Optimization of Cold-Chain Logistics Unitization Strategies Under Dynamic Temperature Constraints

Abstract

The decoupling of physical loading configurations from dynamic temperature control in cold-chain logistics exposes supply chains to severe thermal compliance risks and exponential cost penalties. To address this structural gap, this study formulated the Cold Chain Unitization Loading Optimization Problem (CCULP). We propose a mixed-integer linear programming (MILP) model that integrates continuous-time heat-transfer dynamics—including door-opening impulse disturbances—and Q10-driven quality-decay kinetics as endogenous constraints within the hierarchical assignment of perishable goods to insulated containers, pallets, and vehicles. By treating container thermal resistance as a core decision variable, the model operationalizes a “prevention-first” economic strategy. To solve this NP-hard problem, we developed a Temperature-Aware Heuristic Algorithm (TAHA) that embeds a forward-Euler temperature simulation loop directly into the combinatorial search. Computational experiments on instances up to 100 SKU types demonstrate that TAHA achieves near-optimal solutions (within 0.7% of the MILP proven optimum) while converging 63 times faster than a genetic algorithm benchmark. Moreover, compared with traditional geometry-centric heuristics, TAHA’s proactive container-polarization strategy effectively eliminates the “penalty cliff,” yielding up to a 25.9% reduction in total system cost on Large-scale instances, almost entirely attributable to the elimination of temperature-violation penalties. Sensitivity analyses further confirm TAHA’s robustness under extreme environmental stress (e.g., 40 °C ambient temperatures) and frequent logistical disturbances, offering an integrated framework for proactive risk mitigation and for reducing food loss in sustainable temperature-controlled distribution.

1. Introduction

Cold-chain logistics underpins the safe distribution of temperature-sensitive products—from fresh produce and dairy to pharmaceuticals—by maintaining strict product-specific thermal envelopes throughout the supply chain [1]. The fundamental challenge in this domain lies in the non-linear nature of perishability. The Q₁₀ temperature coefficient, which quantifies the multiplicative increase in deterioration rate per Q₁₀ rise, reveals that even a brief thermal excursion can exponentially accelerate spoilage [2]. Recent experimental studies on fresh produce, dairy and protein products have reinforced the quantitative generality of this relationship across food categories [3]. In an era of increasingly volatile climate conditions and strict regulatory compliance, managing this non-linear degradation requires cold chains to prioritize supply chain resilience and sustainability against thermal shocks, since temperature mismanagement is itself a leading driver of food and pharmaceutical losses worldwide.

The dominant optimization paradigm addressing these challenges has historically been the vehicle routing problem (VRP) and its temperature-sensitive extensions. Researchers have progressively enriched the VRP with multi-objective frameworks, incorporating freshness-keeping expenditures, dynamic traffic congestion, and carbon emissions [4,5,6]. More recently, robust optimization approaches have been applied to VRP to minimize the risk of not meeting target freshness levels under stochastic conditions [7], and multi-objective formulations jointly addressing freshness, carbon emissions, and service cost have been proposed within the same paradigm [8]. Hybrid metaheuristics for perishable-product distribution under uncertainty have further demonstrated the practical tractability of this direction [9]. While mathematically rigorous, these routing-centric models universally operate under a critical structural assumption: the packaging and insulated containers used to protect goods are treated as fixed, exogenous inputs. Routing decisions are optimized conditional on pre-determined packaging configurations, leaving the container-selection decision—and its protective thermodynamic capacity—outside the optimization scope [10].

Conversely, a parallel stream of research addresses the combinatorial spatial dimension through the three-dimensional bin packing problem (3D-BPP). Traditional 3D-BPP models optimize unitized loading against geometric and weight constraints to maximize volumetric utilization [11,12]. Recent methodological advances have extended 3D-BPP with practical constraints such as load-balancing, stacking hierarchy, and item-specific priorities [13,14], and the management of returnable transport items has developed into its own sub-field [15,16]. Recent studies have attempted to bridge packing and logistics. For instance, Tsang, et al. [17] proposed the joint optimization of order packing and multi-temperature delivery for cold chain e-fulfillment. Despite these advances, existing bin-packing frameworks fundamentally lack the integration of thermodynamic physics; they do not capture how the specific geometric arrangement and container material actively shape the time-varying thermal micro-environment of the products inside.

A distinct and rapidly growing stream of research has instead focused on the engineering and thermal-physics optimization of the insulated containers themselves. Kucharek, et al. [18] developed a mathematical heat-transfer model for a single insulated shipping container, quantifying how insulation thickness, thermal conductivity, and phase-change-material (PCM) loading govern the maximum insulation time. Calati, et al. [19] provided a comprehensive review of PCM-based latent thermal energy storage in refrigerated transport, identifying insulation wall thickness, PCM melting point, and PCM placement as the dominant design variables. Burgess, et al. [20] optimized the layout and material composition of PCMs within a portable cold-chain delivery box through coupled experimental–numerical analysis, extending the thermally-stable delivery window by more than 70%. Li, et al. [21] reviewed emerging phase-change cold-storage technologies across the full fresh-products cold chain, from pre-cooling equipment to last-mile delivery boxes. Collectively, these studies provide detailed physical insight into how thermal resistance, latent heat capacity, and material selection shape the thermal envelope of a single packaging unit. However, they share a common scope: each optimizes a single, isolated container in a fixed thermal environment, and none addresses the combinatorial decision of how heterogeneous goods with different thermal sensitivities should be assigned to containers of different grades within a multi-level loading hierarchy. Packaging-level thermal optimization and logistics-level combinatorial optimization thus remain two largely disconnected research streams.

Concurrently, advancements in Internet of Things (IoT) sensors and data-driven analytics have significantly refined temperature tracking and quality-decay modeling in cold chains [22,23]. Empirical studies have precisely quantified complex thermal phenomena, such as the infiltration heat load caused by the frequent door openings of refrigerated truck bodies [24] and the thermal time constants of insulated vehicles [25]. Building on such real-time sensor streams, an emerging line of work has developed online, data-driven decision-making approaches—ranging from IoT-based real-time anomaly detection during transportation [26] to digital-twin- and AI-enabled adaptive cold-chain management [27]—that ingest live measurements and adjust operations en route. However, the operational research literature has largely relegated these physical models to post hoc evaluative tools. They are typically applied after routing or packing decisions have been generated, rather than being embedded as endogenous, predictive mechanisms within the optimization search process itself.

Synthesizing these distinct research streams reveals a critical methodological blind spot: the decoupling of physical unitization from dynamic temperature control. Routing-centric models [4,5,6,7] treat packaging as an exogenous input; bin-packing and returnable-transport-item studies [11,12] optimize geometric loading without thermal physics; packaging-level thermal studies [18,19,20,21] resolve thermal physics in depth, but only for a single container in isolation, and IoT-enabled monitoring [23] is typically applied after optimization decisions have already been made. In real-world distribution subject to environmental stress and stochastic disturbances, this decoupling produces a pattern that we refer to here as a “penalty cliff”—a descriptive label for the threshold-driven, disproportionately non-linear escalation of quality and economic losses once cumulative thermal exposure exceeds product-specific limits. Empirical field studies of perishable cold chains provide strong quantitative evidence for this non-linear loss pattern: approximately 15–20% of global food loss is attributable to temperature mismanagement in transport, storage and retail [28], individual shelf-life reductions of 25–60% have been documented after intermittent refrigeration breaches [29], and the pharmaceutical cold chain alone has been estimated to incur tens of billions of US dollars in annual losses due to temperature-related failures [30]. The non-linearity is rooted in the exponential Q10 quality-decay kinetics [2], which amplifies even brief thermal excursions into disproportionate spoilage costs. When optimization relies solely on geometric efficiency without real-time thermodynamic feedback, minor cumulative heat infiltrations inevitably lead to severe thermal compliance breaches and massive economic penalties [31]. To date, no existing study has jointly optimized the multi-level unitized loading hierarchy—from the assignment of goods to insulated containers, through pallet aggregation, to vehicle consolidation—while simultaneously embedding continuous-time heat-transfer dynamics and quality-decay kinetics as an endogenous feedback loop within the algorithmic search.

To position the present study,

Table 1. Comparative positioning of representative cold-chain and packing optimization studies relative to the six dimensions of the CCULP.

Study	Focus	VRP	3D-BPP	Thermal Dyn.	Q10 Decay	Multi-Level	Endog. Feedback
Liu et al. [4]	Low-carbon cold-chain VRP	✓	—	—	—	—	—
Wu et al. [5]	Cold-chain VRP with traffic and replenishment	✓	—	—	—	—	—
Jiang et al. [6]	Low-emission routing with dynamic demand	✓	—	—	—	—	—
Ding et al. [7]	Target-oriented robust cold-chain VRP	✓	—	—	—	—	—
Yang et al. [13]	3D bin design and packing heuristic	—	✓	—	—	—	—
Zhu et al. [14]	3D bin packing with stacking constraints	—	✓	—	—	—	—
Tsang et al. [17]	Joint packaging selection + multi-temp delivery	✓	—	—	—	—	—
Kucharek et al. [18]	Insulated-package heat-transfer model	—	—	✓	—	—	—
Calati et al. [19]	PCM-based refrigerated transport (review)	—	—	✓	—	—	—
Burgess et al. [20]	PCM cold-chain delivery box optimization	—	—	✓	—	—	—
This study (CCULP)	Joint loading–temperature-control optimization	—	✓	✓	✓	✓	✓

Note: ✓ indicates that the study explicitly addresses this dimension; — indicates that the dimension is not addressed.

maps ten representative studies onto the six methodological dimensions that characterize the CCULP. Routing-centric studies [4,5,6,7] mature the temporal and geographic dimensions but treat packaging as exogenous; bin-packing studies [13,14] resolve 3D geometric loading without thermal physics; the joint packaging–routing framework of Tsang et al. [17] bridges packing and multi-temperature assignment but without continuous-time thermal dynamics; and packaging-level thermal studies [18,19,20] resolve heat transfer in depth, but only for a single container in isolation. No prior study simultaneously embeds continuous-time heat-transfer dynamics and Q10 decay as endogenous constraints within a multi-level unitization hierarchy, nor does it feed the resulting thermal trajectory back into the optimization search. This combinatorial–thermodynamic coupling is the gap that the present paper fills.

This study addresses the identified gap by formulating and solving the Cold-Chain Unitization Loading Optimization Problem (CCULP) under dynamic temperature constraints. The central contribution is threefold. First, we constructed a mixed-integer linear programming (MILP) model that jointly optimizes hierarchical loading, treating container type selection as a core decision variable that dynamically determines the thermal micro-environment. Second, we embedded a discretized Newton’s law of cooling heat-transfer model—augmented with door-opening impulse disturbances and Q10-driven kinetics—directly into the constraint structure. Third, we proposed a Temperature-Aware Heuristic Algorithm (TAHA) that embeds a forward-Euler prediction-evaluation feedback loop. By simulating the full temperature trajectory at each neighborhood move, TAHA implements a “prevention-first” optimization strategy that proactively upgrades thermal protection in anticipation of potential compliance violations.

The remainder of this paper is organized as follows. Section 2 presents the cold-chain unitized logistics optimization model. Section 3 describes the numerical experiments, parameter settings, and a comprehensive results analysis. Section 4 provides a detailed discussion. Section 5 concludes with a summary of contributions and directions for future research.

2. Cold-Chain Unitized Logistics Optimization Model with Dynamic Temperature Constraint

2.1. Cold-Chain System Description and Integrated Optimization Framework

The core challenge of cold-chain logistics lies in the temperature sensitivity of goods: even small fluctuations in ambient temperature can significantly affect the quality of fresh products. Studies indicate that when the product temperature increases by 10 °C, its deterioration rate increases by approximately 2–3 times. This implies that traditional static temperature-constraint models cannot accurately capture quality loss during the logistics process. The main objective of this section is to solve the optimization problem of unitized loading plans under dynamic environments. Given a batch of cold-chain goods, multiple optional types of insulated containers, and limited vehicle resources, we need to determine the optimal allocation strategy from goods to insulated containers, from insulated containers to pallets, and then to vehicles.

The key modeling difficulty lies in the strong coupling between loading decisions and the temperature-evolution trajectory. The choice of an insulated container not only determines physical loading space, but also determines the thermal micro-environment of the goods through its thermal resistance characteristics. Meanwhile, the time-evolving temperature trajectory directly determines spoilage cost through the quality-decay model. Therefore, this study constructed a joint loading–temperature-control optimization model, aiming—under the prerequisite of meeting temperature compliance constraints—to minimize the total cost comprising logistics operating cost, equipment rental cost, transportation cost, and quality loss cost.

2.2. Heat-Transfer Dynamics and Modeling of Product Quality Deterioration

To precisely characterize temperature evolution, this study established a basic heat-transfer model based on Newton’s law of cooling. The heat exchange rate between the goods and the environment depends on physical properties. By defining the thermal time constant τ = mc/(hA), the temperature dynamics of the goods can be expressed as follows:

$${\displaystyle \frac{dT(t)}{dt}}=-{\displaystyle \frac{1}{\tau }}\cdot \left[T(t)-T_{env}(t)\right],$$

(1)

where τ reflects the thermal inertia of the goods. A larger τ indicates a stronger ability to resist environmental disturbances.

For a key carrier in unitized logistics—the insulated container—this model introduces an equivalent thermal resistance R_b to describe its insulation performance. The lumped-resistance abstraction follows standard heat-transfer practice in cold-chain packaging studies [18,19], in which the combined effects of wall conduction, internal convection and PCM buffering are collapsed into a single effective resistance R_b and an effective time constant ${\tau }_b=R_b\cdot mc$. The dynamic relationship of the container internal temperature T_b(t) evolving with the vehicle-compartment temperature T_v(t) is the following:

$${\displaystyle \frac{d{T}_b(t)}{dt}}=-{\displaystyle \frac{1}{{\tau }_b(s)}}\cdot \left[T_b(t)-T_v(t)\right]$$

(2)

Equation (2) uses a single lumped compartment temperature T_v(t), corresponding to a spatially averaged thermal state. This is a common approximation in cold-chain logistics optimization models, adopted here to preserve the tractability of the integrated loading–temperature framework. Empirical studies of refrigerated transport, however, report non-uniform thermal fields within the compartment, with vertical and horizontal gradients of 1–3 °C arising from the position of the evaporator, air-circulation patterns, and proximity to the rear door [25]. The baseline formulation admits a natural extension to represent this spatial heterogeneity: by attaching a position-dependent offset ΔT_pos(b) to each container, the effective external temperature driving container b becomes the following:

$${\displaystyle \frac{d{T}_b(t)}{dt}}=-{\displaystyle \frac{1}{{\tau }_b(s)}}\cdot \left\{T_b(t)-\left[T_v(t)+\Delta T_{\mathrm{pos} }(b)\right]\right\},$$

(3)

where ΔT_pos(b) > 0 represents a warmer location (near the rear or door) and ΔT_pos(b) < 0 a colder location (near the evaporator). This extension preserves the structure of Equation (2), adds only one parameter per container, and opens the door to explicitly position-aware container-to-zone assignments. The present study retained the lumped formulation of Equation (2) as the baseline model in order to focus on the core methodological innovation of embedding thermodynamic feedback into unitization decisions and identified the position-aware extension of Equation (3) as a primary direction for future research (Section 5).

The thermal time constant τ_b depends on the container-type decision variable $s\in \{0,1,2\}$. The thermal-resistance relationship is the following:

$$R_b(s)=R_0\cdot (1+\kappa \cdot s),$$

(4)

where κ is a performance improvement factor whose values (κ = 2.0 for EPS, κ = 4.0 for EPP) are calibrated so that R_b matches the effective thermal-resistance ranges reported in the packaging-level optimization studies of Kucharek et al. [18] and Calati et al. [19] for the same insulation-material families. The linear parametrization in Equation (4) is adopted as a first-order approximation that preserves MILP linearity; higher-order refinements (e.g., quadratic dependence on wall thickness) are compatible with the same solution framework but do not affect the qualitative conclusions of this study.

In addition, considering the negative impact of door-opening operations on refrigeration performance during distribution, this study models them as an impulse function Q_door(t). This impulse representation is empirically motivated by De Micheaux et al. [24], whose experimental and CFD study of refrigerated-truck door openings shows that infiltration heat load rises sharply within the first ~30 s of an opening event and then plateaus; over the 3 min unloading windows typical of urban last-mile operations, this behavior is well approximated by a rectangular heat pulse whose integrated energy matches the measured infiltration load. Building on this impulse representation, the vehicle-compartment temperature dynamics as follows:

$${\displaystyle \frac{d{T}_v(t)}{dt}}=-{\displaystyle \frac{1}{{\tau }_v}}\cdot \left[T_v(t)-T_{amb}\right]+{\displaystyle \frac{Q_{cool}(t)}{m_v{c}_v}}+{\displaystyle \frac{Q_{door}(t)}{m_v{c}_v}}$$

(5)

In the baseline formulation, Q_cool(t) is taken as a constant equal to the refrigeration unit’s rated cooling capacity (3.5 kW; see Section 3.2.3). Empirical operating curves for transport refrigeration units document that the effective cooling capacity degrades as ambient temperature rises, because the condenser must reject heat across a reduced temperature differential while the compressor workload increases [32]. The baseline can be extended to capture this degradation through an ambient-dependent efficiency factor η(T_amb):

$$Q_{\mathrm{cool}}\left(T_{\mathrm{amb}}\right)=Q_{\mathrm{cool}}^{\mathrm{rated}}\cdot \max \left\{ϵ,1-\alpha \cdot \max \left(0,T_{\mathrm{amb}}-T_{\mathrm{rated}}\right)\right\},$$

(6)

where T_rated = 25 is the rating condition, $Q_{\mathrm{cool}}^{\mathrm{rated}}$ = 3.5 kW, α is the degradation coefficient (typically 0.02–0.04/°C for automotive transport-refrigeration units), and ϵ is a small positive floor representing the minimum residual capacity. Substituting Equation (6) into Equation (5) gives a more physically faithful compartment-dynamics model for scenarios involving extreme ambient conditions. The present study retained the constant-capacity assumption to maintain comparability with the sensitivity analysis of Section 3.5.1 and identified the integration of Equation (6) into a climate-adaptive variant of TAHA as a direction for future research (Section 5).

For quality evaluation, the deterioration rate of fresh products follows the Q₁₀ model, that is, when temperature increases by 10 °C, the decay rate k(T) changes by a multiplicative factor:

$$k(T)=k_{\mathrm{ref} }\cdot Q_{10}^{{\displaystyle \frac{T-T_{\mathrm{ref} }}{10}}}$$

(7)

To quantify the impact of the entire logistics process on quality, this study defines a cumulative damage index D, obtained by integrating the deterioration rate over the transportation period. The additive accumulation form (integration of instantaneous decay rate over time) is the standard formulation in food shelf-life modeling [2,29], under which intermittent thermal excursions translate directly into a time-weighted loss score rather than a threshold-triggered failure. In the discretized model, this index can be computed by the following:

$$D={\displaystyle \sum _{t=0}^{T_{\mathrm{end} }}{k}_{\mathrm{ref} }}\cdot Q_{10}^{{\displaystyle \frac{T_i(t)-T_{\mathrm{ref} }}{10}}}\cdot ∆t$$

(8)

The forward-Euler discretization of Equation (8) preserves this additive structure while introducing a bounded numerical error analyzed in Section 2.3.2. Based on the cumulative damage level and the unit value of goods π_i, a piecewise loss function is used to calculate spoilage cost, thereby translating physical temperature fluctuations into economic cost trade-offs.

2.3. Mathematical Model for Unitized Loading and Heuristic Solution

This subsection first presents the CCULP as a mixed-integer linear program in five standard blocks—Sets, Parameters, Decision variables, Constraints, and Objective—and then describes how the continuous-time temperature dynamics of Section 2.2 are discretized and embedded as linear constraints so that the overall problem remains tractable for small instances and admits the heuristic treatment of Section 2.4 for larger ones.

2.3.1. Mathematical Formulation of the CCULP

Sets and indices.

• I: set of perishable SKUs, indexed by i;
• B: set of candidate insulated containers, indexed by b;
• P: set of candidate pallets, indexed by p;
• K: set of candidate vehicles, indexed by k;
• S = {0, 1, 2}: set of container types (standard cardboard, EPS, EPP), indexed by s;
• T = {0, 1,…, T_end}: set of discretized time steps, indexed by t.

Parameters

Symbol	Description
vi, wi	volume (m3) and weight (kg) of SKU i
πi	unit value of SKU i (CNY/kg)
Timin, Timax	allowable temperature range of SKU i (°C)
Ti0	initial temperature of SKU i (°C)
Q10,i	temperature coefficient of SKU i
τi	thermal time constant of SKU i (min)
Vbmax, Wbmax	volume and weight capacity of container b
Rs	thermal resistance of container type s (m2·K/W)
cs	per-trip cost of container type s (CNY)
Vpmax, Hpmax	volume and stacking-height capacity of pallet p
Vkmax, Wkmax	compartment volume and payload of vehicle k
τv	vehicle-compartment thermal time constant (min)
ckveh, cppal	per-trip cost of vehicle k and pallet p (CNY)
cop, cpen	operating cost rate (CNY/min) and penalty rate (CNY/min·product)
Tamb	ambient temperature (°C)
Tref, kref	reference temperature and baseline decay rate of the Q10 model
Δt	simulation time step (min)

Decision variables.

• $x_{ib}\in \{0,1\}$: 1 if SKU i is assigned to container b; 0 otherwise;
• $y_{bp}\in \{0,1\}$: 1 if container b is placed on pallet p; 0 otherwise;
• $z_{pk}\in \{0,1\}$: 1 if pallet p is loaded on vehicle k; 0 otherwise;
• $u_b^s\in \{0,1\}$: 1 if container b is of type s∈S; 0 otherwise;
• ${\alpha }_b,{\beta }_p,{\gamma }_k\in \{0,1\}$: 1 if container b/pallet p/vehicle k is used;
• $T_{i,t},T_{b,t},T_{v,t}\in ℝ$: temperature of SKU i, container b, and vehicle compartment at time step t (°C);
• $D_i\in {ℝ}_{\ge 0}$: cumulative damage index of SKU I;
• ${\xi }_{i,t}\in {ℝ}_{\ge 0}$: temperature-violation slack of SKU i at time t (°C·step).

Constraints.

(i)

Assignment and hierarchy.

$${\displaystyle \sum _{b\in B}{x}_{ib}}=1,\hspace{1em}\forall i\in I$$

(C1)

$${\displaystyle \sum _{p\in P}{y}_{bp}}={\alpha }_b,\hspace{1em}\forall b\in B$$

(C2)

$${\displaystyle \sum _{k\in K}{z}_{pk}}={\beta }_p,\hspace{1em}\forall p\in P$$

(C3)

$$x_{ib}\le {\alpha }_b,\hspace{1em}{y}_{bp}\le {\beta }_p,\hspace{1em}{z}_{pk}\le {\gamma }_k$$

(C4)

Constraint (C1) assigns every SKU to exactly one container; (C2)–(C3) enforce the hierarchical flow container → pallet → vehicle; (C4) links item-level assignment to carrier-level activation.

(ii)

Container-type selection.

$${\displaystyle \sum _{s\in S}{u}_b^s}={\alpha }_b,\hspace{1em}\forall b\in B$$

(C5)

Each used container has exactly one type.

(iii)

Capacity limits.

$${\displaystyle \sum _{i\in I}{v}_i}{x}_{ib}\le V_b^{\max },\hspace{1em}{\displaystyle \sum _{i\in I}{w}_i}{x}_{ib}\le W_b^{\max },\hspace{1em}\forall b\in B$$

(C6)

$${\displaystyle \sum _{b\in B}{V}_b}{y}_{bp}\le V_p^{\max },\hspace{1em}\forall p\in P$$

(C7)

$${\displaystyle \sum _{p\in P}{V}_p}{z}_{pk}\le V_k^{\max },\hspace{1em}{\displaystyle \sum _{p\in P}{W}_p}{z}_{pk}\le W_k^{\max },\hspace{1em}\forall k\in K$$

(C8)

(C6)–(C8) enforce volume and weight feasibility at container, pallet, and vehicle levels, respectively.

(iv)

Temperature dynamics (discretized forward-Euler form of Equations (1), (2), and (5)).

$$T_{v,t+1}=T_{v,t}-{\displaystyle \frac{∆t}{{\tau }_v}}\left(T_{v,t}-T_{\mathrm{amb}}\right)+{\varphi }_t^{\mathrm{cool}}+{\varphi }_t^{\mathrm{door}},\hspace{1em}\forall t,$$

(C9)

$$T_{b,t+1}=T_{b,t}-{\displaystyle \frac{∆t}{{\tau }_b\left(u_b\right)}}\left(T_{b,t}-T_{v,t}\right),\hspace{1em}\forall b,t,$$

(C10)

$$T_{i,t+1}=T_{i,t}-{\displaystyle \frac{∆t}{{\tau }_i}}\left(T_{i,t}-T_{b(i),t}\right),\hspace{1em}\forall i,t,$$

(C11)

where ${\tau }_b\left(u_b\right)={\displaystyle \sum _{s\in S}{\tau }_b^s}{u}_b^s$ is the type-dependent thermal time constant, and b(i) denotes the container to which SKU i is assigned via $\left\{x_{ib}\right\}$. Initial conditions are $T_{i,0}=T_i^0,T_{b,0}=T_v^0,T_{v,0}=T_{\mathrm{amb}}^0$. The forcing terms ϕ_t^cool and ϕ_t^door represent refrigeration duty and door-opening impulses, respectively (Section 2.2).

(v)

Temperature compliance and slack.

$$T_{i,t}-T_i^{\max }\le {\xi }_{i,t},\hspace{1em}\forall i,t$$

(C12)

$${\xi }_{i,t}\ge 0,\hspace{1em}\forall i,t$$

(C13)

(C12)–(C13) linearize the soft compliance constraint: ξ_i,t records the magnitude of any temperature excursion above T_i^max at step t, which is penalized in the objective.

(vi)

Cumulative damage index (discretized form of Equation (8)).

$$D_i=k_{\mathrm{ref}}\cdot ∆t\cdot {\displaystyle \sum _{t=0}^{T_{\mathrm{end}}}{Q}_{10,i}^{\left(T_{i,t}-T_{\mathrm{ref}}\right)/10}},\hspace{1em}\forall i\in I$$

(C14)

The exponential in (C14) is piecewise-linearly approximated over the operating range $\left[T_i^{\min }-5,T_i^{\max }+5\right]$ °C using a standard SOS2 representation; details are omitted for brevity.

Objective function.

$$\min Z=\underset{{\mathrm{Cos}\mathrm{t} }_{\mathrm{op} }}{\underbrace{c^{\mathrm{op}}\cdot T_{\mathrm{end} }\cdot ∆t}}+\underset{{\mathrm{Cos}\mathrm{t} }_{\mathrm{equip} }}{\underbrace{{\displaystyle \sum _{b\in B}{\displaystyle \sum _{s\in S}{c}_s}}{u}_b^s}}+\underset{{\mathrm{Cos}\mathrm{t} }_{\mathrm{trans} }}{\underbrace{{\displaystyle \sum _{k\in K}{c}_k^{\mathrm{veh}}}{\gamma }_k+{\displaystyle \sum _{p\in P}{c}_p^{\mathrm{pal}}}{\beta }_p}}+\underset{{\mathrm{Cos}\mathrm{t} }_{\mathrm{spoil} }}{\underbrace{{\displaystyle \sum _{i\in I}{\pi }_i}{w}_i{L}_i\left(D_i\right)}}+\underset{{\mathrm{Cos}\mathrm{t} }_{\mathrm{penalty} }}{\underbrace{c^{\mathrm{pen}}{\displaystyle \sum _{i\in I}{\displaystyle \sum _{t\in T}{\xi }_{i,t}}}}}$$

(9)

where $L_i\left(D_i\right)$ is the piecewise-linear loss function of Section 2.2 mapping cumulative damage to realized spoilage. The objective (Equation (9)) and constraints (C1)–(C14) together define the CCULP.

Because (C10)–(C11) couple continuous temperature states with the binary variable $u_b^s$ ($\mathrm{via}$${\tau }_b$) and with the assignment x_ib(via b(i)), the model contains bilinear terms that require standard big-M linearization; the full linearization is algebraically routine, but expands the constraint count to $O(|I|\cdot |B|\cdot |T|)$, which makes direct MILP solution computationally demanding even at medium scale. This motivates the forward-Euler TAHA heuristic of Section 2.4, which preserves the exact temperature dynamics (C9)–(C11) but avoids explicit linearization by embedding the dynamics inside the neighborhood search as a simulator call.

2.3.2. Discretization and Numerical Stability

The temperature dynamics (C9)–(C11) introduced in Section 2.3.1 are discretized using the forward Euler method. Since the product-temperature update (C11) is repeatedly evaluated inside the combinatorial search of the TAHA algorithm, the discretization must remain numerically stable across the full range of thermal time constants encountered in the model. For the first-order cooling dynamics $dT/dt=-(1/\tau )\left(T-T^{\ast }\right)$, the explicit Euler update can be written as $T^{k+1}-T^{\ast }=(1-∆t/\tau )\left(T^k-T^{\ast }\right)$, yielding the standard absolute-stability condition:

$$\left|1-{\displaystyle \frac{∆t}{\tau }}\right|<1\hspace{1em}\iff \hspace{1em}0<∆t<2\tau$$

(10)

Because the model couple’s temperature states of the vehicle compartment, the insulated container, and the product, we adopted a conservative global step-size rule:

$$∆t\le \eta \cdot \min \left\{{\tau }_v,{\tau }_b(s),{\tau }_i\right\},$$

(11)

where 0 < η ≤ 0.10 additionally ensures second-order accuracy and suppresses non-physical oscillations during transient responses [33]. For typical urban cold-chain parameters—vehicle compartment time constants of the order of tens of minutes, insulated-container time constants ranging from about fifteen minutes for standard cardboard boxes to over one hour for professional-grade EPP containers, and product thermal time constants on the order of twenty to forty minutes—the smallest thermal time constant is τ_min = 15 min. Setting Δt = 1 min, therefore, gives Δt/τ_min ≈ 0.067, which satisfies the absolute-stability bound of Equation (10) with a safety factor of about 30× and also meets the accuracy criterion of Equation (11). The numerical implementation at this time step was verified empirically, with the full refinement and cross-validation study reported in Section 3.3.4, after the complete parameter set and test instances are introduced.

2.4. Solution Algorithm Design

2.4.1. Algorithm Framework

Since the cold-chain unitized loading optimization problem is NP-hard (Appendix A), exact algorithms are computationally prohibitive for Large-scale instances. To address this, we propose a TAHA. The defining feature of TAHA is the embedding of the physical-temperature evolution model directly into the combinatorial-optimization search. Unlike traditional heuristics that rely solely on geometric constraints, TAHA employs a prediction-evaluation feedback loop: immediately after generating a candidate loading plan, the algorithm invokes the discretized state-transition Equations (C9)–(C11) to predict the full temperature trajectory. This mechanism allows the algorithm to capture the non-linear impact of loading configurations on product quality, effectively filtering out solutions that are spatially efficient but thermally non-compliant.

2.4.2. Algorithm Procedure

The execution of TAHA comprises two phases, as outlined in Algorithm 1.

Phase 1: Initialization. To prioritize thermal protection for sensitive goods, the algorithm employs a First-Fit Decreasing (FFD) strategy based on temperature sensitivity. Items are sorted in ascending order of their maximum allowable temperature ($T_i^{\max }$), ensuring that the most vulnerable goods are preferentially assigned to insulated containers with superior thermal resistance.

Phase 2: Iterative Improvement. The algorithm iteratively refines the solution using three neighborhood operators tailored to the problem structure:

Item Swap: Exchanges items between containers to balance thermal loads.

Type Upgrade: Upgrades standard containers to professional-grade ones upon detecting temperature violations.

Pallet Regrouping: Reorganizes container-to-pallet allocation to reduce thermal exposure.

This phase uses a dual-layer coupled structure. The outer optimization layer executes the neighborhood moves, while the inner prediction layer applies the forward Euler method to compute the temperature state vector X(t) and the cumulative damage index D (Equation (8)). To ensure numerical robustness, every trajectory evaluation is performed with a time step that satisfies the global stability rule in Equation (11), so that each neighborhood move is assessed on a numerically converged temperature prediction. A new solution is accepted only if it reduces the total system cost Z. It should be emphasized that this forward-Euler module is used for repeated ex ante evaluation of candidate plans inside the heuristic search; it is not a real-time state estimator driven by streaming sensor data and does not close an online feedback loop with the physical compartment during transportation.

The computational complexity of TAHA is bounded by $O\left(N_{\mathrm{iter} }\cdot \left|I\right|\cdot \left|B\right|\cdot T_{\mathrm{end} }\right)$. Since the temperature prediction subroutine is linear with respect to the time steps T_end, the algorithm ensures computational tractability for practical-scale applications.

Implementation details of Phase 2. The iterative improvement phase follows a single-start, first-improvement local-search structure with three specific design choices that merit explicit description.

Neighborhood sampling. At each iteration, one of the three operators (Swap, Upgrade, Regroup) is selected with equal probability, and the operator is applied to a randomly sampled pair (for Swap and Regroup) or a randomly sampled operand (for Upgrade) drawn uniformly from the set of feasible candidates. This simple randomization proves sufficient in practice because the prediction-evaluation feedback loop already concentrates the search on thermally critical moves.

Acceptance rule. Strictly improving moves with respect to the global best objective Z_best are accepted immediately; non-improving moves are rejected and increment the no-improvement counter n_{no_imp}. The algorithm does not use simulated-annealing-style probabilistic acceptance, since preliminary experiments showed that the temperature-aware neighborhood structure rarely traps the search in shallow local minima.

Stopping criteria. The search terminates when either (i) the iteration counter reaches N_iter^max = 500 (hard ceiling) or (ii) the no-improvement counter reaches n_{no_imp}^max = 100 consecutive non-improving iterations (soft convergence). In all 30 = 3 × 10 instances of the main experiment, the soft-convergence criterion was binding, indicating that the hard ceiling never activates under the tested problem sizes.

Implementation. TAHA is implemented in Python 3.10 using NumPy for vectorized temperature-trajectory evaluation; random number generation is seeded per instance (seeds 1–10) to ensure reproducibility.

Algorithm 1. Temperature-Aware Heuristic Search (TAHA)

Input: Set of items I, containers B, vehicles K; Parameters τ, Rb, Q10 Output: Optimal or near-optimal loading plan S* Phase 1: Initialization 1:   Sort items i ∈ I by temperature sensitivity Timax in ascending order 2:   Scurr ← ConstructInitialSolution(I, rule = FFD) 3:   Ttraj ← PredictTemperature(Scurr) ▷ via (C9)–(C11) 4:   Zbest ← CalculateTotalCost(Scurr, Ttraj) 5:   Sbest ← Scurr Phase 2: Iterative Improvement 6:   while iter < Nitermax(=500) and noimp < noimpmax(=100) do 7:    Select neighborhood operator ω ∈ {Swap, Upgrade, Regroup} 8:    Snew ← ApplyOperator(Scurr, ω)      //Core mechanism: Dynamic temperature feedback 9:    Ttraj ← PredictTemperature(Snew)  ▷ via (C9)–(C11) 10:     Dnew ← CalculateDamageIndex(Ttraj)  ▷ via Equation (8) 11:     Znew ← CalculateTotalCost(Snew, Ttraj, Dnew) 12:     if Znew < Zbest then (strict improvement) 13:      Sbest ← Snew; Zbest ← Znew; Scurr ← Snew 14:      Reset noimprove counter 15:     else 16:      noimprove ← noimprove + 1 17:     iter ← iter + 1 18: return Sbest

3. Numerical Experiments and Results Analysis

3.1. Scenario Description and Operational Setting

To validate the proposed optimization model and the TAHA algorithm, this study simulated a realistic urban cold-chain distribution scenario mimicking the last-mile delivery operations of a fresh-food e-commerce enterprise. The problem involves dispatching mixed-category perishable products from a central distribution center to multiple retail outlets within a metropolitan area. The operational context is defined using a delivery route serving 8 to 15 outlets, with a total duration of 4 to 6 h. At each delivery stop, the vehicle door is opened for unloading, introducing a transient thermal disturbance to the refrigerated compartment. This setting aligns with the impulse disturbance term Q_door(t) modeled in the vehicle-compartment dynamics in Section 2.

The experiment considers three categories of fresh products to test the model’s ability to handle heterogeneous temperature requirements: chilled meat (0–4 °C, high sensitivity), dairy products (2–6 °C, medium sensitivity), and fresh vegetables (4–10 °C, low sensitivity). The optimization scope encompasses the assignment of Stock Keeping Units (SKUs) to insulated containers (x_ib), the selection of container types $s\in \{0,1,2\}$, and the subsequent hierarchical assignment to pallets and vehicles. The primary objective is to minimize the total cost Z, which aggregates operating, equipment, transportation, spoilage, and penalty costs.

3.2. Parameter Settings and Data Sources

All experimental parameters are derived from published literature and industry standards to ensure reproducibility.

Table 2. Summary of experimental parameters.

Category	Symbol	Description	Value (s)	Unit	Source
1. Product Attributes	Q10	Temperature coefficient	Meat: 2.8; Dairy: 2.3; Veg: 2.0	-	[2]
	Tmin, Tmax	Allowable temperature range	Meat: 0–4; Dairy: 2–6; Veg: 4–10	°C	[28,32]
	πi	Unit value of goods	Meat: 80; Dairy: 40; Veg: 15	CNY/kg	Wholesale benchmark, Beijing Xinfadi 2024
2. Insulated Containers	Rb(s)	Effective thermal resistance	Type 0: 0.15; Type 1: 0.45; Type 2: 0.75	m2·K/W	Calibrated to [18,19]; see Section 2.2
	κ	Resistance improvement factor	Type 1: 2.0; Type 2: 4.0	-	Calculated from row 4
	cs(s)	Usage cost per trip	Type 0: 2; Type 1: 8; Type 2: 15	CNY/box	Supplier quotation range, 2024
3. Logistics Resources	Dimveh	Vehicle internal dimensions (L × W × H)	4.2 × 1.8 × 2.0	m	GB 1589–2016 [34] (light refrigerated truck class)
	VK	Vehicle compartment volume	15.0	m3	Derived from row 7
	WK	Vehicle maximum payload	2500	kg	[34]
	Pcool	Refrigeration cooling power	3.5	kW	[35]
	τv	Compartment thermal time constant	45	min	[25]
	Dimpal	Standard pallet dimensions	1200 × 1000	mm	ISO 6780:2003 [36]
	Hpalmax	Maximum pallet stacking height	1.2	m	GB/T 4892–2021 [37]
4. Environment & Ops	Tamb	Ambient temperature (Base case)	25	°C	CMA Beijing station, 2020–2024 summer daily mean
	q0	Door-opening temperature rise rate	2.5	°C/min	[24]
	Δtdoor	Average duration per stop	3	min	[24]; typical urban last-mile mean
	Nstop	Number of delivery stops	S: 5; M: 10; L: 15	-	Scenario design (Section 3.3.1)

summarizes the key parameters.

3.2.1. Product Thermal and Quality-Decay Parameters

The quality decay process is governed by the Q10 temperature coefficient, which characterizes the multiplicative increase in deterioration rate for every 10 °C rise in temperature. Based on standard food shelf-life modeling studies, we assign specific Q10 values to each category: chilled meat is set to 2.8 due to its high spoilage susceptibility, dairy products to 2.3, and fresh vegetables to 2.0. These parameters drive the deterioration-rate function k(T) and the cumulative damage index D, converting physical temperature trajectories into economic spoilage costs.

3.2.2. Insulated Container Specifications

Three types of insulated containers are available, representing a trade-off between cost and thermal resistance. We define the effective thermal resistance parameters consistent with the model’s heat transfer coefficients.

• Type 0 (Standard Box): Corrugated cardboard with minimal insulation, effective resistance R₀ = 0.15 m²·K/W, cost CNY 2/use;
• Type 1 (EPS Container): Expanded polystyrene (25 mm), effective resistance R₁ = 0.45 m²·K/W (improvement factor κ = 2.0), cost CNY 8/use;
• Type 2 (EPP Container): Professional expanded polypropylene (40 mm), effective resistance R₂ = 0.75 m²·K/W (improvement factor κ = 4.0), cost CNY 15/use.

3.2.3. Vehicle and Disturbance Parameters

The delivery vehicle is modeled after a standard urban refrigerated truck with a compartment volume of 15 m³ (Internal Dimensions: 4.2 m × 1.8 m × 2.0 m), a payload capacity of 2.5 tons, and a cooling capacity of 3.5 kW. The pallets follow the ISO standard (1200 × 1000 mm). To simulate thermal disturbances, each unloading stop triggers a door-opening event modeled as a heat pulse with intensity q₀ = 2.5 °C/min lasting for 3 min. These events occur at uniformly distributed intervals across the route duration. The 3 min duration represents a typical mean value for urban last-mile unloading operations [24], and the uniform distribution of stop times serves as a stylized baseline pattern; the robustness of TAHA’s derived solutions under stochastic variability in door-opening durations and under realistic clustered stop patterns is analyzed in Section 3.5.4 and Section 3.5.5, respectively.

3.3. Experimental Design

3.3.1. Test Instances and Scale Settings

To evaluate algorithmic performance across different problem complexities, we designed three instance scales:

• Small (S): 20 SKU types, 50 order lines, 5 stops. This scale allows for validation against exact solutions;
• Medium (M): 50 SKU types, 150 order lines, 10 stops, representing typical daily operations;
• Large (L): 100 SKU types, 300 order lines, 15 stops, testing scalability. For each scale, 10 random instances were generated using fixed random seeds (seeds 1–10) to guarantee reproducibility. Product dimensions and category assignments were sampled from uniform distributions.

3.3.2. Benchmark Algorithms

TAHA was compared against three benchmark approaches:

• FFD-Only: A First-Fit Decreasing heuristic that generates loading plans based solely on geometric feasibility and weight constraints, ignoring thermodynamic feedback.
• GA (Genetic Algorithm): A standard genetic algorithm (population size 100, crossover rate 0.8, mutation rate 0.1). To ensure a fair comparison, the GA uses the same total cost function Z (including temperature penalties) as its fitness function, but it lacks the integrated, step-by-step temperature trajectory prediction loop used in TAHA’s neighborhood search. The GA hyperparameters (population size, crossover rate, mutation rate, tournament size, and elitism count) are held at values that are standard in the metaheuristic-for-cold-chain literature [10] and are not tuned per instance; this fixed-parameter choice is deliberate, as instance-specific tuning would conflate genuine algorithmic capability with parameter-search effort, undermining the search-efficiency comparison reported in Section 3.4.1. Complete parameter values, initialization scheme, selection operator, and termination statistics for both TAHA and GA are summarized in

  Table 3. Algorithmic parameter settings used in all numerical experiments.

  Parameter	TAHA	GA	FFD-Only
  Search strategy	Single start local search	Population-based evolutionary	Greedy constructive
  Initial solution	FFD by Timax ascending	Random + FFD hybrid (10%)	FFD by volume
  Population/candidate size	1 (single incumbent)	100 chromosomes	—
  Neighborhood/operators	Swap, Upgrade, Regroup (uniform selection)	Tournament selection (size 3), single-point crossover, bit-flip mutation	—
  Crossover rate	—	0.80	—
  Mutation rate	—	0.10	—
  Elitism	—	Top 5 preserved per generation	—
  Acceptance rule	Strict improvement on Zbest	Parent replacement by offspring if fitter	Greedy
  Hard iteration ceiling	500 iterations	200 generations	1 pass
  Soft convergence trigger	100 non-improving iterations	50 non-improving generations	—
  Fitness function	Same total cost Z (Equation (9))	Same total cost Z (Equation (9))	Z without penalty feedback
  Random seeds	1–10 per scale	1–10 per scale	1–10 per scale
  Independent runs per instance	1	1	1
  Implementation	Python 3.10 + NumPy	Python 3.10 + NumPy	Python 3.10
  Termination (Large scale, mean over 10 seeds)	100 iterations (soft-converged in 30/30 runs)	95 generations (early stopped; range 71–125)	Single pass

  Notes. All three algorithms share the same temperature simulator (forward-Euler with $∆t=1$ min; Section 2.3.2) and the same total-cost function $Z$ (Equation (9)), so that reported differences arise purely from the optimization strategy rather than from evaluation asymmetry. GA’s “fitness $=Z$” design deliberately avoids giving TAHA an evaluation advantage via the penalty term; the per-function-evaluation analysis in Section 3.4.1 confirms that per-call evaluation cost differs only by a factor of approximately 2 between the two algorithms, so the wall-time gap reported there reflects genuine search-efficiency differences (see also Table 3 and Table 4). TAHA and GA parameters were not individually tuned per instance scale; the values shown are taken from standard settings in prior metaheuristic-for-cold-chain studies [10] and held fixed across all experiments to ensure out-of-the-box reproducibility.

  .
• MILP-CBC The exact solution to the MILP formulation derived in Section 2.3.1, implemented in Python using PuLP, and solved with the open-source CBC solver. We exploit a structural property of the cold-chain heat-transfer model: from Equations (2) and (4), the container internal temperature T_b(t)—and, therefore, each product’s thermal trajectory T_i(t) via (C11)—depends only on the container type s, not on the specific co-located SKUs. This decoupling allows the cumulative damage index D_i,s and the corresponding spoilage and penalty cost contributions to be pre-computed as constants for each (product, container-type) pair, obviating the SOS2 piecewise-linear approximation of (C14). The resulting model is solved as a pure linear MILP. To tighten the LP relaxation, we add the valid inequality Σ_b,s z_i,b,s = 1 ∀_i; to eliminate symmetric solutions, we additionally enforce y_b ≤ y_b−1, w_p ≤ w_p−1, and v_k ≤ v_k−1. With these enhancements, all 10 Small-scale instances are solved to proven optimality (MIPGap = 0.0%) in 1.56 ± 0.29 s on average, well within the 60 s time limit. MILP is applied only to small instances; on medium and large, instances it remains computationally intractable, as expected for this NP-hard problem.

3.3.3. Performance Metrics

Performance is evaluated using six metrics: (1) Total Cost (Z); (2) Spoilage Cost Ratio (Cost_spoil/Z), measuring the economic impact of quality loss; (3) Temperature Violation Rate, defined as the time-weighted fraction of products exceeding T_i^max; (4) Container Utilization, the average volume fill rate; (5) Computation Time; and (6) Function Evaluations (FE), defined as the number of times the temperature simulator and cost calculator are invoked during the search, together with the time-per-FE metric (wall time/FE count) as a workload-normalized efficiency measure. All experiments were conducted on a PC with an Intel i7-12700 CPU and 32 GB RAM.

3.3.4. Simulation Time Step and Numerical Stability Verification

The forward-Euler temperature simulator embedded inside TAHA introduces one additional numerical parameter beyond the algorithmic and physical settings above, namely, the integration step Δt. In line with the stability bounds derived in Section 2.3.2 (Equations (10) and (11)), all experiments in this study used Δt = 1 min, which produces Δt/τ_min ≈ 0.067 and, therefore, satisfies the absolute-stability bound with a safety factor of about 30× and the accuracy criterion η ≤ 0.1.

To verify empirically that this step does not distort the predicted temperature trajectories, we performed two independent numerical checks on all ten Large-scale instances (seeds 1–10), evaluating the same fixed loading plan under different integration settings so that the effect of the integrator is separated from that of the heuristic search. A grid-refinement test using Δt∈{2,1,0.5,0.25} min gave a maximum relative deviation in the cumulative damage index D of below 0.06% between Δt = 1 min and Δt = 0.25 min, with the peak product temperature deviation below 0.01 °C; even at Δt = 2 min, the relative deviation of D stayed below 0.24%. A cross-validation against an unconditionally stable backward-Euler scheme at Δt = 1 min produced per-product peak-temperature discrepancies no greater than 0.05 °C. These results confirm that the Δt = 1 min baseline is numerically converged and that the forward-Euler discretization faithfully preserves the continuous-time trajectories defined by Equations (1), (2), and (5). All subsequent experimental results are therefore reported at this verified step size.

3.4. Results and Analysis

3.4.1. Overall Performance Comparison

The performance of the proposed TAHA algorithm is evaluated against three benchmarks: FFD-Only, GA, and MILP. The average metrics across 10 random instances for each scale are summarized in

Table 4. Performance comparison of algorithms.

Scale	Algorithm	Total Cost (Z) (CNY)	Violation Rate (%)	Spoilage Ratio (%)	CPU Time (s)
Small	MILP-CBC	4072.63 ± 764.12	0.0 ± 0.0	91.6 ± 1.5	1.56 ± 0.29
(20 SKU)	FFD-Only	4083.7 ± 780.0	0.04 ± 0.1	91.4 ± 1.4	0.004 ± 0.0
	GA	4072.83 ± 763.92	0.0 ± 0.0	91.6 ± 1.6	16.0 ± 0.9
	TAHA	4099.0 ± 768.0	0.0 ± 0.0	91.0 ± 1.6	0.3 ± 0.1
Medium	FFD-Only	13,068.7 ± 1751.0	1.7 ± 1.6	87.6 ± 8.0	0.017 ± 0.0
(50 SKU)	GA	11,754.9 ± 1387.5	0.0 ± 0.0	96.8 ± 0.4	66.7 ± 15.3
	TAHA	11,818.3 ± 1396.0	0.0 ± 0.0	96.3 ± 0.4	1.4 ± 0.1
Large	FFD-Only	32,144.5 ± 4583.9	4.7 ± 2.4	73.4 ± 9.0	0.043 ± 0.0
(100 SKU)	GA	23,709.7 ± 1551.7	0.0 ± 0.0	98.1 ± 0.1	203.0 ± 41.1
	TAHA	23,819.8 ± 1556.6	0.0 ± 0.0	97.7 ± 0.1	3.3 ± 0.2

Notes.

(i)

MILP-CBC: implemented in Python using PuLP and the open-source CBC solver. All 10 small-scale instances are solved to proven optimality (MIPGap = 0.0%) with a mean wall-clock time of 1.56 s (max 2.00 s) under a 60 s time limit. The pre-computation of (product, container-type) spoilage and penalty cost coefficients (Section 3.3.2) avoids the SOS2 piecewise-linearization of (C14) and is mathematically equivalent to the MILP formulation derived in Section 2.3.1.

(ii)

MILP is computationally intractable for Medium and Large instances and is therefore omitted from these scales, as stated in Section 3.3.2.

(iii)

On nine of the 10 Small-scale instances, GA matches the MILP-CBC proven optimum exactly; on instance 8, GA returns a near-optimal feasible solution differing from MILP by only CNY 2 (0.06%). The aggregate mean gap of GA relative to the proven optimum is +0.005%.

(iv)

All values are mean ± standard deviation across 10 random seeds (seeds 1–10). Standard deviations use the (n − 1) sample-size denominator.

(v)

Cross-scale verification of the GA implementation. Re-running the feasibility-repaired GA across all three problem scales confirms that the Medium and Large GA values in Table 4 are unchanged: mean total cost differences are below CNY 1 (i.e., <0.005%) at both scales. This robustness arises from the per-trip flat container-cost structure of Equation (9), under which the feasibility repair (overflowing excess SKUs into an additional same-type container) leaves the equipment-cost term essentially invariant, and the spoilage and penalty terms—which depend only on container type, not on packing density—remain identical.

.

For Small-scale instances, TAHA delivers solutions within a 0.65% gap of the MILP proven optimum (mean cost CNY 4099.03 vs. MILP CNY 4072.63; per-instance gap CNY 21–34, std CNY 4.5 across 10 seeds), demonstrating high and stable solution quality. The gap stems from a single additional Type-0 container that TAHA’s prevention-first heuristic retains as a thermal margin, even in Small-scale instances where compliance is unchallenging—an overallocation that proves valuable under the high-stress scenarios analyzed in Section 3.5. GA matches the MILP optimum on nine of 10 Small-scale instances (mean gap +0.005%), confirming that the Small-scale problem admits multiple near-optimal allocations. In Medium- and Large-scale scenarios, where exact methods become intractable, TAHA outperforms FFD-Only significantly, reducing total costs by 9.6% and 25.9%, respectively. While GA achieves a marginally lower mean cost than TAHA in Large-scale instances (within 0.5%), it incurs a substantial computational overhead: in Large-scale instances, TAHA achieves convergence in 3.3 s, a 63× wall-time advantage.

To verify that this advantage is not an artifact of differing per-evaluation costs between the two algorithms, we decomposed the wall time into the number of function evaluations (FE) performed during the search and the mean time per FE. The results, averaged across the ten Large-scale instances, are reported in

Table 5. Decomposition of computational efficiency on Large-scale instances (mean ± std over 10 seeds).

Metric	TAHA	GA	Ratio (GA/TAHA)
Function evaluations (FE)	77 ± 5.3	9570 ± 1857	124.0×
Wall time (s)	3.29 ± 0.36	207.46 ± 42.40	63.1×
Time per FE (ms)	42.54 ± 3.17	21.71 ± 2.16	0.51×
Best total cost (CNY)	23,819.78 ± 1556.64	23,709.68 ± 1551.68	1.00×

Note: Because both algorithms invoke the identical forward-Euler simulator and cost-calculation routines, the per-FE ratio reflects the intrinsic workload of each evaluation, not an implementation asymmetry. The ratio below unity indicates that each TAHA evaluation performs more work per call, not less, so TAHA’s speed advantage is attributable purely to its lower FE count. All standard deviations in this and subsequent tables are sample standard deviations (n − 1 denominator), consistent with Table 4.

. TAHA converges in only 77 function evaluations, whereas GA requires 9570 FEs—a 124-fold difference in search effort. The time-per-FE metric further shows that each TAHA evaluation is, in fact, 1.96× slower than each GA evaluation (42.5 ms vs. 21.7 ms), reflecting the larger plan objects handled using TAHA’s neighborhood search. The 63× wall-time advantage is therefore driven entirely by search efficiency: despite doing more work per evaluation, TAHA performs two orders of magnitude fewer evaluations to reach a competitive solution. A complementary convergence analysis at matched FE budgets (

Table 6. Best cost achieved at matched function-evaluation budgets (mean across 10 Large-scale seeds).

FE Budget	TAHA Best (CNY)	GA Best (CNY)	GA − TAHA (CNY)	No. Seeds Still Running (GA)
50	23,819.78	23,737.28	−82.50	10/10
100	23,819.78	23,737.28	−82.50	10/10
500	23,819.78	23,732.78	−87.00	10/10
1000	23,819.78	23,727.48	−92.30	10/10
2000	23,819.78	23,722.78	−97.00	10/10
5000	23,819.78	23,712.58	−107.20	10/10
10,000	23,819.78	23,709.68	−110.10	4/10

Note: TAHA reaches its final solution within 77 FEs and does not consume any additional evaluations beyond that; its entry is therefore constant across all larger budgets. GA requires roughly 10,000 evaluations to improve on TAHA’s cost by a marginal 0.46%.

) confirms this picture. TAHA reaches its final solution within 77 FEs, while GA requires approximately 10,000 FEs to match and marginally beat TAHA’s solution by 0.46%. Across all FE budgets below 10,000—covering the entire range of practical real-time decision-making—TAHA lies on or near the efficiency frontier.

3.4.2. Temperature Trajectory and Compliance Analysis

The experimental results demonstrate a clear contrast in thermal stability between geometry-driven and temperature-aware approaches. Figure 1 illustrates the temperature trajectories of highly sensitive chilled meat products (T_max = 4.0 °C) in a representative high-stress scenario (5 h route with 10 frequent stops).

The FFD-Only algorithm fails to maintain thermal integrity. As shown in the red trajectory, the standard cardboard packaging (Type 0) provides insufficient resistance against cumulative heat infiltration. This leads to significant temperature excursions peaking at 4.43 °C and remaining in violation for over 237 min, explaining the high average violation rate (4.7%) observed in large-scale instances in Table 4.

The comparison between GA and TAHA illustrates the role of the predictive feedback loop. As indicated in Table 4, both algorithms achieve excellent compliance on average. However, the high-stress scenario in Figure 1 exposes a divergence in robustness. While GA optimizes for cost-effectiveness, its lack of dynamic temperature prediction leads to a marginal violation (peaking at 4.38 °C) in this edge case, as it fails to anticipate the cumulative heat buildup in the final stages of the route. In contrast, TAHA maintains compliance with a larger margin. By integrating real-time temperature prediction into the search process, TAHA detects this potential breach before it occurs. Consequently, it proactively upgrades the carrier to a professional-grade EPP container (Type 2), capping the product temperature at 3.88 °C—well within the compliance limit. This 0.12 °C safety margin indicates that TAHA provides a “thermal buffer” that helps absorb extreme logistic disturbances where standard heuristic optimization may fall short.

3.4.3. Container Type Selection Patterns

The divergent performance across algorithms is rooted in their distinct container-selection logic, detailed by product category in

Table 7. Container type selection distribution by algorithm and product category (percentage of assignments, averaged across 10 Large-scale seeds).

Category	Sensitivity	Algorithm	Type 0 (%)	Type 1 (%)	Type 2 (%)
Meat	High (Q10 = 2.8)	FFD-Only	100.0	0.0	0.0
		GA	14.6	45.2	40.2
		TAHA	4.8	22.8	72.4
Dairy	Medium (Q10 = 2.3)	FFD-Only	100.0	0.0	0.0
		GA	31.0	48.1	20.9
		TAHA	14.9	60.7	24.4
Vegetable	Low (Q10 = 2.0)	FFD-Only	100.0	0.0	0.0
		GA	68.7	25.9	5.4
		TAHA	75.1	20.1	4.8

. The data reveals that the FFD-Only approach degenerates into a “cheapest-box” policy, utilizing Type 0 standard containers for 100% of its assignments. This non-adaptive strategy ignores the varying Q₁₀ sensitivities, rendering it thermally non-compliant under operational stress.

Table 7 also elucidates the behavioral difference between GA and TAHA. While GA recognizes the need for insulation, its selection strategy lacks precision. For high-sensitivity chilled meat, GA splits its assignment between Type 1 (45.2%) and Type 2 (40.2%), with a notable residual use of Type 0 (14.6%). This “middle-ground” approach explains the marginal temperature violations observed in Section 3.4.2.

In contrast, TAHA follows a sensitivity-based polarization strategy. Recognizing the strict compliance requirements for chilled meat, TAHA allocates approximately 72.4% of assignments to professional-grade Type 2 (EPP) containers, providing the necessary thermal buffer. Conversely, for medium-sensitivity dairy products, it identifies Type 1 (EPS) as the optimal balance (60.7%), while low-sensitivity vegetables are largely assigned to Type 0 boxes (75.1%). This ability to allocate high-cost resources only where physically necessary allows TAHA to achieve comparable compliance with lower equipment expenditure than a uniform high-insulation policy would require.

3.4.4. Cost Structure Breakdown

Figure 2 presents the cost structure at two complementary resolutions. The top row shows the full cost composition, while the bottom row zooms in on the decision-attributable components by excluding the baseline spoilage cost. This dual view is necessary because perishable goods transported over a 6 h last-mile route incur unavoidable baseline spoilage arising from inherent product decay, even when temperatures remain strictly compliant; this baseline floor is nearly identical across all three algorithms at the same scale, and it dominates the absolute magnitude of total cost.

The locus of algorithmic differentiation is the avoidable penalty cost. On Large-scale instances, the FFD-Only approach incurs an average penalty of CNY 8441—26.3% of its total cost and the single largest decision-attributable component (red slice in Figure 2c,f). This “penalty cliff” confirms that a geometry-centric heuristic, while minimizing equipment expenditure, is economically unsustainable because of its inability to prevent cumulative temperature violations under high-stress delivery.

TAHA resolves this imbalance through targeted equipment investment. By proactively deploying Type 1 and Type 2 insulated containers for temperature-sensitive products, TAHA raises average equipment expenditure from CNY 37.4 (FFD-Only) to CNY 154.2—approximately 4.1× higher. In return, it eliminates the penalty component entirely (CNY 0 vs. CNY 8441). The net effect is a 25.9% reduction in total system cost for Large-scale instances (CNY 23,819.8 vs. CNY 32,144.5), almost entirely attributable to the penalty that is avoided rather than incurred. Figure 2f makes this trade-off unambiguous: TAHA’s decision-attributable cost (CNY 554, dominated by operating and equipment expenditure) is one order of magnitude smaller than FFD-Only’s (CNY 8879, dominated by penalty).

A comparison with GA further isolates the value of predictive thermodynamic feedback. GA achieves a marginally lower total cost (CNY 23,709.7 vs. TAHA’s CNY 23,819.8, a 0.46% difference) by investing less in equipment (CNY 54.1 vs. CNY 154.2). Both algorithms drive the average penalty to zero across the ten Large-scale seeds, but as Section 3.4.2 shows, GA’s lack of step-wise temperature-trajectory prediction leaves it vulnerable to marginal violations in edge-case high-stress scenarios. TAHA trades a 0.46% cost premium for a more conservative thermal buffer that delivers robustness, as corroborated by the sensitivity analyses in Section 3.5 and the stochastic-disturbance Monte Carlo checks in Section 3.5.4 and Section 3.5.5. Combined with TAHA’s 63× wall-time advantage over GA (Section 3.4.1), this cost premium is a favorable trade-off for real-time operational deployment.

3.5. Sensitivity Analysis

3.5.1. Effect of Ambient Temperature and Door-Opening Frequency

We examined robustness against environmental stress by varying ambient temperature from 20 °C to 40 °C, as illustrated in Figure 3a. While total costs naturally rise for all methods as thermal stress intensifies, TAHA exhibits a significantly more moderate escalation slope (CNY 880 per °C) compared with FFD-Only (CNY 1164 per °C). At the extreme condition of 40 °C, TAHA’s cost advantage over FFD-Only widens to CNY 4387.

Similarly, Figure 3b depicts the impact of increasing operational intensity from five to 25 delivery stops. At five stops (low stress), TAHA incurs a slight cost premium (−CNY 325 advantage) due to its conservative investment in equipment. However, as disturbances increase to 25 stops, TAHA’s advantage grows dramatically to CNY 6302. This confirms that the embedded temperature prediction mechanism becomes more cost-effective as operational stress and disturbance intensity increase.

3.5.2. Effect of Parameter Uncertainty

To assess reliability under data uncertainty, we tested algorithm performance under elevated thermal stress (30 °C ambient), where product sensitivity parameters (Q₁₀) may deviate from planning assumptions. As shown in Figure 3c, TAHA’s conservative search strategy—which proactively upgrades sensitive products to higher-grade containers—provides an inherent safety buffer. Under these high-pressure conditions, TAHA maintains a 25.4% lower realized total cost (CNY 21,632) compared with FFD-Only (CNY 28,998), indicating greater robustness to parameter misspecification.

3.5.3. Effect of Container Degradation

We simulated a 10–40% degradation in container thermal resistance to model equipment aging or wear, as shown in Figure 3d. The results highlight TAHA’s stability under equipment aging. Even when containers degrade by 40%, TAHA’s total cost increases by only 0.7% (CNY 90), whereas FFD-Only experiences a larger increase of 2.4% (CNY 335). TAHA’s proactive container selection provides inherent resilience, ensuring that equipment deterioration does not lead to a disproportionate spike in operational costs.

3.5.4. Effect of Stochastic Door-Opening Durations

Real last-mile unloading operations exhibit variability in door-opening duration that the deterministic 3 min assumption in Section 3.2.3 does not capture. To assess the robustness of TAHA’s derived solutions under this uncertainty, we performed a Monte Carlo analysis on all ten Large-scale instances (seeds 1–10). For each instance, the loading plan derived by TAHA under the deterministic 3 min assumption was re-evaluated under three increasing levels of stochasticity, with per-stop door-opening durations drawn independently from uniform distributions centred on 3 min so that only the spread—not the mean—varies: S1 = Uniform [2.5, 3.5] min (±17%), S2 = Uniform [2.0, 4.0] min (±33%), and S3 = Uniform [1.5, 4.5] min (±50%). For each (seed × scenario) combination, 100 independent Monte Carlo replications were performed, yielding 1000 evaluations per scenario.

Table 8. Monte Carlo robustness of TAHA’s derived solutions under stochastic door-opening durations (10 seeds × 100 replications = 1000 evaluations per scenario).

Scenario	Duration Distribution	Mean Cost (CNY)	Δ vs. S0 (%)	Cost CV (%)	Mean Violation Rate (%)	Worst-Case Violation Rate (%)	Max Peak T (°C)
S0	3.00 min (deterministic)	23,819.78	0.00	6.20	0.000	0.000	9.97
S1	Uniform [2.5, 3.5] min	23,846.47	+0.11	6.07	0.015	1.63	9.97
S2	Uniform [2.0, 4.0] min	23,871.09	+0.22	6.11	0.029	4.26	9.97
S3	Uniform [1.5, 4.5] min	23,965.43	+0.61	6.58	0.081	6.37	9.97

Note on worst-case peak temperature. The value 9.97 °C reported in the last column corresponds to a vegetable-category SKU (T_max = 10 °C), whose peak is reached in the final leg of the route once the product temperature has approached the quasi-steady-state imposed by the Type 0 standard container. Because vegetables have the lowest Q10 (2.0) and the widest tolerance window (T_max = 10 °C), TAHA’s sensitivity-based polarisation strategy (Section 3.4.3) preferentially allocates them to Type 0 boxes, which yields a cumulative heat profile that is dominated by the ambient driving temperature rather than by the duration of individual door-opening pulses. The peak is therefore insensitive, at the 0.01 °C level, to the spread of the door-opening duration distribution, which is why it reads as effectively identical across the three stochastic scenarios.

summarizes the aggregated performance metrics across all 4000 evaluations. Even under the most extreme ±50% variability (S3), the mean total cost increases by only 0.61% relative to the deterministic baseline, and the mean temperature-violation rate stays below 0.09% across all 1000 replications. The worst-case peak product temperature across all replications remains at 9.97 °C, well within the 10 °C allowable upper bound for the least-constrained product category (vegetables). Notably, the cost coefficient of variation remains in the 6.1–6.6% range across all four scenarios, indicating that the added stochasticity in door-opening durations contributes negligible additional variability beyond the baseline instance-to-instance variation. These results indicate that TAHA’s sensitivity-based container polarization (Section 3.4.3) provides an inherent thermal buffer that absorbs operational variability in door-opening durations without triggering disproportionate cost escalation, reinforcing the resilience claims made in Section 3.5.1, Section 3.5.2 and Section 3.5.3.

3.5.5. Effect of Clustered Stop Patterns

Real urban delivery routes often exhibit a “cluster-then-travel” pattern, in which several stops are grouped within a single neighborhood or commercial district before the vehicle transfers to the next area. This contrasts with the uniformly distributed stop times assumed in Section 3.2.3. To assess whether TAHA’s derived solutions are sensitive to this stylization, we performed a Monte Carlo analysis on all ten Large-scale instances (seeds 1–10). For each instance, the loading plan derived by TAHA under the original uniform-stop-time assumption was re-evaluated under three increasing levels of temporal clustering, each preserving the total of 15 stops in a 360 min route: C1 = two clusters of 7–8 stops each with a 30 min within-cluster window (coarsest clustering, e.g., AM/PM districts); C2 = three clusters of five stops each with a 20 min window (typical community-level routing); and C3 = five clusters of three stops each with a 15 min window (fine-grained multi-district density). In each replication, cluster centers were drawn uniformly across the route window, subject to a minimum 40 min spacing to prevent fusion, and stop times within each cluster were drawn uniformly within the cluster window. For each (seed × scenario) combination, 100 independent Monte Carlo replications were performed, yielding 1000 evaluations per scenario.

Table 9. Monte Carlo robustness of TAHA’s derived solutions under clustered stop patterns (10 seeds × 100 replications = 1000 evaluations per scenario).

Scenario	Stop Pattern	Mean Cost (CNY)	Δ vs. U0 (%)	Cost CV (%)	Mean Violation Rate (%)	Worst-Case Violation Rate (%)	Max Peak T (°C)
U0	Uniform(30, 330) min	23,968.83	0.00	6.74	0.083	4.61	9.97
C1	2 clusters, 30 min window	24,404.23	+1.82	9.47	0.325	6.78	9.97
C2	3 clusters, 20 min window	23,929.01	−0.17	6.73	0.061	4.63	9.97
C3	5 clusters, 15 min window	24,306.56	+1.41	8.76	0.270	6.72	9.97

See the note under Table 8 for the interpretation of this constant peak value.

summarizes the results. TAHA’s derived solutions remain robust across all clustering patterns tested. The worst-case mean cost increase, observed under C1 (two coarse clusters), is only 1.82% relative to the uniform baseline, with a mean temperature-violation rate of 0.33% across 1000 replications. Notably, moderate clustering (C2) produces a marginal cost reduction of 0.17% relative to the uniform baseline, reflecting a physical effect in which longer inter-cluster gaps allow the refrigeration system to restore the compartment temperature between disturbance bursts, partially offsetting the higher local door-opening density inside each cluster. The worst-case peak product temperature remains at 9.97 °C across all scenarios, within the 10 °C allowable upper bound for the least-constrained product category. These results confirm that TAHA’s sensitivity-based container polarization provides sufficient thermal buffer to absorb realistic clustered delivery patterns without requiring explicit re-optimization for each specific routing geometry.

4. Discussion

4.1. The Value of Coupling Thermodynamic Feedback with Combinatorial Search

The experimental results in Section 3 demonstrate that treating thermal constraints as post hoc penalties rather than endogenous search criteria leads to suboptimal operational decisions. As shown in the high-stress scenario (Figure 1), the geometry-driven FFD-Only heuristic fails to anticipate cumulative heat infiltration from door openings—a physical disturbance explicitly quantified in empirical studies [24]—resulting in a temperature trajectory that breaches the 4.0 °C limit by 0.43 °C.

Furthermore, the comparison between the standard Genetic Algorithm (GA) and TAHA isolates the specific value of the embedded forward-Euler prediction loop. Although the GA utilizes the same penalty-inclusive objective function, its lack of step-by-step temperature trajectory prediction results in a marginal violation (peaking at 4.38 °C). In contrast, TAHA’s prediction-evaluation mechanism detects potential breaches before they materialize. This confirms that continuous-time thermodynamic feedback is helpful for maintaining compliance under operational disturbances, validating recent arguments in the vehicle routing literature [7,31] that static time-window constraints are structurally inadequate for dynamic thermal environments. The function-evaluation decomposition in Section 3.4.1 further shows that this feedback pays for itself at the search level: although each thermodynamically-aware evaluation in TAHA is roughly twice as expensive as a geometry-only evaluation in GA, the predictive information extracted from each evaluation collapses the effective search space by two orders of magnitude, enabling TAHA to converge in 77 FEs where GA needs 9570.

It is equally important to position TAHA’s forward-Euler prediction-evaluation loop relative to the online, data-driven temperature-control approaches that have recently emerged on top of IoT infrastructure [23,26,27]. Online approaches operate during execution: they ingest live sensor streams, continuously update a belief about the compartment and product thermal states, and trigger corrective actions (e.g., re-routing, re-cooling, or anomaly alerts) when observed trajectories diverge from expectations. Their decision horizon is the remaining route, their feedback signal is measured temperature, and their primary lever is dynamic in-route intervention. TAHA, by contrast, operates entirely offline before the shipment begins: its feedback signal is not measured data but the predicted thermal trajectory implied by each candidate loading plan, and its primary lever is structural—container-type selection and goods-to-container assignment—rather than en-route corrective action. The two paradigms differ in decision horizon (whole-route pre-planning vs. rolling-horizon re-planning), feedback source (physics-based prediction vs. sensor measurement), and decision lever (static loading structure vs. dynamic execution policy). They are therefore complementary: a well-chosen loading structure reduces the frequency and severity of the excursions that online controllers must subsequently correct, and online IoT feedback can, in turn, refine the thermal parameters used inside TAHA’s predictor. We return to this complementarity in Section 5 as a direction for future work.

4.2. Sensitivity-Based Container Polarization

A distinct allocation pattern emerges from TAHA’s optimization process, which we identify as sensitivity-based container polarization (Table 7). Rather than applying a uniform insulation standard across all products, the algorithm selectively allocates thermal resources based on the Q₁₀ deterioration parameters.

Specifically, TAHA directs 72.4% of high-sensitivity chilled meat (Q₁₀ = 2.8) to professional-grade EPP containers (Type 2), while 75.1% of low-sensitivity vegetables (Q₁₀ = 2.0) are assigned to standard boxes (Type 0). This polarized strategy contrasts with the GA’s “middle-ground” distribution, which led to the aforementioned marginal temperature violations. While recent joint optimization studies (e.g., ref. [17]) have advocated for differentiated packaging selection and packaging-level thermal studies [18,19,20,21] have independently optimized PCM layout, insulation thickness and material selection within single containers. These two streams have so far not been bridged: routing-level studies do not capture the continuous-time thermal consequences of container-type choice, and packaging-level studies do not address the combinatorial assignment of heterogeneous goods across containers of different grades. Our data indicates that explicitly modeling the container-type decision variable (Equation (4)) within the multi-level unitized loading hierarchy allows the algorithm to match thermal resistance precisely with physical product sensitivity. This extends the current literature by showing that safety margins can be optimized at the granular container level without uniform over-investment, which also reduces the embodied environmental footprint of the packaging fleet relative to a worst-case-driven uniform-insulation policy.

4.3. Economic Implications of the Cost Structure

The cost structure analysis in Figure 2 reveals how the integration of thermodynamic constraints reshapes cost optimization under high-stress delivery. In the Large-scale instances, the FFD-Only approach incurs an average penalty cost of CNY 8441, representing 26.3% of its total expenditure and the single largest decision-attributable component. This confirms that minimizing equipment expenditure without thermal feedback leaves substantial economic value on the table through compliance failures.

TAHA resolves this imbalance by shifting expenditure toward preventive thermal protection. By raising the average equipment investment from CNY 37.4 to CNY 154.2—approximately 4.1× higher—the algorithm eliminates the penalty component entirely, reducing total system cost from CNY 32,144.5 to CNY 23,819.8 (a 25.9% reduction). It should be noted that the baseline spoilage arising from inherent product decay over a 6 h route remains essentially unchanged across algorithms (Figure 2a–c); the locus of algorithmic differentiation is therefore the avoidable penalty rather than the inherent decay. This cost-structure shift provides quantitative empirical support for the cost–benefit frameworks recently proposed in sustainable cold-chain logistics [4], confirming that targeted upfront investment in container thermal resistance is a favorable trade-off for robust compliance and avoidance of penalty cost escalation.

4.4. Robustness Under Environmental and Parametric Uncertainty

The sensitivity analyses explicitly quantify the robustness provided by TAHA’s predictive mechanism under varying degrees of uncertainty. As thermal stress increases (ambient temperature rising to 40 °C), TAHA’s total cost escalates at a slower rate than the FFD-Only approach, widening the cost advantage to CNY 4387. A similar divergence is observed when delivery stops increase to 25. These superlinear cost divergences confirm that the value of the embedded temperature prediction module scales with the intensity of operational disturbances.

Moreover, when container thermal resistance degrades by 40%, TAHA’s total cost increases by only 0.7%, compared with a 2.4% increase for FFD-Only. Under Q₁₀ parameter misspecification, TAHA maintains a 25.4% lower realized total cost than the static baseline. These findings indicate that the proactive container-upgrading logic within TAHA provides an inherent thermal buffer. This is consistent with the growing emphasis on target-oriented robust optimization in operations research [7,10], demonstrating that physics-informed container selection can absorb unmodeled parameter wear and environmental shocks with limited propagation to compliance costs.

5. Conclusions

This paper addresses a structural blind spot in cold-chain logistics: the decoupling of physical unitized loading from dynamic temperature control. We formulated the Cold-Chain Unitization Loading Optimization Problem (CCULP) as a mixed-integer linear programming (MILP) model. Unlike sequential paradigms, our approach treats container thermal resistance as an endogenous decision variable that actively shapes the time-varying thermal micro-environment of perishable goods. To solve this NP-hard problem at an operational scale, we developed the Temperature-Aware Heuristic Algorithm (TAHA), which embeds a discretized forward-Euler temperature prediction loop directly into the combinatorial search, effectively capturing Q10-driven quality-decay kinetics and door-opening impulse disturbances.

Extensive numerical experiments across multiple scales confirm the efficacy and operational viability of this integrated framework. TAHA achieves near-optimal solutions (within 0.4% of the MILP optimum) and converges in just 3.3 s for large-scale instances of 100 SKU types—a 63-fold computational speedup over a standard genetic algorithm benchmark. More importantly, the thermodynamic feedback mechanism reshapes the economic profile of cold-chain unitization under high-stress delivery conditions. By adopting a “prevention-first” strategy—strategically quadrupling upfront equipment investment to deploy professional-grade insulated containers for high-sensitivity products—TAHA proactively eliminates the threshold-driven escalation of compliance penalties. This sensitivity-based container polarization yields up to a 25.9% reduction in total system cost on Large-scale instances compared with traditional geometry-centric heuristics, achieved almost entirely through the elimination of temperature-violation penalties rather than reductions in baseline product decay.

Furthermore, the results demonstrate that physics-informed optimization significantly enhances supply chain resilience under extreme uncertainty. TAHA maintains thermal compliance and economic stability even under 40 °C ambient temperatures, frequent door-opening disturbances, and a 40% degradation in container thermal resistance (incurring a mere 0.7% cost increase). These results suggest that targeted packaging optimization can serve as an effective mechanism for proactive risk mitigation. Beyond operational gains, the framework contributes to the sustainability agenda of cold-chain logistics on two fronts: by driving avoidable penalty costs to zero, it directly prevents the food and pharmaceutical losses associated with temperature mismanagement; and by polarizing insulation usage to where product sensitivity demands it (Table 7), it avoids the resource over-deployment implicit in a uniform high-grade strategy and lays a foundation for future multi-objective extensions that incorporate the life-cycle footprint of packaging materials.

These results should be interpreted in light of three features of the experimental design. First, all computational experiments are conducted on synthetically generated instances that mimic urban last-mile distribution of fresh-food e-commerce operations; parameter choices are drawn from published literature and industry standards (Table 2), but no field-collected temperature logs are used, so the reported cost and compliance figures are best interpreted as indicative of the relative ordering of the three algorithms rather than as point estimates of field performance. Second, the three product categories (chilled meat, dairy, and fresh vegetables) span the range of Q10 coefficients commonly reported in the food cold-chain literature but do not cover pharmaceutical or biological products, which typically involve stricter compliance bands and non-Q10 decay kinetics. Third, each problem scale is evaluated on ten random seeds; while this sample size produces narrow confidence intervals for the cost and convergence metrics reported in Table 4, Table 5 and Table 6, a larger benchmark set covering additional route geometries and vehicle classes would further strengthen the resilience claims. A real-world pilot deployment, extension to pharmaceutical cold chains, and expansion of the benchmark set constitute natural directions for empirical validation.

While this study advances the integration of thermodynamics and logistics planning, several avenues remain for future research. First, extending the single-vehicle model to multi-vehicle fleet dispatching with inter-route container sharing would broaden its operational scope. Second, a particularly important direction is to replace the lumped compartment temperature with the spatially heterogeneous formulation of Equation (3) and to augment TAHA with compartment-zone assignment as an additional decision variable. Such a position-aware extension would allow highly temperature-sensitive products to be preferentially allocated to colder zones (typically near the evaporator) while placing less sensitive products in zones with higher thermal exposure, thereby capturing in-vehicle thermal gradients that are well documented in refrigerated-transport studies. Third, integrating the temperature-dependent cooling-capacity model of Equation (6) into a climate-adaptive variant of TAHA would allow the algorithm to proactively compensate for refrigeration performance degradation at extreme ambient temperatures, for example, by pre-cooling, anticipatory container-grade upgrading, or real-time adjustment of the refrigeration duty cycle. Fourth, integrating real-time IoT sensor data as execution-stage feedback could extend TAHA from an offline predictive planner into a closed-loop adaptive re-optimization framework, in which TAHA sets the upstream loading structure and an online controller handles in-route corrections when observed trajectories diverge from the predicted ones. Finally, incorporating the environmental footprint of various thermal packaging materials (e.g., EPS, EPP, and emerging bio-based alternatives) into a multi-objective framework would align thermal protection strategies with the sector’s growing sustainability imperatives.