1-s2.0-S0307904X14002066-main.pdf

Applied Mathematical Modelling 38 (2014) 5592–5608

Contents lists available at ScienceDirect

Applied Mathematical Modelling

journal homepage: www.elsevier .com/locate /apm

Parameters optimization of selected casting processes usingteaching–learning-based optimization algorithm

http://dx.doi.org/10.1016/j.apm.2014.04.0360307-904X/� 2014 Elsevier Inc. All rights reserved.

⇑ Corresponding author. Tel.: +91 261 2201661; fax: +91 2612227334.E-mail address: [email protected] (R.V. Rao).

1 Tel.: +91 261 2201661; fax: +91 2612227334.

R. Venkata Rao ⇑, V.D. Kalyankar 1, G. Waghmare 1

Department of Mechanical Engineering, S.V. National Institute of Technology, Surat, Gujarat 395007, India

a r t i c l e i n f o

Article history:Received 16 August 2012Received in revised form 31 December 2013Accepted 15 April 2014Available online 9 May 2014

Keywords:Parameter optimizationSqueeze castingDie castingContinuous castingMathematical modelsTLBO algorithm

a b s t r a c t

In the present work, mathematical models of three important casting processes are consid-ered namely squeeze casting, continuous casting and die casting for the parameters opti-mization of respective processes. A recently developed advanced optimization algorithmnamed as teaching–learning-based optimization (TLBO) is used for the parameters optimi-zation of these casting processes. Each process is described with a suitable example whichinvolves respective process parameters. The mathematical model related to the squeezecasting is a multi-objective problem whereas the model related to the continuous castingis multi-objective multi-constrained problem and the problem related to the die casting isa single objective problem. The mathematical models which are considered in the presentwork were previously attempted by genetic algorithm and simulated annealing algorithms.However, attempt is made in the present work to minimize the computational efforts usingthe TLBO algorithm. Considerable improvements in results are obtained in all the cases andit is believed that a global optimum solution is achieved in the case of die casting process.

� 2014 Elsevier Inc. All rights reserved.

1. Introduction

Casting is one of the oldest manufacturing processes and with the progress of time, lot of developments took place in thecasting process. It is used for production of various complicated shapes that cannot be easily manufactured by any machiningprocess. Keeping in view the involvement of high temperature and safety aspects of the operators, various manual activitiesof the casting processes are getting replaced by the high end automatic technologies. Based on the type of mold and way offilling the molten metal, various types of casting processes are now available such as sand casting, die casting, continuouscasting, squeeze casting, investment casting, etc. and each type of casting process is having its particular application area.The major success of all these casting processes depends upon the close control over all the input parameters and the propercontrol over the metal solidification. Due to involvement of advanced technologies in the casting process, a small variation inany of the input parameters affects the process output and produce defective castings. Hence the designers and foundry per-sonnel are making lot of efforts to develop various mathematical models in the form of input–output relations so as toachieve exact parameters setting instead of going for trial attempts. However, this can be achieved by using advanced opti-mization techniques as tools for obtaining the optimum parameters setting for the casting processes under consideration.

http://crossmark.crossref.org/dialog/?doi=10.1016/j.apm.2014.04.036&domain=pdf

http://dx.doi.org/10.1016/j.apm.2014.04.036

mailto:[email protected]

http://dx.doi.org/10.1016/j.apm.2014.04.036

http://www.sciencedirect.com/science/journal/0307904X

http://www.elsevier.com/locate/apm

R.V. Rao et al. / Applied Mathematical Modelling 38 (2014) 5592–5608 5593

In this work, efforts are carried out to prove the importance of advanced optimization techniques in the field of param-eters optimization of various casting processes so that the designers and foundry personnel can achieve their objectivesalong with satisfying various constraints and limits of the respective process models. In the past, use of very few optimiza-tion techniques was involved in this field. Three important casting processes are considered in this work namely squeezecasting, continuous casting and die casting process. Optimization of mathematical models of these processes is carried byusing a recently developed advanced optimization algorithm named as teaching–learning-based optimization (TLBO) algo-rithm. The contribution of this paper is the application of the TLBO algorithm to the selected casting processes and to provethe effectiveness of the algorithm.

In the next section the TLBO algorithm is explained followed by the optimization aspects of squeeze casting, continuouscasting and die casting processes using the TLBO algorithm. The importance of parameters optimization of respectiveprocesses is described with detailed literature survey, problem description and result comparison.

2. Teaching–learning-based optimization algorithm

Teaching–learning-based optimization algorithm is a teaching–learning process inspired algorithm recently proposed byRao et al. [1,2] and Rao and Patel [3] based on the effect of influence of a teacher on the output of learners in a class. In thisalgorithm a group of learners are considered as population and different subjects offered to the learners are considered asdifferent design parameters and a learner’s result is analogous to the ‘fitness’ value of the optimization problem. The bestsolution in the entire population is considered as the teacher. The design parameters are actually the parameters involvedin the objective function of the given optimization problem and the best solution is the best value of the objective function.

The working of TLBO algorithm is divided into two parts, ‘Teacher phase’ and ‘Learner phase’. Working of both thesephases is described in detail by Rao et al. [1,2]. The same explanation of teacher phase and learner phase is referred herefor the working of TLBO algorithm. Fig. 1 represents the flowchart of TLBO algorithm [2]. The TLBO algorithm has beenalready tested on several constrained and unconstrained benchmark functions and proved better than the other advancedoptimization techniques [3]. It is also proving better in various field of engineering such as those reported by Niknamet al. [4–7] in the field of electrical engineering, Togan [8] in the field of civil engineering. Similarly, Krishnanand et al.[9] used it for the problems related to economic load dispatch, Rao and Kalyankar [10–12] used it for various fields relatedto manufacturing processes such as machining processes, modern machining processes, laser beam welding process, etc. andRao and Patel [13,14] used it to attempt multi-objective mathematical models in the field of thermal engineering. Eventhough Crepinšek et al. [15] raised some doubts about the algorithm-specific parameter less concept of TLBO algorithmand some other issues, however, Rao and Patel [3] had already cleared all those issues and justified that the TLBO algorithmis an algorithm-specific parameter less algorithm.

In the literature, it is observed that, the TLBO algorithm is not yet used in the field of optimization of mathematical mod-els of casting process. Hence the same is now used for the parameters optimization of various casting processes under con-sideration. In the next three sections, application of TLBO algorithm is presented for the optimization of mathematicalmodels of squeeze casting, continuous casting and die casting process respectively.

3. Parameters optimization of squeeze casting process

Various automotive parts are now getting produced by magnesium, aluminum and their alloys due to their improvedmechanical properties and suitability to various advanced and complicated parts. Products belonging to these materialsalong with other ferrous materials can be easily manufactured by casting processes. However, the products of squeeze cast-ing process are comparatively stronger due to better grain size and less metallic shrinking. Squeeze casting process has num-ber of advantages such as elimination of gas and shrinkage porosities, high ductility, reduction of metal wastage due to theabsence of feeders or risers, etc. over the other casting processes. Various important input parameters involved in thesqueeze casting process are: squeeze pressure, melt temperature, die preheating temperature, squeeze time, melt volumeand quality, time delay before pressurization, etc.

Proper setting of these input parameters will not only help to increase the production but it also minimizes the defectsand rejection level. However, this proper parameter setting should not be based on trial attempts. In the literature, someresearchers had carried out research to study the effects of various input parameters on the process output and also triedto achieve effective parameter setting. Hu [16] reviewed the progress in squeeze casting process and presented the effectof process variables on the cast structure and properties of magnesium alloys and magnesium based composites. Designof experiments was also discussed in their work for optimization of squeeze casting process. The various important inputprocess parameters discussed were melt volume and quality, magnitude and duration of applied pressure, die temperature,pouring temperature, time delay before pressurization, lubrication, etc. Kim et al. [17] analyzed the microstructure of asqueeze cast product. Comparisons of microstructure for a squeeze cast billet and the gravity cast billet were carried outand analysis suggested the squeeze cast process as a best choice. In their other work, Kim et al. [18] carried out the exper-imental investigation to study the effect of die geometry on the microstructure of squeeze cast component.

Zhou et al. [19] compared the results of identical alloy component produced by squeeze cast and high pressure die cast andshowed that the porosity problem was very less in case of squeeze casting. Their study also showed significant improvement

RejectseYoN

X''j,P,i=X'j,P,i+ri(X'j,P,i -X'j,Q,i)

Is new solution better than existing?

Is termination criteria satisfied?

Final value of solutions

Yes

No

Accept

TeacherPhase

StudentPhase

Select two solutions randomly X'total-P,i and X'total-

Is X'total-P,i better than X'total-Q,i

X''j,P,i=X'j,P,i+ri(X'j,Q,i - X'j,P,i)

RejectNo

No Yes

Calculate the mean of each design variable

Identify the best solution (teacher)

Modify solution based on best solution Difference_Meanj,k,i = ri (Xj,kbest,i - TFMj,i)

Is new solution better than existing?

Yes

Keep previous solution

Keep previous solution

Initialize number of students (population), termination criterion

Accept

Fig. 1. Flowchart of TLBO algorithm [2].

5594 R.V. Rao et al. / Applied Mathematical Modelling 38 (2014) 5592–5608

in mechanical properties of squeeze cast component over that of die cast component. Baek and Kwon [20] studied the effect ofprocess parameters on the fluidity of a squeeze cast Al–Si product and showed that the fluidity of a selected material increasedwith the silicon content. Yang et al. [21] investigated the effects of various process parameters such as applied pressure, pour-ing temperature and die temperatures on the macrostructure of a squeeze cast alloy. Run-xia et al. [22] studied the effect ofspecific pressure on the microstructure and mechanical properties of ZA27 squeezed castings. The experimental investigationsuggested that the fine microstructure can be obtained with the increase of pressure. The micro structural study also revealedthat the strength and plasticity of squeeze casting can be increased by homogeneous distribution of Al and Cu elements in thematrix of squeeze casting ZA27 alloy.

Moosa et al. [23] studied the effect of various input process parameters on the ultimate tensile strength of squeeze castcomponents made of carbon fiber Al–Si composites. Various parameters considered in their experimental study weresqueeze pressure, die preheating temperature, pouring temperature, squeeze time and delay time. A set of input processparameter were suggested in their study for obtaining the maximum tensile strength for given combination of material. Zanget al. [24] carried out the experimental investigation to study the influence of applied pressure on the tensile behavior andmicrostructure of a squeeze cast magnesium alloy. The results of their investigation showed that the fracture mode of thealloy changes from brittle to ductile as the applied pressures increases. Senthil and Amirthagadeswaran [25] used thesqueeze casting process for preparing the aluminum alloy castings of a non symmetrical component and carried out exper-imental investigation to study the influence of various process parameters on mechanical properties of the casting. Taguchi’s


orthogonal array was used to conduct the required number of experiments. Various process parameters considered weresqueeze pressure, melt temperature, die preheating temperature, die insert material and compression holding time. Math-ematical models were developed for hardness and tensile strength using MINITAB 14 software. However, no optimizationtechnique was involved in their work.

It is observed from the literature that even though some research work was carried out related to the parameters opti-mization of squeeze casting process, but no optimization technique was used by previous researchers. Hence there is a scopefor using the advanced optimization techniques in the field of squeeze casting process parameters optimization. To provethis, an application example related to squeeze casting process is taken from the literature and the TLBO algorithm is appliedto it to get improvement in the result. The example is described in the following subsection along with the resultcomparison.

3.1. Application example

Senthil and Amirthagadeswaran [25] presented the parameters optimization of squeeze casting process using Taguchimethod. Non symmetrical AC2A aluminum alloy was used in their experimental investigation. Various input process param-eters considered by Senthil and Amirthagadeswaran [25] were squeeze pressure (MPa), melt temperature (�C), die preheat-ing temperature (�C), die insert material and compression holding time (seconds). Each parameter was considered at fourlevels and Taguchi’s L16 orthogonal array was used to conduct 16 experiments. The influence of these input process param-eters were studied on the important mechanical properties such as hardness and tensile strength. MINITAB software wasused by Senthil and Amirthagadeswaran [25] for the analysis purpose and the relationship between the input and outputprocess parameters was presented in the form mathematical model and the same is reproduced below by Eqs. (1) and (2).

Average hardness; H ¼ �3:82542þ 0:8787 � Aþ 0:46587 � C þ 0:30411 � E� 0:00393 � A2 � 0:00116 � C2

þ 0:00097 � E2 þ 0:00051 � A � C � 0:00333 � A � E� 0:00018 � C � E; ð1Þ

Average tensile strength; TS ¼ �11:2606þ 2:5778 � Aþ 1:3316 � C þ 0:7552 � E� 0:0116 � A2 � 0:0034 � C2

þ 0:0031 � E2 þ 0:0015 � A � C � 0:0097 � A � E� 0:001 � C � E; ð2Þ

where, A = squeeze pressure, B = melt temperature, C = die preheating temperature and E = compression holding time. One ofthe input parameters, die insert material, was not involved in both the equations and it was also not described about how tochoose the die insert material. Hence the same die insert material suggested by Senthil and Amirthagadeswaran [25] i.e. hotdie steel is continued in the present work. Same range of process parameters as used by Senthil and Amirthagadeswaran [25]is considered in the present work and is produced below:

Squeeze pressure ¼ 50—125 ðMPaÞ;

Melt temperature ¼ 675—750 ð�CÞ;

Die preheating temperature ¼ 150—300 ð�CÞ;

Compression holding time ¼ 15—60 ðsÞ:

Senthil and Amirthagadeswaran [25] had shown the optimum level of process parameters as squeeze pressure = 100 MPa,melt temperature = 725 �C, die preheating temperature = 200 �C and compression holding time = 45 s. This set of inputparameters gives the average hardness of 100.76 BHN and average tensile strength of 278.45 MPa. However, Senthil andAmirthagadeswaran [25] obtained this optimum set of process parameters by using MINITAB software and no advancedoptimization technique was involved in their work.

In the present work, same mathematical models are used as given by Senthil and Amirthagadeswaran [25] for hardnessand tensile strength and the process parameters optimization is carried out by using the TLBO algorithm. Initially the math-ematical models of hardness and tensile strength are attempted separately to find out the optimum parameter setting foreach of them and then both are attempted simultaneously as a combined objective function and a common parameter set-ting is also obtained. The population size and number of generations required to run the TLBO algorithm are decided by con-ducting trials at the beginning to check for the consistency of result and finally a population size of 10 and number ofgenerations of 20 are used for running the algorithm for this problem. The results obtained by TLBO algorithm for hardnessand tensile strength along with the comparison with the previous results are shown in Tables 1 and 2, respectively.

The maximum hardness and tensile strength reported by Senthil and Amirthagadeswaran [25] are 100 BHN and 278 MPa,respectively. However, the TLBO has increased the hardness from 100 BHN to 103 BHN and tensile strength from 278 MPa to290 MPa which is always beneficial for the metal industry. The convergence of hardness and tensile strength with respect togenerations is shown in Figs. 2 and 3, respectively. This shows that the use of advanced optimization algorithm like TLBO willdefinitely help the casting industries to improve their performance. In order to check for consistency of result, each algo-rithm is run for 50 times and the average result and standard deviation is obtained. In case of hardness, the average result

Table 1Result obtained by TLBO algorithm for hardness.

Parameters Expt. result [25] TLBO result

Best Average Std. deviation

Squeeze pressure (MPa) 100 119Melt temperature (�C) 725 686Die preheating temperature (�C) 200 225Compression holding time (s) 45 15Maximum hardness (BHN) 100.76 103.068 102.738 0.438

Bold indicates the best optimum value.

Table 2Result obtained by TLBO algorithm for tensile strength.

Parameters Expt. result [25] TLBO result


Squeeze pressure (MPa) 100 119Melt temperature (�C) 725 675Die preheating temperature (�C) 200 220Compression holding time (s) 45 15Maximum tensile strength (Mpa) 278.45 290.30 289.22 1.59


97

98

99

100

101

102

103

104

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Har

dnes

s (B

HN

)

Number of genera�ons

Fig. 2. Variation of hardness with generations.

278

280

282

284

286

288

290

292

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Ten

sile

stre

ngth

(MPa

)

Number of generations

Fig. 3. Variation of tensile strength with generations.



of 50 runs is 102.738 and the standard deviation obtained is 0.438. Similarly, in the case of tensile strength, the averageresult and standard deviation of 50 runs obtained is 289.22 and 1.59, respectively. In both the cases the average result of50 runs is observed to be very nearer to the best result and also the deviation is comparatively small.

Effort is also carried out here to obtain the common parameter setting which satisfies both the objectives and gives theimproved result. For this purpose, a combined objective function is prepared by normalizing both the objectives and assign-ing some weightage to each of them. The combined objective function used in the present work is given below by Eq. (3).

MaxðZÞ ¼ w1 � ðH=HmaxÞ þ ð1�w1Þ � TS=TSmaxð Þ; ð3Þ

where, Hmax and TSmax are the maximum values of hardness and tensile strength respectively which can be obtained byattempting an individual objective function and w1 is the weightage assigned to the first objective. In the present case,the weightage w1 is varied in the steps of 0.05 in order to obtain Pareto optimal points. The combined objective functionapproach was not used by Senthil and Amirthagadeswaran [25]; however a common parameter setting was produced intheir work which was suggested for both the objectives. Now a population size of 10 and 20 generations are used for runningthe TLBO algorithm for the combined objective function and a set of Pareto optimal solutions obtained is presented in theform of a graph as shown in Fig. 4.

4. Parameters optimization of continuous casting process

Due to low production cost and higher production rates, continuous casting process is now getting widely used for themass production of various billets, blooms, thin sheets, etc. Regularly used metals in industries such as steel, copper and alu-minum can be easily casted by using this process. In continuous casting process, molten metal is poured from one end andthe solidified slabs come out from other end through various rollers. However, control over various process parameters isvery important in this process to achieve the desired quality product. Various input process parameters involved in contin-uous casting process are: casting speed, flux viscosity, flux density, flux solidification temperature, mold thickness, moldstroke length, cooling water temperature, cooling water flow rate, etc. Each of these parameters directly affects the processoutputs. Non-metallic inclusion in the form of oxides is one of the major problems in continuous casting of steels that canlead to excessive casting repairs or rejected castings. To avoid such problems, close control over the casting speed, coolingwater flow rate, etc. is very important. By setting these various important parameters by trial may lead to more rejectionsand thereby decreasing the production rate and increasing the cost. Selection of optimum process parameters will help toavoid all these problems and this can be achieved by using advanced optimization algorithms.

In the literature it is observed that few researchers had carried out research on the continuous casting process. Zhou et al.[26] used nondominated sorting genetic algorithm for multiobjective optimization of a continuous casting process. Twoobjective functions were attempted in their work which involved maximization of the average value of the monomer con-version of the product and minimization of the length of the film reactor. Seven decision variables were used in the studywhich included the temperature, feed flow rate, the film thickness, the monomer conversion at the output, and three coef-ficients describing the wall temperature used in the film reactor. Effect of various decision variables was discussed and theresult in the form of Pareto set was presented. Cheung and Garcia [27] carried out optimization of the quality of a SAE 1010

280

282

284

286

288

290

292

101 101.5 102 102.5 103 103.5

Tens

ile s

tren

gth

(MPa

)

Hardness (BHN)

Fig. 4. Pareto optimal points for combined objective function of squeeze casting process.


steel billet by using heuristic search technique. A knowledge base was proposed in their work by interacting a numericalheat transfer model and an artificial intelligence heuristic search method. The work was focused on obtaining the optimizedcooling conditions for the continuous casting process which results in defect-free billet production.

Chakraborti et al. [28] used pareto-converging genetic algorithm for optimizing the casting velocity. Two different objec-tives were considered for casting velocity subjected to three constraints due to a number of physical considerations. Thework was restricted only to the mold region. Subsequently, Chakraborti et al. [29] extended their work on the spray and radi-ation cooling regions. In another work, Chakraborti et al. [30] used finite volume approach along with genetic algorithm tosolve the pertinent transport equations of a continuous caster mold and reported improvement in casting velocity and solid-ified shell thickness.

Santos et al. [31] used genetic algorithm and a knowledge base of operational parameters in order to develop a mathe-matical model and a computational algorithm to maximize the quality of steel billets produced by a continuous casting pro-cess. The optimization strategy was concerned with achieving a set of optimal cooling conditions in order to attain highestproduct quality. In another work, Santos et al. [32] developed a numerical code based on genetic algorithm for an industrialdata of a continuous caster machine and determined optimum settings of water flow rates in different sprays zones.

Ghosh et al. [33] discussed the problems occurred in a continuous casting of a thin slab due to meniscus-level fluctuation.Genetic algorithm was used to find the optimum combination of spray cooling. The total bulging was minimized and also thesurface temperature at the slab exit was also maintained minimum. Kulkarni and Babu [34] attempted the multiobjectiveoptimization problem for producing quality products in a continuous casting process. Simulated annealing algorithm wasused in their work to obtain the optimum process parameters in order to satisfy 17 critical quality conditions. Cheunget al. [35] determined an improved cooling condition of mold and spray cooling zones of a continuously billet castingmachine and suggested suitable modification in the secondary cooling zone by proposing provision of two spray zones. Aheuristic search technique supported by a Knowledge Base and a heat flow mathematical model was used for determiningthe results and the suggested modification resulted in a shorter metallurgical length and lower surface reheating comparedwith the set of original spray zones.

Filipic et al. [36] used differential evolution for multi-objective optimization technique for optimizing the coolant flows incontinuous casting of steel in order to satisfy multiple objectives associated with temperature and core length. Miettinen[37] described an interactive classification-based multiobjective optimization method: NIMBUS which converts the originalobjective functions together with preference information coming from the decision maker into scalar-valued optimizationproblems. A real-life problem related to continuous casting of steel was attempted in their work in order to minimize thedefects in the final product.

Bhattacharya et al. [38] used genetic algorithm to obtain the optimum parameters of the mold oscillation system in thecontinuous casting process. The important parameters considered were casting speed, stroke, frequency and deviation fromsinusoid. The main focus of the work was to maximize the lubrication index and minimize depth of oscillation marks andfriction. The comparison of results was made with the conditions described by the original equipment manufacturer. Sub-sequently, Bhattacharya and Sambasivam [39] used differential evolutionary algorithm for the same problem and comparedthe results with those obtained by genetic algorithm.

Ye et al. [40] described an Engineering-Driven Rule-Based Algorithm (ERD) and various challenges related to the bleedsdetection were explained. An attempt was made to solve bleed detection problem using a real case study in which missdetection rate and false alarm rate were optimized by considering the four important process variables related to the geom-etry and two important coefficients of the process. A full factorial three-level experimental design was used for determiningthe process parameters to conduct the experiments. Lopez et al. [41] created a simulator for the continuous casting processusing a computational algorithm based on the numerical method. The results obtained by the simulator were validated forthe three different steel casters produced in an industry. Jabri et al. [42] used particle swarm optimization for tuning theprocess parameters of a real plant in order to improve the bulging effect rejection.

It is observed from the literature that among the various optimization techniques available, simulated annealing (SA)[34], genetic algorithm (GA) [38], differential evolutionary (DE) [39] and particle swarm optimization (PSO) [42] algorithmswere used by some researchers for the parameters optimization of continuous casting process. However, subsequently it isproved by many researchers of different fields that the results given by SA, GA, DE, PSO, etc. are not always optimum. Thus itcan be concluded that there is a good scope for making use of recently developed advanced optimization techniques foroptimizing the continuous casting process parameters.Hence, in the present work, an attempt is made to optimize theparameters of continuous casting process by using the TLBO algorithm.


An example of continuous casting process attempted by Kulkarni and Babu [34] is considered in the present work for theprocess parameters optimization. 17 critical conditions were presented by Kulkarni and Babu [34] as an individual objectiveand finally a combined total loss function was given which represents all the conditions. However, various equations givenby Kulkarni and Babu [34] involved certain process constants, and in some cases the required process constant values werenot mentioned in their work. Moreover, those required values cannot be assumed in order to make the comparisons of result.Hence, in the present work, 10 critical conditions are considered out of 17 whose data is available and the entire result com-parison is made with respect to these 10 conditions only. This approach will not affect the complete meaning of the process


objectives as the individual conditions consist of respective mathematical models and those models are independent of theconditions. Moreover, result of each condition was given by Kulkarni and Babu [34] and the comparison of results obtained inthe present work is strictly made for those conditions only. The conditions considered in the present work are conditions 1 to9 and condition 13. The same equations of these conditions as given by Kulkarni and Babu [34] are considered and explainedbelow by Eqs. (4)–(15).

Condition 1 : Q 1 ¼ gVc; ð4Þ

where g is the viscosity in poise and Vc is the casting speed in m/min. In order to have a fair comparison of results, samevalue of casting speed i.e. Vc = 1.1 m/min is considered in this work as that suggested by Kulkarni and Babu [34] and the con-straint for Q1 was 1 6 Q 1 6 3.

Condition 2 : Q2 ¼ Tsol=g0:0472 ðfor crack sensitive gradesÞ; ð5Þ

Q 2 ¼ Tsol=g0:072 ðfor sticker sensitive gradesÞ; ð6Þ

where Tsol is mold flux solidification temperature in �C. Sticker sensitive grade is considered here and hence Q2 is calculatedsubsequently in this work by using Eq. (6). The constraint for condition 2 as given by Kulkarni and Babu [34] is considered as1025 6 Q 2 6 1075.

Condition 3 : Q 3 ¼ 1:801� 0:2461Vc � 0:044g� 0:00107Tsol; ð7Þ

It is observed that the equation of Q 3 given by Kulkarni and Babu [34] was having slight error mainly related to the con-stant term. The same is rectified now and the constant term is changed from 1.952 to 1.801 and the correct equation of Q 3 isgiven by Eq. (7) which now gives the exact value of L3 = 0.09 as reported by Kulkarni and Babu [34] with their parameterssettings and also satisfies the condition of 0.15 6 Q3 6 0.45.

Condition 4 : Q 4 ¼ 0:70 � ð2=sÞ0:3ð60=f Þ½gðVcÞ2��0:5þ 0:17 For 0:08 < %C < 0:16; ð8Þ

Q 4 ¼ 0:70 � ð2=sÞ0:3ð60=f Þ½gðVcÞ2��0:5þ 0:22 For 0:16 < %C < 0:08; ð9Þ

where Q4 is related to the powder consumption in kg/m2 which is having a constraint of 0.15 6 Q 4 6 0.45 [34], s is the strokelength in mm and f is the frequency in cpm. Even though it is not clear about which equation was used by Kulkarni and Babu[34], but it can be seen that the Eq. (9) is misleading about its limit and also it violates the constraint limit of Q4 if the param-eter limit given by Kulkarni and Babu [34] is substituted in it. Hence, Eq. (8) is considered in this work for condition 4 withslight correction in it which satisfies the results of Kulkarni and Babu [34] also.

Condition 5 : Q 5 ¼ Vc=f ; ð10Þ

where, Vc is the casting speed, as stated earlier, but it should be taken in mm/min (i.e. 1100 mm/min) and the constraint forcondition 5 is Q 5 6 25 mm [34].

Condition 6 : Q 6 ¼ 600 ðs=10 � f Þ0:5: ð11Þ

In this case, the stroke length (s) should be taken in cm and frequency (f) should be taken in cycles per second as men-tioned by Kulkarni and Babu [34]. The condition to be satisfied while parameter design is: Q6 < 400 lm [34].

Condition 7 : Q 7 ¼ ðVm=VcÞ; ð12Þ

where Vm is the mold velocity (m/min) which was not given by Kulkarni and Babu [34]. However, it can be obtained byattempting several equations related to mold velocity given by Kulkarni and Babu [34] and it is obtained as Vm = 2.40 m/min which satisfies the necessary constraints and their results. The constraint to be followed for condition 7 is Q7 > 1.2 [34].

Condition 8 : Q 8 ¼ ðRp � q=VcÞ ðvolume=surface areaÞ; ð13Þ

where Rp is the pool drain rate in m/min, q is the liquid flux density in kg/m3. The volume is in m3 and surface area is in m2.The volume and surface area were not clearly given by Kulkarni and Babu [34], however, it is obtained by doing reverse cal-culations of appropriate equations related to condition 8 so as to get the same value as that of Kulkarni and Babu [34] inorder to have fair comparison of result. The constraint for condition 8 is 0.15 6 Q 8 6 0.45 [34].

Condition 9 : Q 9 ¼ K � ðTsurf =TsolÞðLm=VcÞ ðgÞ0:5ðsÞ�0:25ðf Þ0:25ðVcÞ0:25=7:7

� �; ð14Þ

where Tsurf is the strand surface temperature in �C, Tsol is the powder solidification temperature in �C, Lm is the mold length inmeters, s is the stroke length in meters, f is the frequency in cpm, Vc is the casting speed in m/min, g is the viscosity in poiseand K is a constant (K = 0.251). In this case also the strand surface temperature Tsurf was not given by Kulkarni and Babu [34].However, it is obtained by doing reverse calculations of appropriate equations related to condition 9 so as to get same valueas that of Kulkarni and Babu [34]. The constraint for condition 9 is 0.15 6 Q9 6 0.45 [34].

Table 3Results

Loss

L1

L2

L3

L4

L5

L6

L7

L8

L9

L13

Tota

Bold in


Condition 13 : Q13 ¼ Lm=ðwide face dimensionÞ0:3; ð15Þ

where, Lm is the mold length in meters and wide face dimension (0.269) is in meter. The constraint for condition 13 as givenby Kulkarni and Babu [34] was Q 13 P 1.

All these conditions were having respective limits and constraints. Kulkarni and Babu [34] used quadratic loss functionsto derive the undesirability index for each condition. Finally the total undesirability index was obtained by them by sum-ming all the individual loss function values and the optimum values for the process parameters was correspond to minimumvalue for the total loss function. The individual loss functions given by Kulkarni and Babu [34] for these above mentioned 10conditions are reproduced below by Eqs. (16)–(25) and the total loss function is given by Eq. (26).

L1 ¼ ðQ 1 � 2Þ2; ð16Þ

L2 ¼ 0:0016ðQ 2 � 1050Þ2; ð17Þ

L3 ¼ 44:5 ðQ 3 � 0:3Þ2; ð18Þ

L4 ¼ 44:5 ðQ 4 � 0:3Þ2; ð19Þ

L5 ¼ 0:0016ðQ 5Þ2; ð20Þ

L6 ¼ 6:25� 10�6 ðQ6Þ2; ð21Þ

L7 ¼ 1:44 ð1=Q 7Þ2; ð22Þ

L8 ¼ 44:5 ðQ 8 � 0:3Þ2; ð23Þ

L9 ¼ 44:5 ðQ 9 � 0:3Þ2; ð24Þ

L13 ¼ ð1=Q 13Þ2; ð25Þ

Total loss function; Z ¼ L1 þ L2 þ L3 þ L4 þ L5 þ L6 þ L7 þ L8 þ L9 þ L13: ð26Þ

The individual loss function values vary from 0 to 1 and if any loss functions value is larger, then the respective undesir-ability is higher and the objective is to minimize the undesirability. The loss function values obtained by Kulkarni and Babu[34] for the above 10 conditions is given in Table 3. Simulated annealing algorithm was used by Kulkarni and Babu [34] toobtain the optimum process parameters for the continuous casting process under consideration. Now the same models asexplained above are attempted by the TLBO algorithm to get improvement in the result. The total loss function is of mini-mization type subjected to those 10 conditions with their limits and constraints. The process variables involved in the equa-tion consists of certain range and the same is used in the present work as given by Kulkarni and Babu [34]. As number ofconditions is to be considered simultaneously along with large number of input parameters, hence initially trials are carriedout by running the algorithm with different population sizes and number of generations to decide. Finally consistent resultsare obtained by using the population size of 100 and the number of generations as 50. All the conditions are handled care-fully and consistent results are obtained by TLBO algorithm which is given in Table 3 along with the comparison with theresults of simulated annealing.

obtained by TLBO algorithm for the 10 loss functions under consideration.

function SA result [34] TLBO result


0.24 0.230.80 0.0020.09 0.010.12 0.050.19 0.140.15 0.130.30 0.300.67 0.00040.80 0.770.87 0.91

l loss function (Z) 4.23 2.54 2.57 0.057

dicates the best optimum value.

2

2.5

3

3.5

4

4.5

0 10 20 30 40 50

Tot

al lo

ss fu

nctio

n va

lue


Fig. 5. Variation of total loss function value with generations.


The TLBO algorithm has suggested the following parameter settings which give the optimum results. Viscosity, g = 1.38poise, frequency, f = 116.64 cpm, stroke, s = 11.46 mm, flux solidus temperature, Tsol = 1075.86 (�C), drain rate, Rp = 2.11mm/min, flux density, q = 2349.23 kg/m3 and mold length, Lm = 704.47 mm. All these parameter values are within therespective limits as that given by Kulkarni and Babu [34]. In case of individual loss function L2 and L8, there is considerableimprovement in result and the undesirability has become almost zero from 0.8 and 0.67, respectively. Overall, almost in allcases, the loss function is improved except in L13 where it is slightly increased. This is quite possible because a commonparameter setting is obtained in the present work which satisfies all the conditions simultaneously. However the L13 functionwas also on higher side as given by Kulkarni and Babu [34]. The TLBO algorithm has given a significant improvement in theresult and the total loss function is improved from 4.23 to 2.54 thereby achieving the improvement by above 60%. The aver-age result of 50 runs obtained is 2.57 which is very nearer to the best result and the standard deviation obtained is 0.057. Theconvergence of result with generations is shown in Fig. 5. The comparison between the results for SA and TLBO is clearlymade only for the 10 conditions which are considered in this work. Even though Kulkarni and Babu [34] had attempted theirwork for total 17 conditions, but keeping in view the present improvement in results and success rate, it may be stated thatthe TLBO algorithm can also handle all those 17 conditions simultaneously provided the required data is made available.

5. Parameters optimization of die casting process

Die casting is a versatile process for producing various engineering parts by forcing molten metal under high pressure intoreusable steel molds. Die cast parts are now becoming very popular and finding its wide application ranging from very com-plicated parts used in automobiles to simple toys, but the overall equipment and metal die cost is very high. Also the accu-racy level required to set the proper input parameters in a die casting process is very high which may leads to high volume ofrejection. Hence proper setting of the input process parameters is very important in die casting process. There are large num-ber of input process parameters involved in this process which can be categorized as die casting machine related parameters,shot sleeve related parameters, die related parameters and cast metal related parameters [43]. Out of these, the few impor-tant parameters which can significantly affect the process output are melt temperature, injection pressure, injection time,die temperature, holding pressure, pressure holding time, etc. Any small deviation in the setting of these parameters willaffect the product quality. Hence these parameters must be decided by making use of advanced computational techniques.

Some researchers had attempted to study the effect of various input parameters on the process output and tried to opti-mize the situation. Syrcos [43] presented the effects of process parameters on the casting density of aluminum alloy casting.Various influential process parameters considered were piston velocity, metal temperature, filling time and hydraulic pres-sure. However, any advanced optimization technique was not involved in their work and the optimum parameter settingpresented was obtained by using Taguchi’s method. Yarlagadda and Chiang [44] used artificial neural network to predictthe process parameters for the given conditions of a pressure die casting. An industrial data was taken and the effects of fourinput parameters was discussed in their work namely injection time, injection pressure, melt temperature and die temper-ature. In another work, Yarlagadda [45] applied the artificial neural network for the physical model of a die cast product ofzinc alloy.

Lin and Tai [46] used simulated annealing algorithm for the optimization of various aspects of die casting process such asoptimization of runner design, optimization of position for the injection gate was carried out by Tai and Lin [47] and similarlyoptimal selection of gate location was carried out by Lin [48]. Krimpenis et al. [49] carried out the simulation of pressure die


casting using ProCAST software and finite element approach. Neural network model was then developed and parametersoptimization was carried out using genetic algorithm. Tsoukalas [50] used the combination of multivariable linear regressionand genetic algorithm for minimizing the porosity level of aluminum alloy die casting. Taguchi’s L27 orthogonal array wasused and experiments were conducted by considering five input parameters namely holding furnace temperature, die tem-perature, plunger velocities in the first and second stage and multiplied pressure. Optimized parameter setting obtained byusing genetic algorithm was produced in their work, however, it is observed that the computational efforts involved in theirwork to obtain the optimum parameter setting was comparatively very high. Verrana et al. [51] used the concept of design ofexperiments for analyzing the influence of injecting parameters on the internal quality of die cast part. The important injec-tion parameters considered were slow shot, fast shot and upset pressure and finally studied the effect of these parameters onthe casting density.

Kong et al. [52] studied the die temperature profile using infrared thermograph technology and attempt was made tooptimize the internal cooling system in order to provide even cooling to the components and the die. Commercial compu-tational fluid dynamics code was used in their work for optimizing and redesigning the internal cooling system. Wong andPao [53] emphasized various losses and difficulties faced by foundries due to dependency on skilled foundry men and severaltrial attempts carried out by them to achieve appropriate heat transfer and directional solidification in die casting. Use ofevolutionary algorithms in general and genetic algorithm in particular was highlighted in their work for optimizing the con-ditions of die casting process. Campatelli and Scippa [54] presented a case study of an automotive component, produced byhigh pressure die casting, and optimized the tolerance level of the die geometry. However only the concept of finite elementsimulation was involved in their work and any type of heuristic algorithm was not used by them for the optimization pur-pose. Zhang and Wang [55] used combination of artificial neural network and genetic algorithm for optimizing the processparameters of low pressure die casting in order to improve the quality of product. However, as any experiments were notconducted and only the software simulated data was used, hence the level of relative error was very high.

Even though several advanced optimization techniques were available in the literature, but only genetic algorithm andsimulated annealing were used by few researchers in the past for the parameters optimization of die casting process. Henceattempt is made here to use the TLBO algorithm to achieve the optimum solution for the die casting process. An applicationexample is considered for this purpose from the literature which is described below in detail along with the result.


This example is taken from the literature which is based on the research work of Tsoukalas [50]. In this work, Tsoukalas[50] had attempted to minimize the porosity level of an aluminum alloy component produced by pressure die casting pro-cess. A combined approach of multivariable linear regression and genetic algorithm was used in their work to determine theoptimum condition for the situation under consideration. The input variables considered by Tsoukalas [50] were holding fur-nace temperature ‘F’ (�C), die temperature ‘D’ (�C), plunger velocity in the first stage ‘S’ (m/s), plunger velocity in the secondstage ‘H’ (m/s) and multiplied pressure ‘M’ (bar). L27 orthogonal array of Taguchi method was used in their work and exper-iments were conducted as per the experimental layout. Subsequently the relation between all the input parameters with theresponse i.e. porosity was given in the form of a mathematical model. The same mathematical model as used by Tsoukalas[50] is given below by Eq. (27).

Porosity; P ¼ 1:623� 0:766 � 10�3ðFÞ � 1:301 � 10�3ðDÞ � 0:136ðSÞ þ 0:029ðHÞ � 1:636 � 10�3ðMÞ: ð27Þ

The parameters involved in Eq. (31) were having certain ranges and the same range as used by Tsoukalas [50] is used inthis work and is given below by Eqs. (28)–(32).

Holding furnace temperature ‘F’ ¼ 610—730 ð�CÞ; ð28Þ

Die temperature ‘D’ ¼ 190—270 ð�CÞ; ð29Þ

Plunger velocity in the first stage ‘S’ ¼ 0:02—0:34 ðm=sÞ; ð30Þ

Plunger velocity in the second stage ‘H’ ¼ 1:2—3:8 ðm=sÞ; ð31Þ

Multiplied pressure ‘M’ ¼ 120—280 ðbarÞ: ð32Þ

Tsoukalas [50] used combination of multivariable linear regression and genetic algorithm to optimize the process param-eters for the given conditions and compared his result with the experimental result. The result obtained by genetic algorithmin their work had shown improvement over the experimental result. However, the numbers of generations taken by geneticalgorithm were 1000 and the population size was not mentioned. Also the required GA operator setting was also not given intheir work such as crossover rate, mutation rate, etc. Thus, it indicates that computational efforts taken by Tsoukalas [50] toachieve the optimum parameters setting for the problem under consideration was comparatively more. Hence attempts arecarried out in the present work to find out the optimum solution of the problem with minimum computational efforts.The TLBO algorithm is now used for the parameters optimization of the same model of Tsoukalas [50] along with same

Table 4Result obtained by TLBO algorithm and its comparison with previous results.

Parameters GA result [50] TLBO result


Holding furnace temperature ‘F’, (�C) 729.4 730Die temperature ‘D’, (�C) 269.9 270Plunger velocity in the first stage ‘S’, (m/s) 0.336 0.34Plunger velocity in the second stage ‘H’, (m/s) 1.2 1.2Multiplied pressure ‘M’, (Bar) 275.7 280Number of generation 1000 10Minimum porosity (%) 0.251 0.243 0.2496 0.02


0.2

0.25

0.3

0.35

0.4

0.45

1 2 3 4 5 6 7 8 9 10

Poro

sity

(%)


Fig. 6. Variation of porosity with generations.


parameters range. The above mentioned mathematical model of porosity was attempted by TLBO algorithm by using a verysmall population size of 10 and the number of generations used is 10. The results obtained by TLBO algorithm are given inTable 4 along with the comparison with the results of genetic algorithm.

The same example is tried with the TLBO algorithm several times to check for any further improvement in the result andit is observed that the result given in Table 4 is the global optimum solution for the problem under consideration. Eventhough the number of generations used to run the TLBO algorithm is 10, but the global optimum solution of 0.243% isobtained in fourth generation itself and the consistent result is observed in further generations. The algorithm is allowedto run 50 times in order to check the average result and it is reported as 0.2496 which is almost nearer to the best result.The standard deviation of 50 runs obtained is 0.02 which is also an indication of good and consistent result. The convergenceof result obtained by TLBO algorithm is shown in Fig. 6. In the case of results given by Tsoukalas [50], the number of gen-erations used was 1000 and the result of porosity = 0.251 was converged only after 800 generations as shown by the graph-ical representation in their work. Also the parameter setting given by Tsoukalas [50] is very difficult to set on the machine,whereas the TLBO algorithm has given such a setting which can be easily adjusted on the machine. Hence it is clear from thiswork that the computational effort taken by TLBO algorithm is very negligible compared to genetic algorithm and proved itscapabilities in the field of parameters optimization of die casting process. The TLBO algorithm has also given the believed tobe global optimum solution in this case which can be treated as success by the foundry personnel to make use of thisadvanced algorithm to obtain the accurate optimum setting for the die casting process with less computational efforts.

6. Conclusions

Mathematical models in terms of input–output process parameters of squeeze casting, continuous casting and die castingprocesses are optimized in this work using the recently developed TLBO algorithm. The processes identified are gettingwidely used in various industries for producing very complicated parts of small and big size with variety of materials. Theseprocesses involve various input parameters which affects the output of respective process. The effectiveness of these pro-cesses depends upon the relations of process parameters with each other and the similar relations in the form of variousmathematical models are used in the present work. The detailed literature survey has proved that there is a good scopefor the use of advanced optimization techniques like TLBO in the field of parameters optimization of these casting processes.Multi-objective mathematical model is considered in the case of squeeze casting process and initially the objectives are


attempted individually where the hardness and tensile strength is improved considerably. A common parameter setting isalso obtained for satisfying both these objectives simultaneously. For continuous casting process, a problem having 10 indi-vidual loss functions and a total loss function are considered which was earlier attempted by using simulated annealing algo-rithm. Even though the original problem of continuous casting process was having 17 loss functions but only 10 lossfunctions are considered in the present work whose complete information is available. Fair comparison of these 10 loss func-tions with the results of the previous researchers is justified by considering the 10 complete models of the respective lossfunctions. The TLBO algorithm has effectively handled the problem and has given significant improvement of above 60%in total loss function compared to the previous result. In the case of two loss functions, the undesirability index is almostreduced to zero.

For die casting process, the model under consideration is having five input parameters and it was earlier attempted byusing genetic algorithm using 1000 generations. Whereas, in the present work, the same model is satisfactorily attemptedwith only 10 generations, thereby drastically reducing the computational efforts. Moreover, the solution obtained is alsobelieved to be a global optimum solution for the die casting process. Thus, the TLBO algorithm have effectively handledthe various mathematical models and proved its capabilities in the field of parameters optimization of casting processes withless computational efforts. The algorithm can be attempted on other types of casting processes also.

Appendix A. Code of TLBO algorithm for the parameters optimization of selected casting processes

To run the TLBO code, user has to create separate MATLAB files for each function (i.e., separate .m file for main, mainline,tlbo and fun) and then the main.m file is to be executed.

%%%%%%%%%%%%%%%% Code for Hardness problem %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% main %%%%%%%%%%%%%%%%%%%%%

clear allclcno_of_run = 50;no_of_student = 10; %specify population sizeno_of_iteration = 20; %specify number of iterationstf = 1;for i = 1:no_of_run

[bvf bvx]=mainline(no_of_student, no_of_iteration, tf);bvf1(i,:)=bvf;bvx1(i,:)=bvx;

endbvf = bvf1(:,1);[bvfmin,k0]=max(bvf)bvxmin = bvx1(k0,:)bvfmin = bvf1(k0,:)

%%%%%%%%%%%%%%%%%%% mainline %%%%%%%%%%%%%%%%%%%%%

function [bvf bvx]=mainline(no_of_student, no_of_iteration, tf)no_of_variable = 3; % specify number of variableslowerlimitofa = 50; % specify the lower boundupperlimitofa = 125; % specify the upper boundlowerlimitofb = 150;upperlimitofb = 300;lowerlimitofc = 15;upperlimitofc = 60;for i = 1:no_of_student% initialization of the variablesa = lowerlimitofa + rand⁄(upperlimitofa-lowerlimitofa);b = lowerlimitofb + rand⁄(upperlimitofb-lowerlimitofb);c = lowerlimitofc + rand⁄(upperlimitofc-lowerlimitofc);

x(i,:)=[a b c];end% limit arraylimit=[lowerlimitofa;upperlimitofa;lowerlimitofb;


upperlimitofb;lowerlimitofc;upperlimitofc];parameter=[no_of_student;

no_of_iteration;tf;no_of_variable];

[bvf,bvx]=tlbo(limit,x,parameter);

%%%%%%%%%%%%%%%%%%%%% tlbo %%%%%%%%%%%%%%%%%%%%

function [bvf,bvx]=tlbo(limit,x,parameter)no_of_student = parameter(1);no_of_iteration = parameter(2);tf = parameter(3);no_of_variable = parameter(4);for i = 1:no_of_student

variable = x(i,:);[funxz]=fun(variable);fxx(i,1)=funxz;

endfor ng1 = 1:no_of_iteration;

m = mean(x);[sfx,k0]=max(fxx);bt = x(k0,:);for i = 1:no_of_student

k = 1;for j = 1:no_of_variable

xs(i,j)=x(i,j)+rand⁄(bt(1,j)-tf⁄m(1,j));if xs(i,j)<limit(k) || xs(i,j)>limit(k + 1)

x1(i,j)=x(i,j);else

x1(i,j)=xs(i,j);endk = j + 2;endvariable = x1(i,:);end

for i = 1:no_of_studentvariable = x1(i,:);[funxz]=fun(variable);fxx1(i,1)=funxz;

endfor i = 1:no_of_student

if fxx1(i,1)<fxx(i,1)fxx1(i,:)=fxx(i,:);x1(i,:)=x(i,:);

endend

[sfx1,k1]=max(fxx1);bs = x1(k1,:);for i = 1:no_of_studentk = 1;for j = 1:no_of_variablexs(i,j)=x1(i,j)+rand⁄(bs(1,j)-x1(i,j));if xs(i,j)<limit(k) || xs(i,j)>limit(k + 1)x(i,j)=x1(i,j);else

x(i,j)=xs(i,j);end

k = j + 2;


endvariable = x(i,:);end

for i = 1:no_of_studentvariable = x(i,:);[funxz]=fun(variable);fxx(i,1)=funxz;end

for i = 1:no_of_studentif fxx(i,1)<fxx1(i,1)

fxx(i,:)=fxx1(i,:);x(i,:)=x1(i,:);end

endend[bvff,k2]=max(fxx);bvf(1,:)=fxx(k2,:);

bvx = x(k2,:);

%%%%%%%%%%%%%%%%%%%% fun %%%%%%%%%%%%%%%%%%%%

function funxz = fun(variable)no_of_variable = 3;funxz = �3.82542 + 0.8787⁄variable(1) + 0.46587⁄variable(2) + 0.30411⁄variable(3) � 0.00393⁄variable(1)⁄variable(1)� 0.00116⁄variable(2)⁄variable(2) + 0.00097⁄variable(3)⁄variable(3) + 0.00051⁄variable(1)⁄variable(2) � 0.00333⁄variable(1)⁄variable(3) � 0.00018⁄variable(2)⁄variable(3);;end

%%%%%%%%%%%%%% Code for Tensile strength problem%%%%%%%%%%%%%%The code is similar to that given above for the Hardness problem. The files of tlbo and main remain the same. However,

the fun file is to be replaced by the following file and also make the appropriate changes in mainline file by specifying vari-ables and the bounds.

%%%%%%%%%%%%%%%%%%%% fun %%%%%%%%%%%%%%%%%%%%%%

function funxz = fun(variable)no_of_variable = 3;funxz = �11.2606 + 2.5778⁄variable(1) + 1.3316⁄variable(2) + 0.7552⁄variable(3) � 0.0116⁄variable(1)⁄variable(1)� 0.0034⁄variable(2)⁄variable(2) + 0.0031⁄variable(3)⁄variable(3) + 0.0015⁄variable(1)⁄variable(2) � 0.0097⁄vari-able(1)⁄variable(3) � 0.001⁄variable(2)⁄variable(3);end

%%%%%%%%%%%%%%%%% Code for total loss function%%%%%%%%%%%%%%%The code is similar to that given above for the Hardness problem. The file main remain the same. However, the fun file is

to be replaced by the following file and also make the appropriate changes in the tlbo file for minimization, and mainline fileby specifying variables and the bounds.

%%%%%%%%%%%%%%%%%%% fun %%%%%%%%%%%%%%%%%%%%

function funxz = fun(variable)no_of_variable = 10;L1 = (variable(1)-2)^2;L2 = 0.0016 ⁄(variable(2)�1050)^2;L3 = 44.5 ⁄(variable(3)�0.3)^2;L4 = 44.5 ⁄(variable(4)�0.3)^2;L5 = 0.0016 ⁄(variable(5))^2;L6 = 6.25⁄10^�6 ⁄(variable(6))^2;L7 = 1.44 ⁄(1/variable(7))^2;L8 = 44.5 ⁄(variable(8)�0.3)^2;L9 = 44.5 ⁄(variable(9)�0.3)^2;L13 = (1/variable(10))^2;funxz = L1 + L2 + L3 + L4 + L5 + L6 + L7 + L8 + L9 + L13;


%%%%%%%%%%%%%%%%%%% Code for porosity %%%%%%%%%%%%%%%%The code is similar to that given above for the Hardness problem. The file main remains the same. However, the fun file is

to be replaced by the following file and also make the appropriate changes in the tlbo file for minimization, and mainline fileby specifying variables and the bounds.

%%%%%%%%%%%%%%%%%%%%%% fun %%%%%%%%%%%%%%%%%%%%

function funxz = fun(variable)no_of_variable = 5;funxz = 1.623–0.766⁄10^�3⁄variable(1) � 1.301⁄10^�3⁄variable(2) � 0.136⁄variable(3) + 0.029⁄variable(4) �1.636⁄10^�3⁄variable(5);end

References

[1] R.V. Rao, V.J. Savsani, D.P. Vakharia, Teaching–learning-based optimization: a novel method for constrained mechanical design optimization problems,Comput. Aided Des. 43 (2011) 303–315.

[2] R.V. Rao, V.J. Savsani, D.P. Vakharia, Teaching–learning-based optimization: an optimization method for continuous non-linear large scale problems,Inf. Sci. 183 (2012) 1–15.

[3] R.V. Rao, V. Patel, An elitist teaching–learning-based optimization algorithm for solving complex constrained optimization problems, Int. J. Ind. Eng.Comput. 3 (4) (2012) 535–560.

[4] T. Niknam, A.K. Fard, A. Baziar, Multi-objective stochastic distribution feeder reconfiguration problem considering hydrogen and thermal energyproduction by fuel cell power plants, Energy 42 (2012) 563–573.

[5] T. Niknam, R.A. Abarghooee, M.R. Narimani, An efficient scenario-based stochastic programming framework for multi-objective optimal micro-gridoperation, Appl. Energy 99 (2012) 455–470.

[6] T. Niknam, R.A. Abarghooee, M.R. Narimani, A new multi objective optimization approach based on TLBO for location of automatic voltage regulators indistribution systems, Eng. Appl. Artif. Intell. 25 (2012) 1577–1588.

[7] T. Niknam, F. Golestaneh, M.S. Sadeghi, H-multiobjective teaching–learning-based optimization for dynamic economic emission dispatch, IEEE Syst. J.6 (2) (2012) 341–352.

[8] V. Togan, Design of planar steel frames using teaching–learning based optimization, Eng. Struct. 34 (2012) 225–232.[9] K.R. Krishnanand, B.K. Panigrahi, P.K. Rout, A. Mohapatra, Application of multi-objective teaching-learning-based algorithm to an economic load

dispatch problem with incommensurable objectives, in: SEMCCO (2011) Part I, LNCS, 7076, 2011, pp. 697–705.[10] R.V. Rao, V.D. Kalyankar, Parameter optimization of machining processes using a new optimization algorithm, Mater. Manuf. Processes 27 (9) (2012)

978–985.[11] R.V. Rao, V.D. Kalyankar, Multi-objective multi-parameter optimization of the industrial LBW process using a new optimization algorithm, in:

Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture, 226(6), 2012, pp. 1018–1025.[12] R.V. Rao, V.D. Kalyankar, Parameter optimization of modern machining processes using teaching–learning-based optimization algorithm, Eng. Appl.

Artif. Intell. 26 (2013) 524–531.[13] R.V. Rao, V. Patel, Multi-objective optimization of heat exchangers using a modified teaching–learning-based optimization algorithm, Appl. Math.

Model. 37 (2013) 1047–1062.[14] R.V. Rao, V. Patel, Multi-objective optimization of two stage thermoelectric cooler using a modified teaching–learning-based optimization algorithm,

Eng. Appl. Artif. Intell. 26 (2013) 430–445.[15] M. Crepinšek, S-H. Liu, L. Mernik, A note on teaching–learning-based optimization algorithm, Inf. Sci. 212 (2012) 79–93.[16] H. Hu, Squeeze casting of magnesium alloys and their composites, J. Mater. Sci. 33 (1998) 1579–1589.[17] S.W. Kim, G. Durrant, J.H. Lee, B. Cantor, The microstructure of direct squeeze cast and gravity die cast 7050 (Al-6.2Zn-2.3Cu-2.3Mg) wrought Al alloy, J.

Mater. Synth. Process. 6 (2) (1998) 75–87.[18] S.W. Kim, G. Durrant, J.H. Lee, B. Cantor, The effect of die geometry on the microstructure of indirect squeeze cast and gravity die cast 7050 (Al-6.2Zn-

2.3Cu-2.3Mg) wrought Al alloy, J. Mater. Sci. 34 (1999) 1873–1883.[19] M. Zhou, H. Hu, N. Li, J. Lo, Microstructure and tensile properties of squeeze cast magnesium alloy AM50, J. Mater. Eng. Perform. 14 (2005) 539–545.[20] J.K. Baek, H.W. Kwon, Effect of squeeze cast process parameters on fluidity of hypereutectic Al–Si alloy, J. Mech. Sci. Technol. 24 (2008) 7–11.[21] Y. Yang, L. Peng, P. Fu, B. Hu, W. Ding, B. Yu, Effects of process parameters on the macrostructure of a squeeze cast Mg-2.5 mass% Nd alloy, Mater. Trans.

50 (12) (2009) 2820–2825.[22] L.I. Run-xia, L.I. Rong-de, B.A.I. Yan-hua, Effect of specific pressure on microstructure and mechanical properties of squeeze casting ZA27 alloy, Trans.

Nonferrous Met. Soc. China 20 (2010) 59–63.[23] A.A. Moosa, K.K. Al-Khazraji, O.S. Muhammed, Tensile strength of squeeze cast carbon fibers reinforced Al–Si matrix composites, J. Miner. Mater.

Charact. Eng. 10 (2) (2011) 127–141.[24] Q. Zhang, M. Masoumi, H. Hu, Influence of applied pressure on tensile behaviour and microstructure of squeeze cast Mg alloy AM50 with Ca addition, J.

Mater. Eng. Perform. 21 (2012) 38–46.[25] P. Senthil, K.S. Amirthagadeswaran, Optimization of squeeze casting parameters for non symmetrical AC2A aluminium alloy castings through Taguchi

method, J. Mech. Sci. Technol. 26 (4) (2012) 1141–1147.[26] F. Zhou, S.K. Gupta, A.K. Ray, Multiobjective optimization of the continuous casting process for poly (methyl methacrylate) using adapted genetic

algorithm, J. Appl. Polym. Sci. 78 (2000) 1439–1458.[27] N. Cheung, A. Garcia, The use of a heuristic search technique for the optimization of quality of steel billets produced by continuous casting, Eng. Appl.

Artif. Intell. 14 (2001) 229–238.[28] N. Chakraborti, R. Kumar, D. Jain, A study of the continuous casting mold using a pareto-converging genetic algorithm, Appl. Math. Model. 25 (2001)

287–297.[29] N. Chakraborti, R.S.P. Gupta, T.K. Tiwari, Optimisation of continuous casting process using genetic algorithms: studies of spray and radiation cooling

regions, Ironmaking Steelmaking 30 (4) (2003) 273–278.[30] N. Chakraborti, K.S. Kumar, G.G. Roy, A heat transfer study of the continuous caster mold using a finite volume approach coupled with genetic

algorithms, J. Mater. Eng. Perform. 12 (2003) 430–435.[31] C.A. Santos, J.A. Spim, A. Garcia, Mathematical modelling and optimization strategies (genetic algorithm and knowledge base) applied to the

continuous casting of steel, Eng. Appl. Artif. Intell. 16 (2003) 511–527.[32] C.A. Santos, N. Cheung, A. Garcia, Application of a solidification mathematical model and a genetic algorithm in the optimization of strand thermal

profile along the continuous casting of steel, Mater. Manuf. Processes 20 (2005) 421–434.

http://refhub.elsevier.com/S0307-904X(14)00206-6/h0005
























































[33] S. Ghosh, K. Mitra, B. Basu, Y.A. Jategaonkar, Control of meniscus-level fluctuation by optimization of spray cooling in an industrial thin slab castingmachine using a genetic algorithm, Mater. Manuf. Processes 19 (3) (2004) 549–562.

[34] M.S. Kulkarni, A.S. Babu, Managing quality in continuous casting process using product quality model and simulated annealing, J. Mater. Process.Technol. 166 (2005) 294–306.

[35] N. Cheung, C.A. Santos, J.A. Spim, A. Garcia, Application of a heuristic search technique for the improvement of spray zones cooling conditions incontinuously cast steel billets, Appl. Math. Model. 30 (2006) 104–115.

[36] B. Filipic, T. Tusar, E. Laitinen, Preliminary numerical experiments in multiobjective optimization of a metallurgical production process, Informatica 31(2007) 233–240.

[37] K. Miettinen, Using interactive multiobjective optimization in continuous casting of steel, Mater. Manuf. Processes 22 (2007) 585–593.[38] A.K. Bhattacharya, D. Sambasivam, A. Roychowdhury, J. Das, Optimization of continuous casting mould oscillation parameters in steel manufacturing

process using genetic algorithms, IEEE Congr. Evol. Comput. (2007) 3998–4004.[39] A.K. Bhattacharya, D. Sambasivam, Optimization of oscillation parameters in continuous casting process of steel manufacturing: genetic algorithms

versus differential evolution, Evol. Comput. (2009) 77–102.[40] E.P.L. Ye, J. Shi, T.S. Chang, On-line bleeds detection in continuous casting processes using engineering-driven rule-based algorithm, J. Manuf. Sci. Eng.

131 (2009) 610081–610089.[41] A.R. Lopez, G.S. Cortes, M.P. Pardave, M.A.R. Romo, R.A. Lopez, Computational algorithms to simulate the steel continuous casting, Int. J. Miner. Metall.

Mater. 17 (5) (2010) 596–607.[42] K. Jabri, D. Dumur, E. Godoy, A. Mouchette, B. Bele, Particle swarm optimization based tuning of a modified smith predictor for mould level control in

continuous casting, J. Process Control 21 (2011) 263–270.[43] G.P. Syrcos, Die casting process optimization using Taguchi method, J. Mater. Process. Technol. 135 (2003) 68–74.[44] P.K.D.V. Yarlagadda, E.C.W. Chiang, A neural network system for the prediction of process parameters in pressure die casting, J. Mater. Process. Technol.

89–90 (1999) 583–590.[45] P.K.D.V. Yarlagadda, Prediction of die casting process parameters by using an artificial neural network model for zinc alloys, Int. J. Prod. Res. 38 (2000)

119–139.[46] J.C. Lin, C.C. Tai, The runner optimisation design of a die-casting die and the part produced, Int. J. Adv. Manuf. Technol. 14 (1998) 133–145.[47] C.C. Tai, J.C. Lin, The optimal position for the injection gate of a die-casting die, J. Mater. Process. Technol. 86 (1999) 87–100.[48] J.C. Lin, Selection of the optimal gate location for a die-casting die with a freeform surface, Int. J. Adv. Manuf. Technol. 19 (2002) 278–284.[49] A. Krimpenis, P.G. Benardos, G.C. Vosniakos, A. Koukouvitaki, Simulation-based selection of optimum pressure die-casting process parameters using

neural nets and genetic algorithms, Int. J. Adv. Manuf. Technol. 27 (2006) 509–517.[50] V.D. Tsoukalas, Optimization of porosity formation in AlSi9Cu3 pressure die castings using genetic algorithm analysis, Mater. Des. 29 (2008) 2027–

2033.[51] G.O. Verrana, R.P.K. Mendes, L.V.O.D. Valentina, DOE applied to optimization of aluminium alloy die castings, J. Mater. Process. Technol. 200 (2008)

120–125.[52] L.X. Kong, F.H. She, W.M. Gao, S. Nahavandi, P.D. Hodgson, Integrated optimization system for high pressure die casting processes, J. Mater. Process.

Technol. 201 (2008) 629–634.[53] M.L.D. Wong, W.K.S. Pao, A genetic algorithm for optimizing gravity die casting’s heat transfer coefficients, Expert Syst. Appl. 38 (2011) 7076–7080.[54] G. Campatelli, A. Scippa, A heuristic approach to meet geometric tolerance in high pressure die casting, Simul. Model. Pract. Theory 22 (2012) 109–122.[55] L. Zhang, R. Wang, An intelligent system for low-pressure die-cast process parameters optimization, Int. J. Adv. Manuf. Technol. 65 (2013) 517–524.









































1-s2.0-S0307904X14002066-main.pdf

Documents

Transcript of 1-s2.0-S0307904X14002066-main.pdf