Multi-Objective Optimization of Skd11 Steel Grinding Process using Entropy and RAM Methods

Grinding is a prevalent method employed for machining products demanding high precision within the mechanical processing industry. This study focuses on the multi-objective optimization of the SKD11 steel grinding process on a surface grinding machine. An experimental procedure, encompassing a total of nine trials, was designed utilizing the Taguchi method. Within each trial, variations were introduced to three cutting parameters: workpiece velocity, feed rate, and depth of cut. Concurrently, measurements were conducted for four objectives, also designated as criteria: surface roughness (Ra), cutting force component along the x-axis (Fx), cutting force component along the y-axis (Fy), and cutting force component along the z-axis (Fz). The ENTROPY method was deployed for the computation of criteria weights, while the RAM method was utilized to resolve the multi-objective optimization problem. The results yielded optimal values for workpiece velocity, feed rate, and depth of cut, corresponding to 10 m/min, 4 mm/stroke, and 0.01 mm, respectively. Associated with these optimal cutting parameter values, the objective values for Ra, Fx, Fy, and Fz were determined to be 0.49 mm, 18.4 N, 15.2 N, and 28.4 N, respectively.

Introduction

Grinding is a widely prevalent machining method in mechanical manufacturing [1]–[4]. This method is frequently employed for processing products that necessitate high precision [5]–[7]. To fully capitalize on the advantages of the grinding method, conducting research on optimizing the grinding process is an essential undertaking [8]–[10]. Numerous studies have been carried out to optimize the multi-objective grinding process, aiming to simultaneously ensure that multiple parameters of the grinding process achieve desired values. Published research also demonstrates that scientists have applied various algorithms to address the multi-objective optimization problem, as well as diverse methods for calculating the weights of objectives. Within the scope of this paper, it is not feasible to summarize all published studies; rather, only a selection of recent research on this topic will be considered.

The Nead-Mean algorithm, integrated into DESIGN EXPERT V7.1.3 software, was utilized to optimize the multi-objective grinding process of EN-8 steel, concurrently aiming for minimal surface roughness and maximal material removal rate, with equal weights assigned to both criteria, specifically 0.5 [11]. In Trung et al. [12], the DEAR algorithm was also employed to optimize the multi-objective grinding process of SAE420 steel, where the optimization goal was to simultaneously minimize surface roughness and grinding machine spindle vibration along the x, y, and z axes; in this study, the weights of the objectives were also calculated using the DEAR algorithm itself. The MOORA and COPRAS algorithms were applied to optimize the multi-objective grinding process of SKD11 steel, concurrently aiming for minimal surface roughness and maximal material removal rate, with the weights of these two criteria determined by the Entropy method [13]. The GA algorithm was utilized to optimize the grinding process of Pinus sylvestris material, aiming for minimal surface roughness [14]. The GA algorithm was also employed to optimize the multi-objective grinding process of SKD11 steel, where the weights of three objectives – surface roughness, grinding time, and deviation between actual and desired grinding depth–were equally assigned, specifically 1/3 [15]. The DEAR algorithm was used to optimize the multi-objective grinding process of AISI 4140 steel, simultaneously aiming for minimal surface roughness and maximal material removal rate, with the weights of the objectives calculated using the DEAR algorithm itself [16]. The TOPSIS algorithm was employed to optimize the multi-objective grinding process of DIN 1.2379 steel, concurrently aiming for minimal surface roughness, minimal grinding machine spindle vibration along the x, y, and z axes, and maximal material removal rate, with the weights of the objectives equally assigned, specifically 0.2 [17]. In Danh et al. [18], the Nead-Mean algorithm was also used to optimize the multi-objective grinding process of Hardox 500 steel, concurrently aiming for minimal surface roughness and maximal material removal rate, with the weights of these two criteria equally assigned. The PSO algorithm was utilized to optimize the multi-objective grinding process of D2 tool steel, simultaneously aiming for maximal material removal rate and minimal dimensional error, with the weights of these two parameters not explicitly defined [19]. The desirability functional approach (DFA) algorithm was employed to optimize the multi-objective grinding process of AISI 4140 steel, concurrently aiming for minimal cutting heat, maximal material removal rate, and minimal machining cost, with the weights of these three parameters equally assigned, specifically 1/3 [20]. A meta-heuristic algorithm was used to optimize the multi-objective grinding process of AISI 316 stainless steel, simultaneously aiming for minimal surface roughness, minimal shape deviation, and maximal material removal rate, with the weights of these three parameters assigned based on the subjective perspective of the optimization problem solver [21]. In Xiao et al. [22], the GA algorithm was also employed to optimize the multi-objective grinding process of optical BK7 diamond grinding, where the weights of two objectives, surface roughness and specific energy, were not explicitly addressed, etc.

The brief summary of several studies above indicates that various algorithms and weight calculation methods have been utilized to optimize the multi-objective grinding process. However, there appears to be a gap in the literature regarding the integration of the ENTROPY method for calculating criteria weights and the RAM algorithm for optimizing the multi-objective grinding process. This gap has motivated the current research.

Materials and Methods

Experimental Setup

The experimental specimens were fabricated from SKD11 steel, with dimensions of 40 mm in length, 25 mm in width, and 8 mm in height, respectively. The chemical composition of some major elements of this steel type is summarized in Table I. A surface grinding machine, model APSG-820/8A manufactured by Taiwan, was utilized to conduct the experiments. Surface roughness was measured using a roughness tester, model SJ-201, produced by Japan. Cutting force components were measured using a dynamometer from KISTLER, Germany. To mitigate the influence of random errors on the measurement results, each experiment involved measuring the response parameters (surface roughness and cutting force components) at least three times. The response value for each experiment was determined as the average of these consecutive measurements.

Table I. Chemical Composition of SKD11 Steel

C (%)	Si (%)	Mn (%)	P (%)	S (%)	Cr (%)	Ni (%)	Mo (%)
1.03	0.23	0.31	0.022	0.022	11.71	0.18	0.92

Experimental Matrix

During the experimental process, three parameters—workpiece velocity, feed rate, and depth of cut—were varied in each experiment. These three parameters are easily adjustable by machine operators [23], [24]. In the experimental procedure, each cutting parameter was varied across three levels, corresponding to coded levels 1, 2, and 3, as shown in Table II. The values in Table II were selected based on a review of relevant literature and the technological capabilities of the experimental machine used [23], [24].

Table II. Input Parameters

Parameter	Unit	Symbol	Value at level
			1	2	3
Workpiece velocity	m/min	v	5	10	15
Feed-rate	mm/stroke	f	4	6	8
Depth of cut	mm	t	0.005	0.01	0.015

The experimental matrix was structured using the Taguchi method, comprising nine experiments, as detailed in Table III. This experimental design is widely used in optimization experiments and has been frequently applied in the field of mechanical engineering in recent times [23], [24].

Table III. Experimental Matrix

Experiment	Code value			Real value
	v	f	t	v (m/min)	f (mm/stroke)	t (mm)
#1	1	1	1	5	4	0.005
#2	1	2	2	5	6	0.01
#3	1	3	3	5	8	0.015
#4	2	1	2	10	4	0.01
#5	2	2	3	10	6	0.015
#6	2	3	1	10	8	0.005
#7	3	1	3	15	4	0.015
#8	3	2	1	15	6	0.005
#9	3	3	2	15	8	0.01

Experimental Results

Conducting the experiments in the order specified in Table III, the parameters Ra, Fx, Fy, and Fz were measured for each experiment. Table IV presents a summary of the experimental results.

Table IV. Experimental Results

Experiment	Input parameters			Responses
	v (m/min)	f (mm/stroke)	t (mm)	Ra (μm)	Fx (N)	Fy (N)	Fz (N)
#1	5	4	0.005	0.82	21.7	11.3	27.1
#2	5	6	0.01	0.62	34.5	20.5	24.3
#3	5	8	0.015	0.75	39.4	16.4	26.2
#4	10	4	0.01	0.49	18.4	15.2	28.4
#5	10	6	0.015	0.51	22.5	20.6	30.4
#6	10	8	0.005	0.41	29.6	19.8	31.2
#7	15	4	0.015	0.94	31.7	22.7	22.8
#8	15	6	0.005	0.82	32.7	28.6	30.6
#9	15	8	0.01	0.73	28.1	18.4	31.5

The data in Table IV indicate that Ra has the smallest value of 0.41 mm in experiment #6, Fx has the smallest value of 18.4 N in experiment #4, Fy has the smallest value of 11.3 N in experiment #1, and Fz has the smallest value of 22.8 N in experiment #7. Thus, it is evident that no single scenario (experiment) simultaneously yields the smallest values for Ra, Fx, Fy, and Fz. Instead, it is only possible to identify a scenario where all four criteria are considered “minimal.” Naturally, this cannot be achieved by merely observing the data in Table IV; it requires ranking the scenarios to select the best one. For this reason, the RAM algorithm will be used to rank the scenarios in this study, but first, the weights of the criteria must be calculated using the ENTROPY method.

Entropy Method

Assuming that m experiments have been conducted, with n output parameters measured in each experiment, and denoting xij as the value of the j-th output parameter in the i-th experiment, where j ranges from 1 to n and i ranges from 1 to m, the calculation of weights for the j-th parameters using the Entropy method is performed in the following sequence [25]–[27]:

• Step 1: Determine the normalized values for the criteria using (1).

(1) $$\begin{array}{r}{n}_{\text{ij}}=\frac{x_{\text{ij}}}{m+{\sum }_{i=1}^m{x}_{\text{ij}}^2}\end{array}$$

• Step 2: Calculate the Entropy measure for each parameter j using (2).

(2) $$\begin{array}{r}{e}_j=\sum _{i=1}^m[n_{ij}\times ln\left(n_{ij}\right)]-(1-\sum _{i=1}^m{n}_{ij})\times ln(1-\sum _{i=1}^m{n}_{ij})\end{array}$$

• Step 3: Calculate the weight for each parameter using (3).

(3) $$\begin{array}{r}{w}_j=\frac{1-e_j}{{\sum }_{j=1}^n(1-e_j)}\end{array}$$

RAM Algorithm

The steps for using the RAM method to rank the scenarios are as follows [28]–[31]:

• Step 1: Similar to step 1 of the ENTROPY method.

• Step 2: Normalize the data using (4).

(4) $$\begin{array}{r}{r}_{ij}=\frac{x_{ij}}{{\sum }_{i=1}^m{x}_{ij}}\end{array}$$

• Step 3: Calculate the normalized values considering the weights of the criteria using (5). Where wj is the weight of the j-th criterion.

(5) $$\begin{array}{r}{y}_{ij}=w_j\cdot r_{ij}\end{array}$$

• Step 4: Calculate the total normalized scores considering the weights of the criteria using (6) and (7). The letters B and C are used to denote benefit and cost criteria, respectively.

(6) $$\begin{array}{r}{S}_{+i}={\sum }_{j=1}^n{y}_{+ij}ifj\in B\end{array}$$

(7) $$\begin{array}{r}{S}_{-i}={\sum }_{j=1}^n{y}_{-ij}ifj\in C\end{array}$$

• Step 5: Calculate the score of each scenario using (8).

(8) $$\begin{array}{r}R{I}_i=\sqrt[2+S_{-i}]{2+S_{+i}}\end{array}$$

• Step 6: Rank the scenarios in descending order of their scores.

Results and Discussion

Applying (1) through (3), the weights of parameters Ra, Fx, Fy, and Fz were calculated to be 0.364, 0.208, 0.218, and 0.210, respectively.

Applying (4) through (8), the RIi scores for each experiment were calculated and summarized in Table V. The last column of this table also displays the ranking of the experiments based on their scores.

Table V. Scores and Rankings of Experiments

Experiment	Ra (μm)	Fx (N)	Fy (N)	Fz (N)	RIi	Rank
#1	0.82	21.7	11.3	27.1	1.390	4
#2	0.62	34.5	20.5	24.3	1.389	5
#3	0.75	39.4	16.4	26.2	1.387	7
#4	0.49	18.4	15.2	28.4	1.394	1
#5	0.51	22.5	20.6	30.4	1.391	3
#6	0.41	29.6	19.8	31.2	1.391	2
#7	0.94	31.7	22.7	22.8	1.385	8
#8	0.82	32.7	28.6	30.6	1.383	9
#9	0.73	28.1	18.4	31.5	1.388	6

The calculation results identified experiment #4 as the best among the conducted experiments. Accordingly, the optimal values for workpiece velocity, feed rate, and depth of cut are 10 m/min, 4 mm/stroke, and 0.01 mm, respectively. When grinding with these optimal cutting parameter values, the values of the criteria Ra, Fx, Fy, and Fz are 0.49 mm, 18.4 N, 15.2 N, and 28.4 N, respectively.

Conclusion

This study successfully executed the multi-objective optimization of the SKD11 steel grinding process. The RAM algorithm and the Entropy weighting method were integrated for the first time to address the surface grinding optimization problem in this research. Utilizing the Entropy method, the weights of the criteria Ra, Fx, Fy, and Fz were calculated to be 0.364, 0.208, 0.218, and 0.210, respectively. Employing the RAM algorithm, the optimal values for workpiece velocity, feed rate, and depth of cut were determined as 10 m/min, 4 mm/stroke, and 0.01 mm, respectively. Corresponding to these optimal cutting parameter values, the surface roughness (Ra) and the force components Fx, Fy, and Fz were determined to be 0.49 mm, 18.4 N, 15.2 N, and 28.4 N, respectively.