Comparative Study of Application of Production Sequencing and Scheduling Problems in Tire Mixing Operations with ADAM, Grey Wolf Optimizer, and Genetic Algorithm

Abstract

Scheduling and sequencing problems in manufacturing are complex and challenging to solve. Effective process planning is fundamental to optimizing production time and resource utilization in process-type manufacturing environments such as tire manufacturing. This research focuses on an existing tire manufacturing process. The scheduling problem in the compound mixing stage, which is considered the most challenging and vital stage of tire manufacturing, has been solved in this study. Adaptive Moment Estimation Optimizer (ADAM Optimizer), Grey Wolf Optimizer (GWO), and Genetic Algorithm (GA) are selected as solution methodologies. A comparative analysis is performed to evaluate the effectiveness of these algorithms based on critical performance metrics, including completion times, machine utilization, and setup numbers. The results of this study show that ADAM and algorithmic methods optimize machine utilization by 1.28% and save 32.6% production time, outperforming the traditional manual allocation strategies mainly used by industrial companies, as well as GWO and GA.

1. Introduction

The world of manufacturing is moving fast. Client demand has shifted from mass production to small batches of many types, and complexity in the production environment has increased. Tire manufacturing is a complex process that requires the sequencing and scheduling of resources and processes to be productive and cost-effective. Since environmental concerns are prominent in manufacturing, mass production in tire manufacturing is no longer an option. Under these circumstances, modern, flexible manufacturing systems allow the production of many products on one machine while avoiding setup changes, even when faced with product specification changes.

One of the critical operations in tire manufacturing is the compound mixing process, in which different raw materials are combined to produce rubber compounds with specific properties. Fortunately, advancements in computing technology have enabled the tracking and optimizing of production and logistics facilities to control production at the product level. Scheduling contributes to modern manufacturing by reducing waste, waiting time, and inefficiency in inventory and logistics, as well as maximizing the advantage of flexible manufacturing systems. Scheduling has gained much interest and has been studied for decades because of its complex nature, including in tire manufacturing.

The compound mixing stage in tire manufacturing significantly affects the final product’s performance by ensuring that all materials, such as rubber, fillers, oils, sulfur, and accelerators, are appropriately mixed. These materials need to be distributed homogeneously and consistently [1]; the proper distribution of material fillers directly [2] affects the tire’s efficiency and targeted properties, such as hardness, flexibility, wear resistance, and wet grip [3].

One type of scheduling production and manufacturing problem is the parallel machine scheduling problem (PMSP). The PMSP has n jobs and m machines, each requiring a specified processing time. The mixer stage is a fundamental case in a manufacturing environment since it ensures the homogeneous distribution of materials and determines the rheological properties, influencing the final product’s flow behavior within the mold and its vulcanization characteristics [4]. In the mixer stage, multiple machines for a product are placed in one workstation to avoid bottlenecks, as well as different machines for other types of products. So, this stage needs to solve non-identical PMSP.

This paper explores the application of advanced optimization algorithms, including ADAM Optimizer, Grey Wolf Optimizer (GWO), and Genetic Algorithm (GA), in solving the production sequencing and scheduling problem in tire mixing, or, in other words, the compound mixing operation. The first contribution of this work is the algorithmic innovation in discrete optimization, where the ADAM Optimizer is used for the first time in process-type production and tire mixer scheduling. ADAM Optimizer, a gradient-based optimization algorithm widely used in machine learning, addresses a discrete mixer assignment problem. ADAM Optimizer, which is typically used in deep learning applications, is adapted in this study to minimize the makespan: maximizing the processing time between mixers by taking capacity constraints into account.

The second contribution of this study in terms of algorithms is the finite difference gradient approach. Due to the discrete nature of the assignment problem, the gradient cannot be obtained directly. So, the finite difference method is used to calculate the gradient of the fitness function approximately. This method slightly perturbs each decision variable to calculate the rate of change in the objective function, ensuring the applicability of optimization in a discrete environment.

A novel use of ADAM is its employment in discrete optimization. This study employs an uncommon methodology using the ADAM, which is often utilized in continuous areas like deep learning, to solve a discrete scheduling and assignment problem. Using that, the gradient approximation through finite differences illustrates the connection between machine learning (ML) methods and traditional operations research. This study facilitates the integration of data-driven and rule-based methodologies.

The proposed computational model improves production efficiency by reducing idle time caused by setup times, shortening overall production cycles, and enhancing capacity usage. Instead of manual and random assignments, intelligent scheduling enables higher production volumes, better resource allocation, and effective use of machine capabilities. This innovative scheduling and sequencing approach offers a practical and scalable solution for high-tech manufacturing environments striving for operational excellence.

This study discusses the literature on scheduling-related problems in Section 2. Section 3 consists of the problem statement of this work. Section 4 details the methodologies of all three algorithms used. Section 5 and Section 6 are the results and discussion parts of the study.

2. Literature Review

Scheduling methodologies are commonly classified as exact, enumerative, and approximation models. Exact methods explore the entire solution space to prove optimality, while enumerative methods have computational complexity bounded by an exponential function of the input size. Approximation methods, by contrast, provide suboptimal solutions within polynomial computational resources [5,6]. The importance of systematically identifying process bottlenecks and resolving system constraints to improve scheduling efficiency has been emphasized in the literature [7].

Scheduling in compound mixing can be interpreted in two ways: as a routing problem, where jobs are assigned to machines, and as a sequencing problem, where the order of jobs on each machine is decided. The parallel machine scheduling problem (PMSP) involves both machine-assignment and job-sequencing decisions. Various neighborhood operations have been used to improve PMSP solutions, such as job swaps between machines, sequence swaps within a machine, and job insertion from one machine to another in unrelated-machine settings [8]. PMSP is a combinatorial optimization problem that is typically NP-hard, so exact solutions become computationally intensive for large instances [9,10,11]. In particular, the single-machine variant was shown to be NP-hard by Sun et al. [12]. In compound mixing environments, mixer-stage research in tire production has primarily focused on mixing conditions, equipment, and material properties [13,14,15,16,17]. For example, the coupling reaction between TESPT and silica fillers requires temperature control (a minimum of 130 °C is typically needed for effective coupling), while temperatures above about 160 °C can produce scorching; sulfur-free coupling agents have been suggested to mitigate premature reactions [18]. The mixer fill factor influences mixing efficiency and quality, and an optimal fill ratio is necessary for proper dispersion and distribution of compound constituents [19]. Mixer additives have been identified as challenging from a scheduling perspective [20]. The mixing process is highly complex, and parameters that are energy-intensive, such as rotor shaft torque and temperature, must be closely monitored to produce quality rubber mixes [21]. While the mixing process’s physical and chemical demands require precise control for quality, scheduling algorithms must prioritize performance objectives to boost production efficiency.

Makespan—the completion time of the last job—is the most common objective in PMSP [22]. However, other objectives that arise in practical manufacturing systems, such as tardiness and earliness/tardiness trade-offs, are also important [23,24,25]. Three classical parallel machine models are used in the literature: identical parallel machines (machine-independent processing times), uniform parallel machines (different processing rates), and unrelated/non-identical parallel machines (processing times depend on the machine allocation) [26,27,28]. Studies integrating process planning and scheduling with due-date assignment and slack have been reported [29], and earliness/tardiness objectives have been addressed using GA, Ant Colony, and hybrid evolutionary strategies [30,31,32]. Manufacturing systems are subject to random events such as machine failures, workforce fluctuations, material shortages, and other disruptions. Variability appears in interarrival times of materials, processing and repair times, and changeover durations, among others. Flexible scheduling is therefore required to minimize the impact of downtime [33,34,35]. From this perspective, the scheduling and sequencing of the mixing process is a core concern in tire production because of its influence on product quality, production costs, and process efficiency. In light of these scheduling challenges and the need for flexibility in dynamic manufacturing environments, effective management of setup times emerges as a key factor in maintaining operational resilience and meeting production deadlines.

Minimizing setup time is critical to exploiting the benefits of processing different jobs on the same machine. Two setup paradigms are widely studied: sequence-independent and sequence-dependent setup times [36,37]. In the sequence-independent case, setup time is added to the processing time of a job regardless of the predecessor; in the sequence-dependent case, setup time is affected by the previously scheduled job, which increases problem complexity. Timeliness is especially crucial in multi-stage assembly environments where strict due dates are required; as delays propagate through subsequent stages, robust scheduling that can react to changes in setup times is needed [38,39]. Amid these sequence-dependent complexities and the need for adaptability, metaheuristics offer robust frameworks for optimizing uncertain scheduling decisions.

Metaheuristic optimization techniques have received considerable attention for solving global optimization problems. Representative algorithms include the Grey Wolf Optimizer (GWO), which mimics wolves’ social hierarchy and hunting behavior [40], and Genetic Algorithms (GAs), which are inspired by natural selection. Metaheuristic iterations are typically decomposed into selection, search (exploration/exploitation), and update (survival/elimination) phases [41,42,43]. Because of their flexibility and robustness, metaheuristics have been widely applied to multi-modal, non-linear, and complex engineering optimization problems. GWO has been reported as an efficient and robust alternative for non-linear systems [44,45]. The advocated approach in this study incorporates stochastic variables to increase operational robustness.

Research on discrete manufacturing scheduling has overwhelmingly relied on combinatorial heuristics and metaheuristics (e.g., GA) rather than gradient-based optimizers. Job shop scheduling is typically solved by Genetic Algorithms, simulated annealing, and particle swarm optimization because of its NP-hard combinatorial nature [46]. ADAM Optimizer, by contrast, is a stochastic gradient method originally developed for neural network training and is most commonly used as a training optimizer inside machine learning subroutines (for example, as the default optimizer for neural networks in deep RL frameworks) [47,48]. To our knowledge, only one recent study, Xia and Ji [49], has attempted to embed Adam-like ideas into discrete scheduling for flow shop problems, indicating that direct ADAM-based methods remain novel in production scheduling. The literature also lacks direct ADAM vs. GA/GWO comparisons on production scheduling benchmarks with sequence-dependent setups or makespan objectives. Identified gaps in the literature emphasize the advantages of hybrid optimization paradigms. The current study evaluates ADAM-derived adaptations alongside established metaheuristic frameworks.

Mixer-oriented scheduling (e.g., for tire or rubber mixing) has been understudied with respect to robustness under uncertainty. Existing mixer scheduling models are typically deterministic; for example, a Banbury mixer schedule was formulated as a resource-constrained project scheduling problem with fixed capacities but nominal, known parameters [50]. Few, if any, mixer-specific scheduling studies explicitly incorporate machine availability uncertainty, variable setup durations, or stochastic job releases. Contemporary models disregard uncertainties intrinsic to mixer scheduling. The advocated approach incorporates stochastic variables to increase operational robustness. This gap underscores the necessity for innovative integrations of AI/ML with discrete scheduling paradigms. The current investigation addresses this gap by comparing gradient-based optimizers with combinatorial methodologies.

Overall, a gap exists in integrating AI/ML methods with discrete scheduling. Surveys indicate that traditional scheduling methods rely on fixed heuristic rules and lack adaptability; deep RL frameworks have been used to embed ML within scheduling, but usually do not replace combinatorial solvers with gradient-based optimizers [51]. Industry 4.0 reviews urge the adoption of adaptive, data-driven approaches (e.g., RL) for scheduling [52], and future-direction studies identify “integrating machine learning techniques” as a priority area [46]. Despite these calls, practical examples that unify ML-driven and traditional combinatorial approaches remain scarce. This selection tackles ADAM’s ability to deliver viable solutions. The proposed framework validates its extension to industrial scheduling contexts. ADAM was selected in this work as an innovative ML algorithm to obtain fast, feasible solutions. Although ADAM was initially designed for deep learning applications, its potential to optimize scheduling parameters in industrial contexts has been suggested [48].

The ADAM Optimizer was adapted for discrete mixer assignment and sequencing to minimize makespan under capacity constraints.

A finite difference gradient approximation was introduced to enable gradient-based optimization in the discrete assignment setting.

A hybrid data-driven and rule-based scheduling framework was proposed to improve robustness against variable setup durations and machine availability.

The ADAM Optimizer method was compared to the Grey Wolf Optimizer (GWO) and Genetic Algorithm (GA) on parallel machine instances with sequence-dependent setups, reporting solution quality and computational effort.

Practical benefits were demonstrated on representative mixing scenarios, including reduced idle time, improved resource utilization, and shorter production cycles.

3. Problem Statement

This study addresses scheduling and resource allocation in the mixing stage of the tire industry, which is characterized by heterogeneous mixers, sequence-dependent setup times, and inter-stage readiness by adapting the ADAM Optimizer via a finite difference gradient approximation for discrete mixer assignment and sequencing to minimize makespan under capacity constraints.

Specifically, we examined whether the ADAM Optimizer reduces makespan relative to GWO and GA on sequence-dependent case studies, whether it yields lower makespan variance under stochastic setup and machine availability concerns, and whether solution quality improvements are reached at computational costs comparable to those of GWO and GA.

Figure 1 shows the workflow of the Banbury mixer machine. Raw materials are selected and prepared for the relevant compounds. In detail, the first step in tire manufacturing is mixing the main ingredients in specified quantities before manufacturing. Then, they come to Banbury, where receipts are called from a controller. After that, the mixing machine is filled with prepared raw materials. The purpose of this mixing machine is to blend the components. This study inspects a real problem’s workflow in the production line: Banbury mixers mix and knead the rubber compound. The rotors inside them blend the materials to form a compound. Modern mixers have auto controls. These controls manage the temperature, pressure, and other variables during the process to determine the quality of the compound. The mixing procedure creates a homogeneous combination. The quality control examines the compound’s accuracy and homogeneity, implements any required modifications, and classifies the output compounds. After the outputs are controlled, if approved, the outputs are placed in stock as semi-products. Then, semi-products are sent to extruders to be cut. Mixing is the base of tire manufacturing and is required to specify the final product’s mechanical properties, durability, driving performance, and safety.

Figure 2 shows the working environment. The executory product count for these 10 mixers is 2709 unique compounds. When the demand is processed, there are 650 different compounds in this problem since the demand determines the compounds. Consequently, demand can change daily. This study considers a day when 381 compounds, which are also named semi-products, total 7198 units of semi-products. Each compound has a predetermined Banbury mixer. There are 10 Banbury mixer variants, each having different specifications and capacities. These compounds need an optimized schedule and sequence since they cannot be mixed jointly. What the company wants to achieve right now is to lower the setup counts by merging the same compounds produced separately. In this problem, each machine has different features but can operate in parallel.

4. Methodology

At this point, the assumptions established for the three distinct models and the parameters employed in each model are identified. The mathematical method, ADAM Optimizer, GWO, and GA solutions will be detailed in that order. The algorithms are written in Python 3.12, and the software used is PyCharm Professional 2024.3 with Processor AMD Ryzen 7 5825U with Radeon Graphics, 2000 MHz, eight Cores, and 16 Logical Processors.

4.1. Model Assumptions

Every compound is considered indivisible and allocated to only one machine;

Every machine operates simultaneously, though with separate capabilities;

The outputs of all machines should be temporally aligned so that they can be mixed simultaneously;

Each machine’s setup time depends on the last sequence;

The processing and setup times for compounds and machines are fixed;

As soon as a compound starts processing, it cannot stop until it is complete;

Nevertheless, each compound can only be assigned to machines that satisfy specific capacity and operational feasibility conditions.

4.2. Mathematical Models and Integrated Optimization Processes for Problem Optimization

In this section, mathematical models are developed for ADAM Optimizer, GWO, and GA, and the demand for each compound will be allocated economically to the mixers. The following notations in Table 1 are commonly used for algorithms in this study.

4.2.1. Objective Function and Constraints

The main objective is to minimize the production system’s total makespan of all mixers. Table 1, Table 2, Table 3 and Table 4 give the parameter definitions of the model, which aims to reduce the following objective function:

(1)$min\left\{\underset{\mathrm{Makespan}}{\underbrace{\underset{m\in M}{max}{y}_m}}+\lambda \sum _{m\in M}max(0,z_m-C_m)\right\}$

where

(2)$y_m=\sum _{p\in P}{Q}_p·T_{pm}·x_{pm},z_m=\sum _{p\in P}{Q}_p·x_{pm}$

and $x_{pm}\in \{0,1\}$ is the decision variable that indicates whether product p is assigned to mixer m, and $\lambda ={10}^{10}$ is set.

Constraints:

Single Assignment Rule: For each product, only one mixer should be selected from the defined mixer set $F_p$:

(3)$\sum _{m\in F_p}{x}_{pm}=1,\forall p\in P.$

Capacity Limit: The amount of product assigned to each mixer should not exceed the capacity $C_m$ of that mixer:

(4)$z_m\le C_m,\forall m\in M.$

4.2.2. ADAM Optimizer

Adaptive Moment Estimation Optimizer (ADAM Optimizer) is a widely used optimization algorithm combining the benefits of AdaGrad and RMSProp; it employs per-parameter adaptive learning rates based on the exponential moving averages of the first and second moments of gradients, enabling efficient computation and faster empirical convergence on many large-scale problems. In this work, we quantify performance using (i) the iterations (and corresponding function evaluations) required to reach a gradient-norm tolerance $\parallel {g_t\parallel }_2<{10}^{-6}$, (ii) the final objective gap relative to the best-known solution, (iii) wall-clock runtime per instance, and (iv) capacity violation metrics (e.g., ${max}_m\{{\sum }_p{x}_{pm}{u}_{pm}-C_m,0\}$). When we report “improved performance,” we refer explicitly to measurable improvements in one or more of these metrics (fewer iterations/lower objective/smaller feasibility violation/reduced runtime). ADAM Optimizer has shown strong empirical results for scheduling-parameter tuning and in deep architectures such as GoogleNet, ResNet101, and EfficientNet-b0 [53,54].

For mixer allocation, we optimize a continuous assignment vector $\theta \in {\mathbb{R}}^{\left|P\right|}$ whose components ${\theta }_p$ represent positions within the ordered feasible-mixer list $J_p=(j_{p,1},\dots ,j_{p,|F_p|})$. The optimizer initializes ${\theta }_0\left[p\right]=\mathcal{U}(1,|F_p\left|\right)$ for all $p\in P$ to promote balanced exploration; moment vectors start at zero, $m_0={\mathbf{0}}^{\left|P\right|}$, and $v_0={\mathbf{0}}^{\left|P\right|}$, providing unbiased initial moment estimates.

Table 2 lists the ADAM Optimizer hyperparameters used in our experiments.

Figure 3 shows the outline of the algorithm. Direct optimization of discrete assignments is non-differentiable, so we adopt an optimization-by-relaxation approach that (i) relaxes discrete indices into bounded real domains, (ii) constructs a differentiable surrogate via soft assignments, (iii) estimates gradients using robust finite difference (FD) schemes, and (iv) recovers feasible discrete solutions by projection, annealing, rounding, and, if needed, deterministic repair. Specifically, for product p, we define kernel-based soft assignment weights with temperature $\tau >0$:

(5)$w_{pm}({\theta }_p;\tau )=exp\left(-{\displaystyle \frac{{({\mathrm{pos}}_p\left(m\right)-{\theta }_p)}^2}{2{\tau }^2}}\right)⊮\{m\in F_p\},x_{pm}({\theta }_p;\tau )={\displaystyle \frac{w_{pm}({\theta }_p;\tau )}{{\sum }_{k\in F_p}{w}_{pk}({\theta }_p;\tau )}}.$

This soft-assignment construction admits the differentiability of the surrogate objective. As an alternative reparameterization for categorical variables, the Gumbel–Softmax trick may be employed [55].

The relaxed surrogate objective combines the soft–cost evaluation and differentiable penalties:

(6)$\overline{f}(\theta ;\tau ,\rho )=f_{\mathrm{soft}}\left(X(\theta ,\tau )\right)+\rho \mathcal{P}\left(X(\theta ,\tau )\right),$

where $f_{\mathrm{soft}}$ evaluates expected costs under soft assignments, $\mathcal{P}$ is a differentiable penalty for constraint violations, and $\rho >0$ controls penalty strength. Typical penalty choices include quadratic hinge penalties for capacity violations, e.g.,

(7)${\mathcal{P}}_{\mathrm{cap}}\left(X\right)=\sum _m{\left[max\left(0,\sum _p{x}_{pm}{u}_{pm}-C_m\right)\right]}^2,$

with $u_{pm}$ denoting product loads and $C_m$ mixer capacities, standard references for penalty and augmented-Lagrangian methods [53,54].

Gradients of $\overline{f}$ are obtained via coordinate-wise finite differences with an adaptive perturbation scale. For the basis vector $e_i$,

(8)$g_t\left[i\right]={\displaystyle \frac{\overline{f}({\theta }_t+{\delta }_i{e}_i;{\tau }_t,{\rho }_t)-\overline{f}({\theta }_t;{\tau }_t,{\rho }_t)}{{\delta }_i}},$

where the perturbation ${\delta }_i$ is chosen adaptively as

(9)${\delta }_i=max\left({\delta }_{\mathrm{abs}},{\delta }_{\mathrm{rel}}·max(1,|{\theta }_t\left[i\right]\left|\right)\right),$

with ${\delta }_{\mathrm{abs}}$ (e.g., ${10}^{-6}$) as an absolute lower bound and ${\delta }_{\mathrm{rel}}\in [{10}^{-4},{10}^{-2}]$ as a relative scaling parameter. Forward differences halve the evaluation cost relative to central differences while retaining deterministic reproducibility; for theory and randomized zeroth-order, see [56,57].

With gradient estimate $g_t$, the ADAM moment updates and bias corrections proceed in the standard manner:

(10)$\begin{array}{c}{m}_t={\beta }_1{m}_{t-1}+(1-{\beta }_1)g_t,v_t={\beta }_2{v}_{t-1}+(1-{\beta }_2)g_t^2,\end{array}$

(11)$\begin{array}{c}{\widehat{m}}_t={\displaystyle \frac{m_t}{1-{\beta }_1^t}},{\widehat{v}}_t={\displaystyle \frac{v_t}{1-{\beta }_2^t}},{\theta }_{t+1}={\theta }_t-\alpha {\displaystyle \frac{{\widehat{m}}_t}{\sqrt{{\widehat{v}}_t}+ϵ}}.\end{array}$

After each update, we project ${\theta }_{t+1}$ back to feasible bounds,

(12)${\theta }_{t+1}\left[p\right]\leftarrow min\left(max({\theta }_{t+1}\left[p\right],1),\left|F_p\right|\right),\forall p\in P,$

and we apply annealing schedules to harden assignments and enforce feasibility over time:

(13)${\tau }_{t+1}={\gamma }_{\tau }{\tau }_t({\gamma }_{\tau }\in (0,1)),{\rho }_{t+1}={\gamma }_{\rho }{\rho }_t({\gamma }_{\rho }>1),$

with typical choices of ${\gamma }_{\tau }\approx 0.98$ and ${\gamma }_{\rho }\approx 1.05$. At termination the discrete assignment is recovered by modal selection $m_p^{\star }=arg{max}_{m\in F_p}{x}_{pm}({\theta }_T;{\tau }_T)$ or, for sufficiently small ${\tau }_T$, by rounding $\mathrm{round}\left({\theta }_T\left[p\right]\right)$ to the nearest index in $J_p$.

If the rounded modal assignment violates feasibility, we apply a deterministic greedy repair routine that prioritizes reassignments with minimal objective increase while restoring feasibility. For any overloaded mixer m, compute the set $S_m=\{p:x_{pm}=1\}$ and overload $O_m={\sum }_{p\in S_m}{u}_{pm}-C_m>0$. For each $p\in S_m$, identify the best alternative mixer $m^{\prime }=arg{min}_{k\in F_p\setminus \left\{m\right\}}{\Delta }_{p\to k}$ where ${\Delta }_{p\to k}=f(\mathrm{assign}p\to k)-f\left(\mathrm{current}\right)$ is the incremental objective increase. Sort $S_m$ by the ratio ${\Delta }_{p\to m^{\prime }}/u_{p,m^{\prime }}$ (objective increase per unit load alleviated) and reassign top items to their best alternatives until $O_m$ is eliminated or no feasible alternatives remain. If overloads remain, iterate across mixers or apply a secondary global minimal-impact reassignment. Precomputing best alternatives and using heap yields repair complexity $O\left(\right|S_m|log|S_m|+|S_m|·d)$, where d is the average number of alternatives per product; since typically $d\ll \left|M\right|$, the repair is inexpensive relative to global FD evaluations and is deterministic (no random tie-breaking).

To mitigate finite difference noise we (i) reuse shared computations across perturbed evaluations, (ii) include $ϵ$ in denominators to prevent division instabilities, (iii) optionally clip gradients, and (iv) exploit parallel evaluation of the $\left|P\right|+1$ surrogate evaluations on multi-core hardware. Each iteration therefore requires approximately $\left|P\right|+1$ surrogate evaluations (one base plus $\left|P\right|$ perturbations), each costing proportionally to the average number $\left|M\right|$ of feasible mixers; this yields a per-iteration time $\mathcal{O}\left(\right|P|\times |M\left|\right)$ and total time $\mathcal{O}\left(\right|P|\times T\times |M\left|\right)$, with space complexity $\mathcal{O}\left(\right|P\left|\right)$ for core vectors and temporary buffers.

High-dimensional and noisy instances deserve explicit caution. The linear per-iteration dependence on $\left|P\right|$ renders FD-based ADAM expensive as $\left|P\right|$ grows, and FD noise—especially with poorly scaled ${\delta }_i$ or high measurement noise—can slow convergence or bias moment estimates. Randomized zeroth-order schemes (e.g., randomized directional estimators or SPSA) reduce per-iteration evaluations to $\mathcal{O}\left(1\right)$ at the cost of increased variance [56,57]. Practical mitigations for very large $\left|P\right|$ include (i) randomized gradient approximations, (ii) block-coordinate FD perturbations (perturbing a subset of coordinates per iteration), (iii) hybrid strategies that use ADAM-based warm starts followed by exact local refinement (mixed-integer solver), or (iv) stronger regularization and gradient clipping.

Hyperparameters such as ${\tau }_0$, ${\rho }_0$, ${\delta }_{\mathrm{rel}}$, and ADAM’s learning rate significantly affect convergence and discretization timing. Typical robust starting values in our experiments are ${\tau }_0\approx 1.0$, ${\rho }_0\approx 0.1$, ${\gamma }_{\tau }\approx 0.98$, and ${\gamma }_{\rho }\approx 1.05$, but these should be tuned per instance using simple model-selection approaches (grid search on representative validation subsets, randomized search, or lightweight Bayesian optimization) to trade off exploration, FD noise sensitivity, and discretization speed.

The algorithm terminates when any of the following are met: iteration limit $t_{max}=1000$, gradient norm $\parallel g_t{\parallel }_2<{10}^{-6}$, objective change $|\overline{f}\left({\theta }_t\right)-\overline{f}\left({\theta }_{t-1}\right)|<{10}^{-8}$, or stagnation for 50 consecutive iterations (where stagnation is defined as no reduction in $\overline{f}$ exceeding ${10}^{-8}$):

(14)$\mathrm{Termination}=\left\{\begin{array}{cc}\mathrm{True},\hfill & \mathrm{if}t\ge t_{max}\mathrm{OR}{\parallel g_t\parallel }_2<{10}^{-6}\hfill \\ \hfill & \mathrm{OR}|\overline{f}\left({\theta }_t\right)-\overline{f}\left({\theta }_{t-1}\right)|<{10}^{-8}\hfill \\ \hfill & \mathrm{OR}\mathrm{stagnation}\mathrm{count}\ge 50,\hfill \\ \mathrm{False},\hfill & \mathrm{otherwise}.\hfill \end{array}\right.$

Figure 4 shows the pseudo code, which constructs the dataset from the company database and then computes mixer and compound assignments. This implementation preserves ADAM’s per-parameter adaptivity and computational efficiency while enabling effective gradient-guided search in discrete scheduling problems via surrogate relaxation, finite difference gradients, annealing, and deterministic repair.

4.2.3. Grey Wolf Optimizer (GWO)

GWO performs a global search with social hierarchy and hunting dynamics inspired by nature and mainly inspired by the hunting strategy of gray wolves, and is used for global optimization [58]. The GWO is a metaheuristic optimization algorithm inspired by gray wolves’ social hierarchy and hunting behavior based on their pursuit of prey. It is part of the swarm intelligence algorithms, resembling particle swarm optimization or Ant Colony Optimization. Metaheuristics are commonly used for NP-hard problems because they can find near-optimal solutions without exhaustive search. Metaheuristics like GA, simulated annealing, Tabu Search, and, more recently, GWO or others have been applied to address scheduling issues. These methods are popular because they can handle complex constraints and large search spaces. GWO was developed by [40]; GWO mimics the hunting behavior of gray wolves and has been successfully utilized in manufacturing scheduling problems [59,60]. GWO is used for optimization problems: The best solution is called alpha, and the other solutions are called beta and delta, while the others in GWO are omegas. This scenario can be created by updating the wolves’ positions over their leaders (alpha, beta, delta). Leaders consist of random vectors and coefficients that decay over time to switch between exploring and exploiting.

GWO initializes a population of $N_{\mathrm{pop}}$ wolves, each representing a potential solution to the mixer allocation problem through position matrices $X_i\in {\mathbb{R}}^{\left|P\right|\times \left|M\right|}$ randomly generated with uniform distribution across the interval $[0,1]$. The matrix structure enables the representation of allocation probabilities for each product–mixer pair, facilitating the exploration of the solution space through continuous optimization. Following population initialization, the social hierarchy is established through fitness evaluation, where the alpha wolf represents the best solution; beta and delta wolves constitute the second- and third-best solutions, respectively; and the remaining population members form the omega wolves. This hierarchical structure guides the search process through leadership-based position updates throughout the optimization procedure.

(15)$X_i^0\left[p\right]\left[m\right]=\mathcal{U}(0,1),\forall i\in [1,N_{\mathrm{pop}}], \forall p\in P, \forall m\in M$

Table 3 gives the parameter definitions of the GWO model.

Figure 5 shows the outline of the algorithm.

The position matrices are defined as $X_i\in {\mathbb{R}}^{\left|P\right|\times \left|M\right|}$. The decreasing parameter a decreases linearly from 2 to 0 as the iteration progresses.

Calculation steps using random numbers $r_1,r_2\in [0,1]$:

(16)$A=2a·r_1-a,C=2r_2.$

Distance vector for each leader, for example, the $\alpha$ leader:

(17)$D_{\alpha }=|C·X_{\alpha }-X|.$

An updated position:

(18)$X_1=X_{\alpha }-A·D_{\alpha }.$

Similar operations are performed for the other leaders $\beta$ and $\delta$,

(19)$X(t+1)={\displaystyle \frac{X_1+X_2+X_3}{3}}.$

GWO convergence is controlled through multiple mechanisms, ensuring comprehensive optimization termination. The algorithm employs a maximum iteration limit of $T_{max}=500$, leadership stability assessment preventing alpha wolf improvement for 25 consecutive iterations, population convergence verification through average fitness variance below ${10}^{-4}$, and natural convergence indication through the exploration parameter approaching zero as $a\to 0$. The population convergence criterion is particularly significant, as it measures solution clustering through variance analysis defined by ${\sigma }_{\mathrm{pop}}^2={\displaystyle \frac{1}{N_{\mathrm{pop}}}}{\sum }_{i=1}^{N_{\mathrm{pop}}}{(f\left(X_i\right)-\overline{f})}^2<{10}^{-4}$, where $\overline{f}$ represents the average population fitness. This comprehensive convergence framework ensures solution quality and computational efficiency while preventing premature termination.

The computational complexity of GWO is characterized by time complexity $\mathcal{O}(N_{\mathrm{pop}}\times |P|\times |M|\times T_{max})$, where position updates and fitness evaluations dominate the computational cost. Each iteration requires three position updates per wolf for leadership following, resulting in $3\times N_{\mathrm{pop}}$ position computations per generation. The space complexity is $\mathcal{O}(N_{\mathrm{pop}}\times |P|\times |M\left|\right)$, primarily attributed to population position matrix storage; leadership solution preservation for alpha, beta, and delta wolves; and auxiliary computation vectors for distance and coefficient calculations. The substantial memory requirements reflect the population-based nature of the algorithm, where maintaining diversity across multiple solution candidates necessitates proportionally increased storage allocation compared to single-solution optimization approaches.

4.2.4. Genetic Algorithm (GA)

Genetic Algorithm (GA) provides diversity and local improvement in the solution space using a population-based evolutionary process. Several studies have applied metaheuristic algorithms to improve scheduling and allocation efficiency in industries [61]. For instance, industries widely apply GA to enhance scheduling efficiency. Many studies have shown GA to effectively find optimal or near-optimal solutions to various scheduling problems, especially in parallel machine [62,63], flow shop [64,65], and flexible flow shop scheduling areas [66]. GA was introduced by [67] as a robust optimization technique inspired by natural selection principles. It has been widely applied to job shop scheduling and sequencing problems [68]. Various studies have used GA and obtained better results than traditional methods [69,70]. Furthermore, Çelikbilek et al. [71] stated that GA finds an optimal solution and proposes alternative schedules for single- and multi-objective mathematical models. Koca and Avcı [72] studied different GA models for scheduling and stated that GA requires only some tuning to achieve better results, like advanced GA techniques like NSGA-II or NSGA-III.

GA creates an initial population of candidate solutions through randomized permutation generation, where each chromosome represents a product sequence $[p_1,p_2,\dots ,p_{381}]$ defining the processing order for allocation decisions. The population initialization procedure generates $N_{\mathrm{pop}}$ chromosomes using random permutation functions applied to the product set, followed by greedy assignment for mixer allocation to ensure feasibility constraints. The greedy assignment operates through minimization of capacity utilization ratios defined by $x_{pm}^{\ast }=arg{min}_{m\in F_p}\left\{{\displaystyle \frac{y_m+Q_p·T_{pm}}{C_m}}\right\}$, ensuring that each product assignment respects mixer capacity limitations while maintaining solution diversity across the population. Initial fitness evaluation assesses population quality using the makespan objective function augmented with penalty terms for constraint violations, establishing the foundation for subsequent evolutionary operations.

Table 4 gives the parameter definitions of the GA model.

Figure 6 shows the outline of the algorithm. GA is an evolutionary algorithm that simulates natural selection principles to optimize scheduling decisions. The population consists of N candidate solutions, which are called chromosomes. The crossover and mutation rates are determined as $p_c=0.8$ and $p_m=0.1$.

Order crossover: A randomly selected subsequence is inherited from one parent; the remaining genes are inherited sequentially from the other parent.

Mutation/swap:

(20)$\mathrm{Mutated}\mathrm{Seq}=\mathrm{swap}\left(\mathrm{seq}\right[i],\mathrm{seq}[j\left]\right),i,j\sim U(1,|P\left|\right).$

Tournament selection: The next generation inherits the most suitable among the randomly selected k chromosomes.

GA employs generational convergence monitoring through multiple termination criteria, ensuring solution quality and computational efficiency. The algorithm implements a maximum generation limit of $G_{max}=300$, fitness stagnation detection preventing best fitness improvement for 30 consecutive generations, population diversity assessment using Hamming distance metrics below five percent of the chromosome length, and an optional target fitness achievement for predetermined makespan thresholds when available. Population diversity measurement utilizes the Hamming distance formula $\mathrm{Diversity}={\displaystyle \frac{1}{N_{\mathrm{pop}}(N_{\mathrm{pop}}-1)}}{\sum }_{i=1}^{N_{\mathrm{pop}}}{\sum }_{j=i+1}^{N_{\mathrm{pop}}}{\displaystyle \frac{H(C_i,C_j)}{\left|P\right|}}$, where $H(C_i,C_j)$ represents the Hamming distance between chromosomes $C_i$ and $C_j$. This comprehensive convergence framework prevents premature termination and excessive computational expenditure while maintaining solution diversity throughout the evolutionary process.

The computational complexity of GA is characterized by time complexity $\mathcal{O}(N_{\mathrm{pop}}\times |P|\times G_{max})$, where selection, crossover, and mutation operations scale linearly with population size, and fitness evaluation dominates the per-generation computational cost. Tournament selection requires $\mathcal{O}(k\times N_{\mathrm{pop}})$ comparisons per generation, where k represents the tournament size parameter. The space complexity is $\mathcal{O}(N_{\mathrm{pop}}\times |P\left|\right)$, encompassing population chromosome storage, the temporary offspring generation space during crossover operations, and elite solution preservation across generations. The relatively linear scaling concerning the problem size makes GA particularly attractive for large-scale industrial scheduling applications, where the chromosome representation efficiently captures the essential scheduling decisions while maintaining computational tractability.

4.2.5. Company Method: Manual Allocation

Research on manufacturing optimization demonstrates the superiority of metaheuristic algorithms over traditional manual approaches in solving complex NP-hard problems. Parallel Crane Scheduling in Automated Storage/Retrieval Systems requires concurrent optimization of crane assignments, request sequencing, and location selection, targeting total tardiness minimization [73]. Unrelated PMSP incorporates job splitting, inventory constraints, and renewable resources while pursuing makespan minimization [22]. Advanced algorithms achieve superior performance in large-scale environments [73], Discrete GWO handles the constraints of real-world Unrelated PMSP [22], Variable Neighborhood Search achieves 90.19% optimal solution rates [74], and GA optimizes Total Net Profit through integrated production–delivery coordination [75].

Manual allocation constitutes the established practice of mixer-compound assignment in tire manufacturing. This Excel-based methodology combines heuristic expertise with organizational knowledge, maintaining adoption through operational transparency and adaptive capabilities [69,70].

The manual process generates priority sequences $[p_1^{\ast },p_2^{\ast },\dots ,p_{381}^{\ast }]$ via weighted scoring. Priority calculation follows

(21)$\mathrm{Priority}\left(p\right)=w_{\mathrm{deadline}}·{\displaystyle \frac{1}{\text{days\_to\_deadline}}}+w_{\mathrm{customer}}·\text{tier\_weight}\left(p\right)$

integrating delivery urgency with customer stratification. Machine initialization incorporates current workloads $y_m^0$, available capacity $C_m-y_m^0$, and compatibility constraints $F_p$, defining feasible product–mixer relationships.

Compound mixing optimization provides a critical evaluation context for algorithmic advancement. ADAM Optimizer implementation achieves 32.6% production time reduction versus manual allocation, demonstrating quantified algorithmic superiority while advancing the efficiency principles established across contemporary optimization research.

Manual allocation employs a priority-driven greedy strategy wherein products are sequenced according to delivery deadlines and customer importance hierarchies before assignment to qualified production units. The decision-making framework incorporates a weighted evaluation scheme that balances existing machine workloads against processing efficiency metrics. Parameter specifications are established through structured interviews with senior production managers and scheduling supervisors, leveraging their collective expertise accumulated over multiple production cycles. Table 5 summarizes the parametric configuration of the manual allocation framework.

Figure 7 shows the workflow employed in the manual allocation approach. The methodology operates through five distinct phases, integrating quantitative decision criteria with qualitative expert judgments. The weighting coefficients (0.6 for workload balance and 0.4 for processing efficiency) reflect organizational priorities established through management consensus. The optimization objective is formulated as:

(22)$x_{pm}^{\ast }=arg\underset{m\in F_p}{min}\left\{0.6·{\displaystyle \frac{y_m}{C_m}}+0.4·T_{pm}\right\}$

Subject to the flexible capacity constraint:

(23)$z_m+Q_p\le C_m·(1+B_{\mathrm{penalty}})$

Manual allocation follows a deterministic procedure with clearly defined termination conditions, ensuring comprehensive assignment completion and operational feasibility. The methodology terminates upon completing product assignment to feasible mixers, capacity validation within soft constraint tolerances through penalty allowance mechanisms, production supervisor approval through expert validation processes, and successful schedule finalization with production control system integration. The termination condition is mathematically formalized as a deterministic completeness verification where $\mathrm{Completion}=\mathrm{True}$ if ${\sum }_{m\in M}{\sum }_{p\in P}{x}_{pm}=\left|P\right|$ and for all mixers m the condition $z_m\le C_m(1+B_{\mathrm{penalty}})$ is satisfied, ensuring both assignment completeness and capacity feasibility within defined tolerance limits. This deterministic convergence framework eliminates stochastic termination uncertainties while providing transparent validation criteria for industrial implementation.

The implementation begins with product prioritization, where order sequences are determined using delivery urgency and customer tier classifications, which are implemented through Excel sorting algorithms. This initial phase establishes the processing hierarchy that guides subsequent allocation decisions. Machine evaluation is conducted systematically, wherein production units undergo comprehensive capacity and capability assessment utilizing VLOOKUP functions for equipment qualification verification. The evaluation process considers both current workload status and technical specifications to ensure assignment feasibility.

The greedy assignment phase represents the core optimization component, implementing sequential product-to-machine allocation that minimizes the weighted objective function, incorporating workload distribution and processing time considerations. This heuristic approach makes locally optimal decisions at each allocation step, balancing computational efficiency with solution quality. Capacity verification subsequently validates assignment feasibility through conditional formatting mechanisms that provide visual capacity monitoring while permitting controlled constraint violations for critical orders through the penalty parameter.

The methodology concludes with expert refinement, wherein production supervisors conduct manual schedule adjustments that incorporate operational knowledge and unforeseen constraints not captured in the algorithmic framework. This phase utilizes pivot table analysis to facilitate load-balancing decisions and leverages tacit knowledge accumulated through operational experience. The integrated approach maintains a target capacity utilization of 90% while accommodating strategic overrides through the penalty parameter mechanism, enabling a flexible response to high-priority production requirements.

The computational complexity of manual allocation is characterized by time complexity $\mathcal{O}\left(\right|P|\times |M\left|\right)$ through sequential product processing via priority lists, linear mixer evaluation per product using spreadsheet functions, and constant-time operations for lookup and conditional formatting procedures. The space complexity is $\mathcal{O}\left(\right|P|+|M\left|\right)$, encompassing product priority list storage, machine status arrays for workload and capacity monitoring, qualification matrix representation, and assignment matrix storage within spreadsheet environments. The manual approach achieves the lowest computational complexity among all evaluated methods while potentially sacrificing solution optimality compared to metaheuristic alternatives. The deterministic nature eliminates stochastic convergence issues while providing the operational transparency essential for industrial implementation. It is particularly suitable for environments where solution interpretability and real-time adaptability outweigh optimization precision requirements.

The comprehensive comparison of computational characteristics across all implemented algorithms reveals fundamental trade-offs between solution quality and computational efficiency, as presented in Table 6. Manual allocation provides the fastest execution with linear complexity, making it suitable for real-time applications where immediate response is prioritized over optimization precision. ADAM Optimizer achieves efficient gradient-based optimization with moderate computational overhead, balancing convergence speed with solution quality. Population-based methods, including GWO and GA, require higher computational resources but offer superior exploration capabilities for complex solution spaces. GWO exhibits higher memory requirements due to matrix-based position representation. At the same time, GA maintains more efficient chromosome-based encoding.

The following are the typical stages of different algorithms for the problem of mixer-compound assignment: Data Preparation is executed for all algorithms, and P, M, $Q_p$, $T_{pm}$, and $C_m$ are loaded from data sources such as Excel. After the first step, all algorithms have their algorithm stages. Algorithms each have their unique approaches to the problem. After the handiwork is obtained from the algorithms, the repair operations are written. Repair operations add all algorithms to check for more optimal results and if the constraints are followed.

4.2.6. Experimental Study

The final solution is obtained by iteration in each algorithm, and the solutions are stated in Table 7.

The analysis indicates that MIX02 consistently exhibits the highest utilization values. In the Company Allocation method, its utilization exceeds the nominal capacity by reaching over 111%, while in the algorithmic techniques, it hovers around 97–100%. Since bottlenecks in production systems are typically characterized by the resource with the highest relative load—thereby constraining overall throughput—MIX02 can be identified as the bottleneck mixer. To address the over-capacity of MIX02, the company applies overtime on that mixer during the study period, effectively increasing its available capacity for the analyzed runs.

Moreover, MIX10 is a dedicated mixer that handles a single product family; those products could also be allocated to other feasible machines, which explains why MIX10 shows an identical 72.52% utilization across all methods.

Although the compound demand dataset spans 15 days, the Company Allocation schedule completes production in approximately $14.34$ days (≈20,643 min); by contrast, the algorithmic methods produce shorter makespans (ADAM Optimizer $\approx 10.81$ days/ $\mathrm{15,571}$ min; GA $\approx 12.31$ days/$\mathrm{17,728}$ min; GWO $\approx 12.04$ days/$\mathrm{17,350}$ min).

Moreover, execution time analysis confirms ADAM’s clear computational advantage, completing optimization significantly faster than GWO and GA. Its efficiency highlights strong algorithmic convergence and practical suitability for real-time industrial scheduling, whereas the Company Allocation method remains the most time-consuming.

5. Results and Discussion

The corresponding application analysis interprets the allocation outcomes reported in Table 7 and Figure 8 and Figure 9, compares the historical Company Allocation with the algorithmic methods (ADAM, GA, GWO), highlights operational implications such as identified bottlenecks (e.g., MIX02) and dedicated-machine behavior (e.g., MIX10), and provides practical recommendations for production planning and digitalization based on the observed scheduling and makespan results.

The data obtained from the algorithms presents a comparative evaluation of four different allocation methods in terms of machine utilization rate and production time.

As seen in Figure 8, ADAM Optimizer maximized the utilization rate of machines while minimizing the production time by reaching the lowest final maximum makespan value. In contrast, Company Allocation, the historically based manual allocation method of the thought-out company, fell behind ADAM Optimizer in machine utilization rate by 1.28% and in production time by 32.6%. GA and GWO, other algorithmic methods, achieved almost equal results with ADAM Optimizer regarding machine utilization rate, but achieved longer results in makespan values. Algorithmic methods for scheduling and sequencing in production planning, especially ADAM, are more effective than traditional manual allocation methods in maximizing resource utilization and decreasing production time, as illustrated in the findings.

Figure 9 shows the performances of four different allocation methods, ADAM, Company Allocation, GWO, and GA, regarding mixer utilization percentages. According to the observed values, the utilization percentage of mixers in the ADAM, GWO, and GA methods is generally around 98–100%, and no significant deviation is observed between the methods. On the other hand, the Company Allocation method shows substantial deviations in some mixers; for example, there is a utilization rate of around 111.03% in MIX02 and 71% in MIX06. The company cannot fulfill the demand for the compounds assigned to MIX02. In addition, while MIX07 draws attention with lower utilization rates than other mixers in all methods, MIX10 shows consistency by having a constant value (approximately 72.5%) in all methods. These findings prove that algorithmic assignment methods are superior to the history-based manual assignment strategy in balancing usage across mixers and supplying durable performance.

To conclude, several operations in tire manufacturing need to be digitalized instead of being performed by hand. With machine scheduling and sequencing, product lifecycle management, connections to product recipe and inventory management systems, production planning and scheduling, single-handed management of information, and automation of the monitoring and reporting of all metrics, the facility is expected to have more flexible decision-making capabilities. The uncertainties in the current situation will be eliminated. This way, the efficiency targets of processes can be achieved by digital transformation.

By evaluating the complexity of real-world planning processes, algorithmic optimization methods offer optimal solutions compared to traditional methods. This situation reveals that advanced algorithms can integrate more flexible and compatible solutions into production processes. When examining the tire mixing production sequence, it is determined that ADAM Optimizer significantly reduces production time and increases resource efficiency. The expert-based Company Allocation method remains inadequate in this uncertain and dynamic environment. Although GWO and GA achieve stable results in distributing the workload between mixers, undesirable deviations in the distribution occur from time to time. In summary, the Company Allocation method stands out with its ease of application, but it can lead to severe imbalances in production time and resource distribution.

6. Conclusions and Future Work

The analyses in this study indicate that ADAM Optimizer consistently delivers superior performance in rubber mixing scheduling by achieving lower makespan and higher resource utilization. At the same time, GWO and GA exhibit robust yet slightly more variable performance in load balancing. Despite the effectiveness of all three methods, subtle differences in computational directions and sensitivity to production constraints suggest that each algorithm suits operational scenarios.

More research opportunities exist for hybrid optimization methods in areas where combinatorial optimization can use ML techniques. Finite difference gradient and repair mechanisms can be tested for efficiency and robustness to better understand optimization in constrained spaces. Hybrid strategies that exploit the complementary strengths of the solvers warrant investigation: the use of GA/GWO for global combinatorial exploration and warm-start ADAM for fast, local continuous refinement. Such hybrids should be evaluated against standalone methods on makespan and robustness metrics under variable setup times and machine availabilities to quantify gains in convergence speed and solution stability.

Adopting hybrid approaches that connect different algorithms’ strengths is a practical strategy for improving production sequencing and scheduling solutions for future work. More optimal solutions can be achieved by combining the fast convergence capacity of the ADAM Optimizer with the GWO and GA. Thus, creating a more efficient and compatible production system will be possible by eliminating the disadvantages of separate algorithms. Furthermore, future research can focus on integrating deep reinforcement learning with other optimization techniques to enhance scheduling and sequencing efficiency and handle complicated trade-offs in production environments.

Author Contributions

Conceptualization, B.D. and E.Y.; methodology, E.Y. and B.D.; software, E.Y.; formal analysis, E.Y.; investigation, E.Y.; writing—original draft preparation, E.Y.; writing—review and editing, B.D.; supervision, B.D. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Data is unavailable due to privacy.

Conflicts of Interest

The authors declare no conflicts of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Tables

Figure 1 Banbury mixer machine workflow diagram.

Figure 2 The structural overview of the dataset and mixer–compound relationships. Panels: (a) daily load profiles; (b) mixing-time variability across mixers; (c) relative mixing speed by compound; (d) compound counts by tire variant; (e) allocation comparison highlighting MIX02; (f) feasible mixer counts for representative compounds.

Figure 3 The schematic diagram of the ADAM optimizer.

Figure 4 Mixer allocation pseudo code for ADAM optimizer.

Figure 5 The schematic diagram of GWO.

Figure 6 The schematic diagram of GA.

Figure 7 Flowchart representation of the manual allocation methodology.

Figure 8 The sum of machine utilization in percentage and the sum of the final makespan of the scheduling and sequencing problem across methods.

Figure 9 The machine utilization rate of mixers across methods used.

Table 1

Parameters for the optimization models.

Parameter	Value/Range
Set of products/compound types	381
Set of mixers/machines	10
Feasible mixers for the product	Given per product
Quantity of the product (L)	Given per product
Processing time in the mixer	Given per product–mixer pair
Capacity of the mixer (L)	Given per mixer
The total workload of the mixer	Computed
The total assigned quantity in the mixer (L)	Computed
Penalty coefficient for capacity violation	${10}^{10}$

Table 2

ADAM parameters.

Parameter	Description	Value/Range
$\theta$	Continuous assignment vector	${\mathbb{R}}^{\left|P\right|}$
$\alpha$	Learning rate	0.01
${\beta }_1$	First-moment decay rate	0.9
${\beta }_2$	Second-moment decay rate	0.999
$ϵ$	Small constant for stability	${10}^{-8}$
$\delta$	Finite difference step (adaptive; see text)	${10}^{-6}$ reference
$g_t$	Gradient estimate at iteration t	Computed
$m_t$	First-moment estimate	Computed
$v_t$	Second-moment estimate	Computed
${\widehat{m}}_t$	Bias-corrected first moment	Computed
${\widehat{v}}_t$	Bias-corrected second moment	Computed

Table 3

GWO parameters.

Parameter	Description	Value/Range
$X_i$	Wolf i’s position matrix	$\left|P\right|\times \left|M\right|$
a	Exploration–exploitation parameter	$2\to 0$
A	The coefficient for position update	$2a·r_1-a$
C	The coefficient for position update	$2r_2$
$r_1,r_2$	Random values for search dynamics	$\sim U(0,1)$
$X_{\alpha },X_{\beta },X_{\delta }$	Best solutions (leaders)	Computed
$D_{\alpha },D_{\beta },D_{\delta }$	Distance from the prey	Computed
$X(t+1)$	Updated position	${\displaystyle \frac{X_1+X_2+X_3}{3}}$

Table 4

GA parameters.

Parameter	Description	Value/Range
N	Population size	500
$p_c$	Crossover probability	0.8
$p_m$	Mutation probability	0.1
k	Tournament selection size	0.3
Crossover	Order crossover (OX)	0.5
Child[a:b]	Subsequence from Parent 1	Selected randomly
Remaining Genes	Filled from Parent 2	Keeping relative order
Mutation	Swap mutation	–
$i,j$	Random swap positions	$\sim U(1,|P\left|\right)$
Selection	Tournament selection	–
Repair	Feasibility correction	Adjusts infeasible solutions

Table 5

Manual allocation parameters.

Parameter	Description	Value/Range
$w_1$	Workload factor weight	0.6
$w_2$	Processing time factor weight	0.4
$\theta$	Capacity utilization threshold	0.9
$B_{\mathrm{penalty}}$	Capacity violation allowance	0.1
$F_p$	Feasible machine set for product p	Qualification matrix
$y_m$	Current machine workload	Real-time data
$C_m$	Machine capacity (L)	Equipment specifications
$T_{pm}$	Processing time matrix	Historical records
$Q_p$	Product quantity requirement (L)	Customer orders

Table 6

Comparative complexity analysis of optimization algorithms.

Method	Time Complexity	Space Complexity	Convergence Type	Parameters
ADAM	$\mathcal{O}\left(\right|P|\times T\times |M\left|\right)$	$\mathcal{O}\left(\right|P\left|\right)$	Gradient-based	$T=1000$
GWO	$\mathcal{O}(N_{\mathrm{pop}}\times |P|\times |M|\times T_{max})$	$\mathcal{O}(N_{\mathrm{pop}}\times |P|\times |M\left|\right)$	Population-based	$N_{\mathrm{pop}}=50,T_{max}=500$
GA	$\mathcal{O}(N_{\mathrm{pop}}\times |P|\times G_{max})$	$\mathcal{O}(N_{\mathrm{pop}}\times |P\left|\right)$	Evolutionary	$N_{\mathrm{pop}}=500,G_{max}=300$
Manual	$\mathcal{O}\left(\right|P|\times |M\left|\right)$	$\mathcal{O}\left(\right|P|+|M\left|\right)$	Deterministic	Excel-based

Table 7

The results of the problem.

Method	Mixer	Allocated Qty(L)	Capacity(L)	Utilization %	Makespan(min)	Total Exec.Time—10 Runs(h:mm:ss.ms)
ADAM	MIX01	672.07	673.50	99.79		0:01:15.26
	MIX02	580.42	580.43	100.00
	MIX03	639.58	653.30	97.90
	MIX04	550.10	556.30	98.89
	MIX05	505.81	506.10	99.94
	MIX06	408.45	409.70	99.69
	MIX07	1184.89	1324.30	89.47	15,571.001
	MIX08	1316.46	1325.30	99.33
	MIX09	1231.10	1297.10	94.91
	MIX10	108.78	150.00	72.52
GWO	MIX01	672.53	673.50	99.86		0:32:50.06
	MIX02	559.06	580.43	96.32
	MIX03	646.71	653.30	98.99
	MIX04	556.15	556.30	99.97
	MIX05	501.88	506.10	99.17
	MIX06	408.31	409.70	99.66
	MIX07	1130.42	1324.30	85.36	17,349.793
	MIX08	1316.86	1325.30	99.36
	MIX09	1296.98	1297.10	99.99
	MIX10	108.77	150.00	72.51
GA	MIX01	667.68	673.50	99.14		1:21:33.0
	MIX02	580.20	580.43	99.96
	MIX03	652.14	653.30	99.82
	MIX04	554.08	556.30	99.60
	MIX05	506.05	506.10	99.99
	MIX06	400.89	409.70	97.85
	MIX07	1109.90	1324.30	83.81	17,727.685
	MIX08	1323.68	1325.30	99.88
	MIX09	1294.29	1297.10	99.78
	MIX10	108.77	150.00	72.51
CompanyAllocation	MIX01	673.35	673.50	99.98		5:00:00.00
	MIX02	644.49	580.43	111.04	Bottleneck
	MIX03	651.24	653.30	99.69
	MIX04	554.86	556.30	99.74
	MIX05	500.68	506.10	98.93
	MIX06	293.80	409.70	71.71
	MIX07	1167.89	1324.30	88.19	20,643.303
	MIX08	1323.61	1325.30	99.87
	MIX09	1278.99	1297.10	98.60
	MIX10	108.77	150.00	72.51

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