Multi-attribute optimization and influence assessment methodology of casting process parameters combined with integrated MADM and Taguchi method

Many practical casting process optimization problems are ascribable to multi-attribute optimization problems with multiple conflicting quality attributes. This paper proposes a reasonable multi-attribute optimization and influence assessment methodology of casting process parameters combined with integrated multi-attribute decision-making (MADM) and Taguchi method. The proposed methodology consists of the following steps: (1) design experiment arrangement based on Taguchi orthogonal array, and measure multiple quality attributes of castings at every experimental trials, (2) calculate comprehensive quality score (CQS) values of the experimental trials using some MADMs, (3) calculate final CQS (FCQS) values of the experimental trials by integrating the CQS values from some MADMs using integrated MADM, (4) calculate mean FCQS values of the casting process parameters at different levels, (5) calculate ranges of mean FCQS values and influence indices of the casting process parameters, and (6) determine optimal casting process parameters to maximize the comprehensive quality of the castings. To demonstrate its usage and effectiveness, it is applied to one illustration example to determine optimal four casting process parameters such as pouring temperature (PT), degasser amount (DA), holding time (HT) and mould type (MT) for optimizing three quality attributes such as density of casting (D), ultimate tensile strength (UTS) and elongation to fracture (Ef) of A356 Al alloy sand castings. The proposed methodology could be widely applied to not only casting process optimization but also various advanced manufacturing process optimization problems.


Introduction
In casting processes, there are various casting process parameters, and quality of castings are affected from the casting process parameters.Optimization of casting process is a very important problem in foundry practice.
To optimize the casting process parameters, different approaches have been used.Trial and error method demands abundant experiments, and it requires a huge amount of funds, labor and time.To overcome these drawbacks and optimize the casting process parameters using modern optimization methods, Taguchi method and intelligent heuristic optimization methods such as genetic algorithm (GA), simulated annealing (SA) method and swarm particle optimization (SPO) method have been widely used.Among the methods, Taguchi method has been most widely used to solve the engineering optimization problems as it is simple, effective and practical.
Syrcos [1] determined the reasonable die casting process parameters (piston velocities at first and second stages, metal temperature, filling time and hydraulic pressure) to optimize the casting density of AlSi9Cu13 aluminum (Al) alloy castings using Taguchi method.Vijian et al. [2] optimized the squeeze cast parameters of LM6 Al alloy for improving surface roughness using Taguchi method.Verran et al. [3] optimized three injection process parameters (slow shot, fast shot and up set pressure) for the density of the die casting SAE 305 alloy parts using Taguchi method.Kumar et al. [4] optimized the green sand process parameters (green strength, moisture content, pouring temperature and mould hardness) for the cast iron differential housing cover to minimize the casting defects using Taguchi method.Hsu et al. [5] determined the pressure die casting process (molten alloy temperature, die temperature, plunger velocities in the first and second stage, and multiplied pressure in the third stage) to minimize the porosity formation using Taguchi method.Senthil et al. [6] determined the optimal process parameters (die temperature, die insert materials, pouring temperature, squeeze pressure and pressure duration) for improving the yield strength of squeeze cast AC2A alloy using Taguchi method and GA.Apparao et al. [7] optimized the die casting process parameters (injection pressure, molten metal temperature, plunger velocities at first and second stages, and die temperature) using quality function deployment and Taguchi method in order to yield the optimum casting density of the A380 alloy.
Many practical casting process optimization problems are ascribable to multi-attribute optimization problems with multiple conflicting quality attributes.
Cai et al. [8] performed the modeling and optimization of the chemical composition (C, Cr, Si and Mn) of as cast chromium white cast iron for hardness and impact toughness of castings using a green sand mould by multiple quadratic regressions, radial basis function ANN and Taguchi method.Senthil et al. [9] improved the ultimate tensile strength and hardness of AC2A alloy by controlling the process parameters such as die temperature, die insert materials, pouring temperature, squeeze pressure and pressure duration using Taguchi method.Mohiuddin et al. [10] determined the optimal process parameters for the density, ultimate tensile strength and percentage elongation of Al7SiMg alloy castings using Taguchi method.Santhosh et al. [11] optimized the ductile iron casting process parameters such as such as pouring temperature, inoculation, carbon equivalent, moisture content, green compression strength, permeability and mould hardness for various casting defects and the rejection rate using Taguchi method.Karthik et al. [12] optimized the squeeze casting process parameters (squeeze pressure load, die and melt temperatures) of AA2219 alloy for density and hardness using Taguchi method.Refaai et al. [13] optimized the stir casting variables (contents of multiwalled carbon nanotubes and magnesium, and stirring time) for the hardness and ultimate tensile strength of AA7149 alloy in production of multiwalled carbon nanotubes AA7149 composite using Taguchi method.
As above mentioned, in the almost practical casting process optimization problems, there are multiple responses (performance attributes) of castings to be optimized, not only one response.Therefore, it is surely necessary to optimize the multiple responses, simultaneously.It becomes multi-objective optimization (MOO) or multi-response optimization (MRO) problem.The MOO or MRO problem could be solved by converting it to single objective optimization (SOO) or single response optimization (SRO) problem.
To convert the multiple responses into single overall response, weighted combination method, desirability function method, loss function method, fuzzy logic approach, and multi-attribute decision making (MADM) methods have been used [14].Bashiri et al. [14] reviewed some methods for MRO and described the method for converting the multiple responses to single response using four MADMs such as VIse Kriterijumska Optimizacija Kom,.,promisnoResenje (VIKOR), preference ranking organization method for enrichment evaluations (PROMETHEE), elimination and et choice translating reality (ELECTRE) and technique for order preference by similarity to ideal solution (TOPSIS), and compared the optimization results from individual MADMs about one numerical example.They calculated the scores of the alternatives using the four MADMs, and then respectively developed the final multiple quadratic regression models between the controllable variables and scores.Based on the final regression models, they obtained and compared the optimal solutions from the individual MADMs.Sun et al. [15] proposed the optimization method for design of gating system parameters of a cylindrical magnesium casting using Taguchi method.Four gating system parameters (ingate height, ingate width, runner height and runner width) were determined to optimize the multiple performance characteristics (filling velocity, shrinkage porosity and product yield).To convert the multiple characteristics into a single response, the objective function was developed by weighted combination of S/N ratios for the individual characteristics.Patel et al. [16] optimized the squeeze casting process parameters such as squeeze pressure, die and pouring temperatures for both responses such as surface roughness and casting density using Taguchi method and grey relational analysis (GRA).They obtained the single process parameter from both responses using GRA, and then optimized the squeeze casting process parameters using Taguchi method.Pattnaik et al. [17] conducted the modeling and optimization of the investment casting process by combining the desirability function approach and fuzzy logic.The injection temperature, injection pressure and die temperature were considered as process parameters, and the linear shrinkage and surface roughness as the responses for the simultaneous optimization.A fuzzy model was developed with desirability values of responses as inputs and multi-performance index (MPI) as output.The optimal levels of the process parameters were determined by means of the MPIs.Kittur et al. [18] carried out the multi-response optimization of pressure die casting process (fast shot velocity, injection pressure, phase changeover point and holding time) for porosity, surface roughness and hardness using desirability function approach.In the desirability function-based optimization method, the individual responses were converted into individual desirability values using desirability function, and then the overall desirability was calculated by geometric mean of the individual desirabilities.Krishna et al. [19] optimized the influential parameters (type of reinforcement, particle size and weight percentage) on the tensile strength, impact strength and density of AlMg1SiCu hybrid metal matrix composites fabricated by stir casting technique using Taguchi method and fuzzy approach.The input varaibles of the fuzzy logic system were the tensile strength, impact strength and density, and the ouuput variable was overall fuzzy grade.The optimum levels of influential parameters were determined based on overall fuzzy grades.Perumal et al. [20] optimized the tribological properties of hybrid Al matrix composites fabricated by stir casting technique based on the multiple performances such as wear rate, specific wear rate and coefficient of friction using GRA.The GRA was used to convert the multi-responses to a single overall response.Anbuchezhiyan et al. [21] determined the optimal process parameters (particle size, mass fraction and stirring speed) of hollow glass microsphere reinforced magnesium alloy syntactic foams under vacuum die casting in consideration of four mechanical properties (hardness, compressive strength, porosity and density of the syntactic foams) using Taguchi method and GRA method.The GRA was used to convert four responses into single response (grey relational grade).Murugarajan et al. [22] carried out the MRO of the pressure die casting process (injection pressure, shot velocity and furnace temperature) for A413 Al alloy in consideration of two responses such as micro-hardness and surface roughness using central composite face centered design, multiple quadratic regression model and desirability function approach.Papanikolaou et al. [23] calculated the comprehensive performance of the gating systems from four attributes such as air entrainment, surface defect concentration, process energy and mould cost using TOPSIS and selected the optimal casting gating system design with respect to sustainability of the process.Logesh Kamaraj et al. [24] developed the regression model based on composite desirability function and artificial neural network (ANN), and carried out the MRO of the ultrasonic-assisted stir-casting process parameters such as stirring speed, ultrasonic vibration time and depth of ultrasonic vibration for improving the porosity, tensile strength and wear rate properties of the Al hybrid composites using the desirability function method.The composite desirability values for each experimental run were calculated using geometric mean value of the individual desirability values for each output responses.
Pattnaik et al. [25] applied utility concept and Taguchi method for the MRO of wax patterns fabricated by investment casting process.Pattnaik et al. [26] also applied the approach based on Taguchi method and GRA for MRO of wax patterns made by the investment casting process.
For multiple response optimization, some previous works performed the optimizations for individual responses, respectively.As the result, they had inconsistent and conflicting optimization results with one another, and it is a challengeable problem.
To address this problem, many works first converted the multiple responses into single comprehensive response, and then determined the optimal process parameters to optimize the single response.To convert the multiple responses into single comprehensive response, the previous works have applied different MADMs such as TOPSIS, GRA, VIKOR, PSI and so on [27][28][29][30][31][32].
Although many MADMs [33][34][35] are applicable to convert the multiple responses into single comprehensive response, the different MADMs may generate nonnegligible differences in the single responses in a given problem, and therefore the optimization results may differ with one another, and it may generate different influence assessment results of the process parameters.As the individual MADMs have their own principles and features, it is impossible to point out and conclude which is the better MADM method and which is worse one.
It may be more reasonable to determine final results by aggregating the results obtained from the individual MADMs for the given problem by using a reasonable aggregating method, and then perform the optimization of process parameters and influence assessment based on the integrated final results.However, the previous aggretaging methods can determine only final evaluation ranks based on the ranks obtained from individual MADMs, but they lack in determining final evaluation values [36,37].
In this work, we propose an integrating MADM method by combining with the results obtained from different MADMs and considering priority weights of each MADMs, and then we propose a reasonable multi-attribute optimization and influence assessment methodology of casting process parameters using integrated MADM and Taguchi method.We applied the proposed methodology to optimize the sand casting process parameters and evaluate their influences for A356 aluminum alloy (Al7%SiMg) casting to demonstrate the effectiveness.

Methods
This section proposes a reasonable multi-attribute optimization and influence assessment methodology of casting process parameters combined with integrated MADM and Taguchi method.
The details of the proposed methodology are as follows: Step 1: Design experiment arrangement based on Taguchi orthogonal array (OA), and measure multiple quality attributes of the castings at every experimental trials.
(1-1) Select the multiple quality attributes of castings and main controllable casting process parameters that affect the quality of castings.
Let the casting quality attributes be y 1 , y 2 ,..., and y L , and the main controllable casting process parameters be x 1 , x 2 ,..., and x p (L≥2, p≥2), where L and p are the numbers of casting quality attributes and casting process parameters, respectively.
(1-2) Set the levels of the casting process parameters.
Table 1 shows the casting process parameters and their levels.
In Table 1, LV lj is the value of l-th level for j-th casting process parameter ( j = 1, p, l = 1, H ), and H is the number of levels.
(1-3) Select the appropriate OA suited to the number of the casting process parameters and their levels, and arrange the casting process parameters on each column of the OA.
Table 2 shows the selected OA and arrangement of casting process parameters.
In Table 2, n is the number of experimental trials, LI ij is j-th level index of i-th experimental trial, and it is integer between 1 and H.
(1-4) Measure the values of casting quality attributes at every trials by implementing the experimental trials according to the OA.
Table 3 shows the experimental arrangement of the casting process parameters according to OA and corresponding measurement values of quality attributes of the castings.
In Table 3, x ij is the value of j-th casting process parameter for i-th experimental trial, and y ik is the corresponding measurement value of k-th quality attribute The values of casting process parameters at every experimental trials construct a matrix X = (x ij ) n × p , and the values of casting quality attributes construct a matrix (decision matrix) Y = (y ik ) n × L .
Step 2: Calculate the comprehensive quality score (CQS) values of the castings by considering multiple quality attributes for the experimental trials using some MADMs.
(2-1) Construct a normalized decision matrix Z = (z ik ) n × L from the decision matrix For the normalization of the decision matrix, the following linear min-max normalization formula is used.
where SB and SC are the sets of the indices for the benefit and cost attributes, and y kmin and y kmax are the minimum and maximum values of k-th attribute, respectively.The element u ik is calculated using the following formula: w h e r e w k i s t h e we i g h t o f k-t h a t t r i b u t e (w 1 + ⋯ + w k + ⋯ + w L = 1), and they can be determined using entropy weighting method.The formula to calculate the attribute weights using entropy weighting method is as follows ( k = 1, L): where (2-3) Calculate the CQS values {V 1 , ⋯, V i , ⋯, V n } for the castings at every experimental trials using the MADM method.
The CQS values consist of the simple weighted sum values in SAW, the relative closeness values in TOPSIS method, the grey relational degrees in GRA method, the VIKOR indices in VIKOR method, the net outranking flows in PROMETHEE method, and the rank sum ratios in rank sum ratio (RSR) method [33-35, 38, 39].
The CQS value comprehensively reflects the multiple quality attributes of the castings.
Commonly, the CQS values from almost MADMs belong to [0, 1], and the higher the value, the better the quality of the casting.However, the CQS values from a few MADMs such as PROMETHEE and VIKOR don't satisfy this condition, and it should be transformed that it belongs to [0, 1].
To do this, we introduce a modified PROMETHEE method, where the net outranking flow is modified as follows: where F + i and F − i are respectively positive and negative outranking flows.
Also, we introduce a modified VIKOR method, where the VIKOR index is modified as follows: (2) where U i and R i are the utility and regret measures respectively, and The modified net outranking flow and modified VIKOR index belong to [0, 1], and the higher the value, the better the quality of the casting.
Step 3: Calculate the final CQS (FCQS) values of the castings for every experimental trials by aggregating the CQS values from M MADMs (Integrated MADM method).
Let V m1 , V mi ,..., V mn be the CQS values of the casting for n experimental trials using m-th MADM method, where V mi is the CQS value of the casting from i-th experimental trial using m-th MADM method ( m = 1, M,i = 1, n ), and M is the number of the MADMs.They consist of CQS matrix The method to calculate the FCQS values in consideration of the CQS values V m1 , ⋯ , V mi , ⋯ , V mn ;m = 1, M from M MADMs is as follows: (3-1) Construct the comprehensive quality rank (CQR) matrix R = (r mi ) M × n from the CQS matrix V = (V mi ) M × n , where r mi is the rank of CQS value of the casting for i-th experimental trial using m-th MADM method.(3-2) Calculate the mean values of Spearman's rank correlation coefficients (RCCs) between the CQR vectors using each MADM method and other MADM ones as follows: where which is the Spearman's RCC between the CQR vectors r m = (r m1 , ⋯, r mi , ⋯, r mn ) and r k = (r k1 , ⋯, r ki , ⋯, r kn ) from mth and k-th MADMs.The larger the mean of the RCCs is, the more the CQRs from the corresponding MADM method are similar to the CQRs from the other MADMs.
(3-3) Determine the priority weights m ;m = 1, M of the MADMs by normalizing the mean values of the RCCs as follows: (3-4) Calculate the final comprehensive quality scores (FCQSs) of the castings from every experimental trials as follows: (3-5) Determine the final comprehensive quality ranks (FCQRs) of the castings from every experimental trials from FCQSs.The above-mentioned mothod to calculate the FCQS values combined with M MADMs is called integrated MADM method.
Step 4: Calculate t he mean FCQS values S lj ;j = 1, p, l = 1, H of the casting process parameters at the different levels.
S lj is the mean FCQS value of j-th casting process parameter in its l-th level, and it is calculated as follows: where δ ij (l) is the status variable if the level index of j-th casting process parameter at i-th experimental trial is l, and it is expressed as follows: where LI ij is j-th level index of i-th experimental trial.
Step 5: Calculate the ranges of mean FCQS values and influence indices of the casting process parameters.
The ranges of mean FCQS values are calculated as difference between maximum FCQS and minimum FCQS as follows: ( j = 1, p) For each casting process parameter, the ranges of mean FCQS values represent the influences of the casting process parameter on the final comphrehensive quality of the castings.
Therefore, the influence index of the casting process parameter is defined as follows: ( j = 1, p) The higher the value, the higher the influence of the casting process parameters.
Table 4 shows mean FCQS values of casting process parameters at the different levels, their ranges and influence indices.
Step 6: Determine optimal casting process parameters to maximize the comprehensive quality of the castings with FCQS as optimization function. (10) (6-1) Determine the optimum level-combination H, which is the set of levels to maximize the mean FCQS values.
(6-2) Determine the optimal values of the casting process parameters x 1 *,..., x j *,..., x p * as follows: where LV h j j is the value of h j -th level for j-th casting process parameter.
The above-mentioned method is called multi-attribute optimization and influence assessment methodology of casting process parameters combined with integrated MADM and Taguchi method.
Figure 1 shows the flowchart of the proposed methodology.
To compare the results from different MADMs, mean rank correlation coefficient (MRCC) and mean absolute rank deviation (MARD) are introduced.
MRCC is the mean vaue of the rank correlation coefficients between the CQSs obtained from each MADM method and other MAMDs, and it is calculated using the following formula: where R mk is the rank correlation coefficient between CQSs C m1 ,…, C mi ,…, C mn obtained from m-th MADM and CQSs C k1 ,…, C ki ,…, C kn obtained from k-th MADM.
MARD is the mean value of the absolute deviations between the CQRs obtained from each MADM and other MCMDs, and it is calculated using the following formula: The larger the value of MRCC (the smallar the value of MARD), the better CQR from the MADM is coincided with CQRs from the other MADMs, and the higher is the performance of the MADM.

Results and discussion
In this section, the proposed methodology is applied to optimize the sand-casting process parameters and evaluate their influences for A356 aluminum alloy (Al7%SiMg) casting.(18) In the sand casting process, the parameters that affect the quality of castings are classified into moulding sandrelated parameters (type of sand, grain size, grain shape, size distribution), mould-related parameters (mould type, moisture content, binder content, number of strokes), gatingrelated parameters (casting design, gating & risering, pouring height, pouring time), and melting-related parameters (pouring temperature, holding time, metal composition, degassing) [10].
This work selects four melting-related parameters (pouring temperature (PT), degasser amount (DA) and holding time (HT)), and one mould-related parameter (mould type (MT)) as the main controllable process parameters, and keeps the other parameters to constant values.DA is the amount of degasser used for degassing of the castings, and it is percentage of the amount of metal used for melting.For evaluating the quality of the castings, three quality attributes such as density of casting (D), ultimate tensile strength (UTS) and elongation to fracture (E f ) are selected [10].
Table 5 shows the casting processes parameters and their levels.
In Table 5, the mold types 1, 2 and 3 respectively indicate the sodium silicate, dry, and air-set sand moulds.Calculate mean FCQS values of the casting process parameters at different levels

Calculate ranges of mean FCQS values and influence indices of the casting process parameters
Determine optimum level-combination, which is the set of levels to maximize the mean FCQS values Determine optimal values of the casting process parameters according to the optimum level-combinations As the number of process parameters is 4 and the number of levels is 3, Taguchi OA L 9 (3 4 ) is selected for design of experiment.Table 6 shows Taguchi OA L 9 (3 4 ).At each experimental trail according to Taguchi OA L 9 (3 4 ), two test samples are prepared and the average value of D, UTS and E f of the samples are taken.
Figure 2 shows the sketch with main dimensions of the casting and mould for experiment.
Three types of moulding sands are used to prepare the moulds for experiment.The sodium-silicate sand mould is prepared using silica sand (SiO 2 ) and 1.5-6% of sodium silicate solution (water glass or silica gel) as binder.It is prepared and hardened by blowing CO 2 gas at a pressure of 1.5 kg/cm 2 for a few seconds.The dry sand mould is prepared using silica sand, bentonite powder (binder) -6% of sand, water (moisture) -7% of sand, and left to dry for 48-72 h.The air-set sand mould is prepared using silica sand and fast curing chemical mixture in three parts: 1 st part (alkyd resin binder) -2% of sand, 2 nd part (accelerator) -3-8% of 1 st part, and 3 rd part (cross link agent) -15-20% of 1 st part.Required quantity of 1 st and 2 nd parts are premixed, and added to the sand and mulled for 2-4 min.Then 3 rd part is added and mulled for 2 min again, and the mould is immediately made.
For experiment, 9 moulds are prepared according to Taguchi OA L 9 (3 4 ).The A356 alloy is melted in lift out crucible type coke fired pit furnace for pouring.After melting, the degassing is performed using C 2 Cl 6 tablets which are added to the molten metal which liberates Cl gas to flush out the dissolved hydrogen by creating the partial pressure over the melt.The moulds are poured and allowed to solidify.After solidification, the moulds are broken to get the castings.
The chemical composition of A356 alloy used for experiment is as follows: Si-7.42%,Mg-0.568%,Fe-0.22%,Zn-0.019%,Pb-0.012%,Ti-0.007%,Cu-0.005%,Mn-0.003%,Sn-0.002%,Ni-0.001%,Al-remainder For measuring UTS and E f using Universal Testing Machine, the produced castings are machined using CNC machine to prepare the test specimens according to ASTM A370 standard (12.5 mm round tension test specimen with gauge length of 50 mm and grip distance of 160 mm). Figure 3 shows the sketch with main dimensions of the test specimen.
The density of the casting is determined based on Archimedes principle (by weighing test samples in air and water).
Table 7 shows the experimental data according to OA L 9 (3 4 ).
The weights of casting quality attributes are calculated using entropy weighting method.The calculated weights of D, UTS and E f are respectively 0.343465, 0.297001 and 0.359533.
To convert the three quality attributes into single CQS, the six popular MADMs [33-35, 38, 39] such as SAW, TOP-SIS, GRA, modified VIKOR, modified PROMETHEE and RSR methods are used in this work.
Table 8 shows the normalized decision matrix.Tables 9  and 10 show the CQSs and CQRs of nine experimental trials  obtained from six MADMs.Figure 4 shows the bar graph of CQSs of nine experimental trials using six MADMs.Tables 9, 10 and Figure 4 demonstrate that the CQSs and CQRs from different MADMs are not coincide with one another.
Table 11 shows the MRCC values between the CQR vectors using each MADM and other MADMs, and the priority weights of six MADMs.
Table 12 shows the FCQSs and FCQRs of nine experimental trials using integrated MADM method.Figure 5 shows the bar graph of FCQSs of nine experimental trials.
Table 13 shows the MRCCs, MARDs of the MADMs and their ranks.Table 13 illustrates that the FCQS from the proposed integrated MADM method has the maximum MRCC and minimum MARD compared with the results from other MCDMs.It demonstrates that the integrated MADM method is a reasonable method to determine the FCQSs of the castings by combining some MCDMs.
Table 14 shows the casting process parameters and the FCQS values at nine experimental trials.
Table 15 and Figure 6 show the mean FCQSs of the casting process parameters at the different levels and their ranges.
In Table 15, the bold typed numbers indicate the maximum mean FCQSs at four casting process parameters.
From Table 15 and Figure 6, we can find that the ranking of the influences of the cating process parameters on the comprehensve quality of the casting is as follows: DA (51.743%) > PT (19.623%) > HT (17.732%) > MT (10.902%).
Table 15 demonstrates that DA is the most effective process parameter (51.743%) on the comprehensive quality of the Al7SiMg alloy sand castings, and the next are PT, HT and MT.From Table 15 and Figure 6, we can also find that the optimal levels of the casting process parameters are PT at 1st level, DA at 2nd level, HT at 1st level and MT at 2nd level, and the optimal level-combination is as follows: Therefore, the optimal values of the casting process parameters using integrated MADM and Taguchi method are as follows: PT: 690 o C, DA: 1 %, HT: 2 min, MT: dry sand mould.
To confirm the optimal casting process parameters (PT: 690°C, DA: 1%, HT: 2 min, MT: dry sand mould), we develop three multiple quadratic regression models for D, UTS and Ef based on data from Table 7, and then calculate the values of D, UTS and Ef by substituting the optimal process parameters (PT= 690, DA=1, HT=2, MT=2) into the regession models.The calculated values of D, UTS and Ef are respectively 2634 kg/m 3 , 89.214 MPa and 5.12%.Moreover, the value of quality index Q is 195.604MPa.The quality index Q, which is used for evaluating the quality of the cast Al alloy based on tensile mechanical properties, is calculated using the following formula [40,41]: where the coefficient k is equivalent to 150 MPa for A356 Al alloy.For nine experimental trials in OA, the values of quality index Q are respectively 195. 23, 193.56, 129.45, 168.67, 165.66, 132.73, 171.37, 192.64 and 113.18.The confirmation experiment demonstrated that the quality level of the castings according to the optimal casting process parameters were better than nine experimental trials.
The PT of the molten metal should be lower under the condition that satisfies a sufficient fluidity.It is because the lower the PT, the smaller the shrinkage amount and the gas content become.Therefore, at the PT of 690°C, the shrinkage amount of Al alloy and the gas content during melting may become smaller than both PTs of 720 and 750°C.
The degassing is performed to flush out the absorbed hydrogen gas from the molten metal by creating the partial pressure [10].For degassing, the hexachloroethane (C 2 Cl 6 ) tablets are added to the molten metal.The following equation shows the absorption of hydrogen gas in the molten aluminium.
The hydrogen gas liberated in the reaction occupies the interstitial spaces of metal leading to porosity if not removed completely.The following equation shows the chemical reaction that liberates the stable chlorine gas, thus flushing hydrogen from the molten metal [10].
Hence, the DA added to the melt plays an important role in removing the hydrogen gas.More increase or more decrease of DA leads to porosity formation, and it affects the quality of the castings.From the result of this work, the optimal DA for degassing is 1%, and it enables to produce the castings which are dense with an excellent mechanical properties and quality.
One of the important factors affecting to the content of gas is the water content in the molding mixture in the sand-casting process.Therefore, the dry sand mould with lower water content may have better mechanical properties of the castings than other types of MT.
The illustration example considered four casting process parameters PT, DA, HT and MT to improve three quality attributes D, UTS and E f for the sand casting process.
When we have to take into account other important casting process parameters such as filling speed and preheating temperature, and other important casting quality attributes such as surface roughness, hardness, porosity, impact toughness, wear rate and filling ability for various casting processes, we have to select them as the main controllable process parameters and quality attributes at Step 1-1 in Section 2.Then, we can determine the optimal casting process parameters to improve the quality attributes using the proposed methodology for the given casting process.

Conclusions
This paper proposed a reasonable multi-attribute optimization and influence assessment methodology of casting process parameters combined with integrated MADM and Taguchi method, and applied it to optimize the casting process parameters and assess their influences for A356 alloy sand casting to illustrate its effectiveness.
As the result, the following conclusions could be drawn: The integrated MADM method can evaluate reasonable final comprehensive quality scores of castings in full consideration of multiple quality attributes by combining the results obtained from multiple MADMs and the priority weights of the individual MADMs.
The proposed methodology can determine optimal casting process parameters and influence indices of the process parameters by converting the multiple attributes optimization problem to single object optimization problem using the integrated MADM.
The advantages of the proposed methodology compared with other methods are as follows: The proposed methodology can evaluate the final comprehensive quality of castings more accurately and more reasonably using the integrated MADM combined with different MADMs, while the previous methods use only one MADM.Although some previous works tried to apply different MADMs to a given casting process optimization problem, they only compared the results with one another, and didn't determine final results aggregated the results from individual MADMs.
The proposed methodology can determine not only the final comprehensive evaluation ranks but also final comprehensive evaluation scores by aggregating the results obtained from individual MADMs, while the previous aggregating methods produce only final comprehensive evaluation ranks.
The proposed methodology can consider the priority weights of each MADMs to improve the reasonability and scientific accuracy of the final evaluation results, while the previous works didn't consider the priority weights.
The proposed methodology can exactly evaluate the influences of the casting process parameters using the influence indices, quantitatively.
The proposed methodology could be widely applied to not only casting process optimization but also various advanced manufacturing process optimization problems in practice.When we have to optimize the other manufacturing process, we have to select appropriate process parameters and quality attributes related to the manufacturing process at Step 1-1 in Section 2, and then we can determine the optimal manufacturing process parameters by using the proposed methodology.

Fig. 1
Fig. 1 Flowchart of the proposed methodology

Fig. 2
Fig. 2 Sketch with main dimensions of the casting and mould

Table 1
Casting process parameters and their levels

Table 4
Mean FCQS values of casting process parameters at the different levels, their ranges and influence indices mi ,…, Cr mn obtained from m-th MADM and CQRs Cr k1 ,…, Cr ki ,…, Cr kn obtained from k-th MADM.

Table 5
Casting process parameters and their levels

Table 12
Fig. 5 Bar graph of FCQSs of nine experimental trials using integrated MADM method

Table 14
Casting process parameters and FCQS values at nine experimental trials