear regression approach was used to derive this model (Eq. 11). (11) Figure 3. Proposed optimization approach 2 2 2 2 0.625 0.0505 0.0317 0.0677 0.0244 0.0343 0.0445 0.112 0.0315 0.0003 . 0.0166 . 0.0049 . 0.114 . 0.0131 . 0.0411 . D V Wf Tr N V Wf Tr N V Wf V Tr V N Wf Tr Wf N Tr N                Step 1: Selection of appropriate desirability function Step 2: Transformation of response data into invidual desirability values Step 3: Calculation of overall desirability Step 4: Mathematical modeling of overall desirability using RSM Step 5: Check for model adequacy Step 6: Optimization PSO algorithm Step 7: Validation and recommendation 49 The Journal of Engineering Research Vol. 7, No. 1, (2010) 42-52 Sl. No. DP DR DW D 1 0.4953 0.3280 0.8223 0.5112 2 0.4315 0.3666 0.7624 0.4941 3 0.3552 0.3837 0.6712 0.4506 4 0.2574 0.7428 0.9411 0.5646 5 0.2442 0.7492 0.9191 0.5519 6 0.2183 0.7342 0.8657 0.5177 7 0.0574 0.8360 0.9842 0.3615 8 0.0946 0.8103 1.0000 0.4248 9 0.1192 0.7631 0.9845 0.4474 10 0.6845 0.2476 0.7024 0.4919 11 0.6145 0.3076 0.6306 0.4921 12 0.5319 0.3462 0.5275 0.4597 13 0.4404 0.6656 0.8669 0.6334 14 0.4208 0.6935 0.8329 0.6241 15 0.3886 0.6999 0.7677 0.5933 16 0.2341 0.7621 0.9556 0.5545 17 0.2650 0.7578 0.9595 0.5776 18 0.2833 0.7320 0.9321 0.5782 19 1.0000 0.0000 0.6268 0.0000 20 0.9237 0.0815 0.5431 0.3444 21 0.8347 0.1415 0.4281 0.3698 22 0.7495 0.4212 0.8370 0.6417 23 0.7237 0.4705 0.7911 0.6458 24 0.6852 0.4984 0.7140 0.6247 25 0.5369 0.5209 0.9714 0.6477 26 0.5615 0.5380 0.9635 0.6627 27 0.5735 0.5338 0.9242 0.6565 28 0.3621 0.5380 0.6122 0.4923 29 0.3237 0.5916 0.5719 0.4784 30 0.2726 0.6238 0.5002 0.4398 31 0.1621 0.9207 0.8145 0.4954 32 0.1741 0.9421 0.8120 0.5107 33 0.1735 0.9421 0.7782 0.5029 34 0.0000 0.9818 0.9411 0.0000 35 0.0625 0.9711 0.9765 0.3898 36 0.1123 0.9389 0.9806 0.4693 37 0.5199 0.4255 0.4975 0.4792 38 0.4751 0.5005 0.4452 0.4731 39 0.4177 0.5541 0.3617 0.4375 40 0.3136 0.8114 0.7455 0.5745 41 0.3192 0.8542 0.7311 0.5842 42 0.3123 0.8757 0.6854 0.5723 43 0.1451 0.8757 0.9178 0.4886 44 0.2013 0.8864 0.9413 0.5517 45 0.2448 0.8757 0.9334 0.5849 46 0.8038 0.1458 0.4271 0.3685 47 0.7527 0.2422 0.3630 0.4045 48 0.6890 0.3173 0.2676 0.3882 49 0.5912 0.5348 0.7208 0.6108 50 0.5905 0.5991 0.6946 0.6264 51 0.5773 0.6420 0.6370 0.6181 52 0.4164 0.6024 0.9388 0.6175 Table 3. Individual desirability values and calculated overall desirability 50 The Journal of Engineering Research Vol. 7, No. 1, (2010) 42-52 The extent of the significance of presence of factors (and interaction of factors) within the model was checked with the Analysis of Variance method (ANOVA) (Table 4). Based on the calculated P-value (probability of signifi- cance) of the terms (from Table 4) under considerations, insignificant terms (P-value less than 0.05) were excluded and the final reduced model consisting of significant terms was derived (Eq. 12). This model was optimized (maximized) finally using PSO a algorithm. (12) 53 0.4662 0.6345 0.9504 0.6551 54 0.5035 0.6452 0.9307 0.6712 55 0.2896 0.5981 0.2950 0.3711 56 0.2763 0.6667 0.2742 0.3697 57 0.2505 0.7138 0.2222 0.3412 58 0.1274 0.9486 0.5809 0.4126 59 0.1647 0.9850 0.5980 0.4595 60 0.1893 1.0000 0.5837 0.4798 61 0.0032 0.9775 0.7910 0.1346 62 0.0909 0.9818 0.8460 0.4226 63 0.1659 0.9646 0.8696 0.5182 64 0.4158 0.4534 0.1855 0.3270 65 0.3962 0.5434 0.1529 0.3205 66 0.3640 0.6120 0.0889 0.2706 67 0.2473 0.8071 0.5170 0.4691 68 0.2782 0.8650 0.5223 0.5009 69 0.2965 0.9014 0.4961 0.5100 70 0.1167 0.8392 0.7729 0.4230 71 0.1981 0.8650 0.8160 0.5190 72 0.2669 0.8692 0.8277 0.5769 73 0.6681 0.1415 0.1203 0.2249 74 0.6423 0.2529 0.0758 0.2310 75 0.6038 0.3430 0.0000 0.0000 76 0.4934 0.4984 0.4976 0.4964 77 0.5180 0.5777 0.4909 0.5277 78 0.5300 0.6356 0.4529 0.5343 79 0.3565 0.5338 0.7991 0.5337 80 0.4315 0.5809 0.8303 0.5926 81 0.4940 0.6066 0.8302 0.6289 DP (Desirability of penetration), DR (Desirability of reinforcement), DW (Desirability of bead width), OD (Overall desirability) Predictor Coefficient P-value Comment Constant 0.6250 0.000 Significant V -0.0505 0.000 Significant Wf 0.0317 0.003 Significant Tr 0.0677 0.000 Significant N 0.0244 0.019 Significant V2 -0.0343 0.054 Insignificant Wf*Wf -0.0445 0.013 Significant Tr*Tr -0.1120 0.000 Significant N*N -0.0315 0.076 Insignificant V*Wf -0.0003 0.983 Insignificant V*Tr 0.0166 0.185 Insignificant V*N 0.0049 0.694 Insignificant Wf*Tr 0.1140 0.000 Significant Wf*N -0.0131 0.292 Insignificant Tr*N 0.0411 0.001 Significant S = 0.0742512 R-Sq = 78.4% R-Sq (adj) = 73.8% Table 4. Check for significance of the constant and coefficients in the model 2 2 0 581 0 0505 0 0317 0 0677 0 0244 0 0445 0 112 0 114 0 0411 D . . V . Wf . Tr . N . Wf . Tr . Wf .Tr . Tr.N          P - value (probability of significance), S, R-sq (determine to what extent the model can predict well) 51 The Journal of Engineering Research Vol. 7, No. 1, (2010) 42-52 6.3 Particle Swarm Optimization (PSO) In the present study, the reduced mathematical model for overall desirability (Eq. 12) was optimized using a PSO algorithm. It is a constrained optimization problem since the experimental domain was defined by the bounds on the process variables (V, Wf, Tr and N). The objective is to maximize (Eq. 12) subject to the bounds on the process variables. The values of the parameters of the PSO algorithm used here as follows: Population size= 50, Range of Velocity Variation vmax = +4, vmin = -4, Maximum number of iteration = 100, Weighting factor = 0.8, Decrement factor (alpha) = 0.9 and Social parameters C1 = 2.0 and C2 = 2.0. By trial and error the values of the aforesaid parameters were chosen so as to improve an objective function value (overall desirability) at the optimal setting. After optimization the optimal setting becomes: (Optimal value of overall desirability becomes 0.707). Figure 4 shows the convergence curve in PSO. Due to non-availability of optimal factors value within equip- ment's provision, a compromise has to be made. The opti- mal setting should be modified and set to: After evaluating the optimal parameter settings, the next step is to predict and verify the enhancement of qual- ity characteristics using the optimal parametric combina- tion. Table 5 reflects the satisfactory result of confirmato- ry experiment. It indicates that the quality of the weld has improved. 7. Conclusions Weld quality in SAW depends on features of bead geometry, mechanical-metallurgical characteristics of the weld as well as on weld chemistry. The weld quality improvement is treated as a multi-factor, multi-objective optimization problem. The practical application of SAW requires efficient optimization methodology because process parameters are expected to interact in a complex manner. Therefore, any optimization algorithm must seek to identify interaction effects of input factors and be incor- porated in the course of an optimization procedure in a convenient way for developing an efficient methodology. The developed methodology based on RSM, desirability function and PSO algorithm can be applied in practice for continuous quality improvement and off-line quality con- trol. The desirability Function approach converts each of the responses (objectives) into their individual desirabili- ty value. Corresponding to each objective, these individ- ual desirability values are then accumulated to compute the overall/composite desirability function, which is to be optimized (maximized) finally. RSM has been applied to derive a mathematical model of overall desirability repre- sented as a function of process control parameters. This mathematical model has been optimized within an exper- imental domain. Although the paper considers SAW, the procedure is quite generic and can be applied to any process where complex relations among input and output parameters are difficult to predict. References Correia, D.S., Goccalves, C.., Sebastiao, S.C.Junior and Ferraresi, V.S., 2004, " GMAW Welding Optimization using Genetic Algorithms", Journal of Brazilian Society of Mechanical Science and Engineering, 26(1) , DOI: 10.1590/S1678-58782004000100005. Dongcheol, K., Sehun, R. and Hyunsung, P., 2002, "Modeling and Optimization of a GMA Welding Process by Genetic Algorithm and Response Surface Methodology", International Journal of Production Research, Vol. 40(7), pp. 1699-1711. Fuller, D. and Scherer, W., 1998, "The Desirability Function: Underlying Assumptions and Application Implications, Systems, Man, and Cybernetics", 1998 IEEE International Conference, Vol. 4, pp. 4016- 4021. 0 384 0 916 0 982 0 936 .V .Wf Tr . N .                 0 1 1 1 V Wf Tr N                 Figure 4. PSO convergence curve Optimal setting Prediction Experiment Level of factors 0 1 1 1V Wf Tr N 0 1 1 1V Wf Tr N Overall desirability 0.769 0.789 Table 5. Results of confirmatory experiment N.B. Subscripts on factors notation represents factor levels 52 The Journal of Engineering Research Vol. 7, No. 1, (2010) 42-52 Hsien-Yu Tseng, 2006, "Welding Parameters Optimization for Economic Design using Neural Approximation and Genetic Algorithm", International Journal of Advanced Manufacturing Technology, Vol. 27(9-10), pp. 897-901. Leticia, C., Cagnina and Susana, C. Esquivel., 2008, "Solving Engineering Optimization Problems with the Simple Constrained Particle Swarm Optimizer", Informatica, Vol. 32, pp. 319-326. Price, W.L., 1977, "A Controlled Random Search Procedure for Global Optimization", The Computer Journal, Vol. 20(4), pp. 367-370. Pasandideh S.H.R. and Niaki, S.T.A., 2006, "Multi- Response Simulation Optimization using Genetic Algorithm within Desirability Function Framework", Applied Mathematics and Computation, Vol. 175(1), pp. 366-382. Praga-Alejor, R.J., Torres-Trevino, L.M. and Pina- Monarrez, M.R., 2008, "Optimization Welding Process Parameters through Response Surface, Neural Network and Genetic Algorithms", Proceedings of the 2008 Electronics, Robotics and Automotive Mechanics Conference, Vol. 00, pp. 393- 399. Tay, K.M. and Butler, C., 1996, "Modeling and Optimizing of a MIG Welding Process - a Case Study using Experimental Designs and Neural Networks", Quality and Reliability Engineering International, Vol. 13(2), pp. 61 - 70. Zhao, X., Zhang, Y. and Chen G., 2006, Model Optimization of Artificial Neural Networks for Performance Predicting in Spot Welding of the Body Galvanized DP Steel Sheets, Advances in Natural Computation, Vol. 4221/2006, DOI 10.1007/11881070.