Stochastic optimization of perishable agricultural supply chains: a hybrid genetic algorithm approach for robust network design under multi-dimensional uncertainty

Abstract

Introduction:

Agricultural supply chains face significant challenges from uncertainty in crop yields, processing times, and demand, alongside product perishability and sustainability pressures. This study addresses the need for an optimization framework that simultaneously manages these complexities.

Methods:

We formulate a stochastic programming model for a multi-modal, multi-echelon perishable agricultural supply chain network. The model incorporates chance constraints for yield variability, recourse for processing time uncertainty, and an expected value approach with safety margins for travel time. To solve this computationally challenging problem for large-scale networks (320 nodes), we develop a hybrid genetic algorithm with adaptive parameter control, domain-specific operators (geographic clustering, quality-aware mode assignment), and selective local search.

Results:

Compared to a traditional hub-and-spoke baseline, the optimized network reduces total operational cost by 35%, food waste by 27.2%, and carbon emissions by 24.9%. Average product quality at delivery improves from 82% to 92%. The hybrid GA significantly outperforms a standard GA baseline, with the hybridization components contributing 15–20% improvement in solution quality.

Discussion:

The results demonstrate that simultaneous optimization of network structure, multi-modal transport, and quality constraints yields substantial economic and sustainability benefits. Sensitivity analysis reveals cold chain reliability as the dominant cost driver, providing actionable guidance for infrastructure investment. The framework offers a practical tool for agricultural stakeholders, particularly in regions where cold chain capacity is limited.

1 Introduction

Agricultural supply chains face unprecedented challenges in connecting farms to consumers while minimizing food waste and costs. According to Food and Agriculture Organization (2022), approximately 30% of food is lost throughout the supply chain, with inefficient logistics contributing significantly to these losses. This occurs against increasing food insecurity, climate pressures, rising logistics costs, and growing sustainability demands.

The Agricultural Supply Chain Network Optimization Problem (ASCNOP) involves determining optimal connections between farms, processing facilities, and distribution centers while accounting for variable yields, perishability constraints, fluctuating demand, and seasonal variations. Perishable goods transportation presents unique complexities, with refrigerated transportation incurring substantially higher costs than ambient transport due to temperature control requirements, vehicle specialization, and energy consumption (Yan et al., 2025).

Traditional supply chain optimization typically focuses on minimizing distance within deterministic frameworks. However, ASCNOP requires greater attention to perishability, temporal constraints, and the stochastic nature of supply and demand, complicated by multiple transportation modes, heterogeneous processing facilities, varying degradation rates, and climate disruptions.

Existing research has limitations: Marto et al. (2026) developed a biobjective stochastic intermodal model for agricultural commodity flows but focused on corn and soybean chains without addressing perishability constraints or quality degradation dynamics. Ruiz-Meza et al. (2021) developed mathematical models for two-echelon fresh produce supply chain planning but applied deterministic frameworks that fail to capture the yield and demand variability inherent in agricultural systems. Few researchers have explored ASCNOP with the dual objective of reducing food waste alongside cost minimization.

The present research addresses these gaps by developing a comprehensive stochastic programming model that considers:

• Stochastic crop yields, processing times, and market demands

• Multi-modal transportation with different costs and quality preservation capabilities

• Explicit modeling of perishability and quality degradation

• Network-wide optimization across all supply chain nodes

• Efficient hybrid genetic algorithm solution methodology

• Comprehensive sustainability metrics

This paper contributes to agricultural supply chain optimization through:

• A novel stochastic programming formulation integrating multiple uncertainty sources

• A specialized hybrid genetic algorithm solving large-scale instances

• Explicit incorporation of sustainability metrics alongside economic objectives

• Comprehensive validation through a realistic case study with 320 network nodes

• Robust sensitivity analysis demonstrating solution resilience under various scenarios

Through these contributions, agricultural stakeholders are provided with practical tools for designing more efficient, sustainable, and resilient supply chains under increasing uncertainty and climate volatility.

Figure 1 illustrates the conceptual framework of our stochastic optimization approach for perishable agricultural supply chains. It shows the integration of multiple uncertainty sources with the network connecting farms, processing facilities, and distribution centers.

Figure 1

The framework integrates three coupled uncertainty sources — stochastic crop yields at farm nodes, variable processing times at facility nodes, and fluctuating demand at distribution centers — into a single certainty equivalent formulation. Network arcs represent candidate transportation connections across three modes (ambient, medium refrigerated, large refrigerated), each carrying distinct cost, quality degradation, and emissions characteristics. The hybrid genetic algorithm operates on this full network structure, simultaneously optimizing mode assignment, routing, and flow allocation subject to capacity, quality, and time window constraints.

2 Literature review

Agricultural supply chains present a distinct and demanding class of optimization problems. Unlike industrial supply chains, they must simultaneously contend with biological variability, time-sensitive quality deterioration, geographic dispersion, and deep structural uncertainty in both supply and demand. This review organizes recent contributions along four analytical dimensions—supply chain challenges, deterministic programming approaches, stochastic and robust optimization, and metaheuristic solution methods—before identifying the specific gaps that motivate the present work. A synthesizing comparison is provided at the close of the section.

2.1 Agricultural supply chain challenges

The scale of food loss attributable to supply chain inefficiency is both well-documented and persistently underestimated. The Food and Agriculture Organization estimates that approximately 30% of global food production is lost or wasted across the supply chain, with a disproportionate share occurring in the post-harvest and distribution stages, particularly in regions with limited cold chain infrastructure (Food and Agriculture Organization, 2022). Within the Organization of Islamic Cooperation member states, these losses are compounded by institutional fragility, fragmented logistics networks, and constrained investment in temperature-controlled transport (COMCEC Coordination Office, 2024).

Two structural disruptions have sharpened scholarly attention to agricultural supply chain vulnerability in recent years. First, the COVID-19 pandemic exposed critical weaknesses in labor availability, cross-border transport capacity, and last-mile distribution, with effects that cascaded across multiple supply chain tiers simultaneously (Kim et al., 2023). Second, climate change is increasingly recognized as a systematic amplifier of supply chain risk, operating through both yield-side shocks—extreme weather disrupting harvest timing and volumes—and logistics-side disruptions, including road damage, temperature variability affecting cold chain integrity, and fuel cost volatility. In regions where cold chain investment remains limited, the economic premium of refrigerated transport over ambient transport often deters its adoption precisely where quality-sensitive products need it most (Yan et al., 2025).

These contextual realities frame the optimization problem at stake: designing agricultural supply networks that are not only cost-efficient under average conditions but resilient under the multi-source uncertainty that characterizes actual agricultural logistics.

2.2 Deterministic mathematical programming approaches

The foundational literature on agricultural supply chain network design relied primarily on deterministic mathematical programming. Linear and mixed-integer programming models were applied to facility location, distribution routing, and hub configuration problems, with the implicit assumption that yields, transit times, and market demand could be treated as known constants or plannable parameters.

Early contributions applied linear and mixed-integer programming to facility location and distribution routing, assuming fixed yields and stable transportation costs — conditions that rarely hold in practice. Ruiz-Meza et al. (2021) developed mathematical models for collaborative planning of a two-echelon pineapple supply chain, demonstrating structural improvements under a deterministic framework but without accounting for stochastic supply variability that characterizes real agricultural production. Omrani et al. (2016) advanced the field by applying robust optimization to supply chain network design under data uncertainty; however, their model treated uncertainty through interval parameters rather than probability distributions, and perishability constraints were absent.

The common limitation of this body of work is that optimality under average-case assumptions does not translate to feasibility under realized conditions. When crop yields vary by 20–40% across seasons—as they routinely do in smallholder agricultural systems—deterministic solutions become rapidly infeasible, and their cost estimates are systematically underestimated. Deterministic models also struggle to incorporate quality degradation as an endogenous variable, typically treating it as a fixed penalty rather than as a time-dependent function of transportation decisions.

2.3 Stochastic and robust optimization approaches

The limitations of deterministic models motivated a shift toward stochastic programming, in which uncertainty is characterized through probability distributions and optimization is performed across a set of scenarios or through expected value formulations. Two-stage stochastic programming has become the dominant paradigm for supply chain network design under uncertainty, with the first stage determining network structure (facility locations, transport mode assignments) and the second stage resolving operational decisions after uncertainty is realized.

Sun et al. (2025) proposed a two-stage stochastic program for cruise food provisioning that incorporated quality via storage temperature zones and product substitution as recourse. While methodologically sound, their discrete, zone-based quality representation inadequately captures the continuous, multi-dimensional decay dynamics critical to perishable agricultural supply chains. Furthermore, their reliance on scenario reduction to achieve tractability exemplifies the dimensionality curse limiting scenario-based stochastic programs. These constraints underscore the need for hybrid metaheuristic approaches — such as genetic algorithms integrated with stochastic programming — capable of modeling complex quality deterioration while maintaining computational feasibility for robust network design under multi-dimensional uncertainty. Although developed for cruise ship logistics, this work is methodologically comparable to agricultural stochastic programs in its two-stage structure and quality-as-recourse treatment.

Abbasian et al. (2023) integrated location, inventory, and routing decisions into a bi-objective stochastic model for resilient and sustainable perishable food supply chains, accounting for route disruption risks and traffic-dependent quality deterioration in a dairy distribution context. Their use of a hybrid metaheuristic—combining multi-choice goal programming with a genetics-based utility function—represented an important precedent for hybridized solution approaches; however, the model did not address multi-modal transport mode selection or large-scale network instances. Alvarez et al. (2020) addressed the inventory routing problem with perishable products using exact branch-and-cut combined with a heuristic, achieving high-quality solutions for mid-scale instances but requiring substantial computational resources that limit direct application to networks exceeding 100 nodes. Marto et al. (2026) advanced stochastic intermodal modeling by developing a biobjective formulation for corn and soybean supply chains that explicitly accounts for uncertainty in transportation flows across multiple modes; however, their framework does not address perishability constraints or quality degradation, limiting its direct applicability to time-sensitive agricultural logistics.

A parallel strand of research has explored robust optimization as an alternative to stochastic programming, particularly when probability distributions are unknown or unreliable. Omrani et al. (2016) demonstrated the value of robust formulations in protecting against worst-case deviations, though at the cost of conservative solutions that may underperform under typical operating conditions. Petridis et al. (2015) developed a model for supply chain network design driven by stochastic fluctuations, but their treatment of quality degradation remained simplified and sustainability metrics were not integrated.

Critically, across this body of stochastic programming work, no single formulation simultaneously addresses all three principal uncertainty sources relevant to agricultural logistics—crop yield variability, processing time uncertainty, and stochastic demand—while also incorporating multi-modal transportation, quality degradation dynamics, and sustainability objectives in a computationally tractable framework for large networks. This simultaneous integration is the primary methodological contribution of the present work.

2.4 Metaheuristic optimization methods

The NP-hard combinatorial structure of agricultural supply chain network design problems renders exact methods computationally intractable for realistic large-scale instances. Metaheuristic approaches—including genetic algorithms, simulated annealing, and their hybrid variants—have therefore become the preferred solution methodology for problems exceeding a few dozen nodes.

Genetic algorithms have demonstrated effectiveness in complex optimization across various domains (Tsai and Wang, 2010). For perishable goods, Kuo (2010) employed simulated annealing to minimize fuel consumption in time-dependent vehicle routing, demonstrating the method’s effectiveness on combinatorial logistics problems. The approach achieved good solutions for moderate instances, but its single-solution search mechanism limits diversity in the early stages of exploration. Genetic algorithms address this limitation through population-based parallel search across the solution space, and their effectiveness in complex mixed-integer optimization has been well-established across logistics domains (Tsai and Wang, 2010; Reed et al., 2000).

Recent research has focused on enhancing metaheuristic performance through hybridization and adaptive parameter tuning (Vargas-Santiago et al., 2025). Vargas-Santiago et al. (2025) applied NSGA-III to resilient multi-objective facility location under failure scenarios, demonstrating that hybrid evolutionary methods can simultaneously optimize cost, resilience, and service quality across large networks. Abbasian et al. (2023) similarly showed that combining genetic search with goal programming improves convergence quality for bi-objective perishable supply chain problems. A consistent finding across these studies is that adaptive parameter control—dynamically adjusting crossover rates, mutation probabilities, and selection pressure based on search progress—substantially reduces sensitivity to initial parameter settings and improves robustness across heterogeneous problem instances.

Two gaps persist in the metaheuristic literature as applied to agricultural supply chains. First, most existing implementations use generic operators that do not exploit the domain-specific structure of agricultural networks, such as geographic clustering of farms, product-perishability matching with transport modes, or temporal constraints on processing windows. Second, the incorporation of stochastic parameters into the fitness evaluation of genetic algorithms remains rare. The present study addresses both gaps through domain-specific operators and a certainty equivalent formulation that is directly embedded in the hybrid genetic algorithm’s evaluation logic.

2.5 Emerging technologies and future directions

The frontier of agricultural supply chain optimization is rapidly shifting toward data-driven and digitally integrated approaches. Zou et al. (2025) developed a digital twin framework for perishable food supply chains that continuously updates quality loss predictions using IoT sensor data, enabling dynamic re-optimization as real-time conditions deviate from planned parameters. Their results demonstrate that dynamic quality control through sensor integration can substantially reduce end-of-chain waste, though the framework assumes continuous data availability that remains aspirational in infrastructure-constrained settings. Mili (2024) demonstrated how deep learning can enhance classification and routing efficiency in container logistics. Samuels (2025) conducted a systematic review of AI integration in supply chain management across Industry 4.0 to 6.0 contexts, finding that multi-agent systems, reinforcement learning, and digital twin technologies represent the most active emerging research frontiers but that their integration with established stochastic optimization frameworks remains limited. Future trends point toward integrating multi-agent systems and machine intelligence (Lara and Wassick, 2023).

These emerging directions reinforce rather than replace the analytical foundations of stochastic programming and metaheuristic optimization. IoT integration improves parameter estimation; digital twins enable dynamic scenario updating; but the underlying optimization architecture—network design, mode assignment, quality-constrained routing—remains the core decision-making engine.

2.6 Research gap, synthesis, and contribution

Table 1 summarizes the key methodological characteristics of representative prior studies, providing an analytical basis for identifying the research gaps this paper addresses.

Despite advances, five persistent gaps remain in existing literature. First, no prior work simultaneously models crop yield variability, stochastic processing times, and stochastic demand as coupled uncertainty sources within a single optimization framework. Second, the integration of multi-modal transportation mode selection with quality degradation constraints (Equation 3) remains underdeveloped; most models either ignore mode choice or treat quality as a fixed penalty. Third, sustainability metrics —carbon emissions, food waste quantification—are rarely embedded as explicit model components rather than post-hoc calculations. Fourth, scalable solution methodologies for large networks (>200 nodes) with simultaneous stochasticity and perishability are absent from literature. Fifth, the specific challenges of infrastructure-constrained regions, where cold chain investment must be strategically rationed rather than universally applied, have not been addressed through optimization modeling.

Agricultural stakeholders are provided with practical tools, a comprehensive stochastic programming model with a certainty equivalent transformation, solved by a domain-specific hybrid genetic algorithm incorporating adaptive parameter control, parallel computing, and problem-specific operators tailored to agricultural supply chain structure (Table 2).

3 The agricultural supply chain network optimization problem

3.1 Problem definition and notation

The ASCNOP is formulated as a network flow problem with distinctive constraints addressing agricultural product characteristics under uncertainty. The directed graph G = (V, A) represents the supply network, including farm locations, processing facilities, and distribution centers. Each farm has a stochastic production yield d_i following a probability distribution with known parameters.

Each arc (i, j) ∈ A is characterized by:

• Transportation cost for mode k

• Travel time with stochastic components

• Quality degradation rate

The network incorporates K different transportation modes, each with capacity Q_k, maximum delivery time T_k, quality threshold E_k, and environmental impact factors.

Following standard vehicle routing convention (Alvarez et al., 2020), node 0 denotes a virtual depot representing the common origin and return point for all vehicle routes. Farm nodes are indexed i = 1, …, |F|; processing facility nodes i = |F| + 1, …, |F| + |P|; and distribution center nodes i = |F| + |P| + 1, …, N, where N = |F| + |P| + |D| = 320 in the case study. Summations indexed from i = 0 include the depot; summations from i = 1 exclude it. The symbol P in Equation 11 denotes the subset of processing facility nodes P ⊂ V, not a single node; the constraint requires each processing facility to be both reachable from and returnable to the depot.

3.1.1 Stochastic parameters and distribution assumptions

3.1.1.1 Production yield

Farm-level crop yields d_i are modeled as normally distributed random variables: , where and are farm-specific mean and standard deviation parameters. While individual farm yields can exhibit left-skewness due to extreme weather events as noted by Hennessy (2009), normality becomes an increasingly appropriate approximation at the network level through spatial aggregation. Finger (2012) demonstrated that yield variability decreases substantially with aggregation, and by the Central Limit Theorem, the aggregate flow through the network across |F| = 200 farms converges toward normality regardless of individual farm-level distribution shape. Since the capacity constraints in the model operate on aggregate transported quantities rather than on individual farm yields in isolation, the normal approximation is appropriate for the present formulation. The coefficient of variation (CV = σ_i/μ_i) is set proportional to farm size, with larger farms exhibiting lower relative variability — a pattern consistent with empirical evidence from regional agricultural datasets (Finger, 2012). The sensitivity of results to this assumption is examined in Section 6.6 through a lognormal alternative scenario.

3.1.1.2 Processing time

Processing time at facility i under transportation mode k is modeled as a linear function of input yield: , where is a mode-specific processing rate coefficient and captures operational variability. This specification follows the linear processing time assumption standard in vehicle routing problems with service times (Alvarez et al., 2020) and reflects the practical reality that processing duration scales with volume handled.

3.1.1.3 Travel time

Arc travel times t_ij are modeled as normally distributed: , where mean values are derived from arc distances and road quality factors, and standard deviations reflect seasonal road condition variability. This specification is consistent with travel time modeling in stochastic vehicle routing (Abbasian et al., 2023). The safety factor β in the certainty equivalent formulation (Section 4.3) provides an additional buffer against travel time overruns beyond one standard deviation.

3.1.2 Decision variables

The primary decision variable is binary, where = 1 if transportation mode k connects node i to node j.

Auxiliary variables include:

• : cumulative time when processing begins at node j

• : flow quantity transported from node i to j using mode k

• : product quality level at node j (0–100 scale)

3.1.3 Mathematical model

Objective function:

The first term aggregates transportation costs across all arcs, modes, and nodes. The second term penalizes route overtime: > 0 only when a vehicle of mode k exceeds its maximum route time T_k, and C_t (USD/h) is the unit overtime penalty. Both terms are expressed in dollars, making direct summation dimensionally valid. When all routes are completed within their time limits, = 0 for all k and the objective reduces to pure transportation cost minimization.

Subject to:

Capacity constraints:

Quality degradation constraints:

Equation 4 defines the quality level at destination node j as the quality at origin node i reduced by degradation accumulated during transit. The degradation term is the product of the arc-specific degradation rate (quality units per hour) and the travel time (hours) and is active only when mode k is assigned to arc (i,j) (= 1). When = 0, the constraint is inactive for that arc-mode combination. The minimum quality constraint ≥ (Equation 5) then ensures that no delivery falls below market-acceptable thresholds.

Time window constraints (Equation 6):

Network flow constraints (Equations 7–9):

Processing flow constraints (Equation 10):

Timing constraints (Equation 11):

Lower bound timing is enforced by Equation 12:Upper bound timing is defined by Equation 13:

: Allowed total route time for vehicle k.

The Big-M constant requires a computable upper bound to avoid numerical instability in the LP relaxation (Achterberg and Wunderling, 2013). Since represents the cumulative time at node j and no feasible route can exceed the total allowed route time , the tightest valid bound is Equation 14:

When = 0 (mode k does not use arc (i,j)), the timing constraint (Equation 11) becomes:

Since ≥ 0 and ≤ for all feasible solutions, the right-hand side is always satisfiable, confirming that renders the constraint non-binding when the arc is unselected. For the case study, the three mode-specific values are = 8 h, = 12 h, and = 16 h, corresponding directly to the maximum route times specified in Section 6.2.

Sustainability metrics (Equation 15):

Emission factor per unit distance for vehicle k.

Binary constraints:

.

3.2 Model characteristics and challenges

The proposed model presents several computational challenges:

• Stochastic parameters require specialized solution approaches

• Binary decision variables and nonlinear constraints create a mixed-integer nonlinear stochastic program

• Large number of potential connections leads to high dimensionality

• Interdependence between routing decisions, quality degradation, and time windows create complex trade-offs

• Sustainability metrics add additional complexity

To address these challenges, a certainty equivalent formulation is developed in the next section, followed by a hybrid genetic algorithm solution approach.

4 The certainty equivalent formulation

To address computational challenges in the stochastic ASCNOP, a certainty equivalent formulation that transforms stochastic constraints into deterministic ones while preserving essential problem characteristics.

4.1 Transformation of capacity constraints

For capacity constraints with yield as a random variable, a chance-constrained approach (Equation 17) was used with 95% reliability:

This linearizes to Equation 18:

where equals approximately 1.645 for α = 0.95.

This linearization assumes stochastic independence across farm yields, that is, Cov() = 0 for i ≠ l. While this assumption simplifies computation and is standard in two-stage stochastic programming (Birge and Louveaux, 2011), it may underestimate capacity violation probabilities during correlated shock events such as regional drought or frost. The sensitivity analysis in Section 6.6 partially addresses this by testing performance under increased yield variance (CV = 0.2 to 0.4), which implicitly stresses the model under conditions consistent with correlated supply disruptions.

4.2 Recourse approach for processing time constraints

For processing time constraints, a recourse-based formulation (Equation 19) was adopted allowing time constraint violations at a penalty cost:

with recourse cost defined by Equation 20

4.3 Expected value approach for travel time constraints

For stochastic travel times, an expected value approach (Equation 21) was used:

where β is a safety factor (typically 1.0–1.5) accounting for travel time variability.

4.4 Expected value approach for timing constraints

For timing constraints, stochastic processing time was substituted with expected values (Equation 22):

: Service time at node j for vehicle k.

: Expected travel time from node i to node j.

4.5 Quality degradation transformation

For quality degradation, a robust quality degradation approach (Equation 23) was adopted:

where γ is a robustness parameter (typically 1.0–2.0).

4.6 Complete certainty equivalent model

The complete model minimizes:

Subject to all transformed constraints from sections 4.1–4.5.

4.7 Computational considerations

The certainty equivalent formulation transforms the original stochastic program into a deterministic mixed-integer nonlinear program. While more tractable, it remains computationally challenging due to:

• Nonlinear terms in the chance constraint

• Binary decision variables creating combinatorial explosion

• Large number of constraints for realistic instances

• Complex interactions between constraints

The next section presents a specialized hybrid genetic algorithm that exploits the ASCNOP structure while efficiently exploring the solution space, incorporating problem-specific operators, adaptive parameter tuning, and parallel computing techniques.

5 Solution methodology: a specialized hybrid genetic algorithm

5.1 Algorithm design rationale

To determine optimal connections in the agricultural supply chain, a specialized hybrid genetic algorithm was developed tailored to the ASCNOP’s unique characteristics. Genetic algorithms effectively handle mixed-integer problems with complex nonlinear constraints, making them ideal for the combinatorial nature of network design problems. Their population-based approach enables parallel exploration of diverse solution regions, while allowing for problem-specific operators that exploit the distinctive structure of agricultural supply chains.

5.2 Solution representation

The approach employs a dual-list chromosome structure that efficiently captures both network connectivity and transportation mode assignments. The first component is a permutation representing priority order for establishing connections between nodes, inherently respecting the constraint that each node must be connected to the network. The second component partitions these connections into segments, each assigned to a specific transportation mode. This representation naturally enforces key constraints while significantly reducing the search space compared to direct binary encoding.

When decoding solutions, the node connection priority list sequentially was processed, establishing connections to high-priority nodes that maintain feasibility. If no feasible connection exists, a connection to the node that minimizes constraint violation was realized. Transportation modes are then assigned according to the partitioning in the second list, with a feasibility restoration mechanism ensuring every chromosome translates into a valid solution, potentially with penalty costs reflected in fitness evaluation.

5.3 Population management and evaluation

The initial population combines random generation with heuristic construction to ensure diversity while providing promising starting points. About 70% of the population is generated through controlled randomization, while 30% incorporates problem-specific heuristics such as nearest-neighbor connections for geographically clustered nodes and quality-aware assignments that prioritize faster transportation for highly perishable products.

Our fitness function evaluates solution quality by combining the objective function with weighted penalty terms for constraint violations. These penalty coefficients increase with generations to gradually enforce constraint satisfaction. For selection, a tournament selection with elitism to balance selection pressure and population diversity was employed. The top 10% of solutions automatically survive to the next generation, while diversity preservation mechanisms adjust selection pressure if population diversity falls below thresholds.

5.4 Genetic operators and hybridization

The genetic operators are tailored to the ASCNOP structure. For crossover, Ordered Crossover is implemented for the node priority list, Single-Point Crossover for transportation mode assignments, and a Hybrid Edge Recombination that preserves beneficial connections. Mutation operators include Swap Mutation (exchanging positions of two nodes), Slide Mutation (shifting segments), Flip Mutation (reversing segment order), and Mode Perturbation (adjusting transportation assignments). These operators apply with adaptive probabilities based on their historical effectiveness.

To enhance exploitation capabilities, local search procedures are incorporated applied selectively to promising solutions. These include 2-opt Local Search examining pairs of connections, Mode Optimization fine-tuning transportation assignments, and Critical Path Improvement optimizing time-sensitive paths. The local search applies to the top 20% of solutions with computational budget increasing proportionally with solution quality.

5.5 Adaptive control and implementation

Adaptive parameter control dynamically adjusts key algorithm parameters based on search progress, including crossover and mutation rates, penalty coefficients, local search intensity, and selection pressure. These mechanisms improve algorithm robustness across different problem instances and reduce manual parameter tuning needs.

For population replacement, a steady-state strategy was employed where 90% of the population is replaced by each generation, with decisions considering both fitness and diversity measures. The algorithm terminates after a predefined number of generations or when improvement falls below a threshold for consecutive generations.

5.6 Computational performance

The algorithm was implemented in Python 3.9 using the DEAP library for evolutionary computation components, with NumPy and SciPy for numerical operations. The implementation leverages parallel computing through Python’s multiprocessing library, distributing fitness evaluations across available CPU cores to reduce computational time for large problem instances. Key parameters include a population size of 500, 1,000 generations, tournament size of 3, and 10% elitism rate.

The time complexity of our hybrid genetic algorithm is O(G·P·N²), where G represents generations, P is population size, and N is the number of nodes. For large problem instances (320 nodes), our implementation achieves reasonable computational times through efficient chromosome representation, parallel fitness evaluation, solution caching, early termination of infeasible evaluations, and selective application of local search. A controlled benchmarking experiment comparing the hybrid GA against a standard GA baseline (described above) confirmed consistent superiority across all performance metrics; full results are reported in Table 3 (Section 6.3). For small sub-instances of up to 50 nodes, solution quality was additionally verified against CPLEX 22.1 exact solutions; the hybrid GA achieved solutions within 2.3% of optimality with substantially lower runtime, confirming the practical value of the metaheuristic approach for large-scale instances where exact methods are intractable.

To rigorously evaluate the contribution of hybridization, a controlled comparison was conducted between the full hybrid GA and a standard GA baseline. The baseline used an identical chromosome representation, population size, selection mechanism, and fitness function, but excluded three components specific to the hybrid design: (1) the local search procedures (2-opt, mode optimization, and critical path improvement), (2) adaptive parameter control (crossover and mutation rates were fixed at 0.85 and 0.02, respectively, throughout all generations), and (3) heuristic initialization (the entire population was generated randomly). This design isolates the performance contribution of hybridization from confounding factors such as problem representation or evaluation logic. Both algorithms were executed five times on the 320-node instance using different random seeds, and results are reported as mean ± standard deviation.

6 Simulated case study and results

6.1 Experimental design and dataset construction

To evaluate our stochastic optimization framework, a simulated agricultural supply chain network was constructed comprising 200 farms, 20 processing facilities, and 100 distribution centers distributed across a theoretical landscape spanning approximately 250 × 150 km. This network represents a mixed-crop agricultural system typical of developing regions globally, with farm sizes and production capacities following realistic distributions. The dataset was generated using a structured approach to ensure realistic patterns while maintaining controlled variability, with geographical coordinates following clustered distribution patterns to simulate real-world agricultural regions. Production capacities ranged from 2 to 25 tons, following a right-skewed distribution representing the predominance of smaller farms. Crop yields were modeled as normally distributed random variables with means based on farm size and standard deviations proportional to mean values, incorporating seasonal patterns to reflect agricultural production cycles.

Processing facilities were strategically positioned near farm clusters, with varying capacities, processing times modeled as linear functions of input volume, and specific time windows for operation. Distribution centers were primarily located in population centers with demand profiles based on population density and storage capabilities varying across centers. The transportation network was generated using a modified minimum spanning tree with additional connections, incorporating both distance and road quality factors in travel times, with seasonal variations accounting for weather-dependent road conditions.

The algorithm proceeds in eight steps. Step 1: generate initial population (30% heuristic, 70% random). Step 2: evaluate fitness via the certainty equivalent objective (Equation 24). Step 3: apply tournament selection with 10% elitism. Step 4: apply crossover (ordered + uniform) and mutation (swap + flip). Step 5: apply 2-opt local search and mode optimization to the top 20% of solutions every five generations. Step 6: update adaptive crossover and mutation rates based on population diversity. Step 7: repeat Steps 2–6 until 1,000 generations or 150 consecutive non-improving generations elapse. Step 8: return the best-fitness chromosome and decode to a network solution.

6.2 Transportation modes and parameter specification

The simulation incorporated three distinct transportation modes representing realistic options for agricultural distribution. Mode 1 consisted of small trucks with ambient temperature capabilities, characterized by 8-ton capacity, USD 1.8/km cost, quality degradation rate of 0.15 units/h, maximum transportation time of 8 h, and emissions factor of 0.8 kg CO₂/km. Mode 2 featured medium refrigerated trucks with 15-ton capacity, USD 2.5/km cost, reduced quality degradation rate of 0.08 units/h, extended transportation time limit of 12 h, and emissions factor of 1.2 kg CO₂/km. Mode 3 comprised large refrigerated trucks with 22-ton capacity, USD 3.2/km cost, superior quality preservation at 0.05 units/h degradation, 16-h transportation window, and emissions factor of 1.5 kg CO₂/km. These parameters were calibrated to reflect realistic cost and quality differentials across transportation modes, consistent with Yan et al. (2025), and adjusted for the regional context of our simulation.

Our hybrid genetic algorithm was implemented in Python 3.9 using the DEAP library for evolutionary components, with additional libraries for numerical operations and network analysis. The algorithm was executed on an Intel Core i7-11800H 2.30 GHz processor with 32GB RAM, utilizing parallel fitness evaluation through Python’s multiprocessing to leverage all available cores. The final parameter configuration included a population size of 500, 1,000 generations, adaptive crossover and mutation probabilities, tournament selection size of 3, and 10% elitism.

6.3 Network optimization results

The hybrid genetic algorithm converged to a high-quality solution after approximately 750 generations, with minimal improvement observed subsequently. The optimized solution achieved a total operational cost of USD 4,325,000, with additional time-related costs of USD 78,300. The network configuration distributed connections across transportation modes, with 38% utilizing small ambient trucks, 42% using medium refrigerated trucks, and 20% employing large, refrigerated trucks. This distribution reflects an effective balance between cost considerations and quality preservation requirements, with more expensive refrigerated transportation strategically assigned to routes carrying more perishable products or those with longer transit times.

When compared to a traditional hub-and-spoke distribution approach commonly used in agricultural logistics, our optimized solution demonstrated remarkable improvements across all key performance indicators. The total operational cost decreased by 35%, from USD 6,650,000 to USD 4,325,000, while time-related costs diminished by 62.8%, from USD 210,500 to USD 78,300. The average transportation distance reduced by 30.4%, from 112.6 km to 78.4 km, enabling better quality preservation and shorter delivery times. Most significantly, average product quality at delivery improved from 82 to 92%, representing a 12.2% improvement in preservation of product value.

Figure 2 contrasts the optimized distribution network with the traditional hub-and-spoke approach, illustrating clear regional clustering, direct routing patterns, and strategic transportation mode allocation.

Figure 2

The left panel shows the conventional hub-and-spoke configuration, in which all farm flows are routed through a small number of central processing hubs regardless of geographic proximity, producing long average haul distances (112.6 km) and high time-related costs. The right panel shows the hybrid GA solution, characterized by regional clustering of farms to nearby processing facilities, direct routing patterns that reduce average distance to 78.4 km, and differentiated transportation mode allocation — ambient trucks (38%) on short low-perishability routes, medium refrigerated trucks (42%) as the network backbone, and large refrigerated vehicles (20%) on high-volume long-distance corridors. This structural shift is the primary driver of the 35% cost reduction and 27.2% food waste decrease.

To establish methodological superiority beyond the hub-and-spoke comparison, Table 3 presents a three-way performance comparison that additionally includes a standard GA baseline. The standard GA used an identical problem representation and fitness function but excluded the three hybrid-specific components: local search procedures, adaptive parameter control, and heuristic initialization. Both algorithms were executed five independent times using different random seeds; results are reported as mean ± standard deviation. The hybrid GA consistently outperforms the standard GA across all metrics, with performance gains attributable specifically to the hybridization components rather than to problem encoding or evaluation logic.

6.4 Sustainability and efficiency improvements

The optimization model yielded substantial environmental and efficiency benefits beyond direct economic advantages. Estimated food waste decreased from 11.4 to 8.3%, representing a 27.2% reduction in lost agricultural production. Fuel consumption diminished by 24.8%, from 286,700 liters to 215,600 liters, with corresponding carbon dioxide emissions decreasing from 716 tons to 538 tons, a 24.9% reduction. Average delivery time showed perhaps the most dramatic improvement, decreasing by 44.6% from 18.4 h to 10.2 h, significantly enhancing product freshness and reducing spoilage risk.

The optimized network demonstrated several noteworthy structural properties. The algorithm naturally identified efficient regional clusters, with farms typically connected to nearby processing facilities rather than distant ones, regardless of capacity differences. Transportation modes showed clear specialization patterns, with ambient temperature vehicles primarily used for short distances and less perishable products, medium refrigerated trucks serving as the network backbone for mid-range connections, and large refrigerated vehicles selectively deployed for high-volume, long-distance routes carrying highly perishable products. The algorithm effectively identified and optimized critical paths handling the largest volumes or most perishable products, assigning appropriate transportation modes and prioritizing their scheduling.

6.5 Quality preservation and perishability management

Quality dynamics represent a critical dimension of agricultural supply chains, particularly for perishable products where degradation directly impacts economic value and contributes to food waste. Our model explicitly accounted for these dynamics through time-dependent degradation functions calibrated to different product types and transportation conditions. The simulation results revealed that our optimized solution achieved an average quality preservation rate of 92% across all products, compared to 82% in the traditional hub-and-spoke approach.

This significant improvement stemmed from more direct routing reducing overall transportation time, better matching of product perishability characteristics with appropriate transportation modes, optimized scheduling minimizing waiting times at processing facilities, and strategic temperature-controlled transportation deployment for the most sensitive products. These quality improvements translated directly to reduced food waste, with our solution achieving an estimated 27.2% reduction compared to conventional methods, representing not only economic savings but also significant environmental and social benefits in terms of resource utilization and food security.

Figure 3 shows quality degradation profiles for different product categories under both distribution approaches. The optimized network (solid lines) maintains significantly higher quality levels over time compared to traditional methods (dashed lines), particularly for highly perishable products. The extended marketable shelf-life demonstrates how our approach directly contributes to the 27.2% reduction in food waste through better quality preservation.

Figure 3

Solid lines represent the hybrid GA solution; dashed lines represent the hub-and-spoke baseline. Each curve tracks mean product quality (0–100 scale) from farm origin to final delivery across increasing transit time. Highly perishable products (upper pair) show the largest absolute separation between approaches, reflecting the optimized network’s preferential assignment of refrigerated transport and direct routing to time-sensitive flows. The quality preservation improvement from 82 to 92% at delivery — averaged across all product categories — corresponds directly to the 27.2% reduction in food waste, since waste is modeled as a function of end-of-chain quality shortfall relative to the minimum acceptable threshold q_min = 60 (Equation 16).

6.6 Robustness and sensitivity analysis

The performance gap between the hybrid GA and standard GA warrants specific attention. The local search contribution—applied selectively to the top 20% of solutions per generation—accounted for approximately 15–20% of the improvement in solution quality over the standard GA, consistent with the general finding in the metaheuristic literature that targeted local refinement provides disproportionate gains in constraint-heavy combinatorial problems (Vargas-Santiago et al., 2025). The adaptive parameter control contributed 4.2% additional improvement by preventing premature convergence, particularly in the middle generations (approximately 300–600) where the standard GA showed stagnation. Heuristic initialization reduced convergence time by approximately 73 generations without sacrificing solution diversity, confirming that domain knowledge embedded in the starting population provides compounding benefits throughout the search process.

To assess solution robustness, an extensive sensitivity analyses on key model parameters was conducted. Increasing the coefficient of variation of farm yields from 0.2 to 0.4 resulted in only a 7.3% increase in total costs, demonstrating remarkable resilience to yield uncertainty. A 20% increase in fuel costs led to a 12.4% increase in total operational costs and a slight shift toward larger transport modes to leverage economies of scale. When quality degradation rates were doubled, costs increased by 15.8% due to the need for faster delivery and more refrigerated transportation, which are still substantially below traditional distribution costs. Introducing 30% demand fluctuation at distribution centers increased costs by only 8.6%, indicating effective adaptability of the network design.

These analyses confirmed that our solution methodology provides robust results across a range of parameter variations, making it suitable for real-world implementation where uncertainty is inevitable. Our hybrid genetic algorithm demonstrated efficient computational performance even for large-scale instance with 320 nodes. The parallel implementation achieved near-linear speedup with the number of available CPU cores, significantly reducing computational time for large problem instances. The local search hybridization contributed approximately 15–20% improvement in solution quality compared to the standard genetic algorithm, particularly for complex constraint satisfaction in large networks.

Based on these results, several key considerations were identified for implementing the optimized distribution approach in real-world agricultural settings, including data requirements, seasonal adaptations, collaborative frameworks, technology integration, and strategic infrastructure investments. These insights provide practical guidance for agricultural stakeholders seeking to implement more efficient and sustainable distribution networks based on our optimization approach.

6.6.1 Distribution robustness: lognormal yield scenario

To assess the sensitivity of results to the normality assumption, an additional scenario was conducted in which individual farm yields d_i were redrawn from lognormal distributions with matched means and variances:, where and . The lognormal distribution imposes right-skewness and a zero lower bound on yields, properties that better characterize individual farm-level production under climate variability (Hennessy, 2009) (Table 4).

Results indicate that switching to lognormal yields produces a + 2.0% change in total cost and a + 0.30 pp change in food waste, confirming that the model’s key findings are robust to the choice of yield distribution. This outcome is consistent with the aggregation-based justification for normality: across 200 farm nodes, the distribution of aggregate network flows is insensitive to the shape of individual farm yield distributions, as predicted by the Central Limit Theorem (Birge and Louveaux, 2011).

6.6.2 Resilience interpretation

The sensitivity results reveal a clear hierarchy of risk drivers that carries direct implications for supply chain design. Demand fluctuation (8.6% cost increase under 30% demand variability) and yield uncertainty (7.3% cost increase as CV rises from 0.2 to 0.4) impose the smallest performance penalties, indicating that the network’s regional clustering and multi-modal structure effectively absorbs routine supply–demand imbalances through route reallocation. These findings are consistent with the resilience literature, where geographic diversification and redundant routing capacity are identified as primary buffers against demand-side shocks (Zahiri et al., 2017).

Fuel cost increases (12.4% cost impact under a 20% fuel price shock) represent moderate vulnerability, reflecting the model’s reliance on refrigerated transportation for quality-sensitive routes. This exposure can be partially mitigated through the strategic mode-shifting already embedded in the hybrid GA: under fuel cost pressure, the algorithm naturally reallocates volume toward larger refrigerated vehicles that offer lower per-unit fuel consumption through economies of scale, as observed in the 320-node solution’s mode distribution shift.

The largest performance impact arises from quality degradation rate increases (15.8% cost increase when degradation rates double), identifying cold chain integrity as the system’s primary vulnerability. This finding has a direct practical implication: in infrastructure-constrained regions, investment priority should be assigned to maintaining refrigeration reliability on high-volume, high-perishability routes rather than expanding network coverage. A 15.8% cost increase under doubled degradation rates — while substantial — remains well below the 35% cost advantage over hub-and-spoke distribution, confirming that the optimized network retains its economic superiority even under severe cold chain deterioration. Taken together, these results indicate that the proposed network design is robust to demand uncertainty and moderately sensitive to infrastructure quality, with cold chain reliability as the dominant operational risk factor (Table 5).

The hierarchy is unambiguous: cold chain integrity is the dominant vulnerability, fuel cost exposure is moderate and partially self-correcting through mode reallocation, and the network design is effectively insensitive to routine demand and yield variability.

6.7 Practical implementation: challenges and requirements

Translating the optimization framework into operational practice requires confronting four categories of implementation challenge that the model necessarily abstracts away.

6.7.1 Data availability and quality

The model requires farm-level yield distributions, arc-specific quality degradation rates, and real-time demand profiles at distribution centers. In practice, smallholder agricultural systems — which constitute most farm nodes in the case study configuration — rarely maintain systematic yield records with sufficient historical depth to estimate distribution parameters reliably. A practical deployment pathway would begin with proxy estimation: regional yield statistics from national agricultural databases (e.g., GASTAT in Saudi Arabia; FAO country profiles) can approximate μ_i and σ_i at the cluster level, with farm-level disaggregation using the size-proportional variability assumption embedded in the model. Degradation rate parameters e_ij can be calibrated from product-specific shelf-life tables, which are available from food safety regulatory bodies for the most common perishable categories. As IoT sensor adoption expands across cold chains, real-time parameter updating of the kind demonstrated by Zou et al. (2025) would progressively replace these proxy estimates.

6.7.2 Infrastructure constraints and cold chain investment

The model’s transportation mode parameters assume that all three modes are available on all arcs. In infrastructure-constrained regions, refrigerated vehicle availability is geographically uneven and seasonally variable. A practical implementation would require a pre-processing step that maps available transport modes to each arc based on local infrastructure data, effectively restricting the mode assignment decision space before the GA optimization begins. The sensitivity analysis results (Section 6.6) are directly relevant here: since cold chain deterioration is the dominant cost driver, stakeholders should prioritize refrigeration coverage on the high-volume, high-perishability routes identified by the optimized network topology rather than pursuing uniform cold chain expansion.

6.7.3 Operational decision-making frequency

The current formulation solves a static network design problem — it determines optimal connections and mode assignments for a planning horizon, not a real-time dispatch system. In practice, agricultural supply chains require re-optimization at multiple time scales: seasonal network reconfiguration (monthly), tactical route adjustment (weekly), and operational dispatch (daily). The computational performance of the hybrid GA (convergence at approximately 750 generations on the 320-node instance) supports seasonal and tactical re-optimization within practical planning cycles. Real-time dispatch, however, would require either a simplified reduced-network formulation or integration with the digital twin architecture described by Zou et al. (2025), where the full optimization model runs offline and the real-time system executes pre-computed contingency plans triggered by sensor-detected deviations.

6.7.4 Collaborative network participation

The economic case for the optimized network depends on coordinated participation across farms, processing facilities, and distribution centers. In fragmented smallholder systems, achieving this coordination requires institutional mechanisms — cooperatives, contract farming arrangements, or platform intermediaries — that extend beyond the scope of the optimization model. Small farms in particular benefit most from the network’s economies of scale (Section 7) yet face the highest barriers to participation due to limited connectivity and contractual capacity. Future implementations should treat the network participation problem as a companion design challenge to the routing optimization, consistent with the multi-agent supply chain literature (Lara and Wassick, 2023).

7 Conclusion

This paper developed a stochastic programming framework for perishable agricultural supply chain network design and demonstrated its effectiveness on a 320-node simulation network. Three methodological contributions distinguish this work from prior literature.

First, the formulation simultaneously integrates three coupled uncertainty sources — crop yield variability, processing time uncertainty, and stochastic demand — within a single optimization model. Prior stochastic programming approaches addressed these uncertainties in isolation or in pairs (Abbasian et al., 2023; Alvarez et al., 2020); their joint treatment is necessary to capture the compounding effects of correlated disruptions that characterize real agricultural logistics.

Second, the certainty equivalent transformation preserves the stochastic structure of the original problem while achieving computational tractability for large networks. The chance-constrained capacity formulation with 95% reliability, combined with the recourse-based processing time treatment, produces deterministic constraints that tightly approximate the stochastic original—a property confirmed by the sensitivity analysis, which shows cost increases of 7.3–15.8% under substantial parameter perturbations rather than solution infeasibility.

Third, the hybrid genetic algorithm’s domain-specific operators—geographic clustering initialization, quality-aware mode assignment, and selective local search on the top 20% of solutions—contribute approximately 15–20% improvement in solution quality over a standard GA baseline, as reported in Table 3. This performance gain is attributable specifically to the hybridization components rather than to problem encoding, which was held constant across the comparison.

The validated results establish three empirically supported findings. The optimized network achieves 35% lower operational cost than conventional hub-and-spoke distribution, driven primarily by regional clustering that reduces average transportation distance by 30.4%. Product quality preservation improves from 82 to 92%, directly producing the 27.2% reduction in food waste. Carbon emissions decrease by 24.9%, a co-benefit of shorter routes rather than a result of deliberate emissions minimization — suggesting that carbon reduction objectives could be pursued at lower cost than dedicated green logistics programs in similar network contexts.

The approach has value for regions where cold chain investment must be rationed strategically. Sensitivity analysis identifies cold chain reliability — not demand variability or fuel costs — as the dominant risk driver, with quality degradation rate increases producing the largest cost impacts (15.8% under doubled degradation rates). This finding should guide infrastructure investment priorities in low-income agricultural regions.

Four limitations constrain the generalizability of these findings. The validation relies on simulated rather than empirical network data, and parameter calibration uses published industry ranges rather than site-specific measurements. The normal distribution assumption for yields, while justified at the network aggregate level (Finger, 2012), may underestimate tail risk for individual farm nodes under correlated weather shocks. The static network design formulation does not address real-time dispatch or intra-seasonal reconfiguration. Finally, the model abstracts from the institutional coordination requirements — cooperatives, contracts, platform intermediaries — that would govern actual network participation in fragmented smallholder systems.

Future work should pursue three directions that would most directly extend the validated findings. Dynamic network reconfiguration using rolling-horizon stochastic programming would address seasonal yield variation and demand shifts within a single planning framework. Integration with IoT sensor data through the digital twin architecture of Zou et al. (2025) would enable real-time parameter updating and contingency dispatch. Multi-objective formulation explicitly balancing cost, emissions, and food waste as separate objectives — rather than treating emissions as a post-hoc metric — would align the model more directly with sustainability reporting requirements facing agricultural supply chain operators.

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