2.1 Advances in teaching-learning-based optimization algorithm

The Teaching-Learning-based Optimization (TLO) algorithm was proposed by Rao in 2011 [14], which is a metaheuristic algorithm that mimics the teaching and learning behaviors in a classroom setting. Due to its simple structure and excellent performance, the TLO algorithm has been widely applied to practical problems since its inception. In the context of optimization issues in turning processes, Lin et al. proposed a multi-objective TLO algorithm that employs an improved random key encoding method and uses solutions from the non-dominated set as the “teacher solution” in the teaching phase. Li et al. [15] defined the optimal solution based on the dominance relationship and further combined the Sigma method to rank solutions in the non-dominated set, thus distinguishing the quality of non-dominated solutions. Shao et al. [16] developed a discrete TLO algorithm that directly encodes solutions in the discrete space and introduced a probabilistic model to control the learning intensity during the learning phase. Currently, the application of TLO algorithms to discrete problems is mainly achieved by defining a mapping relationship between the discrete and continuous spaces for solution encoding and subsequent evolutionary operations.

Recent advancements have further demonstrated the potential of TLO and similar evolutionary mechanisms in complex scheduling. Tan et al. [17] proposed an enhanced multi-objective TLBO for integrated production and distribution scheduling, verifying that incorporating heuristic rules and specific neighborhood searches can significantly boost performance in bi-objective problems. Similarly, in the broader context of evolutionary computation, Ding et al. [18] integrated a fluid master-apprentice evolutionary algorithm with deep reinforcement learning to solve the FJSP with multiplicity, highlighting the growing trend of utilizing learning mechanisms to guide the evolutionary process. Furthermore, Luo et al. [19] developed an affinity propagation hierarchical memetic algorithm for multimodal multi-objective FJSP, employing problem-specific neighborhood structures to enhance convergence. These studies collectively suggest that generic operators are often insufficient for complex constraints.

However, due to the difficulty in reflecting the structural information of the discrete space in the continuous space, the standard mapping-based approach often disrupts the precedence constraints of scheduling. At the same time, utilizing problem knowledge to guide the search direction of the TLO algorithm, thereby reducing the randomness of the evolutionary direction, is of great significance for solving the multi-objective collaborative optimization problem of the tool and processing equipment in the discrete manufacturing environment.

2.2 Brief summary

Despite the advancements, there are notable deficiencies in the current research. First, The lack of research on job shop environments is a significant oversight compared to parallel machine settings, given their prevalence in modern manufacturing. The existing models have made significant strides in incorporating tool degradation effects and optimizing maintenance alongside production scheduling. However, these models often rely on complex assumptions that may not fully capture the nuances of real-world manufacturing scenarios where tool magazine capacity is strictly finite and strongly coupled with machine routing flexibility. The integration of tool wear, energy consumption, and production efficiency into a cohesive model remains a challenge, with the potential for oversimplification or over complication hindering practical applicability. The literature pertinent to scheduling problem with tool allocation published in recent years are summarized in Table 1.

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Table 1. Summary of literature pertinent to scheduling problem with tool allocation published in recent years.

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Additionally, the reliance on mapping techniques in TLO algorithms for discrete problems limits the effectiveness of these solutions. There is a clear need for more research that can bridge the gap between theoretical models and practical applications, particularly in developing algorithms that can effectively handle the discrete nature of tool allocation problems without losing critical structural information. Furthermore, the incorporation of problem-specific knowledge to guide the search process in TLO algorithms could significantly enhance the solution quality and convergence speed, a direction that current research has yet to fully explore.

3.1 Definition of BFJSP-TA

The scenario of the proposed BFJSP-TA is illustrated in Fig 1, which can be defined as follows. There is a set of machines M and a set of tools K. Each tool is treated as an individual tool copy and is assumed to be compatible with all machines. During the planning horizon, each tool copy is assigned to at most one machine, and the tool magazine configuration is assumed to remain unchanged. There is a set of jobs J need to be processed. The release time and due time of job i are A_i and D_i, respectively. The job contains a set of operations with precedence constraint. Notably, three important characters of real-world manufacturing scenarios are considered as follows:

• Machine m has a limited tool slot capacity Q_m.
• The punishment on tardiness of jobs are different.
• Operation O_ij has compatible machine set M_ij and tool set K_ij.

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Fig 1. Scenario of the proposed BFJSP-TA.

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Notations of the proposed BFJSP-TA are listed in Table 2. To simplify the problem at hand, the following predefined hypothesis should be satisfied: a) transportation and setup time is negligible, b) preemption of jobs is not allowed, c). all machines and tools are available at the beginning, d) breakdown is not considered throughout the planning horizon. These assumptions are commonly adopted in the flexible job-shop literature [24,25] to focus on the coupled decisions of tool allocation and production scheduling, while avoiding additional uncertainties and auxiliary times that would require extra parameters and decision variables. Therefore, the BFJSP-TA contains three sub-problems: tool assigning to machine, operation assigning to machine and tool, operation sequencing. The goals of BFJSP-TA are minimizing the tool wear cost and weighted sum of tardiness, simultaneously.

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Table 2. Notations of BFJSP-TA.

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3.2 Mathematical model of BFJSP-TA

a. Formulation of tool wear cost (TWC)

Using a tool for machining will inevitably result in tool wear [8], and the cost associated with this wear is directly proportional to the duration of the tool’s usage. Therefore, the cost of tool wear can be calculated using Eq. 1.

(1)

b. Formulation of weighted sum of tardiness (WST)

In this study, tardiness is defined as the positive deviation of a job’s completion time from its due time, i.e., the amount of time a job finishes after its due date. To reflect that different jobs may have different priorities in a practical machine shop, the weighted sum of tardiness is adopted, where each job i is assigned a tardiness weight and higher-priority jobs incur larger penalties when delayed. Accordingly, the weighted sum of tardiness aggregates the weighted tardiness of all jobs in the schedule, capturing the overall impact of delays under their respective priorities.

The calculation of the weighted sum of tardiness involves determining the tardiness for each job by subtracting the due date from the completion time, assigning a weight to each job based on its priority or the cost of delay, and then multiplying the tardiness by its weight to obtain the job’s weighted tardiness. The completion time is obtained by Eq. 2, and therefore, the weighted sum of tardiness is calculated by Eq. 3.

(2)

(3)

c. Mixed-integer programming (MIP) model of BFJSP-TA

The MIP model of BFJSP-TA can be formulated as follows.

(4)

Subject to:

(5)

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

(14)

(15)

(16)

(17)

(18)

Eq. 4 is the objective function. Constraint 5 denotes that each operation is processed only once. Constraint 6 indicates that each tool copy is assigned to at most one machine during the planning horizon. Constraints 7 and 8 guarantee that the operation can only be processed by the compatible machine and tool, respectively. Constraints 9 - 11 denote that the operations are sequenced on the machine one by one. Constraint 12 guarantees the tool slot capacity of each machine. Constraints 13 and 14 ensure no conflicts between the decision variables. Constraint 15 links the completion time of each operation to its selected processing time under the chosen machine–tool pair. Constraint 16 enforces the precedence relationship of the operations belonging to the same job. Constraint 17 indicates that the job can not be processed before the release time. Constraints 18 enforces machine capacity by ensuring that two operations assigned to two consecutive positions on the same machine are processed without overlap.

4.1 Solution encoding and decoding

The encoding method must ensure the feasibility of solutions and facilitate subsequent evolutionary operations of the algorithm. For the BFJSP-TA problem, this paper designs a three-row vector encoding method. The three-row vector includes: the Operation Sequence (OS) vector, the Tool Allocation (TA) vector, and the Machine Allocation (MA) vector. As illustrated in Fig 2, the encoding example and specific descriptions of these vectors are as follows:

• OS vector: This vector represents the sequence of operations from left to right, with its length equal to the number of operations to be processed. Each element is an operation, denoted by the workpiece index, and elements with the same operation index are arranged from left to right for all operations of the same workpiece.
• TA vector: The elements of this vector correspond one-to-one with the operations to be processed, and this mapping relationship remains unchanged throughout the entire evolutionary process of the algorithm. The element value represents the tool allocated to the corresponding operation, thus the length of the TA vector is equal to the number of operations to be processed.
• MA vector: The elements of this vector correspond one-to-one with the available tools, and this mapping relationship remains unchanged throughout the entire evolutionary process of the algorithm. The element value represents the machine allocated to the corresponding tool, thus the length of the MA vector is equal to the number of available tools.

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Fig 2. Illustration of the proposed coding method.

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This paper adopts an “active decoding” method [17] for decoding, which can be described as follows: Arrange the operations according to the OS vector from left to right; for operation O_ij, allocate O_ij to tool K_k according to the corresponding element value in the TA vector, thus determining its processing time, and determine the machine M_m for O_ij according to the corresponding element value in the MA vector (i.e., the machine allocated to tool), placing the operation at the earliest possible start time without affecting the start time of other operations and without violating constraints.

4.2 Chaotic initialization

Generally, enhancing the diversity of solutions in the initial population helps to reduce the probability of the algorithm falling into a local optimum during the evolutionary process [26]. Chaotic mapping has a series of characteristics such as initial value sensitivity and pseudo-randomness, and has been widely used in existing literature to enhance the diversity of the algorithm’s population [27,28]. Therefore, this paper combines the Tent chaotic mapping [29] to generate initial solutions. The specific steps are as follows:

Step 1:

Randomly generate a number g within the range of 0 to 1.

Step 2:

Using g as the initial value of the chaotic sequence, generate a chaotic sequence S₁ with the same length as the OS vector using the Tent chaotic mapping, and rearrange the preset OS_Pre vector order according to the size of S₁; as shown in Fig 3, the smallest number in S₁ corresponds to 2 in OS_Pre, so the first element of the OS vector is 2, and similarly, the initial OS vector is obtained.

Step 3:

Using the last number of S₁ as the initial value of the chaotic sequence, generate a chaotic sequence S₂ with the same length as the TA using the Tent chaotic mapping, and use a linear mapping method to associate the chaotic sequence with the optional tool number corresponding to the element, thus obtaining the initial TA vector; if using the tool configuration available for the operation in Fig 2, the initial TA vector can be obtained as shown in Fig 3, for example: the third number of S₂ is 0.938, corresponding to the second optional tool of O₁₂, then it denotes k₃.

Step 4:

Using the last number of S₂ as the initial value of the chaotic sequence, generate a chaotic sequence S₃ with the same length as the TA using the Tent chaotic mapping, and the subsequent process is very similar to the initialization of OS, which is not repeated here.

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Fig 3. Illustration of the proposed chaotic initialization method.

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4.3 Crossover-based teacher and learner operators

The traditional Teaching-Learning Optimization (TLO) algorithm is designed for single objective and continuous optimization problems, and its specific process can be found in the literature [14]. To effectively apply the evolutionary mechanism of the TLO algorithm to the multi-objective discrete optimization problems presented in this paper, two aspects require to be emphasized:

• How to determine the teacher solution? In the traditional TLO, the solution with the best fitness in the current population is chosen as the teacher solution. However, in multi-objective optimization problems, there are usually multiple non-dominated front solutions, making the original approach inapplicable.
• How to exchange information between solutions? In this paper, the encoding of solutions consists of discrete element combinations, which makes it difficult to use the numerical operation methods in traditional TLO for information exchange.

To address these issues, this paper uses the crowding degree in the objective space [30] to rank solutions that are on the same non-dominated front and introduces a teacher ratio parameter rT to determine the proportion of teacher solutions in the population. A multi-point crossover (MPX) operation is used to achieve information exchange between two solutions, and a crossover intensity parameter is introduced. The schematic diagram of the MPX operation is shown in Fig 4, where the OS vector uses a position-based multi-point crossover method, and the TA and MA vectors use a direct multi-point crossover method, with the number of crossover points being the crossover intensity multiplied by the vector length. The overall process of the discrete teaching and learning operator is described in the pseudo-code of Algorithm 2.

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Fig 4. Scheme of MPX.

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Algorithm 2 Procedure of the crossover-based teaching and learning operator.

Require: Population P, number of population members npop, teacher ratio r_T, crossover intensity ζ

1: Ps Non-dominated sorting of P to determine the non-dominated front, stored in Ps

2: Ranking solutions in the same non-dominated front of Ps by crowding distance, stored in Pr

3: Calculating the number of teacher solutions

4: The first nT in Pr form teacher solution set, the rest form student solution set

5: for do

6:   Randomly select a teacher solution te from the set of teacher solutions Te

7:   A student learns from a teacher via the MPX operator, and its offspring is stored in Cte

8: end for

9: for do

10:   Randomly select a student solution st from the set of student solutions St such that

11:   A student learns from a teacher via the MPX operator, and its offspring is stored in Cst

12: end for

13: Merge offspring solutions from the teaching and learning phases, and discard duplicates

14: return C

4.4 Neighborhood search

Critical operations are one of the key pieces of knowledge in the collaborative optimization of production processes, and traditional movement of critical operations can quickly reduce completion time [29]. In response to the two objectives of BFJSP-TA, KTLBO has improved upon the traditional movement of critical operations, designing an efficient local search strategy.

The basic idea behind the traditional movement of critical operations is to remove the critical operation from its current critical operation block, and the schematic diagram of its movement is shown in Fig 5. The specific process has been described in detail in the literature [29] and will not be reiterated here. This paper has made improvements in two aspects:

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Fig 5. Illustration of the proposed neighborhood structures.

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Population Grouping: Local search can easily reduce the diversity of the population, leading to the algorithm falling into a local optimum. To balance the intensity of search and the diversity of the population, the population is evenly divided into three groups, namely: one that only executes the Neighborhood Structure 1 (NS1), one that only executes neighborhood structure 2 (NS2), and one that executes NS1 and NS2 with equal probability;

Critical Operation selection: To balance the production cost and completion time objectives during the evolution, the selection of the critical operation to be moved will be biased towards operations that can quickly reduce their production cost. The specific approach is to identify those operations whose current processing cost is greater than their average production cost (the average of production costs corresponding to all possible tool allocation schemes), and then randomly select one of them for movement.

5.1 Parameter tuning

The KTLBO algorithm includes three important parameters: population size (npop), teacher solution ratio (rT), and crossover intensity (). We used orthogonal experiments to optimize these parameters, selecting 4 values within a reasonable range for each parameter, which were determined based on preliminary tests. The experiments are arranged using the orthogonal table . To eliminate the impact of randomness inherent in meta-heuristics, each experimental configuration in the orthogonal array was executed 20 times independently to obtain the average IGD value.

The experiments are conducted on the #DP18 test case, using IGD as the comprehensive evaluation metric. The design of the orthogonal experiments and the results are shown in Table 4.

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Table 4. Orthogonal experiment results for parameter tuning and range analysis.

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To determine the detailed impact of each parameter, the Taguchi method is employed. As presented in the bottom of Table 4, K_i denotes the average IGD value for a specific factor at level i, and the Range () indicates the sensitivity of the parameter. A smaller K_i signifies better performance, while a larger R implies greater influence on the results.

According to Table 4, the optimal parameter combination corresponds to the minimum K values (highlighted in bold): npop=30, rT=0.3, and ζ=0.05. Furthermore, the R values reveal that the population size (npop) has the most significant impact on the algorithm’s performance (R = 0.0033), followed by crossover intensity (R = 0.0031) and teacher ratio (R = 0.0028). These optimal parameters are adopted for all subsequent experiments.

5.2 Effectiveness validation

1) Effectiveness validation of the proposed strategies

To validate the necessity and contribution of the proposed knowledge-driven strategies, this paper designed two comparative variants, named KTLBO1 and KTLBO2. KTLBO1 is a version of KTLBO without the crossover-based teaching-learning operatorto simulate the lack of structural topology preservation, while KTLBO2 is a version of KTLBO without the neighborhood searchto simulate the absence of critical-path guidance. The Wilcoxon rank-sum test is employed to verify the significance of differences between optimal and general results. The results of the effectiveness experiment using evaluation metrics are presented in Table 5. The best values within the same category are displayed in bold, and the symbols “+”, “≈”, and “-” indicate that KTLBO is significantly better than, statistically equivalent to, and significantly worse than its rival methods.

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Table 5. Statistical results of evaluation metrics by KTLBO with its variants.

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It can be observed that in terms of GD and IGD, KTLBO achieved the best values in all test cases except for #DP17, and is significantly superior to KTLBO1 and KTLBO2 in most cases. Regarding , KTLBO obtains the best values in all test cases except for #DP17, #MK12, and #MK14, and is significantly better than KTLBO1 in most cases. The analysis of these results indicates that the discrete recombination mechanism is crucial; without the crossover-based operator (KTLBO1), the algorithm fails to preserve excellent gene structures of scheduling sequences, leading to a significant deterioration in convergence accuracy. Furthermore, KTLBO2 exhibits premature convergence, validating that the neighborhood search utilizing critical path knowledge is essential for breaking the deadlocks caused by tool capacity constraints. In summary, the proposed strategies are not merely incremental additions but are proven to be essential for addressing the specific complexity of BFJSP-TA.

2) Comparative analysis of algorithm performance

To verify the comprehensive performance of the proposed KTLBO algorithm, three well-known multi-objective meta-heuristics are employed as comparisons, which are NSGA-II [30], MOGWO [32], and SPEA2 [33]. The results of the comparative analysis of comprehensive algorithm performance using evaluation metrics are shown in Table 6. To further visualize the distribution and stability of the experimental results, the violin plots of the IGD metrics across all test cases are illustrated in Fig 6. To validate the statistical significance of the proposed method, the non-parametric Friedman test and Wilcoxon signed-rank test are conducted. The best values within the same category are displayed in bold, and the symbols “+”, “≈”, and “-” indicate that KTLBO is significantly better than, statistically equivalent to, and significantly worse than its comparison methods.

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Table 6. Statistical results of evaluation metrics by KTLBO and its comparisons.

https://doi.org/10.1371/journal.pone.0342585.t006

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Fig 6. Violin plot of IGD metrics.

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Table 7 presents the average rankings obtained from the Friedman test. It can be observed that KTLBO achieves the lowest ranking values (1.00, 1.29, and 1.00) across all three metrics (GD, Δ, and IGD), indicating the superior overall performance of the proposed algorithm. Furthermore, Table 8 summarizes the results of the Wilcoxon signed-rank test with a significance level of 0.05. The p-values for all pairwise comparisons are less than 0.05, which rejects the null hypothesis and confirms that the performance improvement of KTLBO is statistically significant.

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Table 7. Average ranking values of algorithms by Friedman test.

https://doi.org/10.1371/journal.pone.0342585.t007

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Table 8. Statistics results by the Wilcoxon signed-rank test ( = 0.05).

https://doi.org/10.1371/journal.pone.0342585.t008

In terms of GD and IGD, KTLBO achieved the best values in most instances and is significantly better than the comparative algorithms in almost all cases (14/0/0). As observed in Fig 6, the IGD distribution of KTLBO consistently lies at the lowest position with a more compact shape compared to other algorithms. This indicates that KTLBO not only achieves higher convergence accuracy but also demonstrates superior robustness and stability against random variations in independent runs. For , KTLBO obtained the best values in all test cases except for #DP17 and #DP18, and is significantly superior to the comparative algorithms in most cases. Therefore, supported by the statistical tests, KTLBO demonstrates a clear advantage in comprehensive performance among state-of-art meta-heuristics.

Regarding the scalability of the algorithm, the test instances used in this study include varying scales (e.g., small-scale #MK06 and large-scale #DP18). As shown in Table 6 and the statistical tests, KTLBO maintains a consistent lead (Rank 1 and p < 0.05) across all instances regardless of the problem size. This suggests that the proposed knowledge-driven strategies can effectively handle the increased complexity of the solution space in larger problems, indicating good scalability of the proposed algorithm.

5.3 Comparison of KTLBO and Gurobi

To validate the correctness of the proposed MIP formulation for the BFJSP-TA and assess the capability of the KTLBO algorithm in obtaining high-quality solutions, the commercial exact solver Gurobi is employed as comparison. Due to the NP-hard nature of the problem, the comparison is conducted on a set of randomly generated small-scale instances (denoted as #Small1 to #Small5). In these instances, the processing times of operations are generated following a uniform distribution in the range [2,20].

Since Gurobi is designed primarily for single-objective optimization and cannot directly generate a Pareto front, the two objective functions of BFJSP-TA, namely Total Tool Wear Cost (TWC) and Weighted Sum of Tardiness (WST), are minimized separately to verify the algorithm’s accuracy in finding extreme values. KTLBO runs 20 independent times for each instance, and the best solution found is recorded. The time limit for Gurobi is set to 3600 seconds for each run to find the global optimum or a lower bound.

The comparison results are reported in Table 9. The performance gap is defined as in Eq. (19).

(19)

where a gap of 0 indicates that KTLBO successfully finds the same optimal solution as Gurobi. A positive gap implies that KTLBO finds a worse feasible solution than the continuous execution of Gurobi within the time limit. The superscript ^⋆ marks values proven optimal by Gurobi.

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Table 9. Comparison results of KTLBO and Gurobi on small-scale instances.

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As can be seen from Table 9, KTLBO demonstrates excellent convergence capabilities on small-scale instances. For the smallest instances (#Small1 and #Small2), KTLBO successfully finds the optimal solutions (Gap=0%) that are certified by Gurobi. For the slightly larger instances (#Small3 and #Small4), although the problem complexity increases, the gaps for the TWC objective remain strictly within 1.2%, and the gaps for the WST objective are kept within a reasonable range (2.8% to 6.7%). Even in instance #Small5, where the search space expands significantly, KTLBO maintains a competitive performance with a TWC gap of only 2.5%, while the WST gap inevitably increases slightly due to the conflict between objectives and the NP-hard nature of the problem. It is worth noting that KTLBO uses much less time in solving #Small3, #Small4 and #Small5. These results confirm that the proposed algorithm is capable of effectively approaching the global optimum, validating the reliability of its optimization mechanism.

5.4 Necessity of collaborative optimization

To validate the application effectiveness of the collaborative optimization of tool allocation and production scheduling proposed in this paper, a comparative experiment is conducted using the traditional sequential optimization method as a control. In the sequential optimization, tools are first randomly and evenly allocated to the machining equipment, followed by the optimization of operation assignment and machining sequence. To reduce the impacts of randomness, the sequential optimization experiments independently run five times (denoted as SO1-5). The experiments are carried out on test cases #DP13, #MK12, and #MK15.

As shown in Fig 7, the first front solutions obtained by the collaborative optimization method (CO) and the five sequential optimization methods (SO1-5) are depicted. It can be clearly seen that the front of CO completely dominates the front of SO.

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Fig 7. Pareto front obtained by collaborative optimization (CO) and sequential optimization (SO1-5).

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The superiority of the collaborative strategy can be attributed to the inherent coupling between tool allocation and operation scheduling capabilities. In the sequential optimization framework, the decision space is artificially severed, as tool allocation is fixed prior to scheduling. This creates a resource rigidity constraint. For example, if the random allocation phase fails to assign a necessary tool to a high-performance machine, the subsequent scheduling phase is permanently blocked from utilizing that machine for the corresponding operation, even if it would have significantly reduced the tardiness. The scheduling algorithm is forced to search within a strictly limited subspace defined by the initial allocation.

Conversely, the collaborative optimization explores the joint search space of tool configuration and operation sequencing simultaneously. This allows the algorithm to adaptively resolve bottlenecks by reconfiguring tool locations to match scheduling needs, effectively assigning tools to machines that are identified as critical for minimizing delays or costs during the evolutionary process. The clear gap between the CO and SO fronts in Fig 7 visually quantifies the opportunity cost incurred by the sequential separation of these coupled decisions, demonstrating that high-quality solutions are structurally inaccessible to the sequential approach.