Biased multi-objective particle swarm optimization with simulation for stochastic multi-echelon supply chain

Abstract

This paper studies production and distribution planning in a multi-level supply chain for industrial gas. The proposed solution uses a simulation-driven, agent-guided biased multi-objective particle swarm optimization (MOPSO) framework combined with discrete-event simulation (DES). This approach balances competing goals like cost, work time, and service performance under uncertain demand and production variability. The model considers mode-dependent production with different operating modes for each facility and jointly examines routing and scheduling choices. Biasing through balancing, routing, distribution, and priority agents influences initialization and search. An adaptive evaluation scheme focuses simulation efforts on promising areas of the Pareto front. The framework was tested on a real-world case in the process industry with realistic uncertain parameters. Compared to an unbiased particle swarm baseline, the proposed method achieves faster results and better Pareto-front quality within a similar simulation budget. This leads to effective production and distribution schedule that enhance operational performance and lowering cost while still meeting service targets. The findings indicate that the biased approach outperforms others in finding the best trade-offs between time, cost, and simulation efficiency, showing its potential to improve decision-making in complex and unpredictable supply chain environments.

Keywords

Multi-echelon supply chain stochastic optimization discrete event simulation multi-objective optimization multi-objective particle swarm optimization (MOPSO)scheduling problem

1. Introduction

In the context of globalization, supply chains have evolved into intricate networks defined by highly interconnected operations and interdependencies. Multi-echelon supply chains, comprising manufacturing facilities, distribution centers, and customer nodes, present substantial management complexities.¹ These challenges arise from the necessity to balance multiple competing objectives, such as achieving cost efficiency, minimizing lead times, and enhancing customer satisfaction.² The management of such systems is further complicated by inherent uncertainties, including fluctuating demand patterns, production variability, and logistical disruptions.³ Effectively addressing these multifaceted challenges is essential for organizations seeking to maintain a competitive edge in dynamic and volatile market environments.

An increasing volume of research highlights the effectiveness of multi-objective optimization in managing trade-offs inherent in supply chain decision-making.⁴ By enabling decision-makers to evaluate Pareto-optimal solutions across conflicting objectives, such as cost and time, these approaches offer a comprehensive perspective often absent in single-objective methods.⁵ This led to the constant use of multi-objective evolutionary algorithm (MOEA) in effectively solving problems with large decision space, rather than using branch-and-bound or branch-and-cut method used in integer programming problems. That is because branch-and-bound or branch-and-cut methods are mostly used to solve deterministic problems and have limitations in which it requires high computational power for large scale problems.^6,7

A prominent algorithm in this domain, multi-objective particle swarm optimization (MOPSO), has gained recognition for its capability to efficiently identify Pareto fronts in complex, high-dimensional problem spaces.⁸ MOPSO has been applied successfully to a range of supply chain optimization problems, including vehicle routing, inventory management, and production scheduling. Standard MOPSO, while effective and compatible in solving many multi-objective optimization problems including multi-echelon supply chain scheduling, faces significant limitations. One major issue is its difficulty in maintaining diversity in the Pareto front, often resulting in a poorly distributed set of solutions.⁹ Its performance is highly sensitive to parameter tuning, requiring careful calibration of inertia weight, cognitive and social coefficients, and swarm size to avoid suboptimal results.^8,10 These challenges are further exacerbated in dynamic and high-dimensional problems, where MOPSO’s static nature and inefficiency in adequately exploring large solution spaces limit its adaptability and effectiveness.¹¹ As such, standard MOPSO requires enhancements, such as hybridization with domain-specific heuristics, intelligent agents, and simulation techniques, to address these shortcomings and improve its performance in real-world applications.

To overcome these challenges, numerous researchers have explored hybrid frameworks that merge evolutionary algorithms with domain-specific heuristics. A key approach in this area is the integration of discrete-event simulation (DES), which models the randomness and dynamics of real-world processes such as sudden demand spikes, production delays, and logistical bottlenecks.¹² DES is particularly suited for capturing the inherent stochasticity and variability in supply chain operations,^13–15 offering more realistic evaluations compared to traditional mathematical programming methods such as mixed-integer programming (MIP). While MIP excels in solving deterministic problems with defined constraints and objectives,¹⁵ it often struggles to incorporate stochastic elements or dynamically adjust to real-time variability.¹⁶ In contrast, DES allows for the simulation of dynamic and uncertain environments, making it a more appropriate choice for modeling and optimizing stochastic supply chain problems.^13,15,17

However, integrating DES with MOPSO comes with its own set of challenges. The computational demand increases significantly as each candidate solution requires repeated simulations to evaluate its performance.^18,19 This computational burden is further amplified in stochastic environments, where the variability in demand, production, and logistics necessitates multiple simulation runs to ensure robust evaluations. To address this, researchers have focused on developing intelligent frameworks that reduce unnecessary simulations and allocate more computational resources to the most promising solutions, thereby enhancing both efficiency and effectiveness.²⁰

Simultaneously, advances in intelligent agent-based methods have demonstrated the benefits of embedding domain knowledge directly into the optimization process.²¹ For example, intelligent agents can prioritize customers facing near-stockout conditions or initiate vehicle routes strategically, steering the algorithm toward more promising solution areas with greater efficiency than random searches.^22,23 These targeted mechanisms help mitigate the slow convergence challenges of standard MOPSO in complex scenarios, enabling faster optimization while preserving solution diversity.

Despite these advancements, there remains a significant research gap in developing integrated frameworks that effectively balance computational efficiency with the quality of solutions for multi-echelon supply chains operating under uncertainty. Many existing studies focus on either improving simulation accuracy or refining optimization algorithms, but few have explored the combined potential of these approaches.²⁴ Given the increasing complexity of supply chain operations, bridging this gap is crucial for both theoretical research and practical applications. Such progress would provide decision-makers with effective tools to manage uncertainty while optimizing multiple objectives.

The motivation behind this research lies in the need to develop an optimization framework and algorithm capable of addressing the complexities of multi-objective, large-scale, stochastic supply chain problems. Existing methods often fall short in balancing computational efficiency with solution quality, particularly in scenarios where real-world uncertainties play a significant role. By incorporating stochasticity into the problem formulation, this research seeks to capture the variability and unpredictability inherent in supply chain operations, ensuring that the proposed solutions are not only optimal but also practical and robust in real-world contexts.

Simulation-based multi-objective optimization holds considerable potential for addressing the growing complexities of supply chains.²⁵ This study aims to create a hybrid optimization framework that integrates DES with a biased MOPSO algorithm to improve convergence rates and computational efficiency. The objective is to enable decision-makers to identify Pareto-optimal solutions in minimal simulation time, making the framework viable for real-world applications. The integrated DES-MOPSO approach embeds domain-guided biasing to steer initialization and search and uses an adaptive evaluation scheme to allocate simulation effort to promising regions of the Pareto front, thereby improving convergence and diversity under a fixed simulation budget. This approach addresses the research gap in developing integrated methods that balance computational demands with the quality of solutions, providing a scalable and robust tool for optimizing multi-echelon supply chains under uncertainty.

2. Literature review

2.1. Application of multi-objective optimization in supply chain problems

The application of multi-objective optimization in supply chains has garnered significant interest due to the complex trade-offs inherent in these systems.^4,26–28 Objectives such as cost minimization, service-level optimization, and sustainability often conflict, necessitating advanced techniques to identify Pareto-optimal solutions.²⁹ For instance, multi-objective approaches have been employed in vehicle routing problems to minimize fuel costs while ensuring timely deliveries.³⁰ Similarly, inventory optimization models frequently aim to strike a balance between holding costs and stock-out risks.³¹ Recent studies have highlighted the importance of incorporating real-world constraints, such as stochastic demand and variable production rates, into multi-objective frameworks,^32,33 thereby enhancing their practical applicability in dynamic supply chain environments. In addition, the integration of advanced computational tools has improved the scalability of these approaches, enabling their application in large-scale industrial systems.

MOEAs have emerged as a powerful tool for addressing multi-objective optimization problems due to their inherent ability to explore large, complex solution spaces effectively.³⁴ In particular, MOPSO, an extension of the particle swarm optimization (PSO) algorithm, is one of the prominent algorithms specifically tailored to address the complexities of multi-objective optimization problems.³⁵

Over the years, various MOPSO variants have been developed to address limitations such as premature convergence and limited diversity in solution spaces. Various approaches and techniques³⁶ such as archive-based approaches, adaptive inertia weights, and mutation operators have been integrated to enhance performance. For example, dynamic MOPSO incorporates time-varying parameters to improve adaptability,³⁷ while hybrid MOPSO combines elements of other optimization methods, such as Genetic Algorithms,³⁸ to improve convergence and diversity. In addition, constrained MOPSO variants have been designed to handle specific real-world problems where solutions must satisfy strict feasibility criteria.^39,40

2.2. Biased and hybrid MOEAs

Biased and hybrid MOEAs enhance traditional MOEAs by incorporating domain-specific knowledge and integrating complementary techniques to address complex optimization challenges. Biased MOEAs use intelligent biases, such as weighted objectives, priority-based heuristics, and constraint-driven penalties, to guide the search process toward promising regions of the solution space, improving convergence speed and solution quality.^41,42 Hybrid MOEAs, on the other hand, combine evolutionary algorithms with additional methods like local search (e.g. Tabu Search, Simulated Annealing) or surrogate models (e.g. Kriging, neural networks) to refine solutions and reduce computational costs.^43–45 Together, these approaches enhance the scalability, diversity, and efficiency of MOEAs, making them effective in high-dimensional, dynamic, and real-world optimization problems. By embedding intelligent biases into hybrid frameworks, researchers can achieve faster convergence and better solution quality, particularly in constrained or stochastic environments.

2.2.1. Common hybrid MOEA approaches and applications

Hybrid MOEAs combine the strengths of evolutionary algorithms with problem-specific heuristics to enhance optimization performance.⁴⁶ These hybrid approaches often incorporate techniques such as local search, greedy algorithms, or machine learning-based predictors to refine solution quality.⁴⁷ Applications in supply chain management include hybrid algorithms for integrated production and distribution planning, which leverage problem-specific knowledge to reduce computation time and improve solution accuracy. For instance, hybrid models combining Genetic Algorithms with Simulated Annealing have been used to optimize fleet routing and depot allocation simultaneously.^48,49 Such integrations not only enhance solution accuracy but also enable the handling of large-scale, computationally intensive problems.

2.2.2. Intelligent agent bias

Intelligent agents have increasingly been integrated into MOEAs to guide the optimization process through domain-specific biases.⁵⁰ These agents leverage contextual knowledge, such as prioritizing near-stockout customers or identifying efficient vehicle routes,^51,52 to steer the algorithm toward promising regions of the solution space. By embedding these biases, hybrid MOEAs can overcome challenges related to slow convergence and poor solution diversity. For example, agents designed to monitor inventory levels can dynamically adjust replenishment schedules,⁵² ensuring robust performance under fluctuating demand. The ability to incorporate such domain knowledge directly into the optimization framework sets biased hybrid MOEAs apart from conventional approaches.

2.2.3. Hybrid simulation with metaheuristic

Recent work on hybrid simulation–metaheuristics embeds the optimizer inside a simulation loop so candidate policies are evaluated under stochastic dynamics, while the search maintains a well-spread Pareto set and tracks convergence with hypervolume (HV) and diversity. At the scheduling level, a DES coupled with deep reinforcement learning uses proximal policy optimization to learn dispatching policies under random arrivals and variable processing times, with the simulator providing state transitions and multi-objective feedback (setup changes and tardiness).⁵³

In project settings, a sim heuristic links a stochastic simulator of emissions and costs with an improved NSGA-II (including magnet-based crossover and mode reassignment), showing measurable reductions in make span, cost, and greenhouse gas emissions across benchmarks and a case study.⁵⁴

Together these examples reflect the current hybrid pattern. Simulation captures real operational variability, the metaheuristic (MOEA or learning-based) navigates the search, and practical devices such as adaptive resampling, ranking-and-selection, common random numbers, and archive/diversity maintenance keep the search efficient and statistically stable as the Pareto front is refined.

2.3. DES in supply chain optimization

DES is a powerful tool for modeling the dynamic and stochastic nature of supply chains. By simulating real-world processes such as production delays, demand spikes, and transportation bottlenecks, DES provides a detailed understanding of system behavior under varying conditions.⁵⁵ In the context of optimization, DES is often integrated with evolutionary algorithms to evaluate candidate solutions more accurately. For instance, DES models have been used to assess the impact of inventory policies on service levels and cost efficiency, enabling decision-makers to identify robust strategies.^14,56–58 Despite its computational intensity, DES remains a critical component of hybrid optimization frameworks due to its ability to capture real-world uncertainties effectively. Its role in validating and refining optimization results ensures that proposed solutions are practical and implementable.

2.4. Vendor-managed inventory and inventory routing problems

Vendor-managed inventory (VMI) is a supply chain strategy where suppliers take responsibility for managing inventory levels at customer sites, ensuring timely replenishment. This approach has been shown to reduce inventory holding costs and improve supply chain responsiveness.⁵⁹ In parallel, inventory routing problems (IRPs) focus on optimizing both inventory management and transportation schedules, aiming to minimize overall supply chain costs.⁶⁰ The integration of VMI and IRP into optimization frameworks has led to innovative solutions for complex supply chain problems. For example, models combining VMI with IRP have been used to design efficient replenishment routes that minimize transportation costs while meeting customer demand.^61,62 These approaches highlight the potential of multi-objective optimization to balance competing priorities in modern supply chains. Furthermore, their adaptability to varying operational contexts makes them critical tools in designing resilient and cost-effective supply networks.

2.5. Research gap and contribution

The literature highlights significant progress in supply chain optimization through various methodologies. However, gaps remain in integrating real-world uncertainties and stochastic behaviors into optimization models. For example, Marchetti et al.⁶³ proposed a mixed-integer linear programming (MILP) model to optimize production and distribution in industrial gas supply chains. While their approach effectively coordinated production and distribution, it did not account for stochastic factors, such as fluctuating demand or variable production rates, limiting its applicability to dynamic, real-world environments.

Cóccola et al.⁶⁴ introduced a nested column generation method focusing on inventorying and distribution of chemical fluids. Although innovative in handling the inventory-routing problem, their approach was confined to these two aspects, excluding production-related decisions. The absence of production scheduling limits its application to comprehensive supply chain management. In contrast, another study by Cóccola et al.⁶⁵ presented a two-stage approach integrating production, inventory, and distribution. This model incorporated realistic parameters such as vehicle speed and inventory holding costs, offering a more holistic framework. However, the reliance on a commercial solver like CPLEX or GUROBI that uses branch-and-bound or branch-and-cut method and the absence of stochastic elements constrained its adaptability in uncertain scenarios.

This research addresses these gaps by introducing a biased and hybrid MOPSO integrated with DES to tackle stochastic supply chain scheduling that integrates production and distribution. Stochastic elements, such as fluctuating customer consumption, variable production rates, and varying vehicle speed are incorporated to mirror real-world conditions, while DES is employed to dynamically evaluate the performance of candidate solutions under these uncertainties. By embedding intelligent biases into MOPSO, this approach accelerates convergence toward optimal solutions, significantly reducing simulation time compared to traditional optimization techniques. The inclusion of intelligent agents further enhances the optimization process by prioritizing critical routes and near-stockout customers, improving solution quality and practical applicability.

In addition, the integration of DES and biased hybrid MOPSO provides scalability and flexibility, overcoming the computational bottlenecks of commercial solvers. This enables the framework to handle complex, high-dimensional optimization problems while maintaining realism in its evaluation with a shorter simulation time. By achieving faster convergence and incorporating realistic stochastic conditions, this research presents a robust and efficient methodology for solving dynamic supply chain scheduling problems that integrate production and distribution. Table 1 summarizes related studies and compares their optimization types, variables, and solution methods.

Table 1.

Summary of relevant works and comparison.

Relevant work	Type of optimization	Type of supply chain	Uncertain variables	Solution method
Sitek et al. (2014)⁷⁷	Hybrid Multi-Agent with MP and CLP	Multi-modal transport SC	None	Constraint Propagation, Mathematical Programming
Fattahi and Govindan (2017)⁷⁸	Stochastic MILP	Forward/reverse logistics	Demand and return	Simulated Annealing, Latin Hypercube Sampling
Gutierrez-Alcoba et al. (2017)⁷⁹	Monte Carlo, heuristic methods	Perishable product supply	Demand	Silver’s heuristic
Alavi and Jabbarzadeh (2018)⁸⁰	Stochastic and robust optimization	Chemical product supply	Demand	Lagrangian Relaxation
Razavian et al. (2021)⁸¹	MILP with CVaR	Chemical product supply	Demand	CPLEX Solver
Keyvanshokooh et al. (2016)⁸²	Stochastic and robust optimization	Closed-loop supply chain	Transportation cost, demand, return	Latin Hypercube Sampling, Benders Decomposition
Marchetti et al.⁶³	Mixed-Integer Linear Programming	Industrial gas supply chain	None	CPLEX Solver
Cóccola et al.⁶⁴	Nested Column Generation, MILP	Chemical fluids distribution	None	CPLEX Solver
This work	Hybrid Multi-Agent with MOPSO and DES	Industrial gas supply chain	Transportation speed, consumption rate, production quantity	Biased MOPSO

3. Methodology

3.1. Problem summary and formulation

The problem this research tried to tackles originates from the work of Marchetti et al,⁶³ which focuses on the simultaneous production and distribution of industrial gases in a complex supply chain environment (Figure 1). The industrial gases supply chain is characterized by its highly interconnected production and distribution processes. Each plant operates under multiple production modes for two products, liquid nitrogen (LIN) and liquid oxygen (LOX), where production rates and energy consumption levels vary. In addition, fluctuating electricity prices introduce a significant cost-optimization challenge, as aligning production schedules with favorable pricing windows can greatly reduce operational expenses.

On the distribution side, the supply chain is managed under a vendor managed inventory (VMI) framework. In this system, the supplier takes full responsibility for monitoring and replenishing customer inventories. This involves delivering liquid products to a wide range of customers, each with unique consumption patterns, limited storage capacities, and varying delivery requirements. Ensuring timely deliveries is critical to avoid stockouts, which requires precise inventory control and efficient routing. The transportation process is further constrained by vehicle capacities, routing complexities, and operational costs.

Inventory levels at production plants and customer sites must be carefully managed to stay within predefined limits while meeting safety stock requirements. This adds an additional layer of complexity, as inventory management directly influences production and distribution planning. The challenge lies in simultaneously coordinating these decisions across the supply chain over a given time horizon.

The optimization problem involves determining production schedules, inventory levels, and distribution routes for each time period, aiming to achieve multiple objectives. These include minimizing total costs (encompassing production, transportation, inventory holding, and startup costs), reducing total work time, and maximizing inventory success rates by ensuring all inventories remain within acceptable bounds. These objectives are chosen because of their relevance in supply chain optimization. Furthermore, the list of notations, constraints, and decision variables can be found in Appendix 1.

Cost is a critical metric in supply chain scheduling as it directly impacts the profitability and efficiency of the system. It encompasses factors like inventory holding cost, production cost, transportation cost, and utility costs. Businesses aim to lower costs to remain competitive while maintaining service levels, making this objective universally important. Studies such as Chopra and Meindl⁶⁶ emphasize the importance of minimizing costs in supply chain systems. In addition, Toth and Vigo⁶⁷ explore how cost reduction is a fundamental objective in routing and scheduling optimization.

Minimizing work time ensures the efficient utilization of resources such as production facilities and vehicles. Reducing factory working time implies that production processes are well-optimized, leading to lower energy consumption, reduced wear and tear on equipment, and cost saving in human resources. Similarly, minimizing vehicle operating time reduces fuel usage, maintenance costs, and carbon emissions. Pinedo⁶⁸ discusses minimizing work time as a critical component in scheduling problems, particularly in factory and logistics environments. Furthermore, Touzout et al.⁶⁹ highlight the importance of considering time-dependent travel times and routing constraints to achieve efficient inventory and vehicle scheduling.

A high inventory success rate indicates that the generated schedule is feasible and capable of fulfilling inventory requirements without shortages or overflows. This ensures that all inventory conditions, such as storage capacities, production requirements, and delivery deadlines, are met. By maximizing the inventory success rate, the scheduling system demonstrates its robustness and reliability in handling constraints, such as fluctuating demand, supply variability, and production capabilities. In a dynamic and uncertain supply chain environment, a high success rate ensures that the proposed schedules have a high probability of being executable in real-world scenarios. Several studies have highlighted the importance of success rates as a key performance indicator in scheduling problems, particularly for improving system efficiency and reliability.^70–72

In addition, this work introduces more to the integration of real-world constraints from fluctuating vehicle speed, varying consumption rate, and stochastic production rate. The study seeks to develop an optimized framework capable of achieving Pareto-optimal solutions across these objectives, ensuring an efficient and cost-effective supply chain under a VMI system.

The problem of production–distribution coordination in the industrial gases supply chain can be described as follows:

A set of production plants $p \in P$ capable of producing industrial gases.

Each plant $p$ can operate in a set of production modes $m \in M_{p}$ , where each mode specifies a production rate and energy consumption.

A set of customers $i \in C$ , each with forecasted consumption over a time horizon $t \in T$ , storage capacity limits, and initial inventory levels.

A set of liquid products $l \in L$ is considered for distribution and consumption across the supply chain.

Inventory levels at plants and customers must remain within predefined bounds.

A fleet of vehicles $v \in V$ is available for transporting the product from plants to customers, each with specified capacity and transport costs.

The geographical locations of plants, customers, and routes allow for calculating travel distances and times.

A set of consecutive time periods $t \in T$ during which production and distribution occur.

Time-dependent electricity prices $c_{elect, t}$ for operating production plants.

Forecasts of customer demand $d_{t}$ for each time period and customer, as well as the required safety stock levels.

The problem involves determining production schedules, inventory levels, and distribution plans such that:

Total costs are minimized as shown in Equation (1) (including production, transportation, inventory holding, and startup costs),

Total work time is minimized as shown in Equation (2) (considering production and transport),

Inventory success rate is maximized as shown in Equation (3) (ensuring all inventories remain within bounds) (3).

The goal is to determine operational-level decisions for each time period, including the production rates and modes at each plant, inventory levels at plants and customer locations, and distribution quantities and routes for reaching the Pareto optimal in all objectives.

Objective function 1: minimize total cost

\begin{matrix} Z_{1} = \underset{ElectricityCost}{\underset{︸}{\sum_{p \in P} \sum_{t \in T} q_{p, t} \cdot c_{elec, t}}} + \underset{StartupCost}{\underset{︸}{\sum_{p \in P} s_{p} \cdot δ (q_{p, t})}} \\ + \underset{InventoryHoldingCost}{\underset{︸}{\sum_{i \in C} \sum_{t \in T} I_{i, t} \cdot h_{i}}} + \underset{TransportationCost}{\underset{︸}{\sum_{i \in C} \sum_{t \in T} \sum_{v \in V} R_{i, v, t} \cdot c_{v, fuel}}} \end{matrix}

(1)

Objective function 2: minimize total work time

Z_{2} = \sum_{p \in P} W_{p} + \sum_{v \in V} W_{v}

(2)

Objective function 3: maximize inventory success rate

Z_{3} = \frac{\sum_{t \in T} 1 ((\land_{i \in C} (I_{i, \min} \leq I_{i, t} \leq I_{i, \max})) \land (\land_{p \in P} (I_{p, \min} \leq I_{p, t} \leq I_{p, \max})))}{| T |}

(3)

3.2. Research framework

A multi-objective optimization model framework using MOEA, in particular MOPSO, and intelligent agents for biased initialization and converging solutions combined with DES is proposed by this study. This research framework basic is shown in Figure 2. The framework is designed to solve complex supply chain problems involving connected scheduling of production and distribution characterized by real-world stochastic environments. It combines DES using Simio with Python-based optimization and intelligent agents to achieve an efficient multi-objective optimization. The DES simulates the intricate conditions of the supply chain problems, while Python integrates intelligent agents and MOPSO to optimize solutions effectively.

The framework operates by first defining the decision variables involved in the supply chain system, which include production schedules, inventory management, and distribution routes. These variables are modeled within the DES environment, where the dynamic interactions between production, consumption, and transportation are simulated over multiple time periods. The simulation captures the inherent stochasticity in the supply chain, such as varying customer demand and production rates, to reflect real-world conditions.

Population initialization serves as the cornerstone of the framework’s experimentation. Initial populations are generated either randomly or through biased initialization guided by intelligent agents. After processing the simulation, the optimization process is carried out using the MOPSO algorithm. MOPSO is employed to explore the solution space for potential optimal outcomes, considering multiple conflicting objectives such as minimizing cost and work time. To further enhance the search for optimal solutions, the framework integrates intelligent agents that introduce a bias toward certain objectives based on real-time data, such as prioritizing customers who are at risk of stock outs. These agents interact with the system by adjusting the search direction and refining the solutions, improving the efficiency of the algorithm’s convergence process.

The hybridization of MOPSO and intelligent agents allows for an adaptive optimization process that is capable of addressing the dynamic nature of supply chains. The intelligent agents assist in guiding the swarm to better regions of the solution space by incorporating domain-specific knowledge and responding to changes in the system’s environment. This combination of evolutionary optimization and agent-based guidance ensures that the solutions are both computationally efficient and practically relevant. The new solutions generated are then simulated again until a stopping criterion is met. Post processing is done by removing any non-realistic solution such as 0% success rate and evaluating objective values of the Pareto front using HV and diversity metrics. The full framework working process can be seen in Figure 3. The framework generates a set of Pareto-optimal solutions, which are evaluated based on performance metrics such as HV, diversity, and resource utilization. These solutions are then analyzed to determine the most effective strategies for production and distribution, providing valuable insights for decision-making in supply chain management.

3.3. Simulation setup

The simulation model developed in this study is built using Simio, a DES software known for its flexibility in representing complex systems.⁷³ Simio enables the creation of highly customizable models using object-oriented modeling concepts and supports the integration of externally defined datasets. This flexibility is critical for capturing the intricate dynamics of a multi-echelon supply chain and ensuring accurate representation of its stochastic behavior. Furthermore, the software’s ability to perform batch simulations significantly reduces computational time, allowing multiple scenarios to be analyzed efficiently.

Moreover, Simio’s replication tools provide statistically robust results by simulating multiple iterations of the same scenario. This ensures that the output is not influenced by random noise, offering a more accurate representation of the supply chain’s performance under varying conditions. The integration of Simio with Python allows seamless data exchange, enabling the optimization algorithm to refine solutions based on detailed simulation outputs.

Objects in Simio are built using object-orientation concepts.⁷³ They are built using graphical object-oriented modeling and process logic adapted to current dataset. The current model can be seen in Figure 4. Each system components and model representation can be summarized as follows:

Factories: The factories are modeled as sources with production capacities defined based on real-world datasets. Production schedules are deterministic and are governed by predefined production rates. These rates follow a triangular probability distribution to account for variability in successful production outcomes, mimicking real-world conditions such as delays or inefficiencies. Factories are configured to produce multiple product types, with decisions influenced by production priorities, capacity constraints, and material availability.

Warehouses: Warehouse nodes serve as storage nodes within the supply chain, consolidating goods from factories before dispatching them to customer locations. These are modeled using Simio’s transfer nodes and servers with capacities dynamically adjusted based on replenishment schedules and inventory policies. Inventory levels at warehouses are monitored and managed to ensure timely dispatch while minimizing holding costs.

Customers: Customer entities represent retail locations or end consumers. Their demand patterns are modeled stochastically, using normal distributions derived from historical data. Customers model is designed with the combination of two nodes, 1 transfer node and 1 basic node, which act as the receiving and replenishment of material that represents the inventory.

Transporters: Transportation is modeled using Simio’s transporter objects, representing vehicles with finite capacities and routing logic. Vehicle routing is dynamically controlled by Python scripts, which generate order priorities and delivery sequences. Transporters are configured to operate near their maximum carrying capacity for efficiency. Their speed is calibrated to simulate average transportation speeds observed in the United States within multiple environments, ensuring realistic delivery times.⁷⁴

Materials and Bill of Materials (BOM): Materials act as the definition of the model entity’s identity. Material consumption and production processes are modeled using Simio’s hierarchical materials element, which supports traditional BOM structures. Manufacturing steps are defined as material production events, linking production processes to inventory increase at warehouses, while distribution steps are defined as material consumption which led to the depletion of the warehouses.

By incorporating both DES and MOEA-based solver, the system provides a comprehensive framework for evaluating production and distribution strategies in real-world stochastic environments. This setup ensures that the modeled supply chain achieves its objectives of minimizing costs and work times while maintaining high service levels.

3.4. Biased MOPSO architecture

MOPSO was chosen as the optimization algorithm for this study because of its inherent ability to use its original decision variables as a basis and explore nearby solution spaces.⁷⁵ This characteristic aligns with the delicate nature of the stochastic supply chain environment, where decision variables are highly interdependent and constrained. The problem under investigation involves up to over 4000 decision variables, with decisions influencing each other within complex constraints. Such problems, involving production schedules and distribution routing, are theorized to benefit significantly from MOPSO’s adaptive exploration and exploitation capabilities. To further support on the convergence, an architecture similar to the Elitist PSO (MO-ETPSO)⁷⁶ was built and improved with the intelligent agents and simulations budget constraint.

The biased MOPSO algorithm implemented in this study is structured to address multi-objective optimization problems effectively and quickly. The process involves evaluating and updating a population of solutions iteratively and selectively to achieve a well-distributed Pareto front. This architecture balances the exploration and exploitation of the solution space through a systematic process divided into several key steps.

The initialization process begins by generating decision variables $Q, T, R$ representing production schedules, distribution quantities, and routing plans, respectively, within predefined constraints $B$ (inventory bounds) and $C$ (consumption forecast). Intelligent agents are employed during initialization to ensure that the generated solutions are feasible and closer to the Pareto-optimal front $P *$ . This reduces the computational effort required during subsequent optimization iterations by narrowing the search space to promising regions. These solutions are then simulated in a DES model to evaluate the objective functions $F = 〈 F_{1}^{i}, F_{2}^{i}, F_{3}^{i} 〉 : i = 1, 2, \dots, N$ , where $F_{1}, F_{2}, F_{3}$ , represent the objectives, such as total cost, total work time, and inventory success rate. If none of the solutions meet at least one predefined objective threshold within the constrained simulation budget $K_{1}$ : $Fj (si) \geq Tj, for some j \in {1, 2, 3}$ , new solutions are generated and sent for simulation until the threshold condition is satisfied.

After initialization, the evaluated solutions and simulation results are stored for further optimization. Each particle $s_{i}$ in the population is associated with a unique scenario identifier, its corresponding decision variables, objective values $F (s_{i})$ , and an experiment type. This systematic record ensures the identity of each solution is preserved and maintains a diverse population of feasible solutions.

The next step involves non-dominated sorting and crowding distance calculation. The population is ranked into Pareto fronts $F_{1}, F_{2}, F_{3}$ based on dominance relationships. A solution $s_{i}$ is considered non-dominated if no other solution dominates it across all objectives. To enhance diversity, a crowding distance metric $d (s_{i})$ is computed for each particle within the Pareto front, favoring solutions in less crowded regions of the objective space. From these Pareto fronts, the top $\frac{N}{2}$ solutions are selected based on dominance and crowding distance. This introduces another bias by retaining solutions with a higher likelihood of success, which are repeatedly refined in subsequent iterations. Duplicate scenarios are resolved using a mechanism based on crowding distance and randomness to ensure only the most representative solutions are preserved.

The final step involves generating new solutions through velocity in Equation (4) and position updates in Equation (5), following the PSO formula:

v_{i} = ω v_{i} + ρ r_{1} (p_{i}^{best} - x_{i}) + ρ r_{2} (g^{best} - x_{i}),

(4)

x_{i}^{(t + 1)} = {x_{i}}^{(t)} + {v_{i}}^{(t)}

(5)

Parameter selection for MOPSO was determined based on iterative experimentation and careful adjustment to the problem’s requirements. The inertia weight (ω) was set to 0.7, and the cognitive (c₁) and social (c₂) coefficients were initially set to 2 and 1.5, respectively. These values were obtained after parameter sensitivity analysis and are used to fine-tune the influence of personal bests and global bests.

Updated positions are constrained to predefined bounds by intelligent agents to ensure feasibility. This step introduces new solutions while retaining promising traits from the previous generation, balancing the search for optimal solutions and exploration of the solution space. The top solutions and offsprings are grouped together and is send to be evaluated using DES for $K_{2}$ iterations each. This is repeated until any stopping criterion is met. The full pseudo code can be seen in Figure 5. The stopping criteria include three conditions:

The simulation runtime exceeds one hour.

Generation 5 or 1200 simulations is reached.

The Pareto front scenario solution remains unchanged for three consecutive generations.

Through this structured approach, biased MOPSO iteratively refines the population, driving it toward a diverse and high-quality Pareto front. This architecture provides a robust framework for solving complex multi-objective optimization problems efficiently.

3.5. Agent modeling

In the proposed system, intelligent agents play a crucial role in managing the complexities of production–distribution coordination in a multi-echelon supply chain. Each agent addresses specific challenges within the supply chain, ensuring that the overall system operates efficiently and effectively. The incorporation of agents enables the optimization framework to converge toward feasible and optimal solutions, which was not possible using traditional MOEAs alone. Initial experiments highlighted that MOEA, without the support of agents, resulted in inventory success rates of zero due to infeasible solutions. By incorporating these agents, particularly the Balancing Agent, the inventory success rate improved to at least 20%, providing a viable starting point for optimization. Below is an overview of each agent and its role within the system.

The Balancing Agent is responsible for ensuring the feasibility of production schedules by aligning them with distribution plans in Equation (6) and inventory constraints in Equation (7).

I_{p, t + 1} = I_{p, t} + q_{p, t} - \sum_{q ϵ P, q \neq p} T_{p, q, t}

(6)

I_{p, \min} \leq I_{p, t + 1} \leq I_{p, \max}, \forall p ϵ P, \forall t ϵ T

(7)

It achieves this by projecting inventory levels for each factory after accounting for production, consumption, and transfers, and selecting production modes that maintain inventory within predefined minimum and maximum limits as shown Equation (8). When strictly feasible production modes are unavailable, the Balancing Agent selects the closest alternative. This mechanism prevents the selection of infeasible production plans that could disrupt the system, thereby establishing a strong foundation for further optimization.

min (\sum_{p ϵ P} \sum_{t ϵ T} [\max (0, I_{p, \min} - I_{p, t + 1}) + \max (0, I_{p, t + 1} - I_{p, \max})])

(8)

The Routing Agent minimizes transportation costs by determining the most efficient routes for delivery. Using Euclidean distance calculations, the agent evaluates the shortest paths between delivery points while considering vehicle capacity constraints. It generates combinations of orders, prioritizing those with minimal total travel distance, and assigns rankings to routes to facilitate efficient execution used in Equation (9). By reducing travel distances and optimizing route assignments, the Routing Agent plays a vital role in lowering transportation expenses and improving operational efficiency.

min \sum_{r ϵ R} \sum_{c ϵ r} (\sum_{k = 1}^{n_{c} - 1} \sqrt{{(x_{ik} - x_{ik + 1})}^{2} + {(y_{ik} - y_{ik + 1})}^{2}})

(9)

The Distribution Agent focuses on dynamically adjusts delivery schedules based on customer inventory levels and forecasted consumption, ensuring that stockouts or overstocking are avoided based on Equation (10). It then continues optimizing the allocation of transport resources and scheduling delivery quantities to minimize inefficiencies in the supply chain by adjusting total shipment with individual transporter’s capacity per trip as shown in Equation (11). When delivery quantities exceed vehicle capacities, the agent intelligently splits shipments across multiple trips or vehicles to maintain feasibility. This adaptability enables the supply chain to meet dynamic demands while adhering to operational constraints.

Q_{i, t} = max (D_{i, t} - I_{i, t - 1} + I_{\min, i}, 0)

(10)

Q_{i, t} = min (\max (D_{i, t} - I_{i, t - 1} + I_{\min, i}, 0), C_{trans} \cdot n_{t})

(11)

The Priority Agent ensures that resources are allocated effectively by prioritizing supply chain activities based on urgency and criticality following Equation (12). It assigns higher priorities to customers or factories with inventory levels closer to their minimum thresholds, ensuring that critical demands are addressed first. If priority agent is assigned with Routing Agent, priority will have their demands be fulfilled for the first 10 and afterward are resolved through the Routing Agent decisions on inventory control shown in Equation (13). In addition, the agent optimizes production mode selection, favoring modes that maximize efficiency or reduce costs, such as idle modes when appropriate with Equation (14). This prioritization mechanism ensures that the system can handle high variability in demand without compromising overall efficiency or stability.

I_{c, d} = I_{c, 0} + \sum_{x = 1}^{d - 1} T_{c, x} - C_{c, d} - σ \cdot C_{c, d} + T_{c, d} - R_{c}

(12)

C_{priority} = {c ϵ C : R_{c} \leq 10}

(13)

I_{F, s} (0) = I_{F, s - 1} + T_{F, s}

(14)

Each of these agents contributes uniquely to the optimization process. The Balancing Agent ensures that production schedules are feasible and realistic. The Routing Agent reduces transportation costs by optimizing delivery routes. The Distribution Agent dynamically allocates transport resources to meet customer demands while avoiding inefficiencies. Finally, the Priority Agent ensures that critical demands are met without disrupting the overall supply chain balance. Together, these agents work synergistically within the MOEA framework to enhance convergence toward optimal solutions, addressing the limitations of traditional evolutionary algorithms. This agent-based approach demonstrates the effectiveness of combining heuristic methods with intelligent agents in solving complex supply chain optimization problems.

4. Results and discussion

4.1. Experiment setup

The experiments were conducted to evaluate the performance of the proposed multi-objective optimization framework applied to a multi-echelon supply chain problem. The dataset primarily originates from Marchetti et al,⁶³ with additional considerations for realistic instances inspired by Cóccola et al,⁶⁵ such as transporter speed and inventory holding cost. Two test cases that include simultaneous production and distribution at multiple plants and depots were used for evaluation.

The optimization framework, including the underlying models and algorithms, was implemented in Simio Version 17.261.39190 (64-bit), leveraging .Net Framework 4.8.9282.0. The computational experiments were carried out on a machine running Microsoft Windows 10, equipped with an Intel(R) Core i7-12700 processor, NVIDIA RTX 3070 Ti GPU, and 32 GB of RAM. A summary of the hardware and software environment is provided in Table 2.

Table 2.

Experiment environment.

	Component	Details
Hardware environment	Operating System	Microsoft Windows 10
	Processor	Intel(R) Core i7-12700
	GPU	NVIDIA RTX 3070 Ti
	Memory	32 GB
Software environment	Simio Version	17.261.39190 (64-bit)
	.Net Framework	4.8.9282.0

The optimization process was carried out using the MOPSO approach. Table 3 summarizes the parameters used in the configuration while more detailed data are placed in Appendix 1. The experimental setup incorporated real-world scenarios by introducing stochastic variability in key parameters, such as transporter speed, production rates, and customer consumption rates. Transporter speeds were configured with a mean of 42.4 miles per hour and a standard deviation of 4.6,⁷⁴ derived from average US truck data across various environments. Production variability for industrial gases, liquid nitrogen (LIN) and liquid oxygen (LOX), was modeled using triangular distributions with a 1% error rate,⁸³ consistent with typical gas flow measurement inaccuracies.

Table 3.

Simulation and experiment parameters configuration summary.

Parameter	Value/description
Population size	20
Number of generations	5 optimization generations, starting with an initial population labeled as Generation 0, resulting in a total of 6 evaluated populations.
Inertia weight ( $ω$ )	0.7
Cognitive coefficient ( $c_{1}$ )	2
Social coefficient ( $c_{2})$	1.5
Simulation runs budget	Dynamically varied between 5 and 10, depending on solution quality and convergence criteria

Customer consumption rates included a 5.5% standard deviation to account for demand-side variability,⁸⁴ aligning with established practices in supply chain modeling to capture real-world demand fluctuations. These adjustments enhance the simulation’s realism, enabling evaluation under dynamic and uncertain conditions. The study’s focus on operational efficiency and cost minimization influenced the exclusion of factors like production costs and product pricing. Instead, emphasis was placed on electricity costs, fuel costs, factory setup costs, and inventory success rates, which are directly controllable within the supply chain. This configuration enables a targeted analysis of key operational metrics within the constrained scope of the optimization model.

4.2. Simulation scenarios evaluation

The simulation was conducted using various combinations of intelligent agents to evaluate their comparative effectiveness. Initially, the baseline experiment employed only the DES and MOPSO algorithms. However, this configuration failed to achieve convergence within the defined stopping criteria. To address this limitation, the Balancing Agent with standard MOPSO was introduced as the revised baseline, ensuring that feasible schedules were predominantly selected. Other state-of-the-art MOPSO variants that excel at updating in multiple directions was also implemented and adapted to the problem. Multidirection update-based multiobjective particle swarm optimization (MDU-MoPSO) uses particle update strategy to able guarantee the ability to converge swiftly in multiple directions with good distribution performance.⁸⁵ Another variant, the elitist multi-objective PSO (MO-ETPSO), integrates crowding distance concepts from NSGA-II with swarm elitism and memory, enabling the algorithm to retain high-quality solutions while maintaining simplicity in its operators.⁷⁶ Each of the different MOPSO was given the same initialization value for a baseline.

The integration of different agent combinations enabled a comprehensive comparison of their impact on the performance of the hybridized biased MOEA with DES in scheduling within a stochastic environment. Furthermore, the efficiency of each scenario was assessed based on its ability to meet the stopping criteria, which included simulation runtime, simulation budget, or Pareto front generational convergence. This evaluation provided critical insights into the unique characteristics and performance dynamics of each agent configuration.

4.2.1. Simulation results

The simulation evaluated a total of eight different scenarios with the use of Balancing Agent as the baseline, including three other MOPSO variants. The combinations consist of using routing, priority, and distribution agents. Its routing agent is represented differently as an experiment name as can be seen in Figure 6. Balancing Agent is represented as balance, routing agent is represented as distance, priority agent is represented as priority, and distribution is represented as split.

Each scenario and MOPSO are evaluated from their Pareto fronts values by using known indicator evaluation metrics such as HV and diversity. The HV indicator measures the volume of the objective space dominated by a Pareto front relative to a reference point as shown in Equation (15). HV is widely used in multi-objective optimization to assess the quality of solutions generated by algorithms.⁸⁶ A higher HV indicates better coverage of the objective space and proximity to the ideal Pareto front, which means the higher the value the better the solution is. Diversity measures the distribution and spread of solutions along the Pareto front as it is crucial for MOPSO to ensure that the algorithm explores the objective space well distributed and effectively shown in Equation (16). A higher diversity indicates a wider spread on the Pareto front.

HV = \sum_{i = 1}^{N} Π_{j = 1}^{M} (r_{j} - f_{j} (x_{i}))

(15)

Δ = \frac{\sum_{i = 1}^{N - 1} | d_{i} - \bar{d} |}{(N - 1) \cdot \bar{d}}

(16)

4.2.1.1. Example 1

A preliminary small-scale dataset is introduced, serving as the first example. It involves two production plants and two primary products: liquid nitrogen (LIN) and liquid oxygen (LOX). Each product is available in a single unique grade. The plants operate under two distinct production modes, high LIN and high LOX, each with specific production capacity constraints. Each plant has a unit-specific power consumption of 20 kWh per Mcf, and the startup costs are $7000 for Plant 1 and $4000 for Plant 2.

Each plant and customers maintain an initial inventory level, a maximum storage capacity, and a redline inventory level, which represents the minimum required stock to sustain operations. All quantities are measured in thousand standard cubic feet (Mcf). A depot is co-located with each plant, managing a fleet of trucks dedicated to transporting LIN and LOX. Depot 1 at Plant 1 operates five trucks—three for LIN and two for LOX—while Depot 2 at Plant 2 has four trucks, evenly split between LIN and LOX. The transportation cost is $2.85 per mile, and each trailer has a predefined capacity of 630 Mcf.

The supply chain serves nine customers, five requiring LIN and four requiring LOX, who receive deliveries via truck transport and have consumption rate which can be considered tight to their inventory levels minimum and maximum. The planning horizon spans 1 week, divided into 14 time periods, each representing either a peak or off-peak electricity pricing window with half day duration. Electricity prices fluctuate over time, making energy cost considerations a crucial factor in determining optimal production and distribution strategies. Further details are available in Marchetti et al.⁶³

The optimization results for Example 1 are detailed in Table 4, which highlights the objective function outcomes, including cost breakdowns and performance indicators for the optimized solution. This table enables a direct comparison of various coordination strategies and their impact on system efficiency. Table 5 presents the number of Pareto front solutions obtained across generations, providing insight into the diversity of solutions and the effectiveness of the multi-objective optimization approach. This helps analyze how different trade-offs between cost and operational performance evolve throughout the optimization process.

Table 4.

Objective function results for Example 1.

MOEA	Agents	Average total cost ($)	Average total work time (h)	Average inventory success (%)
MOPSO	Balance Agent	685,356	620	23.33
MDU-MoPSO	Balance Agent	685,707	611	24.00
MO-ETPSO	Balance Agent	748,220	619	67.50
Biased MOPSO	Balance Agent	671,346	602	32.50
Biased MOPSO	Balance, Routing Agent	732,135	606	31.11
Biased MOPSO	Balance, Priority Agent	625,206	610	40.00
Biased MOPSO	Balance, Priority, Routing Agent	685,197	611	50.00
Biased MOPSO	Balance, Distribution Agent	699,980	508	38.00
Biased MOPSO	Balance, Distribution, Routing Agent	659,359	505	23.33
Biased MOPSO	Balance, Distribution, Priority Agent	634,949	511	38.57
Biased MOPSO	Balance, Distribution, Priority, Routing Agent	634,651	505	26.00

Table 5.

Number of Pareto front solutions and generation for Example 1.

MOEA	Agents	No. of Pareto front solutions	Generation
MOPSO	Balance Agent	3	5.0
MDU-MoPSO	Balance Agent	5	4.8
MO-ETPSO	Balance Agent	4	1.0
Biased MOPSO	Balance Agent	4	3.3
Biased MOPSO	Balance, Routing Agent	9	1.4
Biased MOPSO	Balance, Priority Agent	2	2.0
Biased MOPSO	Balance, Priority, Routing Agent	5	2.0
Biased MOPSO	Balance, Distribution Agent	5	2.4
Biased MOPSO	Balance, Distribution, Routing Agent	6	3.0
Biased MOPSO	Balance, Distribution, Priority Agent	7	2.1
Biased MOPSO	Balance, Distribution, Priority, Routing Agent	5	3.8

Further assessment of the Pareto front scenarios is provided in Table 6, which examines key evaluation metrics to determine the quality and distribution of solutions. This assessment is crucial in identifying the most balanced trade-off based on predefined objectives. To visually represent the results, Figure 6 offers a three-dimensional (3D) depiction of the Pareto fronts, illustrating the relationships between objective functions and emphasizing the optimal trade-offs. This visualization aids in understanding the influence of different production–distribution coordination strategies on overall system performance.

Table 6.

Pareto fronts scenarios evaluation metrics for Example 1.

MOEA	Agents	Metric	Value	Simulation time
MOPSO	Balance Agent	Diversity	53376	00:34:33
MOPSO	Balance Agent	Hypervolume	0.041	00:34:33
MDU-MoPSO	Balance Agent	Diversity	96953	00:24:49
MDU-MoPSO	Balance Agent	Hypervolume	0.038	00:24:49
MO-ETPSO	Balance Agent	Diversity	146748	00:28:32
MO-ETPSO	Balance Agent	Hypervolume	0.016	00:28:32
Biased MOPSO	Balance Agent	Diversity	83311	00:23:03
Biased MOPSO	Balance Agent	Hypervolume	0.066	00:23:03
Biased MOPSO	Balance, Routing Agent	Diversity	357443	00:24:11
Biased MOPSO	Balance, Routing Agent	Hypervolume	0.044	00:24:11
Biased MOPSO	Balance, Priority Agent	Diversity	15287	00:24:09
Biased MOPSO	Balance, Priority Agent	Hypervolume	0.052	00:24:09
Biased MOPSO	Balance, Priority, Routing Agent	Diversity	464546	00:26:30
Biased MOPSO	Balance, Priority, Routing Agent	Hypervolume	0.033	00:26:30
Biased MOPSO	Balance, Distribution Agent	Diversity	355127	00:16:58
Biased MOPSO	Balance, Distribution Agent	Hypervolume	0.044	00:16:58
Biased MOPSO	Balance, Distribution, Routing Agent	Diversity	92081	00:32:11
Biased MOPSO	Balance, Distribution, Routing Agent	Hypervolume	0.093	00:32:11
Biased MOPSO	Balance, Distribution, Priority Agent	Diversity	150541	00:22:26
Biased MOPSO	Balance, Distribution, Priority Agent	Hypervolume	0.136	00:22:26
Biased MOPSO	Balance, Distribution, Priority, Routing Agent	Diversity	51780	00:22:30
Biased MOPSO	Balance, Distribution, Priority, Routing Agent	Hypervolume	0.140	00:22:30

4.2.1.2. Example 2

A dataset adapted from a real industrial test case is presented as the second example. This dataset includes three production plants producing two primary products: LIN and LOX. Similar to the first example, each product has a unique grade, and plants operate under two distinct production modes, high LIN and high LOX, each with specific production capacity constraints. Each plant has a unit-specific power consumption of 20 kWh per Mcf, and startup costs remain at $7000 for Plant 1 and $4000 for Plant 2, with an additional third plant incorporated into the network.

The supply chain expands to three depots. Depots D1 and D3 are co-located with Plants P1 and P3, respectively, each managing a fleet of five trucks—three for LIN and two for LOX. Depot D2 operates independently near Plant P2 and has four trucks, evenly split between LIN and LOX. This results in a total distribution capacity of 14 trucks, 8 designated for LIN and 6 for LOX.

The supply chain now serves 50 customers, with 28 requiring LIN and 22 requiring LOX. These customers have varying consumption profiles and require inventory replenishment with a 2-week planning horizon. The total forecasted replenishment demand amounts to 50,896 Mcf for LIN and 28,059 Mcf for LOX. Further details of the dataset available in Marchetti et al.⁶³

The results of the optimization process for Example 2 are summarized in Table 7, which presents the objective function results. This table outlines the cost components and performance metrics associated with the optimized solution, providing a clear comparison of different coordination strategies. Table 8 reports the number of Pareto front solutions and the corresponding generations, reflecting the diversity and efficiency of the multi-objective optimization process. This analysis helps in understanding how different trade-offs between cost and operational efficiency emerge over the iterations.

Table 7.

Objective function results for Example 2.

MOEA	Agents	Average total cost ($)	Average total work time (h)	Average inventory success (%)
MOPSO	Balance Agent	2,064,502	2897.54	84
MDU-MoPSO	Balance Agent	2,102,363	2989.9	80
MO-ETPSO	Balance Agent	2,198,991	2767	63
Biased MOPSO	Balance Agent	2,018,301	2929.38	90
Biased MOPSO	Balance, Routing Agent	2,275,130	2630.93	93.33
Biased MOPSO	Balance, Priority Agent	1,812,713	2367.78	97.5
Biased MOPSO	Balance, Priority, Routing Agent	2,235,233	2634.69	90
Biased MOPSO	Balance, Distribution Agent	1,756,627	1876.93	96.67
Biased MOPSO	Balance, Distribution, Routing Agent	1,897,294	1902.75	100
Biased MOPSO	Balance, Distribution, Priority Agent	1,721,555	1860.2	100
Biased MOPSO	Balance, Distribution, Priority, Routing Agent	1,765,212	1862.4	100

Table 8.

Number of Pareto front solutions and generation for Example 2.

MOEA	Agents	No. of Pareto front solutions	Generation
MOPSO	Balance Agent	5	5.0
MDU-MoPSO	Balance Agent	3	4.3
MO-ETPSO	Balance Agent	3	4.3
Biased MOPSO	Balance Agent	4	3.5
Biased MOPSO	Balance, Routing Agent	6	1.3
Biased MOPSO	Balance, Priority Agent	4	0.0
Biased MOPSO	Balance, Priority, Routing Agent	10	1.8
Biased MOPSO	Balance, Distribution Agent	6	3.7
Biased MOPSO	Balance, Distribution, Routing Agent	4	5.0
Biased MOPSO	Balance, Distribution, Priority Agent	3	2.3
Biased MOPSO	Balance, Distribution, Priority, Routing Agent	2	4.5

Table 9 evaluates the Pareto front scenarios using various metrics, offering insights into the quality and spread of the obtained solutions. This evaluation aids in selecting the most balanced solution based on predefined decision criteria. Figure 7 provides a 3D visualization of the Pareto fronts, illustrating the relationships between objective functions and highlighting optimal trade-offs. This graphical representation is useful for interpreting the impact of different production–distribution coordination strategies on the overall performance of the supply chain.

Table 9.

Pareto fronts scenarios evaluation metrics for Example 2.

MOEA	Agents	Metric	Value	Simulation time
MOPSO	Balance Agent	Diversity	182,073.38	00:37:51
MOPSO	Balance Agent	Hypervolume	0.1824	00:37:51
MDU-MoPSO	Balance Agent	Diversity	269,384.56	00:36:53
MDU-MoPSO	Balance Agent	Hypervolume	0.1892	00:36:53
MO-ETPSO	Balance Agent	Diversity	997,739.73	00:38:13
MO-ETPSO	Balance Agent	Hypervolume	0.105	00:38:13
Biased MOPSO	Balance Agent	Diversity	123,725.51	00:36:41
Biased MOPSO	Balance Agent	Hypervolume	0.206	00:36:41
Biased MOPSO	Balance, Routing Agent	Diversity	2,040,910.77	01:00:00
Biased MOPSO	Balance, Routing Agent	Hypervolume	0.245	01:00:00
Biased MOPSO	Balance, Priority Agent	Diversity	142,222.87	00:33:16
Biased MOPSO	Balance, Priority Agent	Hypervolume	0.3867	00:33:16
Biased MOPSO	Balance, Priority, Routing Agent	Diversity	2,070,787.83	00:56:17
Biased MOPSO	Balance, Priority, Routing Agent	Hypervolume	0.119	00:56:17
Biased MOPSO	Balance, Distribution Agent	Diversity	205,445.56	00:36:37
Biased MOPSO	Balance, Distribution Agent	Hypervolume	0.4589	00:36:37
Biased MOPSO	Balance, Distribution, Routing Agent	Diversity	269,760.74	00:31:56
Biased MOPSO	Balance, Distribution, Routing Agent	Hypervolume	0.4975	00:31:56
Biased MOPSO	Balance, Distribution, Priority Agent	Diversity	195,206.07	00:27:36
Biased MOPSO	Balance, Distribution, Priority Agent	Hypervolume	0.523	00:27:36
Biased MOPSO	Balance, Distribution, Priority, Routing Agent	Diversity	194,638.8	00:30:54
Biased MOPSO	Balance, Distribution, Priority, Routing Agent	Hypervolume	0.5153	00:30:54

4.3. Discussion

Based on the updated results presented in Tables 4 and 7, the raw objective function values indicate a clear differentiation among the tested scenarios, particularly when subjected to non-dominated sorting. Among these, the scenario involving the multi-agent integration continues to demonstrate clear superiority as seen in Figures 8 and 9. This is further substantiated by the indicator evaluation metrics, where the most optimal integration achieves the highest HV and maintains the lowest simulation time. When comparing based on Example 2, it shows the reduction of at least 2.55% in simulation time compared to other MOPSO variants. This incremental improvement demonstrates the potential of introducing biases in the optimization process to achieve greater computational efficiency.

The trade-off between diversity, HV, and simulation time is evident in Figures 10 and 11. Standard MOPSO exhibits balanced values, state-of-the-art MOPSO variants, demonstrates increased diversity due to each of their inherent behavior, and biased MOPSO prioritizes HV by achieving better values across objective functions and reduced simulation time. In realistic real-world applications, the higher HV achieved by biased MOPSO is a decisive advantage, as it corresponds to superior average objective values. However, the most significant breakthrough is observed with the best multi-agent biased MOPSO, which achieves an impressive 241.46% increase in HV in Example 1 and 186.73% increase in Example 2. This is also followed by a simulation time reduction of 35.02% in Example 1% and 27.08% in Example 2.

Because of the small dataset of Example 1, differences or changes may cause bigger impact, hence most comparisons after will use Example 2. In terms of objective functions, this improvement corresponds to a 16.6% reduction in total cost, a 35.8% reduction in total work time, and a 19.05% increase in inventory success. These results highlight the synergistic effect of integrating multiple intelligent agents within the biased MOPSO framework, facilitating more effective decision-making and resource allocation, and enabling faster convergence without compromising solution quality.

HV, as a measure of the Pareto front’s quality, reflects both diversity and proximity to the true Pareto-optimal set. The consistent increase in HV with additional agent integrations underscores the efficiency of exploring the solution space more effectively, delivering broader and higher-quality trade-offs for multi-objective optimization problems. However, integrating too many agents can lead to performance degradation, as observed in Table 6, where adding the Routing Agent reduces HV and significantly increases simulation time. This outcome suggests that conflicting objectives among certain agents can diminish their strengths, resulting in suboptimal performance when all agents are employed simultaneously. This finding reaffirms that effective combinations of specific agents within an MOEA and DES framework can consistently yield Pareto-optimal solutions while maintaining a trade-off in diversity. Identifying and addressing conflicts between agents is critical to avoiding performance deficiencies.

In terms of diversity, the integration of Routing Agents achieves the highest value for expanding the solution search space. This is evident when observing Tables 6 and 9. This agent promotes a diverse spread of solutions in the objective space, enhancing exploration. On the other hand, observation shows degraded performance when too many agents are integrated. A comparison of agent roles, as depicted in Figures 12 and 13, reveals that the Routing Agent while it increases solution space, it is inconsistent in improving metric results. Conversely, the Priority Agent excels in minimizing total cost and inventory success consistently. Further analysis confirms that the Distribution Agent is critical in optimizing work time consistently. Their combination gives increased overall performance in metrics result. This is particularly evident when the Distribution and Priority Agents are combined with the Balancing Agent, as shown in Tables 6 and 9.

Finally, as illustrated in Figures 6 and 7, while also corroborated by Tables 6 and 9, the multi-agent integration was able to consistently synergizes in guiding MOPSO toward Pareto optimality. Each of its ability complement each other to identify solutions closest to the ideal Pareto point in the 3D objective space is shown by their role in reducing simulation time, balancing exploration, and exploitation during optimization. While single-agent configurations such as the Balance Agent demonstrate promising standalone results, multi-agent combinations achieve superior overall performance within simulation time and budget constraints. These findings highlight the inherent trade-offs between diversity and convergence, emphasizing the importance of strategic agent selection and configuration to maximize the efficiency of biased MOEAs.

4.4. Parameter sensitivity analysis

An assessment to the sensitivity of MOPSO to the inertia weight and learning factors was conducted by sweeping representative constant and linearly decreasing schedules for $ω$ , $c_{1}$ , and $c_{2}$ . A suitable configuration for the parameters is important in determining the best solution space it could reach. For consistency, the initial population was fixed across all experiments. Each configuration was executed five independent times using different seeds and evaluated on the resulting Pareto set with HV computed against a fixed reference point and a diversity indicator. Mean values across seeds are used in the re. Based on Tables 10 –13 in Appendix 2, settings that emphasize early exploration (higher initial $ω$ , or larger $c_{1}$ , $c_{2}$ ) tend to increase HV but can compress diversity, revealing a clear HV-diversity trade-off. To further select best configuration, top configurations which have high HV and diversity were selected. Using closest-to-ideal rule, top configurations are plotted and best parameter configuration of $ω = 0.7, c_{1} = 2, and c_{2} = 1.5$ , is selected by minimum Euclidean distance to (1,1) after min–max normalization as seen in Figure 14.

5. Conclusion and future works

5.1. Conclusion

This study has successfully demonstrated that leveraging a carefully designed solution space using compatible intelligent agents and a suitable MOEA, such as biased MOPSO, significantly improves optimization performance. The results reveal a clear advantage of multi-agent biased MOPSO over its baseline counterparts, achieving superior HV, reduced simulation time, and enhanced objective values. This finding confirms that biased factor and agents’ integration which further improved bias are required to achieve optimal performance. Though, combinations of agents must be selected carefully for them to work optimally within the MOEA-DES framework to yield Pareto-optimal solutions that dominate across all objective functions based on the problem. This is because utilizing all agents simultaneously or using non-complement agents may lead to conflicting objectives, reducing overall effectiveness.

The carefully selected combination of Balancing, Distribution, and Priority Agents was able to exhibit the best performance by maintaining high diversity and high HV with minimal simulation time, highlighting its ability to explore a broad solution space while giving high quality solution. The Routing Agent can still be very useful if the problem requires extensive exploration. If combined overall, the complementary nature of the Distribution and Routing Agents—with the former determining efficient delivery quantities and the latter optimizing routing—drives to higher diversity without increasing too much to simulation time. However, the increased diversity comes at the cost of longer simulation times and dispersed average results, prioritizing exploration over immediate convergence.

Further analysis underscores the significant contributions of the Distribution Agent in minimizing work time and the Priority Agent in minimizing total cost and maximizing inventory success. These results emphasize the importance of tailoring agent combinations to specific objectives, achieving balanced optimization across metrics. The findings also validate that the multi-agent bias is the most effective in enhancing MOPSO’s convergence in stochastic supply chain production–distribution scenarios. Its proximity to the ideal Pareto point in the 3D objective space underscores its potential. While single-agent configurations provide foundational benefits, multi-agent combinations, particularly those incorporating different functions, deliver superior results within the constraints of simulation time and budget with their synergy. These results highlight the importance of balancing exploration and convergence, demonstrating that efficient solutions can be achieved by leveraging strategic agent synergies. The enhanced Biased MOPSO framework offers a robust approach to tackling multi-objective optimization problems, underscoring its practical and theoretical significance in advancing the field of supply chain optimization.

5.2. Future work

Future research can enhance the current study by incorporating a complete multi-echelon supply chain model, including suppliers, retailers, and multiple distribution centers, to capture the full complexity of supply chain dynamics. This would allow for the modeling of interactions and dependencies between different echelons, such as lead times, order quantities, and supplier reliability, making the simulation more realistic. In addition, introducing a more stochastic environment to mimic real-world uncertainties, such as random fluctuations in consumer demand in non-VMI supply chain, transportation delays, and machine breakdowns, can further improve the robustness of the model.

Including suppliers as part of the supply chain would provide a more comprehensive analysis, as this experiment focused only on operational costs controllable by production and distribution. Evaluating supplier selection and order allocation strategies could optimize procurement costs and reduce supply risks. Furthermore, utilizing advanced MOEAs, such as NSGA-III, MOEA/D, and SPEA2, or exploring hybrid algorithms, could improve solution diversity and convergence, providing better performance compared to the current MOPSO approach.

Developing more sophisticated agents that integrate priority and routing logic efficiently can significantly enhance distribution efficiency. These agents could also incorporate adaptive learning mechanisms to dynamically adjust their strategies based on environmental changes and historical data. Introducing environmental sustainability objectives, such as minimizing carbon emissions, fuel consumption, and energy use, would enable a more holistic optimization that balances operational efficiency with ecological considerations.

Future work could also include modeling resource constraints, such as limited vehicle capacity, workforce availability, and storage space, to reflect real-world operational challenges. Simulating resource competition across echelons would add further complexity and realism. Finally, exploring dynamic supply chain reconfiguration, where strategies adapt over time to disruptions, seasonal variations, or market trends, and introducing resilience metrics, would help build a more robust and adaptive supply chain model. These enhancements would provide deeper insights and make the proposed framework more applicable to real-world supply chain problems.

Footnotes

Appendix 1

Appendix 2 Acknowledgements

None.

ORCID iD

Shih-Hsien Tseng

Ethics approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Funding

The authors received no financial support for the research, authorship, and/or publication of this article.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Availability of data and materials

Data can be shared upon request.

Author Biographies

Shih-Hsien Tseng is an Associate Professor in the Department of Industrial Management at National Taiwan University of Science and Technology (NTUST). He received his PhD and MS degrees in Industrial and Systems Engineering from The Ohio State University, USA. His research interests include quality engineering, simulation, lean manufacturing, and design of experiments. He has published over 20 journal papers and 25 conference papers, and has led multiple industry-academia collaboration projects focusing on supply chain and manufacturing optimization.

Jason Gozal is an Industrial Engineer with an MBA in Industrial Management from National Taiwan University of Science and Technology (NTUST). His research focuses on simulation-based optimization and supply chain management. His expertise includes data science, scheduling optimization, and process improvement. He has working experience in lean practices, contributing to data-driven efficiency projects that achieved measurable operational improvements.

References

Ferdous

Yousefi

Tosarkani

BM.

A multi-disruption risk analysis system for sustainable supply chain resilience. Int J Disaster Risk Reduct 2025; 116: 105136.

Farhangi

Multi-echelon supply chains with lead times and uncertain demands: a lot-sizing formulation and solutions. Oper Res Forum 2021; 2: 1–25.

Coşkun

Erturgut

How do uncertainties affect supply-chain resilience? The moderating role of information sharing for sustainable supply-chain management. Sustainability 2024; 16: 131.

Jayarathna

Agdas

Dawes

, et al. Multi-objective optimization for sustainable supply chain and logistics: a review. Sustainability 2021; 13: 13617.

Mahrach

Miranda

León

, et al. Comparison between single and multi-objective evolutionary algorithms to solve the knapsack problem and the travelling salesman problem. Mathematics 2020; 8: 2018.

Maher

Ralphs

Shinano

. Assessing the effectiveness of (parallel) branch-and-bound algorithms. arXiv. DOI: 10.48550/arXiv.2104.10025.

Fleming

Göös

Impagliazzo

, et al. On the power and limitations of branch and cut, https://arxiv.org/abs/2102.05019v2 (2021, accessed 24 January 2025).

Ling

Liu

, et al. Multi-objective particle swarm optimization based on particle contribution and mutual information for feature selection method. J Supercomput 2025; 81: 1–34.

Yen

GG.

Adaptive multiobjective particle swarm optimization based on parallel cell coordinate system. IEEE Trans Evol Comput 2015; 19: 1–18.

10.

Coello

CAC

Pulido

Lechuga

. Handling multiple objectives with particle swarm optimization. IEEE Trans Evol Comput 2004; 8: 256–279.

11.

Zhang

Liu

Song

, et al. Density-guided and adaptive update strategy for multi-objective particle swarm optimization. J Comput Des Eng 2024; 11: 222–258.

12.

Vázquez-Serrano

Cárdenas-Barrón

Vicencio-Ortiz

, et al. Hybrid optimization and discrete-event simulation model to reduce waiting times in a primary health center. Expert Syst Appl 2024; 238: 121920.

13.

Lyu

Pons

Palliparampil

, et al. Optimising urban freight logistics using discrete-event simulation and cluster analysis: a stochastic two-tier hub-and-spoke architecture approach. Smart Cities 2023; 6: 2347–2366.

14.

Tako

Robinson

The application of discrete event simulation and system dynamics in the logistics and supply chain context. Decis Support Syst 2012; 52: 802–815.

15.

Wallrath

Franke

MB.

Integrating MILP, discrete-event simulation, and data-driven models for distributed flow shop scheduling using benders cuts. Processes 2024; 12: 1772.

16.

Urbanucci

Limits and potentials of mixed integer linear programming methods for optimization of polygeneration energy systems. Energy Procedia 2018; 148: 1199–1205.

17.

Qin

Jiang

, et al. A data-driven mixed integer programming approach for joint chance-constrained optimal power flow under uncertainty. Int J Mach Learn Cybern 2024; 16: 1111–1127.

18.

Cacereño

Greiner

Galván

Simultaneous optimization of design and maintenance for systems using multi-objective evolutionary algorithms and discrete simulation. Soft Comput 2023; 27: 19213–19246.

19.

Rabe

Deininger

Juan

AA.

Speeding up computational times in simheuristics combining genetic algorithms with discrete-Event simulation. Simul Model Pract Theory 2020; 103: 102089.

20.

Chen

Gong

, et al. A novel hybrid multi-objective particle swarm optimization algorithm with an adaptive resource allocation strategy. IEEE Access 2019; 7: 177082–177100.

21.

Chen

Wang

Zio

, et al. A simulation-based multi-objective optimization framework for the production planning in energy supply chains. Energies 2021; 14: 2684.

22.

Lee

Sikora

Design of intelligent agents for supply chain management. Lect Notes Bus Inf Process 2016; 258: 27–39.

23.

Khan

Al-Mushayt

Alam

, et al. Intelligent supply chain management. J Softw Eng Appl 2010; 03: 404–408.

24.

Yung

Yang

CC.

Intelligent multi-agent for supply chain management. Proc IEEE Int Conf Syst Man Cybern 1999; 2: 528–533.

25.

Zhou

Liu

, et al. Challenges and opportunities in integration of simulation and optimization in maritime logistics. Proc Winter Simul Conf 2018; 2018: 2897–2908.

26.

Gupta

Haq

Ali

, et al. Significance of multi-objective optimization in logistics problem for multi-product supply chain network under the intuitionistic fuzzy environment. Complex Intell Syst 2021; 7: 2119–2139.

27.

Altiparmak

Gen

Lin

, et al. A genetic algorithm approach for multi-objective optimization of supply chain networks. Comput Ind Eng 2006; 51: 196–215.

28.

Luo

Liu

Zhuge

, et al. A multi-objective optimization approach for supply chain design of alum sludge-derived supplementary cementitious material. Case Stud Constr Mat 2022; 17: e01156.

29.

Sarkar

Modak

JM.

Pareto-optimal solutions for multi-objective optimization of fed-batch bioreactors using nondominated sorting genetic algorithm. Chem Eng Sci 2005; 60: 481–492.

30.

Ransikarbum

Wattanasaeng

Madathil

SC.

Analysis of multi-objective vehicle routing problem with flexible time windows: the implication for open innovation dynamics. J Open Innov Technol Mark Complex 2023; 9: 100024.

31.

Hnaien

Delorme

Dolgui

Multi-objective optimization for inventory control in two-level assembly systems under uncertainty of lead times. Comput Oper Res 2010; 37: 1835–1843.

32.

Long

Dong

Wang

, et al. A constrained multi-objective optimization algorithm using an efficient global diversity strategy. Complex Intell Syst 2023; 9: 1455–1478.

33.

Kumar

Ali

, et al. A Benchmark-Suite of real-World constrained multi-objective optimization problems and some baseline results. Swarm Evol Comput 2021; 67: 100961.

34.

Coello

CAC

Lamont

Van Veldhuizen

. Evolutionary algorithms for solving multi-objective problems. New York: Springer, 2007.

35.

Coello

CAC

Lechuga

. MOPSO: a proposal for multiple objective particle swarm optimization. Proc Cong Evol Comput 2002; 2: 1051–1056.

36.

Devi

Jagadev

Dehuri

Comparison of various approaches in multi-objective particle swarm optimization (MOPSO): empirical study. Stud Comput Intell 2015; 592: 75–103.

37.

Aboud

Fdhila

Alimi

AM.

Dynamic multi objective particle swarm optimization based on a new environment change detection strategy. Lect Notes Comput Sci 2017; 10637LNCS: 258–268.

38.

Xie

Yang

, et al. A hybrid algorithm based on NSGA-II and MOPSO for multi-objective designs of electromagnetic devices. IEEE Trans Magn 2023; 59: 1–4.

39.

Gao

Constrained multi-objective particle swarm optimization algorithm. Commun Comput Inf Sci. 2012; 304CCIS: 47–55.

40.

Masuda

Kurihara

A constrained global optimization method based on multi-objective particle swarm optimization. Electron Commun Jpn 2012; 95: 43–54.

41.

Parsonage

Maddock

. Biased dyadic crossover for variable-length multi-objective optimal control problems. In: 2024 IEEE congress on evolutionary computation, CEC 2024—proceedings, Yokohama, 5 July 2024, pp. 1–9. New York: IEEE.

42.

Liu

, et al. MOEA/D with biased weight adjustment inspired by user preference and its application on multi-objective reservoir flood control problem. Soft Comput 2016; 20: 4999–5023.

43.

Wang

Akhtar

Shoemaker

CA.

Efficient multi-objective optimization through parallel surrogate-assisted local search with tabu mechanism and asynchronous option. Eng Optim 2024; 56: 1081–1097.

44.

Liu

Jin

Heiderich

, et al. Surrogate-assisted evolutionary optimization of expensive many-objective irregular problems. Knowl Based Syst 2022; 240: 108197.

45.

Kumar

Singh

PK.

Pareto evolutionary algorithm hybridized with local search for biobjective TSP. Stud Comput Intell 2007; 75: 361–398.

46.

Dutta

Mallipeddi

Das

KN.

Hybrid selection based multi/many-objective evolutionary algorithm. Sci Rep 2022; 12: 6861.

47.

Grosan

Abraham

Hybrid evolutionary algorithms: methodologies, architectures, and reviews. Stud Comput Intell 2007; 75: 1–17.

48.

Agrawal

Yadav

Gupta

, et al. A genetic algorithm model for optimizing vehicle routing problems with perishable products under time-window and quality requirements. Decis Anal J 2022; 5: 100139.

49.

Sun

Wang

Routing design and fleet allocation optimization of freeway service patrol: improved results using genetic algorithm. Phys A Stat Mech Appl 2018; 501: 205–216.

50.

Zaizi

Qassimi

Rakrak

Multi-objective optimization with recommender systems: a systematic review. Inf Syst 2023; 117: 102233.

51.

Ahmad

Ansari

Nadeem

, et al. Inhibition of tyrosine kinase signaling by tyrphostin AG126 downregulates the IL-21/IL-21R and JAK/STAT pathway in the BTBR mouse model of autism. Neurotoxicology 2020; 77: 1–11.

52.

Fu-Gui

Hui-Mei

Bing-de

Agent-based simulation model of single point inventory system. Syst Eng Procedia 2012; 4: 298–304.

53.

Nam

Cho

Woo

JH.

Simulation-based deep reinforcement learning for multi-objective identical parallel machine scheduling problem. Int J Nav Archit Ocean Eng 2024; 16: 100629.

54.

Rabet

Sajadi

Tootoonchy

A hybrid metaheuristic and simulation approach towards green project scheduling. Ann Oper Res 2024; 2024: 1–40.

55.

Babulak

Wang

Discrete event simulation: state of the art. Discrete Eve Simul 2010; 4: 60–63.

56.

Vázquez-Serrano

Peimbert-García

Cárdenas-Barrón

LE.

Discrete-event simulation modeling in healthcare: a comprehensive review. Int J Environ Res Public Health 2021; 18: 12262.

57.

Al-Hawari

Gailan Qasem

Smadi

, et al. The effects of pre or post delay variable changes in discrete event simulation and combined DES/system dynamics approaches in modelling supply chain performance. J Simul 2022; 16: 147–165.

58.

Bottani

Casella

Discrete-event simulation in logistics and supply chain management: a scientometric perspective. Prod Manuf Res 2024; 12: 2415038.

59.

Salas-Navarro

Florez

Cárdenas-Barrón

LE.

A vendor-managed inventory model for a three-layer supply chain considering exponential demand, imperfect system, and remanufacturing. Ann Oper Res 2024; 332: 329–371.

60.

Coelho

Laporte

Improved solutions for inventory-routing problems through valid inequalities and input ordering. Int J Prod Econ 2014; 155: 391–397.

61.

Zhongming

Zhixue

. A strategic inventory routing problem model in VMI TPL. In: 2008 international conference on wireless communications, networking and mobile computing, Wicom 2008, Dalian, 12–14 October 2008, pp. 1–4. New York: IEEE.

62.

Zheng

Guo

Zhang

XD.

Modeling and approach for VMI cyclic inventory routing problem. Proc Int Conf Mach Learn Cybern 2009; 3: 1393–1398.

63.

Marchetti

Gupta

Grossmann

, et al. Simultaneous production and distribution of industrial gas supply-chains. Comput Chem Eng 2014; 69: 39–58.

64.

Cóccola

Méndez

Dondo

RG.

Optimizing the inventorying and distribution of chemical fluids: an innovative nested column generation approach. Comput Chem Eng 2018; 119: 55–69.

65.

Cóccola

Méndez

Dondo

RG.

A two-stage procedure for efficiently solving the integrated problem of production, inventory, and distribution of industrial products. Comput Chem Eng 2020; 134: 106690.

66.

Hofmann

Supply chain management: strategy, planning and operation, Chopra

Meindl

, 5th ed, ISBN:978-0273765226. J Purch Supply Manag 2013; 19: 212–213.

67.

Toth

Vigo

The vehicle routing problem. Philadelphia, PA: University City Science Center, 2001.

68.

Pinedo

ML.

Scheduling: theory, algorithms, and systems, sixth edition. Cham: Springer, 2022.

69.

Touzout

Ladier

Hadj-Hamou

Time-dependent travel-time constrained inventory routing problem. Lect Notes Comput Sci 2020; 12433LNCS: 151–166.

70.

Tamilarasu

Singaravel

Quality of service aware improved coati optimization algorithm for efficient task scheduling in cloud computing environment. J Eng Res 2024; 12: 768–780.

71.

Mohammad Hasani Zade

Mansouri

Javidi

. An improved beluga whale optimization using ring topology for solving multi-objective task scheduling in cloud. Comput Ind Eng 2025; 200: 110836.

72.

Suresh

Keerthika

Manjula Devi

, et al. Optimized task scheduling approach with fault tolerant load balancing using multi-objective cat swarm optimization for multi-cloud environment. Appl Soft Comput 2024; 165: 112129.

73.

Houck

Whitehead

Introduction to Simio. In: Proceedings—winter simulation conference 2019, National Harbor, MD, 8–12 December 2019, pp. 3802–3811. New York: IEEE.

74.

Quiros

Smith

Thiruvengadam

, et al. Greenhouse gas emissions from heavy-duty natural gas, hybrid, and conventional diesel on-road trucks during freight transport. Atmos Environ 2017; 168: 36–45.

75.

Ding

, et al. NC-MOPSO: network centrality guided multi-objective particle swarm optimization for transport optimization on networks, https://arxiv.org/abs/2009.03575v3 (2020, accessed 5 January 2025).

76.

Fitas

. A new simplified MOPSO based on swarm elitism and swarm memory: MO-ETPSO. 2024. ArXiv. DOI: 10.48550/arXiv.2402.12856.

77.

Sitek

Nielsen

Wikarek

A hybrid multi-agent approach to the solving supply chain problems. Procedia Comput Sci 2014; 35: 1557–1566.

78.

Fattahi

Govindan

Integrated forward/reverse logistics network design under uncertainty with pricing for collection of used products. Ann Oper Res 2017; 253(1): 193–225.

79.

Gutierrez-Alcoba

Rossi

Martin-Barragan

, et al. A simple heuristic for perishable item inventory control under non-stationary stochastic demand. Int J Prod Res 2017; 55(7): 1885–1897.

80.

Alavi

Jabbarzadeh

Supply chain network design using trade credit and bank credit: a robust optimization model with real world application. Comput Ind Eng 2018; 125: 69–86.

81.

Razavian

Tabriz

Zandieh

, et al. An integrated material-financial risk-averse resilient supply chain model with a real-world application. Comput Ind Eng 2021; 161: 107629.

82.

Keyvanshokooh

Ryan

Kabir

Hybrid robust and stochastic optimization for closed-loop supply chain network design using accelerated Benders decomposition. Eur J Oper Res 2016; 249(1): 76–92.

83.

Wright

Johnson

Moldover

, et al. Errors in rate of rise gas flow measurements from flow work. https://www.nist.gov/publications/errors-rate-rise-gas-flow-measurements-flow-work (2018, accessed 4 January 2025).

84.

Joshi

Sectoral and temporal changes in natural gas demand in the United States. J Nat Gas Sci Eng 2021; 96: 104245.

85.

Zhang

Hou

, et al. Multidirection update-based multiobjective particle swarm optimization for mixed no-idle flow-shop scheduling problem. Complex Syst Model Simul 2021; 1: 176–197.

86.

Zitzler

Thiele

Multiobjective evolutionary algorithms: a comparative case study and the strength Pareto approach. IEEE Trans Evol Comput 1999; 3: 257–271.