1 Introduction

A significant challenge in the development of materials with high strength to weight ratio in material sciences and engineering remains even today. Following the fast-paced development of science and technology, aluminium and its alloys have become a pertinent area of constant research, consistently improvising and imparting practical applications due to their low density, high strength, and high ductility. Al composite/hybrid composites are fabricated through mixing of two or more materials that chemically and physically exist in distinct phases. Al2O3, SiC, B4C, Mg2, Si, TiC, etc. reinforcement particles are added to the base materials to enhance their properties. Hybrid metal matrix composites (HMMCs) can be sorted into a type of reinforced metal matrix composites and are also extensively used as composite materials [1, 2]. HMMC’s constitute an important member of a group of structural materials that find widespread applications in automotive, defense, and aerospace due to low density, high specific strength and modulus, excellent wear resistance, higher service temperature, and comparatively higher physical and mechanical properties than monolithic and composite materials[3].

However, machining of HMMCs is a very challenging task in the present scenario because of the hard, reinforced, abrasive particles present in their matrix. They pose significant difficulties in machining through traditional machining processes [4]. Milling, drilling, shaping, turning, and sawing are some of the conventional machining methods used for machining of HMMCs. But, none of the conventional machining methods, when used to machine HMMC’s, can yield high dimensional accuracies and can become very expensive to produce such results. Apart from these reasons, defects like cracks and voids on the machined surface of HMMC are commonly found in samples machined through conventional routes. [5, 6]. The replacement for machining HMMCs with unconventional machining methods such as laser beam machining, plasma machining, water jet machining, electrical discharge machining, wire electrical discharge machining, etc. is being widely explored [7].

AAJM is one such adaptable and accepted nontraditional machining process for cutting, trimming, drilling, and deburring hard and brittle materials like ceramics, composites, and HMMC. AAJM is highly advantageous compared to conventional machining due to a high degree of machining flexibility, low machining forces, and less heat-affected zone (HAZ). Another advantage of AAJM is that the cutting cost is nominal compared to other conventional machining processes. Thamizhvalavan et al. studied the machinability characteristics of Abrasive Aqua Jet (AAJ) on Al 6063/ B4C/ZrSiO4 composites with 5% of fixed reinforcement particles added to the matrix material. The results testified that maximum MRR and minimum Ra and kerf TA were obtained at the higher abrasive flow rate, aqua WJP and lower TS [8]. Manoj et al. considered the TiB2 reinforcement particles as reinforcements into Al 7075 matrix and fabricated the Al 7075/TiB2 composite for AWJ machining. WJP, TS and SOD have been selected as machining factors to estimate the response factors like TA, MRR and Ra by adopting the Taguchi-DEAR approach. They found that the significance of WJP was greater than other factors on TA, MRR and Ra of AWJ machining [9].

Dhanawade et al. investigated the effect of AWJM on carbon epoxy composite. RSM approach was used to develop a mathematical model and to find out the significance of control factors to influence the responses. It concluded that the fitted mathematical model for Ra is in good agreement with the experimental result [10]. Siddiqui and Shukla applied a combined approach by Taguchi-PCA to evaluate the performance measures of AAJM with multiple quality characteristics [11]. Sasikumar et al. optimized the AAJM parameters on Al 7075/TiC/B4C using RSM-Box Behnken Design (BBD) approach. The performance parameters of study considered are Ra, kerf top and kerf bottom against the AAJM input factors, namely SOD, TS and WJP. The results confirmed that the most significant contributing factors are TS and WJP respectively and also regression models for kerf characteristics are developed [12]. Kumar et al. examined the titanium-carbon fiber-reinforced plastics-titanium hybrid laminate materials using AWJM with TS, WJP, and SOD as input parameters. They employed the RSM-CCD approach to analyze the influence factors on quality characteristics of MRR and Ra. It has been observed that WJP and TS had maximum influence on the MRR and Ra [13].

Iqbal et al. considered the factorial experimental layout to examine the AAJM process factor’s influence on performance characteristics like % of striation free surface, cutting width, the surface texture of Al 2219 and AISI 4340 [14]. Kumar et al. investigated the parametric effect of AWJM on Ra of Inconel 718 alloy using a combined RSM-BBD approach and successfully formulated quadratic models of Ra in terms of WJP, SOD. The results established an excellent interrelationship between experimental and predicted values at 95% confidence level. They concluded that TS and abrasive flow rate substantially affected the Ra. WJP was insignificant in their studies but WJP's interactions were reported to be highly significant [15]. Ravi Kumar et al. studied the AAJM factors on Al/WC composite using the RSM-CCD approach. In their process, the input factors like SOD, TS and percentage of TiC were considered. They confirmed that MRR was most affected by the TS followed by the percentage of TiC and SOD respectively: at the same time, Ra was highly influenced by the percentage of TiC, followed by TS and SOD [16].

Furthermore, a little to scarce amount of scientific studies were carried out on hybrid optimization techniques of various processes. Tofigh et al. studied compocast process parameters optimization of nano A356/Al2O3 composite with neuro-fuzzy inference system and particle swarm optimization. They have also carried out investigations on the abrasive wear behavior of Al/B4C composite with a pin-on-disc machine using hybrid particle swarm optimization-artificial neural network (PSO–ANN) technique. The results confirmed that the novel search technique has eliminated premature convergence problems that are common with high dimensional problems [17, 18]. Shabani et al. optimized the process parameters of pressure-assisted semi-solid processing of A360/TiC nano composite using the ANFIS-PSO technique. It was noted that the trained model could behave erratically in unseen input conditions and was not easy to interpret in case of inadequate training dataset; since, in such hybrid models the learning was entirely data-driven with stringent requirements on the quality of the training dataset [19].

Akbari et al. examined the microstructural and mechanical properties of A356/B4C composites using FSP. The FSP parameters optimization was selected as an artificial neural network (ANN) and non-dominated sorting genetic algorithm-II (NSGA-II). The results confirmed that a strong relationship was established between the FSP parameters and microstructural and mechanical properties of the composites [20]. Shabani et al. implemented the ANFIS-PSO algorithm to predict the experimental results and optimize the processing parameters of Al/Al2O3 composites and Al composite. ANFIS-PSO algorithm considered for the casting process had resulted in enhancement of the results [21]. Shamsipour et al. evaluated the wear behavior of optimized parameters of Al/TiC nanocomposite using ANFIS and PSO techniques [22].

The literature survey indicates that some research work was published on the machining of Al 7075 composite with the AAJM process [8, 9]. Furthermore, very scarce or no research has been carried out on Al 7075/B4C/Gr with the RSM-CCD approach. The present work investigates the effect of with process parameters of AAJM on hybrid Al 7075/B4C/Gr composite. The main objective of the study is the multi-parametric optimization of AAJM process parameters using the RSM-CCD-TOPSIS approach. WJP, SOD, TS are selected as AAJM process input factors. The AAJM output parameters like TA, Ra and MRR are investigated. The importance of the AAJM input parameters on machining output characteristics was examined by TOPSIS. TOPSIS is one of the multi-criteria decision-makers. It is based on the criteria weights that are assigned by the support on the information and numerical data, to determine the rating of alternatives and for evaluation, prioritization, and selection. This technique compares a set of experimental conditions on predetermined performance characteristics. The best input factors and their percentage values were found out using the ANOVA tests. Furthermore, a mathematical relationship between the measured performance characteristics and the AAJM process input factors is formulated. Finally, using the multi-objective approach of TOPSIS has been carried out within the tested range of AAJM parameters.

2 Experimental procedure

2.1 Material fabrication

The Al 7075 alloy is considered as the parent material. The elemental composition of Al 7075 alloy is presented in Table.1. In the current investigation, B4C and Gr are used as reinforcement particles. B4C and Gr reinforcement particle sizes are below 25–30 µm as depicted in Figs. 1 and 2 respectively. The preparation of hybrid composite material was done by mixing B4C and Gr particles with parent metal, and the B4C incrementally varying at 4% by weight and Gr particles being kept constant at 3% by weight in the matrix. The rotary stir casting technique is used to fabricate the HMMCs, as depicted in Fig. 3. The fabricated HMMCs chemical composition is mentioned in Table.2.

Table 1 Chemical composition of Al 7075 alloy
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Fig.1
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SEM image of graphite powder (Gr) particles

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Fig.2
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SEM image of boron carbide (B4C) powder particles

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Fig.3
figure 3

Stir casting machine for fabrication of composite

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Table 2 Chemical composition of hybrid Al 7075/B4C/Gr alloy
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2.2 Machine tool

The experiments are conducted on an Abrasive Aqua Jet Machine of MJ TT model with a CNC controller (Make: Citizen, India). The input factors like WJP, SOD and TS are considered and maintained at equal levels as mentioned in Table 3. During the entire cutting process, the machining parameters of AAJM such as orifice diameter, impact angle and nozzle diameter are kept unchanged. All AAJM machined samples are cut as a square hole of 15 × 15x10 mm dimension as shown in Fig. 4.

Table 3 Levels of input factors
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Fig.4
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AAJ machining setup

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2.3 Performance evaluation

For this study, the cutting performance evaluations are considered based on the output measurements like TA, Ra and MRR. The TA is measured for taperness of the AAJM machined composite surface, as it is a significant output measurement and is necessary to have a minimum kerf taper maintained during the cutting by AAJM to maintain dimensional stability. The TA is measured (top and bottom) using the Coordinate Measuring Machine (CMM). Ra is measured by using handy surf equipment at various locations on machined samples (Make: Zeiss, India, Model: E-35B). To calculate the MRR by weight, differences of the before and after machined-sample of the composite are measured by an electronic digital weighing machine (Make: Citizen, India, Model: CY 204) with least count of 0.0001 g.

2.4 Design of experiments (DOE)

The experiments are designed using RSM-CCD plan to arrive at the best optimal AAJM parameters for Al7075/B4C/Gr composite.

2.4.1 RSM for single objective optimization

RSM as a subcategory of design of experiments (DOE) technique widely used in the planning, design, and optimization of the manufacturing processes. It is a statistical tool that is highly capable of analyzing and optimizing the problem [23]. One of the essential statistical RSM-CCD techniques is used for the present investigation, which has the accompanying points of interest: CCD approach isn't just compelling in foreseeing the reaction of the fitted model along with a minimal measure of investigations. It assists in the investigation of the collaborations between various factors as well. Besides, it can assess the elements of the quadratic model productively and maintain a strategic difference from treatment mixes at an extreme range. The fundamental reason for the adoption of this approach is to investigate the connection between the input parameters and the output measurements on the hybrid Al 7075/B4C/Gr composite machining.

In the CCD technique, for the construction of a second-order polynomial model, as given by Eq. (1), each factor is varied at three levels (+ , 0, -). The specific advantage of this technique is that it is not necessary to run experiments in all combinations of factors when the number of factors becomes three. The design can be executed using a fraction of the total number of variable combinations. The possible design options can be either regular fractional or minimum run experimental resolution. It has a smooth function that enhances the best performance characteristics of a particular study and hence, it eliminates the unwanted parameters/factors. It reduces the effect of noise and allows for the use of derivative-based algorithms [24, 25].

$$Z = b_{0} + \mathop \sum \limits_{{n = 1}}^{n} \left( {b_{i} Y_{i} } \right) + \mathop \sum \limits_{{i = 1}}^{n} \left( {b_{{ii}} Y_{i} ^{2} } \right) + \mathop \sum \limits_{{i = 1}}^{{n - 1}} \mathop \sum \limits_{{j = 1}}^{n} \left( {b_{{ij}} Y_{i} Y} \right)$$
(1)

The RSM models are developed for the sustainable measures of optimum response values using Design Expert® 11.0 (DOE) statistical software. The aim is to identify the best response values and these are influenced by variable dependent factors from the DOE. Three input factors and three levels are selected with the RSM-CCD layout. The input factors are WJP, SOD and TS. The number of input factors and their values can be observed in Table 3. For the AAJM process, TA and Ra are “the lower the better” characteristics and MRR is the “higher the better” performance characteristic.

In general, the results of single measured characteristics can be compensated by any one of the performance measured characteristics in the AAJM cutting process. For that cause, the best condition of the multi-performance measures is tougher than the single performance measures. In the present work, an attempt was made to investigate the best multiple degree characteristics of the AAJM process from the RSM-CCD plan through the TOPSIS approach.

2.4.2 TOPSIS for multi-objective optimization

In 1995, Hwang and Yoon initially developed the TOPSIS technique to evaluate the preference for an order using similarity for an ideal solution. A theoretical solution to the problem is seen as an ideal solution when the values of all attributes correspond to the minimum values of the attributes. TOPSIS targets to get the best minimum and maximum way of the distance from the positive solution and negative solution (hypothetically best and hypothetically worst) arranged and subsequently the development of the preorder through similitude with the perfect arrangement.

The following steps are involved in the TOPSIS approach [26, 27].

Step 1: Evaluate the RSM-CCD experiments through the performance measurements, namely TA, Ra and MRR.

Step 2: Design a decision matrix as shown below, which consists of ‘m’ alternatives and ‘n’ attributes.

Step 3: Calculate the normalized decision matrix of each response measurement using Eq. (2)

$$r_{{ij}} = \frac{{x_{{ij}} }}{{\sqrt {\mathop \sum \nolimits_{{i = 1}}^{m} x_{{ij}}^{2} } }}\;j = 1,{\text{ }}2, \ldots ,{\text{ }}n$$
(2)

where

\(x_{{ij}}\) represents the actual value of \(i^{{th}}\) value for \(j^{{th}}\) experimental run.

Step 4: Find out the weighted normalized decision matrix considering Eq. 3.

$$V_{{ij}} = ~W_{j} \times ~r_{{ij}}$$
(3)

where

rij is a normalized decision matrix.

Wj has associated weights for matrix.

Vij is constructed weighted normalized decision matrix.

By using equation number (3) i.e., multiplying the normalized decision matrix with its associated weights, the weighted normalized decision matrix is further developed [28, 29]. The standard deviation approach is applied for the present study, to find the unbiased weight of individual measurements. Based on the TOPSIS approach the significant task is to find the distribution of weights. Ram Prasad et al. adopted a measurement level of standardization to calculate weights with help of the standard deviation method [30].

Step 5: The subsequent step is to find out the best (positive Vj+) and worst (negative Vj−) solutions with the help of the following Eqs. 4 and 5.

$$V_{j}^{ + } = \left\{ {\mathop \sum \limits_{{i = 1}}^{{\max }} V_{{ij}} ~/j~ \in J,\mathop \sum \limits_{{i = 1}}^{{\min }} V_{{ij}} ~/j \in J^{l} } \right\}$$
(4)
$$V_{j}^{ - } = ~\left\{ {\mathop \sum \limits_{{i = 1}}^{{\min }} V_{{ij}} ~/~j \in J,\mathop \sum \limits_{{i = 1}}^{{\max }} V_{{ij}} ~/j \in J^{l} } \right\}$$
(5)

Step 6: The next step is to work out the separation of individual alternatives from the best ideal Si+ solution and worst ideal Si− solution using the following Eqs. 6 and 7.

$$S_{i}^{ + } = ~\sqrt {\mathop \sum \limits_{{j = 1}}^{n} \left( {V_{{ij}} - V_{j}^{ + } } \right)^{2} } ,~~i = 1,{\text{ }}2, \ldots ,{\text{ }}m$$
(6)
$$_{i}^{ - } = \sqrt {\mathop \sum \limits_{{j = 1}}^{n} \left( {V_{{ij}} - V_{j}^{ - } } \right)^{2} } ,\;i = 1,{\text{ }}2, \ldots ,{\text{ }}m$$
(7)

Step 7: The Final step is the calculation of closeness coefficient (Ci+) for individual performance measures adopting the following Eq. 8.

$$C_{i}^{ + } = \left( {\frac{{S_{i}^{ - } }}{{\left( {S_{i}^{ + } + S_{i}^{ - } } \right)}}} \right)$$
(8)

3 Results and discussion

This section can be divided into three parts. The first part deals with single performance characteristics optimization (3.1), and the second part discusses multiple objective parameters optimization using the TOPSIS approach (3.2) and the final part verifies and discusses the conformation experiments of Al 7075/B4C/Gr composite material (3.3). The RSM-CCD experimental layout and their results are presented in Table 4.

Table 4 RSM-CCD layout and their experimental values
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3.1 Single performance characteristics optimization

3.1.1 Effect of parameters on material removal rate (MRR)

The ANOVA result for MRR is as shown in Table 5. The compliance of P-value less than 0.05 levels i.e., at 95% significance reveals a significant model. The model F-value at 183.37 endorses that the model is exceptionally significant. The “Lack of Fit” F-Value standing at 1.20 is evidence that the lack of fit is insignificant. The Lack of Fit F-Value has a chance of occurrence due to noise at 43.9%. The R2 value and Adj. R2 are in great agreement with 98.83% and 98.29% respectively. From Table 5 it can be inferred that the F-values of A, B, C and interactions AC and BC are quite significant and influence the MRR to a great extent.

Table 5 ANOVA for MRR of Al/B4C/Gr composite material
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The mathematical model for the MRR is Eq. 9.

$${\text{MRR}} = 25.530 - 0.05669*A - 8.506*B - 0.3371*C + 1.935*B*B + 0.001974*A*C - 0.03284*B*C$$
(9)

The 3D graphs of MRR for the significant interaction between input factors, namely WJP, SOD and TS are shown in Fig. 5a and b, respectively. It testifies that MRR approaches its maxima at the higher levels of SOD, TS and to its minima at WJP. The MRR was observed to be diminishing with the WJP. However, it should be noted that the proper interaction of WJP and TS are necessary to make a significant effect on MRR [31].

Fig.5
figure 5

Response plot for MRR (mm3/min)

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In the present investigation, due to proper interactions between WJP and TS, low levels of WJP were sufficient to induce the impact and kinetic energy that is passed on to the target point of the work-piece material that resulted in B4C and Gr particles being easily pulled out of the composite material. Similar type of results were obtained in the studies led by some of the coauthors, they reported improved cutting by enhancing the water WJP. In the same way it was observed that MRR increases with increasing the value of SOD. This may be due to the high impact energy of the abrasive on the hybrid composite material which results in an ineffective cut of the MMC and enhances the MRR Enhancement in TS directly influences the MRR of the hybrid composite [31, 32]. At the same time, as hard reinforcement particles were eroded rapidly or pushed out at higher TS led to the achievement of maximum MRR [33]. The optimum condition of MRR is obtained at the A1B3C1. It has been confirmed that lower values of WJP (210 bar), higher values of SOD (3 mm), and lower values of TS (10 mm/min) gives the maximum MRR value at 5.3155 mm3/min. On the other hand, the combination of WJP and TS has proven to be most significant in determining the MRR.

3.1.2 Effect of parameters on surface roughness (Ra)

ANOVA results for Ra are presented in Table 6. The agreement of P-value at less than 0.05 levels, i.e., at 95% significance notifies that the model is a significant model. The model F-value at 231.57 validates that the model is exceptionally significant. F-value standing at 4.52 of ‘Lack of Fit’ confirms its insignificance. Furthermore, the lack of fit may occur at a 5.7% probability due to noise. R2 and Adj. R2 are complimentary at 99.07% and 98.65% respectively. The ANOVA Table 6 suggests that WJP and TS have been major factors to control the Ra.

Table 6 ANOVA for Ra of Al/B4C/Gr composite material
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The mathematical model for Ra is formulated Eq. 10.

$$Ra = 62.45 - 0.5770*A - 0.987*B + 0.2960*C + 0.001367*A*A + 0.00600*A*B - 0.001150*A*C$$
(10)

The 3D response plots of Ra with the best interaction input parameters, WJP and SOD, WJP and TS are depicted in Figs. 6a and b respectively. Ra is directly proportional to WJP, SOD and TS. Normally, higher WJP caters more energy to the surface of the composite which influences the desired contact area that leads to the formation of more voids [34]. Whenever the WJP offers suitably high energy to reinforcement particles, the machining process is acceptable to be passed out without stern jet flirtation which will indicate the less waviness pattern on the surface [35]. Similarly, the higher Ra is obtained for higher SOD because of the higher external drag from the surroundings of the target. The enhancement in SOD makes the jet to enlarge before hitting the target material, thus reducing the kinetic energy of the jet. Related to another factor, Ra increased with TS; higher TS will hinder the total process of the kerf wall. It can be inferred that, when TS enhances, less number of hard reinforcement particles catch the part in machining. So, the availability of a large number of hard particles poses a problem of large minor voids leading to rough Ra [12, 31]. The higher TS give less machining action and increases the time of cutting the surface. The AAJM has to penetrate the machining area to improve the cutting area. Finally, interaction plots revealed that at lower conditions of WJP (210 bar), SOD (2 mm) and TS (10 mm/min), the minimum value of the Ra (2.65 μm) was obtained. The optimum condition of the Ra (A1B1C1) was indicated at the lowest levels of input parameters.

Fig.6
figure 6

Response plot for Ra (µm)

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3.1.3 Effect of parameters on taper angle (TA)

ANOVA results for TA are observed in Table 7. The agreement of P-value at less than 0.05 levels, i.e., at 95% significance informs that the model is a significant model. The model F-value at 54.45 theorizes a significant fit. F-value upright at 7.52 of ‘Lack of Fit’ confirms its insignificance. Furthermore, the lack of fit may occur at a 2% probability due to noise. R2 and Adj. R2 are mutually comparable at 99.65% and 95.17% respectively. From the table, it can be inferred that the major influencing factors were SOD and WJP.

Table 7 ANOVA for Taper Angle of Al/B4C/Gr composite material
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The empirical model for TA is formulated as Eq. 11.

$${\text{TA}} = - 28.81 + 0.2742*A - 1.511*B - 0.1321*C - 0.000626*A*A + 0.1962*B*B + 0.00357*A*B + {\text{ }}0.000602*A*C$$
(11)

The 3D graphs of TA with the efficient interaction input parameters, WJP and TS, WJP and SOD, are shown in Figs. 7a and b respectively. From Fig. 7a, it is understood that TA increases with WJP and TS. As increased WJP produces more kinetic energy of the water jet impinging onto the hybrid composite material, the TA generated will be large. The kinetic energy of the inert abrasive particles increases at maximum hydraulic pressure and enhances the capability for machining [36]. The TA increases with increase in SOD as depicted in Fig. 7b. It is clearly observed that higher SOD tends to achieve a higher TA. The higher SOD allows the water jet to expand together with the normal density of abrasive particles on the outer surface of the divergent AAJ before impinging into the work-piece [31, 32]. Higher and moderate SOD results in minimum divergent and maximum sharpening of the jet, respectively and thus reducing the TA [31]. This explains that TA increases with an increase in TS. However, this leads to a reduction in the cutting ability of the jet due to the damage of the backend abrasive particles. TA is proportional to the kinetic and impact energy of the abrasive mixture and HMMC reinforcement percentage. TA had decreased due to the collisions of the reinforcement particles, this is due to high the hardness of the composite and high energy absorption rate by reinforcement particles in the Al matrix material [37, 38]. This in turn absorbs the impact energy and kinetic energy available in the matrix for machining and reduces taper angle. It was observed that at minimum TA of 0.375 was obtained at the lowest levels of WJP and the medium level and higher levels of SOD and TS. The optimum condition of TA was obtained at 210 bar WJP, 2.5 mm SOD and 30 mm/min TS (A1B2C3).

Fig.7
figure 7

Response plot for Taper Angle (00)

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3.2 Multi-objective optimization using TOPSIS approach

For multi-objective optimization, multiple parameters are considered and weights according to their importance are assigned and ranked against each other to find the best solution. This is essentially done by the TOPSIS Approach. The detailed steps of this approach are explained below.

Step1: For this study TA and Ra are considered as lower the better values (non-beneficial attributes), at the same time, MRR is selected as higher the better values (beneficial attributes).

Step 2: To represent in the form of a decision matrix all the experimental data available on the attributes as observed in Table 4.

Step 3: Calculation of the normalized decision matrix is as represented in Table 8.

Table 8 Normalized decision matrix
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Step 4: Weighted normalized decision matrix is calculated as observed in Table 9. The attributed factor weights were selected as TA = 0.6027, Ra = 0.2925 and MRR = 0.103. The weights of attributes sum must be equal to 1.

Table 9 Determination of weighted normalized decision
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Step 5: To achieve the best and worst ideal solutions.

$$V_{j}^{ + } = \left\{ {0.0283,~0.0475,~0.0388} \right\}$$
$$V_{j}^{ - } = \left\{ {0.0175,~0.0827,~0.02082} \right\}$$

Step 6: To achieve the separation measures, the calculations are in Table. 10

Table 10 Separation measures for positive and negative ideal solutions
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Step 7: Finding out the relative closeness to the best condition. From the Table the relative closeness coefficient condition is 0.8958, matching to the pilot condition no 15 is observed, at input factor settings of A = 210 bar, B = 2 mm and C = 30 m/min.

Step 8: The assigned TOPSIS ranking scores (Ci) are presented in Table 11.

Table 11 Relative closeness values and their TOPSIS rankings
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Then ANOVA is applied for the TOPSIS model and it was observed that important input factors are WJP (55.21%), SOD (23.36%), TS (2.56%), with the interaction of WJP and SOD (6.01%) for AAJM. The effect of major influencing input factors on the multi-objective performance of values, ranks and divisions are mentioned in Table 11. Based on the ANOVA hybrid RSM-TOPSIS results, it was found that WJP plays a significant role in AAJM. Hybrid RSM-TOPSIS response plots and ANOVA values are shown in Fig. 8 and Table 12. A1B1C3 settings of AAJM factors were selected as the optimum levels of factors.

Fig.8
figure 8

Response plot for RSM-TOPSIS

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Table 12 ANOVA for TOPSIS model of Al/B4C/Gr composite material
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3.3 Confirmation experiments

Table 13 shows the confirmation test results for the best conditions of AAJM input factors to check the enhancement in the output performance measures. From the confirmation test results, it is verified that the residual value is below 5% of the initially conducted RSM-CCD measured values [39, 40]. The hybrid TOPSIS test results indicating that the best input factor setting is A1B1C3 has been successfully verified. Table 13 indicates the conformation of the single RSM–CCD results and hybrid RSM-CCD-TOPSIS results of AAJM for hybrid Al 7075/B4C/ Gr is represented.

Table 13 Confirmation experimental results
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4 Conclusion

The present work is carried out to find the best solution of AAJM optimum output measurements of Al 7075/B4C/Gr with the RSM-TOPSIS method. The result indicated from the present experimental study is extremely helpful for selecting the optimum machining conditions for Al 7075/B4C/Gr composite, and the following conclusions can be drawn.

  1. 1.

    The optimum MRR was indicated with (A1B3C1) optimum condition concerning lower water jet pressure (210 bar), higher Stand-off distance (3 mm), and lower traversing speed (10 mm/min). The occurred maximum MRR value is at 5.311 mm3/min and corresponding values of R2 value and Adj. R2 are 98.83% and 98.29% respectively.

  2. 2.

    The desirable lower Ra was obtained at 2.65 μm and conditions were (A1B1C1); 210 bar jet pressure, 2 mm Stand-off distance and 10 mm/min traversing speed and correlating values of R2 value and Adj. R2 are 99.07% and 98.65% respectively.

  3. 3.

    The optimum value for taper angle was 0.375°, and dominating input factors are set at (A1B2C3): lower jet pressure (210 bar), Stand-off distance (2.5 mm), and higher traverse speed (30 mm/min), corresponding values of R2 value and Adj. R2 are 99.65% and 95.17% respectively.

  4. 4.

    The corresponding value recommended through the RSM-TOPSIS method indicated that optimum lower input factors were jet pressure and Stand-off distance and higher factors were traversing speed, it will give the best experimental setting for experimental condition 15 and corresponding values of R2 value and Adj. are 99.91% and 95.81% respectively.

  5. 5.

    The RSM-TOPSIS test results indicated that the best input factor setting is A1B1C3 and from ANOVA results, it is observed that distributing important input factors are jet pressure (55.21%), Stand-off distance (23.36%), and traverse speed (2.56%).

  6. 6.

    From the Hybrid RSM-TOPSIS approach results was observed that the TA was 49.6% less than the value predicted by the RSM-CCD approach. These AAJ machining results will be beneficial for applications in the automobile, aerospace, and structural engineering sectors.