Adaptation of Symbolic Discrete Control Synthesis for Energy-Efficient Multi-Pocket Milling

Abstract

In engineering, cost minimization, especially in Computer Numerical Control (CNC) machining like pocket milling, is crucial. Existing tool path definition software often lacks optimization, particularly at critical starting and ending points. This study optimizes CNC machine tool paths for energy-efficient multi-pocket milling, utilizing the Symbolic Discrete Control Synthesis (SDCS) method for formal correctness. In our work, the tool path generation is formulated as a traveling salesman problem. We introduce a modeling framework to adapt SDCS to multi-pocket-milling processes, aiming to enhance precision and efficiency for potential cost savings, including energy and time, in engineering applications. This study reports experimental and comparative results, where comparative evaluations were made using metaheuristic algorithms. Our proposed approach improves CNC machining processes for multi-pocket milling. We experimentally evaluate our control algorithms and demonstrate and validate our approach through case studies.

1. Introduction

Computer Numerical Control (CNC) machining is essential to modern industry and research for the intersection of manufacturing and computer technology [1]. In particular, multi-pocket milling is a procedure that uses tool paths created by Computer-Aided Manufacturing (CAM) software (Tebis 4.0) to allow for the methodical removal of material within a predetermined closed boundary [2]. There is a sizable amount of academic material on multi-pocket milling that has been studied in a variety of research fields [3]. The overall goal of many of these research projects is to improve milling by minimizing processing time while accounting for relevant important factors [4].

One of the greatest challenges in CNC machine multi-pocket operations is the suboptimal tool path definition software. We aim to address these challenges by employing SDCS in this process. This approach targets achieving both optimizations of these paths with formal correctness and saving time and energy. Our proposed approach aims to chart a new and effective path for optimizing tool paths in multi-pocket-milling challenges.

Optimization, which is the methodological search for variable sets to minimize error in problem solving, is important in many different engineering domains [5]. The most used optimization method in recent years has been Metaheuristic (MH) algorithms. Although they do not provide exact solutions, MH algorithms demonstrate the capacity to quickly find near-optimal solutions [6]. While some other MH algorithms have been published in the literature, swarm-based techniques have seen significant advancement in the field of MH algorithms. Figure 1 categorizes well-known MH algorithms.

MH algorithms commonly used in tool path optimization often lack formal correctness. Our research aims to fill this gap by providing a systematic approach to optimizing tool paths in CNC machines with formal correctness. The adaptation of SDCS offers a novel approach for optimization in multi-pocket machining with formal correctness. This adaptation of SDCS to the multi-pocket machining process addresses a critical need for more reliable optimization techniques. MH algorithms are the members of a large family consisting of many branches, including evolutionary, swarm-based and nature-based. Each branch and algorithm has various advantages and disadvantages, such as reaching local minima and having a lower process time. The GA is an evolutionary MH algorithm that is widely known and used in the literature. The BA and ACO are swarm-based algorithms. We combined the advantages of these algorithms to obtain the optimal tool path. However, SDCS has proven its superiority over MH algorithms in optimizing the tool path of the CNC machine.

In the existing body of literature, numerous papers have explored the application of MH methodologies in CNC machine milling. For example, Genetic Algorithms (GAs) have been employed to determine optimal values for cutting parameters, including machining time, cutting forces, metal removal rate and dimensional accuracy during rough pocket milling [7]. Another study utilized GAs to optimize milling parameters, defining the objective function as the machining time in the pocket milling process, with depth of cut and feed rate as independent variables [8]. Additionally, Ant Colony Optimization (ACO) was applied to optimize the multi-pass pocket-milling process, focusing on two objectives: achieving an optimal surface roughness value and minimizing machining time. The independent variables considered in this context were spindle speed, feed rate and depth of cut [9]. Ref. [10] has explored optimal path-planning strategies for pocket milling. Ref. [11] has applied the firefly algorithm to find an optimal solution to the tool path planning for machining processes through the Traveling Salesman Problem (TSP). To conclude, MH algorithms are moderately convenient for CNC operations and widely used to calculate tool paths in such processes [12].

MH algorithms are one of the most effective optimization methods for minimizing cost in the literature; however, they cannot provide formal correctness and the most likely optimal path calculation. Our research aims to optimize CNC machine tool paths for energy-efficient multi-pocket milling by using the Symbolic Discrete Control Synthesis (SDCS) technique. Accordingly, we offer a systematic modeling framework by adapting SDCS for multi-pocket milling. Thus, we achieve energy, time, and cost efficiency with formal correctness by means of our resulting controller.

The term “energy-efficient” in multi-pocket milling refers to less energy usage while maintaining milling quality, with everything remaining constant except for changes in the tool path. This means that as the path shortens, the process is completed faster, resulting in lower energy consumption. Achieving energy efficiency involves minimizing tool paths through strategies employing advanced control techniques. Ultimately, energy-efficient multi-pocket milling balances machining objectives with energy conservation, supporting sustainability and cost-effectiveness in manufacturing.

DCS is a formal framework that synthesizes a controller for desired system behavior by satisfying the given specifications, such as safety constraints or performance requirements. DCS approaches are applied in several disciplines, such as software engineering, robotics, control systems and communication networks, for synthesizing controllers that can effectively handle complex systems. For example, DCS was applied to generate controllers for robotic systems for precise control and steady motion in [13]; Ref. [14] addressed DCS for communication networks that provide dependable and effective data flow; and [15] performed DCS to create controllers that guarantee the safety and correctness of programs. The DCS method is a useful tool for tackling many real-world issues where the creation of controllers that meet particular requirements is essential.

The control theory of discrete event systems (i.e., DCS) was initially approached as a topic in language theory by [16]. In general, the theory revolves around synthesizing a controller for an uncontrolled system, i.e., a plant, to ensure desired system behaviors in line with specified control objectives. Numerous studies have been conducted to develop algorithms, both for finite systems [17,18] and infinite systems [19], to synthesize controllers aiming at objectives such as safety and optimization. Moreover, this method has found extensive application in addressing real-life problems. For instance, recent research, exemplified by [20,21], has successfully achieved energy efficiency in hardware circuits using this approach. Ref. [22] explores coordinating operations in a flexible manufacturing system, including robots, assembly machines, CNC lathes and mills. It emphasizes discrete event systems’ control theory, managed by PLC experts for individual subsystems. The goal is to simultaneously coordinate these subsystems for efficient production while ensuring system integrity and safety.

Ref. [23] presents a case study demonstrating the application of probabilistic model checking in synthesizing a safety controller for a mobile collaborative robot deployed in a manufacturing cell alongside a human operator. This study focuses on the safety controller synthesis for an ICONSYS iAM-R mobile robot engaged in a complex manufacturing process across multiple machine locations. Ref. [24] focuses on discrete controller synthesis for smart manufacturing systems, inheriting advancements from flexible and reconfigurable manufacturing systems. It addresses residual indeterminism in control, emphasizing hierarchical communication between controller layers and operational sequences in manufacturing. In [25], a systematic framework is introduced for an energy-aware configuration manager in reconfigurable manufacturing systems, employing discrete controller synthesis techniques. The approach prioritizes energy consumption reduction and resolving mutual exclusion on shared resources. By dividing control objectives into safety and optimization, resource constraint maintenance and improved system performance, particularly in terms of energy efficiency, are ensured. Ref. [26] introduces regulation control, a method for modeling and controlling complex manufacturing systems using discrete event systems, represented by interpreted Petri nets. It tests this methodology by integrating, communicating and controlling three workstations within a scaled manufacturing cell, demonstrating its practical implementation for safe operation without extensive programming. The manufacturing operations primarily involve CNC machining, turning, milling and handling of metal mechanical tools. Ref. [27] introduces a method rooted in the control theory of discrete event systems to design supervisors achieving optimal throughput in manufacturing systems. By utilizing extended finite automata, it models system activities at two abstraction levels, addressing uncontrollability arising from external inputs and disturbances. Through game-theoretic methods, supervisors are synthesized to optimize throughput, particularly relevant in scenarios such as the milling process within the dice factory, ensuring consistent system performance.

Our research question addressed in this study concerns how to optimize CNC machine tool paths for energy-efficient multi-pocket milling with formal correctness. We propose a novel approach for multi-pocket milling by adapting the SDCS method to achieve energy efficiency. The existing literature has mostly applied metaheuristic algorithms for the optimization of tool path calculations. We offer formal correctness via SDCS for both safety and optimization for multi-pocket milling in CNC machining. When compared with the existing approach, our study provides a unique methodology that brings new knowledge to the field of the optimal generation of trajectories.

The primary contribution of this paper to the existing literature lies in the comparative evaluation of the performance of the SDCS method alongside the MH algorithms. In our research context, the overarching objective is to enhance the efficiency of CNC machine operations by harnessing the advanced optimization capabilities offered by SDCS while concurrently ensuring the attainment of safety objectives stipulated by SDCS. It is noteworthy that this specific research topic has not been previously explored in the literature. Thus, our contribution to the field lies in addressing this gap and providing a comprehensive approach that integrates superior performance and safety objectives through the utilization of SDCS.

Our evaluation process focuses on the optimization of CNC machine tool paths for pocket milling processes, utilizing MH algorithms, specifically ACO, BA, GA, ACO-GA and BA-GA. The hybridized algorithms, ACO-GA and BA-GA, are derived by adjusting the crossover rate and mutation rate parameters of the GA procedure through the incorporation of ACO and the BA. We integrate the starting and ending points of pocket milling into the optimal path generation, with tool paths calculated within the framework of the TSP.

Contributions: We present a framework for integrating SDCS into the Simulink environment, and our contributions within this framework can be outlined as follows:

• Modeling: This involves specifying abstractions for the behaviors of multi-pocket milling, where control objectives are symbolically encoded as the parallel composition of data flow equations using the reactive synchronous language supported by the DCS tool, ReaX (version 0.17.3).
• Tooling: This covers adaptation and code generation, serving as a semi-automated management service that encompasses (i) encoding the abstracted behaviors of multi-pocket milling processes derived from the Simulink platform and their control objectives, (ii) ReaX-DCS compilation of constructed synchronous models and (iii) transferring the resulting controllers back to the Simulink environment.
• Implementation: This involves experimental validation of our approach, ensuring that the specifications of a given multi-pocket processing are met.

The subsequent sections of this work are structured as follows. Section 2 delivers a comprehensive overview of CNC machining, with a specific focus on pocket milling. In Section 3, we provide the necessary background on SDCS. The integration of SDCS into CNC machining for energy-efficient multi-pocket milling is elucidated in Section 4, where we present an illustrative implementation of our modeling framework. Following that, Section 5 conducts an experimental assessment of our framework’s performance. Lastly, Section 6 summarizes key findings, draws conclusions and concludes the paper.

2. Operational Process of Computer Numerical Control

Pocket milling is described as the material removal process within a specified enclosed boundary using Computer-Aided Manufacturing (CAM) software. In the application of any control strategies to pocket milling, 3D models of the workpiece are created through CAD software. The generated CAD file can be imported into CAM software for the simulation of any CNC operation. To simulate a CNC operation in MATLAB software (version 2021a), the design file of the workpiece is exported in XML format and imported into the Simulink environment of MATLAB. Additionally, robot designs such as CNC machines, which are Cartesian robots, are brought into Simulink. The CNC machine is transformed into a Simscape Multibody model, comprising blocks that represent joints, links and interconnections among these components. To achieve the desired movements in the Simscape model of the CNC machine, joint blocks are employed. Output variables such as the position, velocity and acceleration of links can be extracted.

In the case of the tool path not being optimized, it leads to an increase in the milling process time and, consequently, manufacturing costs. To address this issue, the starting and ending points of pocket milling processes were considered instead of the center points. An illustration of non-optimal path planning for pocket milling is provided in Figure 2. The algorithms were implemented to differently shaped pockets such as circle, triangle and hexagon, as shown in Figure 2. The numbers represent the order of the pockets in the non-optimal path. Additionally, Figure 3 depicts a selected pocket and the corresponding tool path generated, taking into account the starting and ending points, particularly in the context of zigzag-type milling.

In our research, we designed the workpiece using Solidworks software (version 2020) and subsequently imported it into Simulink. We applied a MATLAB example featuring a Simscape model of a three-degrees-of-freedom Cartesian robot to implement SDCS in the Simulink environment. To facilitate movement, reference inputs were assigned to the joints of the Cartesian robot, and the current position of the tip point of the robot was obtained from the model. Therefore, no real milling machine, workpiece materials or cutting tools were utilized.

This study focuses on the minimization of the tool path of the CNC machine to reduce the energy consumption during the pocket milling operation. Since energy consumption is directly related to the length of the tool path, minimization of the tool path represents the minimization of the energy consumption as well. However, SDCS can be applied to find the optimal conditions and the parameter values, such as cutting forces, surface quality and material removal rates. To achieve this, a multi-objective cost function model should be derived and then the experiments should be carried out on a real pocket-milling operation. CAM software (Tebis 4.0) includes an optimization section to minimize the tool path. The proposed SDCS method can be integrated into a CAM software easily and can be applied to real milling operations carried out by CNC machines.

3. Symbolic Representation of Discrete Event Systems

In the proposed symbolic system, we consider a set of symbol variables $\mathcal{V}$, where each $v\in \mathcal{V}$ is associated with a data type $\mathcal{T}$. These data types are as follows:

• Boolean: $\mathbb{B}$;
• Integer: $\mathbb{Z}$;
• Rational: $\mathbb{Q}$.

Thus, for each symbol variable $v\in \mathcal{V}$, there exists a corresponding data type $\mathcal{T}$. The grammar rules ${\mathsf{\Gamma }}_{\mathcal{T}}$ that could characterize the symbolic model that we propose are articulated as follows:

• ${\mathsf{\Gamma }}_{\mathcal{T}}=\left({\mathsf{\Gamma }}_{\mathbb{B}}\right)?\left({\mathsf{\Gamma }}_{\mathcal{T}}\right):\left({\mathsf{\Gamma }}_{\mathcal{T}}\right)$;
• ${\mathsf{\Gamma }}_{\mathbb{B}}=\neg {\mathsf{\Gamma }}_{\mathbb{B}}$;
• ${\mathsf{\Gamma }}_{\mathbb{B}}={\mathsf{\Gamma }}_{\mathbb{B}}◊{\mathsf{\Gamma }}_{\mathbb{B}}$, where the unary operator ◊ is a set of $\{\wedge ,\vee ,\Rightarrow ,=\}$;
• ${\mathsf{\Gamma }}_{\mathbb{B}}={\mathsf{\Gamma }}_{\mathbb{B}}$${\mathsf{\Gamma }}_{\mathbb{B}}$, where the unary operator is a set of $\{<,⩽,>,⩾\}$;
• ${\mathsf{\Gamma }}_{\mathbb{Z},\mathbb{Q}}={\mathsf{\Gamma }}_{\mathbb{B}}⊳⊲{\mathsf{\Gamma }}_{\mathbb{B}}$, where the unary operator $⊳⊲$ is a set of $\{+,-,\ast ,/\}$.

In addition to the above, it should be noted that there is a symbol variable v from the set $\mathcal{V}$ for each fixed literal.

Our symbolic system can be conceptualized as a function f with inputs X and outputs Y. In the body of the function, assignments like $q:=e_q$ are used, where q belongs to the set of memory-reserved symbolic variables (i.e., the set of state Q), and $e_q$ represents a corresponding expression with the same data type for the evaluation of discrete events by means of current states, inputs and outputs. Conversely, the variable v acts as a substitute for the expression e in the given assignment $v\stackrel{\mathsf{\Delta }}{=}e$.

Tackling challenges in symbolic control systems bears resemblance to navigating a game-like landscape. Instances involve a sequential interaction between two pivotal players: the environment and the controller. The environment assigns values to a subset of input variables while the controller strategically assigns values to the remaining set. This dynamic interplay propels the progression of the system, shaping its overall state. At the core of these issues lie control objectives, expressed as logical statements involving state and input variables. The central concern is formulating a strategic plan for the controller to effectively meet these objectives. In this interactive dynamic, the environment is tasked with assigning values to non-controllable input variables (U), while the controller manages controllable ones (C). Systems that facilitate the environment in making consistent choices are recognized as deadlock-free. Algorithms designed to tackle control problems must ensure this consistency to foster dynamic progress and the successful achievement of objectives.

The symbolic models that we present aim to identify a sub-state space within the system-state space that characterizes desired system behaviors. This is achieved through binary decision diagrams and the fixed-point calculation method. In the model under consideration, the evaluation of controllable variables (C) is determined in contrast to variables that are not under control (U), where this valuation is denoted by $\left[T\right]$. Our optimization algorithm in the form of a recursion function in an iterative structure is given as the following formal equation:

$$\begin{array}{cc}\hfill {\eta }_1\stackrel{\mathrm{def}}{=}& \mathtt{if}\sigma \mathtt{then}\zeta \mathtt{else}\infty \hfill \\ \hfill {\eta }_{i+1}\stackrel{\mathrm{def}}{=}& \mathtt{if}\sigma \mathtt{then}\left({Max}_U\circ {Min}_C\left({\eta }_i\right)\right)\left[T\right]+\zeta \mathtt{else}\infty ,\hfill \end{array}$$

where the safety objective ($\sigma$) must be always true in order to minimize the optimization objective. The optimal control objective is min/maximizing a cost function, added ($\zeta$) according to the steps (i) specified in the timeline. Therefore, while ${Min}_U$ makes the worst valuation for uncontrollable variables, ${Max}_C$ tries to perform the best for controllable variables.

4. Adaptation of Symbolic Discrete Control Synthesis for Multi-Pocket Milling

The DCS method proves to be a valuable approach for various engineering challenges, encompassing tasks ranging from low-level controller design, such as PID, to high-level controller design, such as robot path planning. We consider the seamless transition of states in scenarios involving CNC operations as a discrete control synthesis problem. This perspective is rooted in the control theory of discrete event systems, denoted as DCS, where the concurrent synchronization of two distinct Mealy machines is encapsulated with a signal and a controller to achieve the desired system behaviors. DCS is guided by the principle of obtaining a controller that effectively realizes the desired system behaviors.

To further elaborate on the working principles of control theory in discrete event systems, let us consider two Mealy machines, each having two distinct states denoted as 0 and 1, with transitions between states facilitated by an input variable. Additionally, let us assume the requirement that both Mealy machines cannot be in states 0 and 1 simultaneously. This condition can be met by introducing a third Mealy machine, acting as a controller, which ensures the desired behavior through the synchronous parallel composition of these three Mealy machines by encapsulating a controllable input variable. The DCS method focuses on generating this controller procedure by working in reverse. It systematically aims to produce this controller, emphasizing its adaptability for any system that can be modeled as discrete events. The main advantage of the DCS technique is its operation as a model-checking tool, always ensuring the fulfillment of desired specifications [19]. However, due to testing all possible scenarios, synthesizing the controller incurs high complexity. Nonetheless, once the controller is synthesized, it operates dynamically. Our modeling approach is systematically elucidated and the adaptation of SDCS for pocket milling is explained via the block diagram outlined below. SDCS is a novel method that minimizes defined cost functions to generate the desired output of any system. Therefore, SDCS can be used in many fields, such as the path planning of mobile robots, control applications in mechatronics, increasing the energy efficiency of hardware chips and optimization of the job shop scheduling.

4.1. Overview

In the adaptation of the SDCS to multi-pocket milling within the synchronous parallel language ReaX environment, we construct our model and objectives as data flow equations. The SDCS method’s model is illustrated in Figure 4. The safety objective is then expressed through propositional formulas encapsulating the state variables of the previously discussed models. A symbolic safety control algorithm proves effective in formulating a strategy. The controllable input variables of our model ensure the fulfillment of the safety objective. The resulting safety objective, integrated with our model, serves as a controller. This controller ensures precise conditions that must be satisfied in CNC operations and inherently adheres to the previously mentioned mutual exclusion constraints. As an alternative for further improvement, our approach involves the application of a symbolic optimal control algorithm to achieve an additional optimization objective. In this context, the cost function defining the new strategy primarily focuses on minimizing processing time, i.e., energy consumption. This integrated approach seamlessly addresses both safety and optimization aspects within the management system.

4.2. Models and Objectives

In our specific scenario, as illustrated below, we have defined each shape processed in each CNC as a state and adorned it as a Mealy machine. Each state transition is under control with a controllable variable and, at each transition, a $\mathsf{\Delta }$ is used to generate the transition distance. Formally, state transitions are encoded as data flow equations as shown below:

$$\begin{array}{cc}\hfill S=& S_i\wedge c_{i,j}\to S_j,\forall i,j\in \left|S\right|,\hfill \end{array}$$

(1)

where S represents each individual shape and $\left|S\right|$ indicates the total number of shapes in the system. The $c_{i,j}$ expression denotes the controllable variable that governs the transition from $S_i$ to $S_j$. Of course, it is necessary to reinforce the control mechanism that we establish for security purposes with optimization goals to ensure that the CNC processes can be carried out as efficiently as possible within a limited time frame. We will achieve this by minimizing a cost function that represents the total costs. The cost function has been encoded as a data flow equation as shown below:

$$\begin{array}{cc}\hfill \sigma =& \sum S_i\wedge c_{i,j}\to {\mathsf{\Delta }}_{i,j},\forall i,j\in \left|S\right|,\hfill \end{array}$$

(2)

where our cost function $\sigma$ is defined with ${\mathsf{\Delta }}_{i,j}$, which represents the numerical value of the length of the path that could be incurred during the transition from $S_i$ to $S_j$.

In our symbolic system, each state has an initial value. In our case, it is the startpoint of the CNC multi-pocket milling process and is specified by the user; the endpoint is determined through the tool SOLIDWORKS CAM. We consider the calculation of the synthesis of a controller that will find the optimal tool path between these two points.

Once we have defined our model as described above, our synthesis algorithms, including safety and optimization algorithms, consistently ensure both safety objectives and minimize our cost function. Subsequently, the obtained controller is transformed into an HDL (Hardware Description Language) code and transferred to the MATLAB environment. This code implements the desired objectives, allowing the CNC to complete its operations as quickly as possible while meeting safety requirements.

4.3. Symbolic Control

In our symbolic model, we have two different control objectives. The first one is the safety objective, which deals with only certain safety goals and mutual constraints that may arise between shared resources. The second one, the optimization objective, works to minimize the cost function generated by tool paths.

In this context, we consider the multi-pocket milling operation on CNC machines as the control of discrete event systems, and we encode it as synchronous parallel data flow equations following our symbolic model approach described above. Subsequently, through the presented safety and optimization algorithms, we obtain a controller that minimizes the tool path while ensuring safety. The obtained controller operates as a power-aware manager, minimizing the tool path and providing time savings simultaneously.

Our approach works systematically in alignment with our research objectives. It enables both concurrent resource sharing and the fulfillment of safety objectives in a systematic manner. Additionally, it guarantees reliability by ensuring the formal correctness of time and energy efficiency while guaranteeing the minimization of the tool path.

4.4. Adaptation Process

Our workflow of the implementation process of SDCS is as shown in Figure 5. As depicted in the block diagram of the system architecture, it consists of four main parts. In the modeling phase, an uncontrolled design is modeled both in the platform to be implemented and in our symbolic model presented in the ReaX environment with the desired system behaviors by a designer. Subsequently, with our safety and optimization algorithms, symbolic control is achieved, resulting in a controller. In the code generation section, this generated controller is transferred to the MATLAB environment with a Verilog or C extension file. Then, simulation can be performed in the MATLAB environment, or MATLAB G code can be obtained and transferred to a physical system, namely a CNC, to execute the desired system behaviors.

The SDCS method does not provide a pre-defined path. The path proposed by SDCS can be seen by implementing it in a simulation environment. Therefore, workpieces, including the pockets that were designed in SOLIDWORKS, are imported to the Simscape environment of MATLAB software. Simscape allows one to import their designs as a multibody. In addition, the links of the machines and robots can be moved and the position of the parts can be obtained as feedback data. The implementation of SDCS using an example Simscape model is as shown in Figure 6.

5. Experimental Evaluation

Our goal in our study is to develop CNC tool paths specifically designed for the energy-efficient multi-pocket-milling process. Our main point was optimizing these tool paths to achieve the most efficient configurations. Notably, in the optimization process, we aimed to consider the starting and ending points of the multi-pocket-milling operation for precision and efficiency. In our work, we focused on obtaining formal correctness for road planning. We employed the SDCS algorithm, a robust tool for addressing optimization challenges, to determine the exact optimal routes for our specific problem. The outcomes derived through the SDCS algorithm offer valuable insights into the efficacy of our approach, particularly in identifying optimal solutions for specific pockets. Consistently producing shorter path lengths, the SDCS algorithm underscores its effectiveness, with these findings holding significant value in comprehending tool path optimization within the context of multi-pocket milling.

In our work, the tool path generation is formulated as a TSP and we applied our symbolic approach to solve the TSP. We evaluated our approach by comparing MH algorithms. Thus, we also applied ACO, BA, GA, ACO-GA and BA-GA to solve the TSP. To ensure equitable comparisons, we maintained a consistent maximum iteration number of 100 across our tests and explored three distinct population sizes of 50, 100 and 150 for the optimization procedure. We illustrate our comparative study of the evaluation in a flow-chart, depicted in Figure 7.

Table 1. Computed tool path lengths and processing times were determined for pocket quantities of 10, 20, 30, 40 and 50, with fixed population sizes of 50, 100 and 150.

Algorithm	Population Size	Pockets
		10		20		30		40		50
		mm	s	mm	s	mm	s	mm	s	mm	s
DCS	-	352.828	-	467.372	-	555.593	-	654.788	-	759.189	-
GA	50	352.828	3.952	467.372	5.238	555.593	9.056	667.634	11.444	842.418	14.006
	100	352.828	5.675	467.372	9.574	555.593	16.613	654.788	22.020	770.118	28.558
	150	352.828	7.554	467.372	16.450	555.593	24.228	654.788	33.332	764.447	42.206
ACO	50	352.828	3.012	467.372	5.644	555.593	8.245	667.634	11.250	842.418	14.259
	100	352.828	5.290	467.372	10.529	555.593	16.012	660.982	22.193	770.118	28.247
	150	352.828	7.637	467.372	15.654	555.593	24.173	654.788	32.926	764.447	41.974
BA	50	352.828	9.402	467.372	9.801	573.218	10.475	734.351	10.987	1074.354	11.580
	100	352.828	9.207	467.372	9.664	555.593	12.351	728.856	10.841	1053.611	12.782
	150	352.828	8.934	467.372	9.473	566.190	10.225	707.910	12.404	1032.026	14.449
ACO-GA	50	352.828	8.506	467.372	24.457	555.593	68.873	654.788	104.442	759.189	188.779
	100	352.828	24.455	467.372	88.525	555.593	254.115	654.788	478.193	759.189	807.708
	150	352.828	48.466	467.372	253.528	555.593	584.175	654.788	1048.583	759.189	1678.658
BA-GA	50	352.828	32.522	467.372	45.478	555.593	89.866	654.788	115.455	842.418	152.884
	100	352.828	45.466	467.372	80.595	555.593	194.188	654.788	245.177	759.189	347.786
	150	352.828	59.457	467.372	161.528	555.593	253.435	654.788	388.178	759.189	587.792

shows our experimental comparative evaluations. The table presents detailed information on computed tool path lengths and processing times for the pairs of different pocket quantities (10, 20, 30, 40 and 50) and population sizes (50, 100 and 150).

We employed MH algorithms, including ACO, BA, GA, ACO-GA and BA-GA, to CNC machine tool paths for multi-pocket milling. Tool paths were formulated as a TSP, and the starting and ending points of milling operations for each pocket were determined as Waypoints (WPs) for the TSP. We obtained results about how various population sizes affect tool path length and processing times by taking into account the starting and ending points for the milling process.

Specifically addressing the generation of optimal tool paths for multi-pocket-milling processes, we utilized swarm-based MH algorithms, ACO and the BA, alongside the evolutionary algorithm GA. Analyzing the results, including those from Table 1, revealed consistent performance closeness between ACO and the GA. Furthermore, the GA, ACO and BA produced identical results for scenarios involving 10 and 20 pockets in terms of tool path length. However, for 30, 40 and 50 pockets, the GA and ACO consistently outperformed the BA, emphasizing their superior suitability for multi-pocket-milling operations. In the evaluation of process times, the BA generally outperforms ACO and the GA, with ACO-GA and BA-GA hybrid algorithms excelling in tool path length but significantly underperforming in processing time compared to non-hybrid algorithms. It is crucial to note that the maximum iteration number was uniformly set to 100 for all tests, acknowledging the trade-off between iterations and processing time in non-hybridized MH algorithms. Although ACO-GA provides optimal solutions, its significantly increased processing time as a hybrid algorithm poses challenges for its selection in pocket-milling operations.

MH algorithms inherently exhibit stochastic behavior, offering no assurances of exact solutions but excelling in uncovering near-optimal solutions within limited time frames. To address potential non-optimal outcomes, our study proposes the SDCS method to enhance multi-pocket-milling operations. Results from SDCS for scenarios involving 10, 20, 30, 40 and 50 pockets align with the GA, ACO and BA for 10 and 20 pockets. However, for more complex scenarios of 30, 40 and 50 pockets, SDCS consistently outperforms the GA, ACO and BA, highlighting its preference in multi-pocket-milling operations. Table 1 includes detailed information on computed tool path lengths and processing times for the different pocket quantities (10, 20, 30, 40 and 50) and population sizes of the MH algorithms (50, 100 and 150). For example, the GA found the optimal tool path length of a workpiece with 50 pockets as 770.118 mm within 28.558 s, while the population size was defined as 100.

In the literature, numerous papers have explored the application of MH methodologies in CNC machine milling [7]. Since MH algorithms inherently possess a stochastic structure, previous studies have successfully obtained near-optimal solutions within limited time frames [8]. Our SDCS-based approach provides exact solutions to CNC machine milling operations. We applied SDCS to problems defined in previous literature. SDCS was compared with Particle Swarm Optimization (PSO), the GA and the Grey Wolf Optimizer (GWO) [28]. Our proposed method, SDCS, yielded results that were 8.79%, 16.18% and 13.97% superior to the mean outcomes obtained with PSO, the GA and the GWO, respectively. MH optimization methods, as observed in the literature, generally provide approximate solutions instead of exact ones. The near-optimal results obtained by MH algorithms in this study are compatible with related studies in the literature, which were conducted with 51 and 30 WPs [29,30].

Our objective function (cost function) is determined as stated in Equation (2), and our optimization (optimal) algorithm always guarantees desired system behavior. We validate our approach in various scenarios, and our verification process is based on model checking [19], which is a verification technique that ensures the alignment of a system or software model with specified properties.

6. Conclusions

In this study, the adaptation of the SDCS method to optimize CNC machine tool paths for energy-efficient multi-pocket milling is presented. In our approach, we construct a cost function (object function) based on route lengths, which we then minimize using our optimization algorithm. Ultimately, the controller achieves time savings by obtaining the minimum route length, thereby ensuring energy efficiency as well. Additionally, a comparative evaluation of our approach with optimization algorithms, including ACO, the BA, the GA and various hybridized approaches, is reported. Compared to traditional MH algorithms, which are commonly employed to optimize CNC machine tool paths in multi-pocket milling, our method provides formal correctness and the results show that our approach outperforms the MH algorithms. Our proposed modeling framework explains in detail how SDCS can be systematically adapted to energy-efficient multi-pocket-milling processes and validates it through case studies.

The results show that our approach matches the evidence presented. It effectively addresses how to optimize CNC machine tool paths for energy-efficient multi-pocket milling with formal correctness and precision. Comparing SDCS with traditional MH algorithms, our research shows SDCS’s superiority in energy and time efficiency while maintaining formal correctness in machining. The experimental results support this by demonstrating SDCS’s enhanced performance over existing methods. Additionally, our study discusses practical applications of SDCS in CNC machining and suggests future research avenues. According to Table 1, our study finds that integrating SDCS into CNC machining improves energy efficiency in multi-pocket milling. Through comparisons with traditional MH algorithms, SDCS proves to be more precise and efficient, offering potential cost savings for manufacturers by reducing energy usage and processing time. Overall, the findings highlight the importance of advanced control theory and optimization techniques in enhancing the sustainability of multi-pocket-milling processes.

We plan to integrate SDCS into SOLIDWORKS add-ins and to develop design guidelines for future research. Through this add-in, the start and end points can be easily extracted in SOLIDWORKS CAM; then, optimum tool paths can be created, and therefore the G code containing optimization obtained from the software can be transferred to physical equipment. This avenue holds promise for further enhancing the practical implementation of SDCS in real-world CNC machining applications. Furthermore, the results obtained in this study are promising for the use of our approach in such engineering applications in terms of cost savings.