The experiment involved end milling Custom 450 stainless steel in a dry, room-temperature environment using cutting inserts that were either untreated (UT), cryo-treated with 24 h of soaking (CT 24 h), or cryo-treated with 36 h of soaking (CT 36 h). This was conducted on a CNC vertical machining centre as shown in Figure 1A. The dimensions of 160 mm × 75 mm × 20 mm (4 numbers) were prepared. The chemical composition and mechanical properties of Custom 450 stainless steel were determined (31). The tool holder used was a standard indexable type (Diameter: 16 mm, Length: 150 mm), and the cutting tool insert was a TiAlN-coated tungsten carbide insert (APMT1135PDR YBG205). The primary cutting parameters employed in this study were spindle speed, feed rate, and depth of cut. (31).

The tungsten carbide inserts underwent cryogenic treatment in a specially designed cryogenic chamber (KRYO 550-16), capable of cooling samples down to deep cryogenic temperatures (-196 °C). The chamber’s temperature was gradually reduced from room temperature (RT) to the cryogenic temperature (CT, approximately -196 °C) at a rate of 2 °C/min as shown in Figure 1B. After being maintained at this temperature (-196 °C) for 24 and 36 h, the temperature was gradually raised back to room temperature (Devi and Mahalingam, 2023). Additionally, the current experiment was conducted to thoroughly examine the performance of cryo-treated cutting tools on Custom 450 stainless steel. The objective was to determine the optimal parameters at various spindle speeds (1,500 rpm, 2,300 rpm, 3,100 rpm), feed rates (0.1, 0.15, 0.2 mm/min), and a constant depth of cut (0.5 mm).

The effectiveness of the system is evaluated using Design of Experiments (DOE), which considers several factors (Shanmugasundar et al., 2019a). DOE is an essential data collection and analysis methodology used across various experimental scenarios. It involves a planned experiment analyzing three aspects: factors, levels of these factors, and responses. The primary goal of DOE is to compare multiple solutions and identify the necessary factors to yield the best output response. It facilitates the examination of the impact of various input factors on a desired response. DOE is capable of identifying significant interactions that might be overlooked when examining a component at a time or dealing with multiple inputs simultaneously.

The foundation of the experiment design is the Taguchi-based orthogonal array, which includes three parameters and corresponding levels. This array can be used to develop optimal parameters, minimize treatments, reduce material costs, and determine the contributing factor of each process parameter in machining. The L27 orthogonal array consists of 13 columns, which can be assigned to test factors and their interactions. For a 3-factor 3-level configuration, the total number of tests required is given by 3 3 = 27 . Therefore, a total of 27 experiments are to be conducted in the L27 OA. Table 1 shows the experimental design.

The Multiple Linear Regression Model (MLRM) can be employed in mathematics to calculate the response value (the output) in terms of several parameters (the inputs) (Shanmugasundar et al., 2021a). This study establishes three variants of the MLRM equations—linear, quadratic, and interaction—using experimental data and MATLAB’s “regress” function. These three models are represented in their standard forms as Eq. 1, Eq. 2, and Eq. 3: F i j = c 1 j + c 2 j X 1 i + c 3 j X 2 i + c 4 j X 3 i (1) F i j = c 1 j + c 2 j X 1 i + c 3 j X 2 i + c 4 j X 3 i + c 5 j X 1 i X 2 i + c 6 j X 1 i X 3 i + c 7 j X 2 i X 3 i (2) F i j = c 1 j + c 2 j X 1 i + c 3 j X 2 i + c 4 j X 3 i + c 5 j X 1 i X 2 i + c 6 j X 1 i X 3 i + c 7 j X 2 i X 3 i + c 8 j X 1 i 2 + c 9 j X 2 i 2 + c 10 j X 3 i 2 (3)

Here, F i j ​ represents the calculated values of the i th experiment for the j th response, c are the coefficients of variables, and X are the variables.

Metaheuristics are a popular set of algorithms that have been used in diverse optimization applications (Shanmugasundar et al., 2019b; Shanmugasundar et al., 2021b; Shanmugasundar et al., 2022a; Shanmugasundar et al., 2023a; Shanmugasundar et al., 2023b). In this study, Grasshopper Optimization Algorithm (Meraihi et al., 2021), Grey Wolf Optimization (Mirjalili et al., 2014), Harmony Search Algorithm (Yang, 2009) and Ant Lion Optimization (Mirjalili, 2015) have been used.

Grasshopper Optimization Algorithm (GHO) is inspired by the nymph and adult stages of grasshopper life (Meraihi et al., 2021). The nymph stage involves short, leisurely steps, while the adult stage features long, rapid movements. These behaviors underlie GHO’s intensification and diversification phases. In the exploration phase, GHO updates the position value of each grasshopper in the swarm and evaluates fitness, symbolizing the search for food sources. During the exploitation phase, it identifies the optimal solution among all options, akin to seeking better food sources.

In GHO, each grasshopper represents a solution, and its position ( X i ) is modeled as follows: X i = O i + B i + R i (4) where X i represents the i th solution’s position, O i the interaction between the solution and other swarms, B i the gravitational pull, and R i the wind advection. The GHO algorithm is shown in Algorithm 1.

Algorithm 1

Pseudocode for Grasshopper Optimization Algorithm.

1. Initialize the grasshopper positions X i i = 1 , 2 , … , n

It should be noted that TOPSIS (Hwang and Yoon, 1981), a commonly used multi-criteria decision-making (Shanmugasundar et al., 2022b; Shanmugasundar et al., 2022c) tool is used to convert the f i into R C V i .

In mathematics, the response value (the output) can be determined in terms of many parameters (the inputs) by using the many Linear Regression Model (MLRM). This study uses experimental data and the “regress” tool in MATLAB to establish three variations of the MLRM equations: linear Equation of cutting force, surface roughness and cutting temperature (Eqs 5, 6, 7) respactively. CF = 237.3 − 0.0272 SS − 285 FR − 0.463 CT (5) SR = − 0.750 + 0.0001172 SS + 9.333 FR + 0.00046 CT (6) TT = 392.2 + 0.00701 SS + 313 FR − 0.552 CT (7)

In mathematics, the Multiple Linear Regression Model (MLRM) can be used to compute the response value (the output) in relation to multiple parameters (the inputs). Using experimental data and MATLAB’s “regress” function, this study establishes three variations of the MLRM equations: interaction Equations 8, (9), and (10) respectively. CF = 869 − 0.2610  SS − 3586  FR − 6.83  CT + 1.409 SSFR + 0.001118 SSCT + 31.5 FRCT (8) SR = − 0.765 + 0.000027 SS + 8.95  FR + 0.0151 CT + 0.00081 SSFR − 0.000002 SSCT − 0.0737 F R CT (9) TT = 351.1 + 0.0379 SS − 95 FR − 1.62 CT − 0.181 S S F R − 0.000184 S S C T + 9.94 F R C T (10)

Equations (11)–(13) depict mathematical models for cutting force, surface roughness, and temperature respectively. The developed mathematical models were verified using ANOVA shown in Table 3. CF = 1480.9 − 0.8107 SS − 4046 FR − 6.013 CT + 0.000119 SS 2 + 1531 FR 2 − 0.02371 CT 2 + 1.4095 SSFR + 0.001118 SSCT + 31.45 FRCT (11)

Figure 2 illustrates the relationship between milling operation factors and cutting forces. Increased spindle speed and feed rate correlate with decreased cutting forces, indicating potential areas for parameter optimization. Feed rate significantly influences cutting forces, affecting tool life.

Surface roughness is impacted by feed rate, cryogenic treatment, and spindle speed. Eq. (12) mathematically represents surface roughness: SR = 2.1159 − 0.001670 SS − 6.239 FR − 0.001916 CT + 0.000000 SS 2 + 50.64 FR 2 + 0.000496 CT 2 + 0.000806 SSFR − 0.000002 SSCT − 0.07367 FRCT (12)

Figure 3 shows that cryo-treated inserts exhibit better surface roughness than untreated inserts. Surface roughness decreases with increased spindle speed and increases with feed rate from 0.1 to 0.15 mm/min and further with 0.2 mm/min.

The mathematical model for temperature (Eq. (13)) includes spindle speed, feed rate, and cryogenic treatment as independent variables: TT = 777.0 − 0.1654 SS − 3069 FR − 4.148 CT + 0.000044 SS 2 + 9911 FR 2 + 0.0737 CT 2 − 0.1812 SSFR − 0.000184 SSCT + 9.94 FRCT (13)

Figure 4 demonstrates that cryo-treated cutting tools outperform untreated inserts in terms of cutting temperature.

The MLRS Linear, Interaction, and Quadratic performance of CF, TT, and SR are shown in Figure 5. The CF and TT quadratic equations respond more favorably than the linear interaction. Compared to the linear interaction, Figure 5 illustrates how well the performs on the SR quadratic equation.

The design factors that most affect the three quality attributes—manufacturing cost, tool life, and workpiece quality—are found using ANOVA. A high F-value indicates a high efficiency level for the parameter. Another important factor is the p-value, which allows statistical analysis to be completed within a 95% confidence interval. Consequently, the parameter affects the response if the computed value is less than 5%. According to Table 3. Cutting force, surface roughness, and temperature all had p values less than 0.05, indicating significant effects from each parameter on the output responses. Similarly, input parameters have a considerable impact on the output responses (square of FR, CT excluded), but no significant influence is seen on cutting force from square of FR, CT and no significant impact is seen on surface roughness from any of the interactions.

The graphical representations of the association between response and cutting variables occur under specific cutting settings that were indicated in the experimental design table. Therefore, it is essential to comprehend how cutting conditions affect cutting variables in order to better clarify these interactions. Figure 6 displayed the combined effect of feed rate and cutting speed on CF fluctuations on three-dimensional graphs. The cutting force reduced as the feed rate and cutting speed increased. The cutting force reduced as the SS increased in the surface plot of CT and SS. Pareto chart (Figure 7) indicates the least significant impact of feed rate repetition on force in the X-direction. The primary effect plot shows reduced cutting force with increased spindle speed. The statistical analysis confirms the significant impact of milling parameters on CF during titanium alloy milling. The response surface plots demonstrate these effects, highlighting improved thermal conductivity due to cryogenic treatment, influencing the heat transfer from the tool’s tip and affecting cutting forces.

When attempting to determine the best pairings between two sets of data, it can be helpful. By drawing an analogy with a topographic map, the color and patterns in Surface Charts show the places that fall within the same range of values. Three variables are often used to generate a 3D Surface Plot: X, Y, and Z. Surface plot for CF.

The surface plot of SR vs FR, SS in Figure 8 illustrates the effect of process parameters on surface roughness. Initially, the surface roughness was higher, but as the cutting speed increased, it decreased and then increased again. Surface roughness rises in dependence on the FR. According to the Pareto chart in Figure 9, spindle speed and feed rate significantly influence surface roughness. The production of built-up edges (BUE) at specific spindle speeds reduces Ra. The SEM image in Figure 10 shows imperfections and micro-voids on machined surfaces caused by carbide particles and BUE. Cryo-treated inserts perform better, reducing heat generation and improving surface finishing. (39).

Effect of process parameters on temperature is shown in Figure 11. Cutting temperature not decreases much. The Pareto diagram (Figure 12) suggests a uniform impact of feed rate and spindle speed on temperature. Residual analysis confirms a consistent pattern across milling operations without specific residue patterns, indicating the reliability of the experimental setup. The results validate the model’s suitability for analyzing cryo-treated cutting tool residues across various milling performance variables. The residual plots demonstrate the significant impact of input parameters on TT during machining.

The efficiency of ALO is compared with GHO, GWO, and HSA. ALO shows superior performance in end milling applications, achieving optimal parameter settings at a lower iteration, as illustrated in the convergence plot (Figure 13). The TOPSIS approach integrated the output responses from 22 runs into a single objective, confirming the normal distribution of output data from all algorithms. The p -value of >   0.05 further substantiates the reliability and consistency of the ALO results.

Figure 14 reveals the impact of high heat loads and cutting zone temperature on chip morphology, with thermal softening facilitating machining at high temperatures. Increasing the feed rate results in more serrated chip teeth, especially with cryo-treated cutting tool inserts at high spindle speeds and feed rates. (36,41).

Table 4 presents the optimal process parameters and response values obtained from 22 runs with a population size of 100 and 100 iterations for all algorithms. Table 5 shows the statistical significance of these algorithms.

The IGD metric is utilized to compare the effectiveness of Grasshopper Optimization (GHO), Grey Wolf Optimization (GWO), Harmony Search Algorithm (HSA), and Ant Lion Optimization (ALO) Algorithms. This measure indicates the computational complexity of the algorithms, as detailed in Khalilpourazari et al. (Khalilpourazari et al., 2020). The IGD is calculated using the Euclidean distance formula: IGD = ∑ i = 1 n   Ed i 2 n (14) where E d i = ∑ j = 1 n o   R i j − R b j 2 (15)

In this formula, R i j represents the i th run j th response value, R b j is the best j th response value, and E d i is the Euclidean distance. The process involves calculating the Euclidean distance for each run and then computing the IGD as the sum of each Euclidean distance squared, divided by the number of runs (27 in this case).

Table 6 shows the IGD values for the selected algorithms. It is evident from the IGD values that the ALO algorithm outperforms GHO, GWO, and HSA.

The Friedman test, a non-parametric alternative to the parametric two-way analysis of variance, is used to identify significant differences in the behavior of the algorithms. This test assumes the null hypothesis of equality of medians between the populations.

The test statistic value is calculated using the method described in (Derrac et al., 2011; Zhang et al., 2023). The Friedman statistic ( F f ) is computed as: F f = 12 r a a + 1 ∑ j R j 2 − a 2 a + 1 2 4 (16)

Here, a is the number of algorithms, r is the number of runs, and R j represents the ultimate rank for each algorithm. The test statistic is compared against the F-distribution table value. If the computed F f is higher than the table value, the result is considered significant.

The process for calculating the ranks involves:

1. Compiling the observed results for each algorithm/problem pair.

The Friedman statistic F f is then calculated under the null hypothesis that all algorithms perform similarly, and their ranks R j should be equal. (36).

According to Table 6, the IGD value for ALO is lower than for the other algorithms, indicating superior performance. The convergence graphs for Cutting Force (CF), Surface Roughness (SR), and Temperature (TT) in Figure 12 further demonstrate the effectiveness of the ALO algorithm compared to GHO, GWO, and HSA. Therefore, it is concluded that ALO outperforms GHO, GWO, and HSA.