Smart Logistics Optimization with Fractional Order Models and Artificial Intelligence Techniques

الموضوعات : مهندسی هوشمند برق

Hosein Esmaili ¹ , Mohammad Ali Afshar Kazemi ² , Reza Radfar ³ , Nazanin Pilevari ⁴

1 - Department of Industrial Management, Science and Research Branch, Islamic Azad University
2 - Department of Industrial Management, Science and Research Branch, Islamic Azad University
3 - Associate Prof
4 - Department of Industrial Management, West Tehran Branch, Islamic Azad University, Tehran, Iran

تاريخ الإرسال : 23 الخميس , رجب, 1446 تاريخ التأكيد : 09 الخميس , ذو الحجة, 1446 تاريخ الإصدار : 17 السبت , محرم, 1447

الکلمات المفتاحية: Multi-modal logistics optimization, Fractional differential equations, Fire Hawk Optimization Algorithm, Stochastic demand modeling, Dynamic network flow analysis.,

ملخص المقالة :

Optimizing dynamic multi-modal logistics networks is vital for efficient resource allocation and cost minimization under stochastic demand. This study develops a robust framework integrating fractional differential equations for modeling temporal flow dynamics and employs the Fire Hawk Optimization Algorithm (FHOA) to address the resulting complex optimization problem. The proposed framework represents logistics networks as directed graphs, incorporating stochastic demand modeled using Gaussian perturbations and fractional derivatives to capture memory effects. Validation using the Barcelona logistics dataset reveals a total cost reduction of 8,835 units, average flow stabilization at 3.05 units, and resilience under demand variance of 52.32. The model effectively identifies critical edges with high flows, balancing throughput and minimizing costs across scenarios. Furthermore, the algorithm enhances decision-making by optimizing transportation policies and resource allocations, ensuring operational efficiency under uncertainty. The study's findings emphasize the significance of integrating advanced mathematical modeling with metaheuristic optimization techniques to tackle the inherent complexities of logistics systems. These insights provide a scalable and adaptive solution for real-world multi-modal logistics networks.

المصادر:

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[7] . Zarbakhshnia, N., Kannan, D., Kiani Mavi, R., & Soleimani, H. (2020). A novel sustainable multi-objective optimization model for forward and reverse logistics system under demand uncertainty. Annals of Operations Research, 295, 843-880. https://doi.org/10.1007/s10479-020-03744-z

[8] . Jiang, J., Zhang, D., Li, S., & Liu, Y. (2019). Multimodal green logistics network design of urban agglomeration with stochastic demand. Journal of Advanced Transportation, 2019(1), 4165942. https://doi.org/10.1155/2019/4165942

[9] . Orozco-Fontalvo, M., Cantillo, V., & Miranda, P. A. (2019, September). A stochastic, multi-commodity multi-period inventory-location problem: Modeling and solving an industrial application. In International Conference on Computational Logistics (pp. 317-331). Cham: Springer International Publishing. https://doi.org/10.1007/978-3-030-31140-7_20

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نص كامل:



143                              International Journal of Smart Electrical Engineering, Vol.13, No.4, Autumn 2025                       ISSN: 2251-9246        

EISSN: 2345-6221                

pp. 1:9

 

 

Fractional Dynamics Fire Hawk Optimization for Smart Multi-Modal Logistics Networks under Stochastic Demand

Hosein Esmaeili, Mohammad Ali Afshar Kazemi*, Reza Radfar, Nazanin Pilevari 

Department of Industrial Management, Science and Research Branch, Islamic Azad University, Tehran, Iran, [e-mail]

Abstract

Dynamic multi-modal logistics networks must remain efficient even when customer demand fluctuates randomly. This study combines fractional-order flow dynamics with the Fire Hawk metaheuristic to build an intelligent decision-making framework that continually reallocates road, rail, and feeder-air capacity in near-real time. Tested on the 2,522 edge Barcelona benchmark (fractional order α = 0.8, 110 origin–destination pairs, five Monte-Carlo demand scenarios), the model cuts total cost by about 8,835 units roughly 16 percent versus the deterministic baseline keeps average flow near 3.05 units, and restricts flow variance to 0.15–0.16 while sustaining demand-variance resilience of 52.3. These results demonstrate that embedding long-memory fractional equations within a nature-inspired optimizer provides a scalable, data-driven tool that relieves congestion, balances throughput, and strengthens robustness for next-generation smart logistics planning.

Keywords: memory effects, metaheuristic search, demand uncertainty, scenario analysis, transportation planning.

Article history: Received 2025/01/23, Revised 2025/05/20; Accepted 2025/06/05, Article Type: Research paper

© 2025 IAUCTB-IJSEE Science. All rights reserved

 

1.       Introduction

 

Over the past decade, the logistics sector has become a proving ground for intelligent decision-making (IDM) systems that blend data-driven perception, optimization and control to cope with real-time complexity. Advanced heuristics, neural rule engines and hybrid expert systems now assist dispatchers in routing, mode choice and capacity allocation, yielding measurable gains in cost and service resilience [1, 2]. Yet most commercial IDM tools still assume static demand and single-modal flows, limiting their ability to react to today’s volatile, multi-actor freight ecosystems.

Multi-modal logistics networks where road, rail, sea and air legs are stitched together offer superior sustainability and flexibility, but also introduce path-dependency, inter-modal transfer constraints and demand uncertainty [3]. Traditional deterministic or single-period models neglect these couplings, often producing fragile policies that fail when volumes swing or congestion propagates [4, 5]. A robust decision framework must therefore capture (i) stochastic, time-varying origin-destination flows, (ii) capacity-induced congestion feedback and (iii) the memory effects that arise when today’s routing choices influence tomorrow’s network state.

Stochastic programming, chance-constrained MILP and metaheuristic search have each been deployed to tackle aspects of this problem. Two-stage formulations hedge against random demand but scale poorly with scenario count [6, 5]; MILP with time windows ensures logical consistency yet explodes combinatorially in large multi-modal graphs [7]. Recent metaheuristics (e.g., modified firefly and genetic algorithms) accelerate search but still rely on integer-order flow dynamics, ignoring the long-range dependence observed in real freight data [8, 9]. Consequently, existing IDM frameworks remain either computationally prohibitive or behaviorally incomplete.

To close this gap, we integrate fractional differential equations well suited for modelling systems with hereditary properties with the Fire Hawk Optimization Algorithm (FHOA), a recent nature-inspired metaheuristic that balances global exploration and local intensification [1]. The fractional terms embed network “memory”, allowing the model to anticipate congestion persistence, while FHOA efficiently searches the high-dimensional flow–capacity space. Validated on the Barcelona benchmark network, the hybrid scheme achieves an 8 % reduction in total operating cost and smoother flow profiles relative to state-of-the-art integer-order baselines [10].

Section 2 reviews related multi-modal optimization literature and positions our work within IDM research. Section 3 formulates the fractional flow and demand dynamics, followed by Section 4, which details the FHOA-based solution procedure. Section 5 presents numerical experiments on synthetic and real-world datasets, while Section 6 discusses managerial insights, limitations and avenues for future research.

2.       Related Work

Multimodal logistics and transportation networks under uncertainty have received significant attention in the literature, covering various topics such as uncertain demands, cost minimization, and complex optimization strategies in dynamic networks. Zhang et al. [11] have proposed an optimization model for multimodal hub-and-spoke transport networks, integrating fuzzy chance-constrained methods to handle demand uncertainty and improve transportation efficiency. Their work provides insights into trade-offs between the number of hubs and cost minimization, emphasizing the impact of hub capacity constraints. Mishra and Lamba [23] have introduced a dynamic multi-modal approach for global supply chain configuration, employing mixed-integer linear programming (MILP) to optimize facility activation, transportation mode selection, and inventory management while considering time-cost trade-offs.

Meng et al. [6] have developed a two-stage stochastic programming model to enhance emergency logistics network resilience, integrating multimodal transport approaches for natural disaster response. Their study demonstrates the significance of dynamic uncertainty management in logistics. Similarly, Peng et al. [12] have proposed a multi-objective optimization model for multimodal transportation using Monte Carlo simulations and data-driven ant colony algorithms. Their model, validated on China’s Belt and Road Initiative, improves computational efficiency and optimizes transit costs and travel time.

For hazardous material (HAZMAT) transportation, Han et al. [13] have developed a multi-objective mixed-integer linear programming model that employs triangular fuzzy random numbers to handle demand fluctuations. The study highlights the influence of confidence levels on transportation risk and economic objectives. Postan et al. [14] have proposed a dynamic optimization model for multi-echelon logistics networks, addressing both deterministic and stochastic demand scenarios over a discrete time horizon.

Sustainability-driven logistics optimization has also been explored in prior studies. Zarbakhshnia et al. [15] have introduced a sustainable multi-objective optimization model for forward and reverse logistics, integrating environmental, social, and economic criteria. Orozco-Fontalvo et al. [17] have examined a strategic inventory-location problem for multi-commodity networks under stochastic demands, demonstrating significant cost reductions through genetic algorithms applied to mixed-integer programming models.

Furthermore, Karimi et al. [18] have addressed multimodal logistics hub location problems by incorporating stochastic demand conditions and commodity splitting to enhance network efficiency. Their study employs discrete chance-constrained programming to improve demand fulfillment accuracy. Similarly, Li et al. [19] have developed a two-stage stochastic programming model for rail-truck intermodal network design, validated on real-world data to optimize cost efficiency under uncertain conditions.

Routing optimization in multimodal logistics has also been widely studied. Desticioğlu Taşdemir and Özyörük [20] have introduced a mathematical model for the multi-depot simultaneous pick-up and delivery vehicle routing problem under stochastic demand, refining non-linear constraints to improve computational efficiency. Al-Ashhab [21] has proposed a stochastic mixed-integer linear programming model for multi-period, multi-product supply chain design, offering a robust framework for maximizing expected profits under stochastic demand. Yu et al. [22] have analyzed logistics distribution network optimization under random demand, integrating Lagrangian relaxation and sub-gradient algorithms to enhance retail store location and distribution path selection.

Lastly, Shahraki and Türkay [23] have presented a bi-level stochastic optimization model for urban logistics networks, incorporating multimodal passenger travel and freight logistics to minimize carbon emissions and traffic congestion. Their findings offer a sustainable framework for urban transportation planning.

Despite rich work on multimodal network design, most studies still linearize flow dynamics and therefore overlook the long-memory effects that dominate real‐time freight movements under volatile demand. Existing fractional-order formulations, meanwhile, have been explored only on small, single-mode test beds and rarely integrated with scalable metaheuristic solvers. Consequently, a comprehensive framework that fuses fractional flow modelling with an adaptive optimization engine for large-scale, stochastic, multi-modal logistics networks remains an open research need.

3.       Problem Formulation

Logistics network optimization is the artwork of efficient transportation to ensure timely deliveries, lower operating costs, and resiliency to demand uncertainties. In this section, we establish a mathematical framework for modeling the dynamics in multi-modal logistics networks acting under stochastic demand using fractional differential equations. Such as the temporal evolution of logistics flows and the stochastic nature of demand processes, that is, the proposed model not only describes logistics processes in a fundamental form but is also significant for real life use.

3.1 Dynamic Multi-Modal Logistics Network Representation

   We represent the logistics network as a directed graph , where:

 is the set of nodes, representing locations such as warehouses, transit hubs, and destinations.

 is the set of edges, representing transportation links (e.g., roads, railways, or air routes) [23,3].

Each node  has:

The time-dependent demand (positive for demand, negative for supply) at node  at time .

 Node weight, representing storage or transfer costs, which may depend on .

Each edge  has the following attributes:

The flow of goods on edge  at time .

The unit transportation cost along the edge.

The maximum capacity of the edge, representing the upper limit of flow.

The time required to traverse edge , which depends on the flow . It is given by the Bureau of Public Roads (BPR) function:

 

where ​ is the free-flow travel time,  and  are empirical constants [23].

 

3.2 Fractional Dynamics of Flows

In dynamic multi-modal logistics networks, the flow on each edge evolves over time. To capture memory effects and time-dependent adjustments, we model the flow dynamics using fractional differential equations (FDEs) [9,24].

 

3.2.1 Fractional Flow Dynamics

The flow on edge  satisfies:

 

(1)

where:

​ is the fractional derivative of order α\alphaα , capturing memory effects.

: Damping coefficient, which models dissipation or resistance in the system.

: External influence (e.g., stochastic demand shocks or system adjustments).

The fractional derivative is defined using the  approximation:

 

(2)

where:

: Discretized time step size.

​ is the binomial coefficient with Gamma functions.

 Number of previous steps included in the approximation.

Flow on edge  at time .

3.2.2 Demand Dynamics

The demand  at node  evolves dynamically according to:

 

(3)

where:

Total outflow from node .

Total inflow to node .

 Stochastic perturbation in demand, modeled as a Gaussian process 

3.3 Objective Function

 Total Cost Function The optimization process to minimize the total cost  uses the Fire Hawk Optimization Algorithm (FHOA) [1], which iteratively refines solutions through attraction and random exploration.

3.3.1 Total Cost Function

The total cost  over a time horizon  is:

(4)

 

Where  represents the transportation cost on edge and  penalizes congestion [9,24].

 

3.3.2 Constraints

-       Flow Conservation

   At each node , the sum of inflows and outflows must balance with the demand:

(5)

 

-       Capacity Constraints

    The flow on each edge  must not exceed its capacity:

 

-       Non-Negativity

   The flows must be non-negative:

 

 

3.4 Numerical Solution of Fractional Dynamics

      To solve the fractional differential equations (FDEs) for flow dynamics and demand evolution, we use numerical approximations. The  Approximation provides a discrete-time approach to approximate fractional derivatives.

 

3.4.1 Discrete-Time Flow Dynamics

   The fractional derivative of the flow  is approximated in Equation (3).

 

The discretized flow dynamics become:

(6)

where:

​ is the flow at the  time step.

 Time-dependent external influences at the  step.

 Damping coefficient.

(7)

 

where:

 

Stochastic perturbation at time step .

 

(8)

Algorithm 1: Fractional Dynamics Solver for Stochastic Demand in Logistics Networks

Input:   Network data (nodes, edges, capacities, costs)   Fractional order   Time step , total time   Initial conditions ,   Damping coefficient   Number of scenarios Output:   Time-evolved flow and demand for all time steps   1. Initialize:                     # Number of time steps           # Flow matrix for edges          # Demand matrix for nodes 2. For each scenario    3. Generate stochastic demand perturbations    4. For time step         For each edge            Compute fractional derivative:                           Update flow:                        For each node            Compute demand update:                           Update demand:               5. Return:   3.4.2 Integration of Stochastic Demand Stochastic demand introduces randomness into the system, requiring robust optimization techniques [4,10,2]. The demand  at node  follows: where:  Deterministic component of demand.  Stochastic component, modeled as a Gaussian process [22].   Scenario-Based Modeling       To handle stochasticity, we generate demand scenarios using Monte Carlo simulation. For each scenario :   (9) where is sampled from  [5].     (10) Algorithm 2: Scenario-Based Stochastic Demand Generation for Logistics Networks Input:   Mean demand matrix   Variance vector   Number of scenarios   Time step h, total time Output:   Stochastic demand matrix for all scenarios 1. Initialize:                   # Number of time steps     2. For each scenario    For each node       For each time step          Generate perturbation:                       Compute demand:              3. Return:     3.4.3 Optimization Framework Objective Function (Discrete-Time Formulation) The continuous-time cost function is discretized as: where: Number of time steps. Travel time at time step , given by the BPR function [8,23].   Decision Variables The primary decision variables are:  Flow on edge  at time step .   Demand at node  at time step .   Flow Conservation (Discrete-Time)     (11) Capacity Constraints     Non-Negativity     The flows must be non-negative:         (11)