Order-Driven Multi-Objective Optimization of a Three-Echelon Low-Carbon Dairy Cold-Chain Network Considering Demand Variability and Shelf-Life Reliability

Abstract

Dairy cold-chain network planning requires coordinated decisions under demand variability, product perishability, and environmental constraints. To address these interrelated challenges, this study formulates an order-driven multi-objective mixed-integer nonlinear programming (MINLP) model for the tactical planning of a three-echelon dairy cold-chain network. The model coordinates distribution-center selection, inventory, transportation allocation, vehicle configuration, and refrigeration decisions to minimize total cost, transportation-related carbon emissions, and the quantity- and importance-weighted average freshness-loss rate. Demand variability is represented through service-level-based safe demand, whereas product freshness is evaluated using Weibull-based shelf-life reliability and inventory–transportation exposure. Transportation congestion is further incorporated to capture its effects on travel time, refrigeration emissions, and freshness deterioration. NSGA-II is employed to generate Pareto solutions, with entropy-weighted TOPSIS used for compromise-solution selection and MOEA/D serving as the benchmark algorithm. Numerical results indicate that NSGA-II achieves favorable convergence performance and comparable solution diversity relative to MOEA/D, while small-scale mixed-integer approximation tests support the quality of the obtained solutions. Multi-scale experiments demonstrate stable computational performance as network size increases. Sensitivity and scenario analyses further reveal distinct effects of service levels, shelf-life characteristics, and road capacity on economic, environmental, and freshness performance. The proposed framework provides tactical decision support for coordinating demand-responsive supply, low-carbon operations, and freshness preservation in dairy cold-chain networks.

1. Introduction

Cold-chain logistics is essential for maintaining the quality and safety of temperature-sensitive products, but it also entails relatively high operating costs, energy consumption, and environmental impacts because appropriate temperature conditions must be maintained throughout storage and transportation. These challenges are particularly pronounced for dairy products, whose quality deteriorates continuously over time. Temperature fluctuations, delivery delays, and inefficient inventory turnover may accelerate deterioration and increase product losses [1]. Therefore, dairy cold-chain planning requires not only efficient transportation and facility configuration but also effective coordination of inventory, refrigeration, environmental performance, and product freshness [2]. With growing sustainability requirements, dairy companies face the practical challenge of ensuring efficient product delivery while reducing environmental impacts and maintaining acceptable freshness at the retail level.

Existing research provides an important foundation for addressing these challenges. Studies on perishable-food supply chains have demonstrated that network configuration, transportation, inventory, and product deterioration should be coordinated rather than considered independently [3,4]. Sustainability and carbon-related factors have also received increasing attention in transportation and cold-chain network optimization [5,6,7]. Meanwhile, multi-objective evolutionary algorithms provide effective tools for generating Pareto solutions when multiple conflicting objectives must be addressed simultaneously [8,9]. Research on demand-responsive supply chains further suggests that production and replenishment decisions should respond to downstream demand information, particularly when demand varies over time [10,11,12]. Together, these developments provide the methodological basis for more integrated dairy cold-chain planning.

Despite these advances, closer coordination among several interrelated decisions remains necessary in dairy cold-chain planning. Retail demand varies across operational periods and directly determines supply and distribution requirements. Products may remain at distribution centers before delivery, making inventory turnover a key determinant of remaining freshness. Transportation conditions and refrigeration operations further influence delivery time, environmental performance, and product deterioration. These factors are closely interdependent: changes in demand alter supply and inventory levels, which subsequently affect transportation requirements and product freshness. Treating these decisions separately may therefore fail to adequately capture the interactions among demand response, network operations, environmental performance, and freshness preservation. An integrated planning framework is needed to coordinate these decisions across multiple operational periods.

To address these challenges, this study develops an order-driven multi-objective optimization framework for the tactical planning of a three-echelon dairy cold-chain network. The framework coordinates demand-responsive supply, network operations, environmental performance, and freshness preservation across multiple products and operational periods. Based on this framework, the main contributions are summarized as follows:

(1)

A three-echelon, two-leg tactical planning model is formulated for the internal production–distribution network of a dairy company. The model jointly coordinates distribution-center location, inventory balance, transportation allocation, vehicle configuration, and refrigeration decisions across multiple products and operational periods. The contribution lies in capturing the interactions among these network and operational decisions within an internal dairy cold-chain setting rather than treating them separately.

(2)

An order-driven coordination structure is incorporated to link downstream demand requirements with upstream supply decisions. Demand variability is transformed into service-level-based safe-demand requirements at the retailer level and propagated upstream through distribution allocation and inventory balance to determine planned supply. Accordingly, “order-driven” refers specifically to the downstream-to-upstream transmission of demand information within the network model rather than to a new inventory-balance mechanism or replenishment theory.

(3)

Freshness evaluation incorporates both inventory turnover and transportation exposure within the tactical planning framework. Weibull-based shelf-life reliability is combined with time exposure from first-leg transportation, an inventory turnover-based equivalent residence time at distribution centers, and second-leg transportation. This formulation provides a tractable means of capturing the joint effects of inventory and transportation decisions on retail-end freshness without explicitly tracking batch-level inventory-age distributions.

The remainder of this paper is organized as follows. Section 2 reviews the relevant literature and positions the present study within the existing research. Section 3 describes the problem and formulates the multi-objective mixed-integer nonlinear programming (MINLP) model. Section 4 presents the solution procedure and compromise–solution selection method. Section 5 reports the computational experiments and corresponding analyses. Finally, Section 6 summarizes the main findings, managerial implications, limitations, and directions for future research.

2. Literature Review

This section reviews the literature related to low-carbon dairy cold-chain network optimization. Existing studies are organized into four closely related research streams: perishable-food supply-chain network design, sustainable supply chains and low-carbon logistics, multi-objective optimization methods for logistics, and order-driven inventory coordination. This review highlights key methodological developments and remaining research gaps, thereby positioning the present study within the existing literature and clarifying its methodological extensions.

2.1. Perishable-Food Supply-Chain Network Design

Perishable-food supply-chain network design differs from conventional network planning because facility location, transportation, inventory, and temperature-controlled operations jointly influence product quality and operational performance. Accordingly, increasing attention has been devoted to incorporating freshness and shelf-life considerations into network configuration decisions. In dairy supply chains, Jouzdani and Govindan [13] incorporated refrigeration decisions and shelf-life uncertainty into network design, using a Weibull distribution to characterize product lifetime. Shafiee et al. [14] subsequently addressed economic, environmental, and social objectives in dairy supply-chain planning. More broadly, integrated location–transportation–inventory decisions have been investigated in sustainable perishable supply chains [4], while multi-objective optimization has been applied to balance cost, environmental impacts, and product quality [3].

Recent research has expanded this stream by addressing operational uncertainty, resilience, and transportation-related impacts. Behzadi et al. [15] investigated robust and resilient strategies for managing supply disruptions in agribusiness supply chains, whereas Sinha and Anand [16] examined network optimization for perishable products using a metaheuristic approach. From a sustainability perspective, Yakavenka et al. [17] developed a multi-objective framework for perishable-food supply-chain design that incorporates transportation and environmental considerations. Collectively, these studies highlight the importance of coordinating network configuration, operational decisions, and product perishability. However, closer integration of tactical network decisions with demand-driven product flows remains necessary in dairy cold-chain planning, particularly when demand variability and shelf-life reliability are considered across multiple operational periods.

2.2. Sustainable Supply Chain and Low-Carbon Logistics

Sustainable supply-chain network design has gradually evolved from traditional cost-oriented optimization toward the joint consideration of economic, environmental, and social performance [18]. Carbon emissions are particularly important in logistics planning because transportation activities directly link facility configuration, product flows, and energy consumption. Demir et al. [6] reviewed the major operational factors affecting fuel consumption and emissions in road freight transportation, including vehicle utilization, travel distance, and operating conditions, while Abdullahi et al. [5] incorporated environmental and fuel-consumption considerations into green vehicle-routing decisions. These studies establish transportation planning as a key component of low-carbon supply-chain optimization.

In cold-chain systems, environmental performance is closely intertwined with refrigeration requirements and product-quality preservation. Musavi and Bozorgi-Amiri [3] jointly considered economic cost, environmental impacts, and product quality in sustainable perishable-food supply chains. For dairy networks, Jouzdani and Govindan [13] incorporated refrigeration decisions and shelf-life characteristics into supply-chain optimization, whereas Yakavenka et al. [17] addressed environmental and transportation-related factors in the multi-objective design of sustainable perishable-food supply chains. These studies underscore the need to coordinate transportation, refrigeration, environmental performance, and product perishability rather than treating emission reduction as an isolated objective.

Recent studies have also addressed uncertainty, disruption risks, and low-carbon policy considerations in supply-chain and cold-chain network optimization. Govindan et al. [19] provided a comprehensive review of supply-chain network design under uncertainty and summarized the major modeling approaches for handling uncertain parameters. Ran et al. [7] examined fresh-product cold-chain network optimization under supply disruptions, demand fluctuations, and dual-carbon policy constraints. From a low-carbon perspective, Fang et al. [20] reviewed advances in logistics network optimization under carbon-related constraints, highlighting relevant developments in environmentally oriented logistics planning. These studies reflect the growing integration of uncertainty management and environmental considerations into sustainable logistics network planning.

2.3. Multi-Objective Optimization Algorithms for Logistics

Cold-chain logistics optimization typically involves conflicting economic, environmental, and product-quality objectives, making multi-objective evolutionary algorithms (MOEAs) well suited to generating alternative trade-off solutions. Among these methods, NSGA-II, proposed by Deb et al. [8], has been widely adopted for its non-dominated sorting, elitist selection, and diversity-preservation mechanisms. MOEA/D, introduced by Zhang and Li [9], follows a different paradigm by decomposing a multi-objective problem into a set of scalar subproblems solved collaboratively. Thus, NSGA-II and MOEA/D represent two established approaches based on Pareto dominance and decomposition, respectively, for addressing complex multi-objective logistics problems.

MOEAs have been increasingly used in cold-chain and perishable-food systems involving network design, routing, inventory coordination, and sustainability decisions. Musavi and Bozorgi-Amiri [3] balanced economic, environmental, and product-quality objectives in perishable supply chains through multi-objective optimization, whereas Sinha and Anand [16] demonstrated the applicability of metaheuristic methods to perishable-product supply-chain networks. Thakur et al. [21] employed NSGA-II with hybrid chromosome encoding to capture trade-offs among routing, energy consumption, and product quality in refrigerated fresh-produce distribution. Related studies have also applied multi-objective and stochastic optimization to sustainable perishable-food supply-chain design and green routing [17], highlighting the applicability of evolutionary and metaheuristic methods to complex cold-chain problems.

2.4. Order-Driven Inventory Coordination

Order-driven supply planning emphasizes that production, replenishment, and distribution decisions respond to downstream demand signals rather than relying solely on upstream capacity or predetermined production schedules. Fisher [12] highlighted the importance of aligning supply-chain strategies with demand characteristics, particularly when demand variability calls for greater responsiveness. Under demand uncertainty, advance demand information can further improve replenishment and inventory planning. Gallego and Özer [10] examined the value of such information in replenishment decisions, while Özer [11] extended this perspective to multi-retailer replenishment systems. More recently, Dai et al. [22] investigated multi-product replenishment and order fulfillment in a two-echelon distribution system subject to storage-capacity constraints. Together, these studies provide a foundation for linking downstream demand signals with upstream replenishment and allocation decisions.

Demand-driven coordination is particularly relevant to multi-echelon networks, where inventory decisions interact with facility configuration, transportation, and product flows. Melo et al. [23] reviewed the integration of facility-location decisions with transportation, inventory, and production planning, while de Kok et al. [24] systematically examined stochastic multi-echelon inventory models involving coordinated inventory and replenishment decisions across multiple stocking locations. For perishable products, Zhang et al. [25] investigated multi-echelon inventory optimization for fresh products with explicit consideration of perishability. Collectively, these studies highlight the importance of coordinating inventory, replenishment, and other network-level decisions across multiple echelons.

2.5. Literature Review Summary

The above literature review reveals four closely related research gaps. First, in perishable-food supply-chain network design, previous studies have incorporated facility location, transportation, inventory, and product deterioration, yet the interactions among inventory turnover, storage exposure, transportation processes, and freshness deterioration remain insufficiently captured in multi-period tactical planning for dairy cold chains. Second, in sustainable and low-carbon logistics, growing attention has been devoted to transportation emissions, refrigeration energy use, and product quality. However, transportation congestion, refrigeration operation, carbon emissions, and freshness preservation are rarely coordinated within network-level planning. Third, in multi-objective optimization, evolutionary algorithms offer effective means of addressing conflicting logistics objectives, but their performance remains problem-dependent, particularly for cold-chain models involving discrete network decisions, nonlinear operational relationships, and multiple constraints. More comprehensive evaluation of convergence, solution quality, computational efficiency, and scalability is therefore required. Fourth, in order-driven inventory coordination, prior research has demonstrated the value of downstream demand information for replenishment and inventory planning, whereas its linkage with network configuration and freshness-sensitive product flows remains underexplored in dairy cold chains. Collectively, these gaps call for an integrated framework that connects downstream demand response with upstream network and operational decisions while balancing economic, environmental, and freshness objectives.

To address these gaps, this study develops an order-driven three-echelon optimization framework for tactical planning of a low-carbon dairy cold-chain network. The model links retailer-side demand requirements with upstream supply, inventory, and distribution decisions. Inventory exposure and Weibull-based shelf-life reliability are incorporated to characterize freshness deterioration, while congestion-aware transportation and refrigeration operations capture their effects on transportation-related carbon emissions and product freshness. The resulting formulation simultaneously minimizes total cost, transportation-related carbon emissions, and the quantity- and importance-weighted average freshness-loss rate. NSGA-II is adopted as the primary optimizer, with MOEA/D serving as a representative benchmark algorithm. Solution quality and computational performance are further assessed through small-scale mixed-integer approximation tests, multi-scale experiments, and sensitivity and scenario analyses.

Table 1. Comparisons between the related literature and this study.

Article	Application	Carbon Emission	Freshness/Quality	Demand Variability	Shelf-Life/ Perishability	Order-Driven Coordination	Refrigeration Decision	Traffic Congestion	Solution Method
Musavi and Bozorgi-Amiri, 2017 [3]	Perishable food	√	√		√				NSGA-II/AUGMECON
Biuki et al., 2020 [4]	Perishable products	√		√	√				Hybrid GA/PSO
Ran et al., 2025 [7]	Fresh products cold chain	√		√					OTS-FICSMA
Jouzdani and Govindan, 2021 [13]	Dairy products	△			√		√	√	RMCGP
Shafiee et al., 2021 [14]	Dairy supply chain	√	△	√	√				Heuristic + augmented ε-constraint
Thakur et al., 2024 [21]	Refrigerated fresh products	√	√		√		√		NSGA-II with hybrid chromosome
Dai et al., 2021 [22]	E-commerce distribution inventory			√		△			Piecewise linear approximation
Zhang et al., 2021 [25]	Fresh products inventory		△		√				GA + Flexsim
Current Paper	Dairy cold chain logistics	√	√	√	√	√	√	√	NSGA-II + Entropy- TOPSIS

Note: √ denotes explicit consideration, △ denotes partial or indirect consideration, and a blank cell denotes no explicit consideration. Freshness/quality covers explicit quality-related measures; shelf-life/perishability covers deterioration or lifetime modeling; and order-driven coordination denotes downstream demand or orders driving upstream supply and allocation decisions.

summarizes the main characteristics of representative studies and positions the proposed framework relative to the existing literature.

3. Problem Description and Model Construction

To provide a comprehensive overview of the research methodology and the logical structure of this study, the overall research framework is illustrated in Figure 1. The proposed framework integrates problem identification, network characterization, mathematical modeling, multi-objective optimization, and computational validation into a unified decision-making process. Specifically, an order-driven three-echelon dairy cold-chain network is formulated by simultaneously considering demand variability, shelf-life reliability, transportation-related carbon emissions, and operational constraints. Based on this formulation, a multi-objective MINLP model is developed, and a solution framework combining evolutionary optimization and systematic computational experiments is established to evaluate solution quality, scalability, and managerial implications.

3.1. Problem Description

This study focuses on the internal cold-chain logistics network of a dairy company, consisting of a company-owned production facility, candidate distribution centers, and retailer demand nodes. The production facility supplies dairy products to selected distribution centers, which subsequently distribute products to retailers according to order requirements. Unlike general supply-chain networks involving external procurement or third-party logistics providers, this study considers an enterprise-owned production–distribution system and focuses on internal coordination decisions. The proposed problem is formulated at the tactical operational-period level rather than the daily routing level, where representative operational periods are used to characterize variations in demand and supply conditions. Accordingly, this study focuses on network configuration, order-driven supply planning, inventory–distribution coordination, transportation allocation, and product freshness preservation, while detailed vehicle routing, customer visit sequences, and outsourced logistics operations are beyond the scope of this study.

Existing studies on perishable-food supply chains have extensively investigated network design, logistics cost reduction, and operational-efficiency improvement. However, several important characteristics of dairy cold-chain operations remain insufficiently addressed. First, dairy demand is affected by seasonal variations and random fluctuations, and planning decisions based only on average demand may result in insufficient service capacity or excessive inventory under varying demand conditions. Previous studies on safety demand and service-level-based planning have highlighted the importance of incorporating demand variability into supply-chain decisions to improve service reliability [26,27,28]. Second, dairy products are highly time-sensitive, and freshness deterioration during storage and transportation directly affects product quality and operational performance. Therefore, shelf-life reliability and freshness preservation should be incorporated into cold-chain optimization decisions [1,29,30]. In addition, the trade-offs among economic cost, transportation-related carbon emissions, and product freshness require an integrated multi-objective optimization framework [13,17].

To address these challenges, this study develops an order-driven multi-objective optimization model for a low-carbon dairy cold-chain network. The proposed model considers multiple dairy products with heterogeneous freshness characteristics, multiple operational periods representing varying demand conditions, and multiple vehicle types with different transportation capacities. Demand variability is incorporated through service-level-based safe-demand requirements, while shelf-life reliability is characterized using a Weibull-based freshness degradation function. The model further integrates transportation congestion effects, refrigeration-related carbon emissions, and inventory–distribution coordination within a tactical operational-period framework. Unlike traditional supply-driven planning approaches, the proposed framework determines upstream production and distribution decisions based on retailer-side order requirements, thereby coordinating demand fulfillment, freshness preservation, and low-carbon operations. The model simultaneously minimizes total cost, transportation-related carbon emissions, and the quantity- and importance-weighted average freshness-loss rate while satisfying capacity, service-level, inventory, and operational constraints.

3.2. Model Assumptions

To formulate a tractable optimization model while capturing the essential characteristics of dairy cold-chain operations, the following assumptions are adopted based on practical planning requirements and previous studies:

(1)

A one-year planning horizon is considered and divided into several representative operational periods, including low-, regular-, and peak-demand periods. The model focuses on tactical operational-period decisions, including production supply, inventory allocation, and distribution planning, rather than daily vehicle routing, customer visit sequences, or detailed delivery scheduling.

(2)

Retailer demand is characterized as a stochastic variable and is assumed to approximately follow a normal distribution based on the statistical fitting of historical sales data from the Dairy Supply Chain Sales Dataset [31]. The distributional assumption is validated through goodness-of-fit comparisons, as detailed in Appendix A.2. The proposed model further considers independent and weakly correlated demand scenarios to characterize demand variability. The applicability of this representation under different correlation levels is evaluated in Appendix A.8.

(3)

To incorporate demand variability while maintaining computational tractability, stochastic demand information is transformed into service-level-based safe-demand requirements using distribution quantiles. Demand risk is controlled ex ante through the selected service-level parameter used to determine safe demand. The resulting safe-demand requirements represent the minimum demand commitment within each operational period and are subsequently treated as deterministic inputs in the optimization model, following previous studies on safe-demand and service-level-based supply-chain planning [26,27].

(4)

Unmet demand, backorders, and lost sales are not explicitly considered in the model. Carbon emissions are mainly associated with vehicle fuel consumption and refrigeration energy consumption, which are converted into CO₂ emissions using predetermined emission factors. The validity of the adopted transportation fuel-consumption parameters is examined through the validation analysis in Appendix A.4.

(5)

The storage-time variable is formulated as an inventory-turnover-based equivalent residence time at the tactical planning level, representing the average storage exposure of products within distribution centers. Freshness evaluation is designed for inventory-age distributions with low to moderate heterogeneity, where average residence time is used to approximate aggregate freshness deterioration. The applicability of this approximation under different inventory-age heterogeneity levels is evaluated in Appendix A.7.

(6)

Product freshness deterioration is characterized using a Weibull-based degradation function under different refrigeration conditions. The Weibull scale and shape parameters are treated as exogenous product-specific characteristics and remain unchanged during optimization. This assumption follows previous perishable-food studies that employ Weibull-type functions to describe shelf-life deterioration and quality degradation behaviors [13]. The Weibull formulation provides a deterministic representation of shelf-life reliability based on cumulative exposure during transportation and storage processes.

(7)

Multiple dairy products and vehicle types are considered. Different products may exhibit variations in demand levels, unit weights, and freshness decay characteristics, whereas vehicle types differ in transportation capacity, operating cost, fuel consumption, and refrigeration energy consumption. All vehicles are equipped with refrigeration systems, and refrigeration activation is determined by a binary decision variable. Activating refrigeration enhances freshness preservation but increases energy consumption and transportation-related carbon emissions.

(8)

Transportation decisions are modeled at an aggregated tactical level with different representations for the two transportation stages. For first-stage transportation from production facilities to distribution centers, travel time is approximated using representative average speeds without explicitly modeling detailed routes or traffic congestion. Large-capacity vehicles are predefined, and multiple dairy products (e.g., Products A and B) can be jointly transported under refrigerated conditions. For second-stage transportation from distribution centers to retailers, each transportation connection is represented as an aggregated link rather than a specific physical road segment. Travel time is calculated using the BPR function with representative traffic parameters [13,32,33]. Detailed route generation, vehicle scheduling, and multi-stop delivery processes are not considered.

3.3. Symbols and Variables

To describe the solution to this multi-objective mixed-integer nonlinear programming (MINLP) problem using mathematical methods, this study defines symbols and variables as shown in

Table 2. Symbols and variables.

Category	Symbol	Meaning and Explanation
Set	$\mathcal{I}$	Set of candidate distribution-center locations, $\mathcal{I}$ = {1, 2,…, I}, containing all possible locations for establishing a distribution center
	$\mathcal{R}$	Set of retailer locations, $\mathcal{R}$ = {1, 2,…, R}, containing all retailer locations requiring distribution services
	$\mathcal{K}$	Set of vehicle types, $\mathcal{K}$ = {1, 2, 3}, containing three different vehicle types
	$\mathcal{P}$	Set of product types, $\mathcal{P}$ = {1, 2}, containing two different types of dairy products
	$\mathcal{T}$	Set of operating periods, $\mathcal{T}$ = {1, 2, 3}, representing the low-demand period, regular-demand period, and peak-demand periods
Computational Variables	$t_i^F$	The transit time for the first leg of transport from the manufacturing plant to distribution center $i$ is calculated as $s_i^F$/$\overline{v}$
	$z_{it}^F$	The number of vehicles for the first leg of transport from the manufacturing plant to distribution center $i$ in period $t$ is determined by rounding up the transport weight and vehicle capacity
	$g_{it}^F$	The refrigeration status for the first leg of transport from the manufacturing plant to distribution center $i$ in period $t$ is determined by whether refrigerated product A is being transported
	$t_{irt}$	Congestion-adjusted representative travel time of the aggregate DC $i$-retailer $r$ transportation connection in period $t$
	$f_{irt}$	Effective aggregate traffic flow of the DC r $i$-retailer $r$ ransportation connection in period, including baseline traffic and the equivalent contribution of logistics vehicles t
	$f_{irt}^0$	Representative baseline traffic flow of the aggregate transportation connection between DC r $i$ and retailer $r$ in period $t$
	$t_{ipt}^I$	Inventory-turnover-based equivalent residence time of product $p$ at distribution center $i$ in period $t$
	$S_{ipt}^F$	Freshness level of product $p$ during the first leg of transport from the manufacturing plant to distribution center $i$ in period $t$
	$S_{iprkt}^D$	Freshness level of product $p$ during the second leg of transport from distribution center $i$ to retailer $r$ via vehicle type $k$ in period $t$
	$F_{ipkrt}$	Freshness level of product $p$ throughout the entire transportation process in period $t$ from the manufacturing plant to distribution center $i$ and then to retailer $r$ via vehicle type $k$
Cost Parameters	$f_i$	Fixed facility costs for establishing a distribution center at location $i$
	$b_k$	Fixed operating costs for vehicle type $k$ (e.g., lease payments, depreciation, etc.)
	$c_{kt}$	Variable cost per unit distance for vehicle type $k$ in period $t$
	$h_{pt}^s$	Unit production and supply cost for product $p$ in period $t$
	$h_{ipt}^{\mathrm{I}}$	Unit inventory holding cost for product $p$ at distribution center $i$
	$c_{ip}^{end}$	Unit disposal cost for the remaining inventory of product $p$ at distribution center $i$ at the end of the planning period
Physical Parameters	$s_i^F$	The first leg of the transport distance from the manufacturing plant to distribution center $i$
	$s_{ir}$	The transport distance from distribution center $i$ to retailer $r$
	$\overline{v}$	The average speed of all vehicle types
	$a_k$	The maximum load capacity of vehicle type $k$
	$m_i$	The maximum storage capacity of distribution center $i$
	$Ca{p}_{pt}^S$	The manufacturing plant’s maximum supply capacity for product $p$ in period $t$
	$w_p$	The unit weight of product $p$
	$l_k$	Average fuel-consumption rate per unit distance for vehicle type $k$ used in the baseline model
Carbon Emissions Parameters	$v_1$	Fuel conversion factor: converts fuel consumption into carbon emissions
	$v_2$	Electricity conversion factor: converts electricity consumption into carbon emissions
	$e_k$	Representative average refrigeration power of vehicle type $k$
Required Parameters	${\tilde{D}}_{rpt}$	Retailer $r$’s stochastic demand for product $p$ in period $t$
	${\mu }_{rpt},{\sigma }_{rpt}$	The mean and standard deviation of the stochastic demand ${\tilde{D}}_{rpt}$
	$D_{rpt}^{safe}$	The safe demand at a given service level
	$S{L}_{rpt}$	Retailer $r$ ’s demand service level for product $p$ in period $t$
	$Z_{SL}$	The quantile of the standard normal distribution at service level $S{L}_{rpt}$
Time and Freshness-Related Parameters	${\bar{I}}_{ip}^{end}$	Maximum allowable ending inventory of product $p$ at distribution center $i$
	${\rho }_p$	Shrinkage rate for product $p$ during storage at the distribution center
	${\alpha }_p$	Maximum acceptable freshness loss rate
	$v_p$	Relative freshness-importance weight of the product $p$
	$t_{irt}^0$	Travel time from distribution center $i$ to retailer $r$ under non-congested conditions
	$d_{pt}^R/d_{pt}^O$	Weibull scale parameter for product $p$ during period $t$ under refrigerated/non-refrigerated conditions
	${\beta }_{pt}^R/{\beta }_{pt}^O$	Weibull shape parameter for product $p$ during period $t$ under refrigerated/non-refrigerated conditions
	$∆t$	a planning period length
Transportation Parameters	$k_{ir}$	Effective traffic capacity of the representative transportation corridor between DC $i$ and retailer $r$
	$u_k$	Weight of vehicle type $k$ in road traffic calculations
	$\mathsf{\alpha }$	Road congestion intensity coefficient; $\mathsf{\alpha }$ = 0.15 represents the baseline congestion level
	$\beta$	Exponential parameter of the traffic congestion function (typically set to 4)
Technical Specifications	$M$	A sufficiently large positive number
	$\epsilon$	A very small positive number, to avoid a denominator of 0
Objective Function	$Z_1$	Economic objective: Minimize the total cost objective function value
	$Z_2$	Environmental objective: Minimize the transportation -related carbon emissions objective function value
	$Z_3$	Quality objective: Quantity- and importance-weighted average freshness-loss rate of delivered products
Decision Variables	$y_i$	Site selection decision variable: whether to establish a distribution center at location $i$
	$g_{ikrt}$	Refrigeration equipment usage decision: in period $t$, for a trip from distribution center $i$ to retailer r using vehicle type $k$, whether to turn on the refrigeration equipment $t$
	$z_{ikrt}$	Number of vehicles of type $k$ used to transport goods from distribution center $i$ to retailer $r$ in period $t$
	$x_{ipkrt}$	Quantity of product $p$ delivered from distribution center $i$ to retailer $r$ using vehicle type k in period $t$
	$I_{ipt}$	Inventory level of product $p$ at distribution center $i$ in period t
	$Q_{ipt}$	Estimated supply requirement for product $p$ from the manufacturing plant to distribution center $i$ in period $t$, driven by projected orders

.

3.4. Construction of the Basic Model

Building on the problem description, model assumptions, and notation definitions, this study formulates a multi-objective mixed-integer nonlinear programming (MINLP) model for the order-driven dairy cold-chain network. The proposed formulation integrates production, inventory, distribution, transportation, and freshness-related decisions within a unified optimization framework. The decision variables determine facility selection, product flows, inventory levels, order fulfillment, vehicle allocation, and refrigeration operations. The model includes three objective functions and a set of operational constraints. The objective functions minimize total cost, transportation-related carbon emissions, and quantity- and importance-weighted average freshness-loss rate. The constraints characterize production capacity, demand fulfillment, inventory balance, transportation feasibility, freshness preservation, and operational requirements.

The objective functions are formulated as follows:

$$\begin{gathered} \mathit{m}\mathit{i}\mathit{n}{\mathit{Z}}_1=\sum _{\mathit{i}\in \mathcal{I}} {\mathit{f}}_{\mathit{i}}{\mathit{y}}_{\mathit{i}}+\sum _{\mathit{i}\in \mathcal{I}} \sum _{\mathit{r}\in \mathcal{R}} \sum _{\mathit{k}\in \mathcal{K}} \left({\mathit{b}}_3+{\mathit{c}}_{3\mathit{t}}{\mathit{s}}_{\mathit{i}}^{\mathit{F}}\right){\mathit{z}}_{\mathit{i}\mathit{t}}^{\mathit{F}} \\ +\sum _{\mathit{i}\in \mathcal{I}} \sum _{\mathit{r}\in \mathcal{R}} \sum _{\mathit{k}\in \mathcal{K}} \left({\mathit{b}}_{\mathit{k}}+{\mathit{c}}_{\mathit{k}\mathit{t}}{\mathit{s}}_{\mathit{i}\mathit{r}}\right){\mathit{z}}_{\mathit{i}\mathit{k}\mathit{r}\mathit{t}} \\ +\sum _{\mathit{i}\in \mathcal{I}} \sum _{\mathit{p}\in \mathcal{P}} {\mathit{h}}_{\mathit{p}\mathit{t}}^{\mathit{s}}{\mathit{Q}}_{\mathit{i}\mathit{p}\mathit{t}}+\sum _{\mathit{i}\in \mathcal{I}} \sum _{\mathit{p}\in \mathcal{P}} {\mathit{h}}_{\mathit{i}\mathit{p}\mathit{t}}^{\mathbf{I}}{\mathit{I}}_{\mathit{i}\mathit{p}\mathit{t}}+\sum _{\mathit{i}\in \mathcal{I}} \sum _{\mathit{p}\in \mathcal{P}} {\mathit{c}}_{\mathit{i}\mathit{p}}^{\mathit{e}\mathit{n}\mathit{d}}·{\mathit{I}}_{\mathit{i}\mathit{p}\mathit{T}} \end{gathered}$$

(1)

$$\begin{gathered} \mathit{m}\mathit{i}\mathit{n}{\mathit{Z}}_2={\mathit{v}}_1\times [\sum _{\mathit{i}\in \mathcal{I}} \sum _{\mathit{t}\in \mathcal{T}} {\mathit{s}}_{\mathit{i}}^{\mathit{F}}{\mathit{z}}_{\mathit{i}\mathit{t}}^{\mathit{F}}{\mathit{l}}_3+\sum _{\mathit{i}\in \mathcal{I}} \sum _{\mathit{r}\in \mathcal{R}} \sum _{\mathit{k}\in \mathcal{K}} \sum _{\mathit{t}\in \mathcal{T}} {\mathit{s}}_{\mathit{i}\mathit{r}}{\mathit{z}}_{\mathit{i}\mathit{k}\mathit{r}\mathit{t}}{\mathit{l}}_{\mathit{k}}] \\ +{\mathit{v}}_2\times \sum _{\mathit{i}\in \mathcal{I}} \sum _{\mathit{r}\in \mathcal{R}} \sum _{\mathit{k}\in \mathcal{K}} \sum _{\mathit{t}\in \mathcal{T}} ({\mathit{g}}_{\mathit{i}\mathit{t}}^{\mathit{F}}·{\mathit{t}}_{\mathit{i}}^{\mathit{F}}·{\mathit{z}}_{\mathit{i}\mathit{t}}^{\mathit{F}}·{\mathit{e}}_3+{\mathit{g}}_{\mathit{i}\mathit{k}\mathit{r}\mathit{t}}·{\mathit{t}}_{\mathit{i}\mathit{r}\mathit{t}}·{\mathit{z}}_{\mathit{i}\mathit{k}\mathit{r}\mathit{t}}·{\mathit{e}}_{\mathit{k}}) \end{gathered}$$

(2)

$$\begin{array}{cc}\mathit{m}\mathit{i}\mathit{n}{\mathit{Z}}_3& ={\displaystyle \frac{{\sum }_{\mathit{i}\in \mathcal{I}} {\sum }_{\mathit{r}\in \mathcal{R}} {\sum }_{\mathit{k}\in \mathcal{K}} {\sum }_{\mathit{p}\in \mathcal{P}} {\sum }_{\mathit{t}\in \mathcal{T}} {\mathit{v}}_{\mathit{p}}·{\mathit{x}}_{\mathit{i}\mathit{p}\mathit{k}\mathit{r}\mathit{t}}·(1-{\mathit{F}}_{\mathit{i}\mathit{p}\mathit{k}\mathit{r}\mathit{t}})}{{\sum }_{\mathit{i}\in \mathcal{I}} {\sum }_{\mathit{r}\in \mathcal{R}} {\sum }_{\mathit{k}\in \mathcal{K}} {\sum }_{\mathit{p}\in \mathcal{P}} {\sum }_{\mathit{t}\in \mathcal{T}} {\mathit{v}}_{\mathit{p}}·{\mathit{x}}_{\mathit{i}\mathit{p}\mathit{k}\mathit{r}\mathit{t}}}}\end{array}$$

(3)

The cost minimization objective in Equation (1) accounts for the total operational cost, including distribution-center fixed costs, first- and second-stage transportation costs, production and supply costs, inventory holding costs, and end-of-period inventory disposal costs. Equation (2) minimizes transportation-related carbon emissions associated with vehicle fuel consumption and refrigeration energy consumption. The emission calculation considers transportation activities, fuel consumption parameters, and refrigeration operation characteristics. Equation (3) minimizes the average freshness-loss rate of products delivered to retailers. Specifically, the freshness retention level, $F_{ipkrt}\in \left[0, 1\right]$, represents the remaining freshness after transportation and storage processes, while $1-F_{ipkrt}$ denotes the corresponding freshness-loss rate. The objective value is normalized by the weighted delivered quantity to obtain the average freshness-loss indicator.

Following the objective functions, the operational constraints of the proposed MINLP model are formulated as follows:

$${\tilde{\mathit{D}}}_{\mathit{r}\mathit{p}\mathit{t}}~\mathit{N}({\mathit{\mu }}_{\mathit{r}\mathit{p}\mathit{t}},{\mathit{\sigma }}_{\mathit{r}\mathit{p}\mathit{t}}^2)$$

(4)

$${\mathit{D}}_{\mathit{r}\mathit{p}\mathit{t}}^{\mathit{s}\mathit{a}\mathit{f}\mathit{e}}={\mathit{\mu }}_{\mathit{r}\mathit{p}\mathit{t}}+{\mathit{Z}}_{\mathit{S}\mathit{L}}{\mathit{\sigma }}_{\mathit{r}\mathit{p}\mathit{t}}$$

(5)

$$\sum _{\mathit{i}\in \mathcal{I}} \sum _{\mathit{k}\in \mathcal{K}} {\mathit{x}}_{\mathit{i}\mathit{p}\mathit{k}\mathit{r}\mathit{t}}\ge {\mathit{D}}_{\mathit{r}\mathit{p}\mathit{t}}^{\mathit{s}\mathit{a}\mathit{f}\mathit{e}},\forall \mathit{r}\in \mathcal{R},\mathit{p}\in \mathcal{P},\mathit{t}\in \mathcal{T}$$

(6)

Equations (4)–(6) transform stochastic demand into deterministic safe-demand requirements. Equation (4) characterizes retailer demand uncertainty using the estimated mean demand, ${\mu }_{rpt}$, and standard deviation, ${\sigma }_{rpt}$. Based on the selected service level, Equation (5) converts stochastic demand into a deterministic safe-demand requirement, where $Z_{SL}$ represents the corresponding standard normal quantile. Equation (6) ensures demand fulfillment by requiring the total quantity delivered from selected distribution centers to each retailer to be no less than the required safe demand. This formulation links downstream service requirements with upstream supply and distribution decisions.

$$\sum _{\mathit{i}\in \mathcal{I}} {\mathit{Q}}_{\mathit{i}\mathit{p}\mathit{t}}\le \mathit{C}\mathit{a}{\mathit{p}}_{\mathit{p}\mathit{t}}^{\mathit{S}},\forall \mathit{p}\in \mathcal{P},\mathit{t}\in \mathcal{T}$$

(7)

$${\mathit{Q}}_{\mathit{i}\mathit{p}\mathit{t}}\le \mathit{M}{\mathit{y}}_{\mathit{i}},\forall \mathit{i}\in \mathcal{I},\mathit{p}\in \mathcal{P},\mathit{t}\in \mathcal{T}$$

(8)

$${\mathit{I}}_{\mathit{i}\mathit{p}1}={\mathit{Q}}_{\mathit{i}\mathit{p}1}-\sum _{\mathit{r}\in \mathcal{R}} \sum _{\mathit{k}\in \mathcal{K}} {\mathit{x}}_{\mathit{i}\mathit{p}\mathit{k}\mathit{r}1},\forall \mathit{i}\in \mathcal{I},\mathit{p}\in \mathcal{P}$$

(9)

$${\mathit{I}}_{\mathit{i}\mathit{p}\mathit{t}}=\left(1-{\mathit{\rho }}_{\mathit{p}}\right)·{\mathit{I}}_{\mathit{i}\mathit{p}\left(\mathit{t}-1\right)}+{\mathit{Q}}_{\mathit{i}\mathit{p}\mathit{t}}-\sum _{\mathit{r}\in \mathcal{R}} \sum _{\mathit{k}\in \mathcal{K}} {\mathit{x}}_{\mathit{i}\mathit{p}\mathit{k}\mathit{r}\mathit{t}},\forall \mathit{i}\in \mathcal{I},\mathit{p}\in \mathcal{P},\mathit{t}\in \mathcal{T}$$

(10)

$$0\le {\mathit{I}}_{\mathit{i}\mathit{p}\mathit{T}}\le {\stackrel{\mathit{‾}}{\mathit{I}}}_{\mathit{i}\mathit{p}}^{\mathit{e}\mathit{n}\mathit{d}},\forall \mathit{i}\in \mathcal{I},\mathit{p}\in \mathcal{P}$$

(11)

Equations (7)–(11) describe the relationship among production supply, distribution allocation, and inventory dynamics. Equation (7) limits the production quantity of each dairy product according to the available production capacity. Equation (8) establishes the logical relationship between production decisions and selected distribution centers through the binary opening variable, $y_i$. Equation (9) determines the initial inventory level at distribution centers based on initial-period production and outbound distribution quantities. Equation (10) represents inventory transitions across consecutive operational periods, where the remaining inventory from the previous period is adjusted by the product-specific storage loss rate, ${\rho }_p$, and the current-period inventory is updated by production supply and distribution quantities. Equation (11) constrains the terminal inventory level within the predefined allowable range.

$${\mathit{t}}_{\mathit{i}}^{\mathit{F}}={\displaystyle \frac{{\mathit{s}}_{\mathit{i}}^{\mathit{F}}}{\overline{\mathit{v}}}},\forall \mathit{i}\in \mathcal{I}$$

(12)

$$\begin{array}{c}{\mathit{z}}_{\mathit{i}\mathit{t}}^{\mathit{F}}=\lceil {\displaystyle \frac{{\sum }_{\mathit{p}\in \mathcal{P}} {\mathit{w}}_{\mathit{p}}·{\mathit{Q}}_{\mathit{i}\mathit{p}\mathit{t}}}{{\mathit{a}}_3}}\rceil ,\forall \mathit{i}\in \mathcal{I},\mathit{t}\in \mathcal{T}\end{array}$$

(13)

$$\begin{array}{c}{\mathit{g}}_{\mathit{i}\mathit{t}}^{\mathit{F}}=\{\begin{array}{c}1,{\mathit{Q}}_{\mathit{i}1\mathit{t}}>0\\ 0,\mathit{o}\mathit{t}\mathit{h}\mathit{e}\mathit{r}\mathit{w}\mathit{i}\mathit{s}\mathit{e}\end{array},\forall \mathit{i}\in \mathcal{I},\mathit{t}\in \mathcal{T}\end{array}$$

(14)

Equations (12)–(14) represent the aggregated first-stage transportation process from production facilities to distribution centers. Equation (12) calculates first-stage transportation time based on the distance between production facilities and distribution centers and the representative average transportation speed. Equation (13) determines the required number of large-capacity vehicles by dividing the total shipment weight, calculated from product quantities and unit weights, by the predefined vehicle capacity, $a_3$. Equation (14) defines the refrigeration activation status according to whether shipment occurs during the corresponding operational period.

$$\begin{array}{c}{\displaystyle \sum _{\mathit{p}\in \mathcal{P}}} {\mathit{w}}_{\mathit{p}}·{\mathit{I}}_{\mathit{i}\mathit{p}\mathit{t}}+{\displaystyle \sum _{\mathit{r}\in \mathcal{R}}} {\displaystyle \sum _{\mathit{k}\in \mathcal{K}}}{\mathit{w}}_{\mathit{p}}· {\mathit{x}}_{\mathit{i}\mathit{p}\mathit{k}\mathit{r}\mathit{t}}\le {\mathit{m}}_{\mathit{i}}·{\mathit{y}}_{\mathit{i}},\forall \mathit{i}\in \mathcal{I},\mathit{t}\in \mathcal{T}\end{array}$$

(15)

$$\sum _{\mathit{p}\in \mathcal{P}} {\mathit{w}}_{\mathit{p}}·{\mathit{x}}_{\mathit{i}\mathit{p}\mathit{k}\mathit{r}\mathit{t}}\le {\mathit{a}}_{\mathit{k}}·{\mathit{z}}_{\mathit{i}\mathit{k}\mathit{r}\mathit{t}},\forall \mathit{i}\in \mathcal{I},\mathit{r}\in \mathcal{R},\mathit{k}\in \mathcal{K},\mathit{t}\in \mathcal{T}$$

(16)

$${\mathit{z}}_{\mathit{i}\mathit{k}\mathit{r}\mathit{t}}\le \mathit{M}{\mathit{y}}_{\mathit{i}},\forall \mathit{i}\in \mathcal{I},\mathit{k}\in \mathcal{K},\mathit{r}\in \mathcal{R},\mathit{t}\in \mathcal{T}$$

(17)

$${\mathit{x}}_{\mathit{i}\mathit{p}\mathit{k}\mathit{r}\mathit{t}}\le \mathit{M}{\mathit{y}}_{\mathit{i}},\forall \mathit{i}\in \mathcal{I},\mathit{p}\in \mathcal{P},\mathit{k}\in \mathcal{K},\mathit{r}\in \mathcal{R},\mathit{t}\in \mathcal{T}$$

(18)

Equations (15)–(18) describe the capacity limitations and logical relationships among distribution-center operations, vehicle allocation, and product transportation. Equation (15) restricts the total weight of stored and distributed products at each distribution center according to its storage capacity, $m_i$. The binary variable $y_i$ links the capacity limitation with the distribution-center selection decision. Equation (16) limits the total shipment weight assigned to each vehicle type based on its capacity, $a_k$. Equations (17) and (18) establish logical relationships between distribution-center activation and transportation allocation, preventing vehicle assignments and product flows from being associated with inactive distribution centers.

$${\mathit{t}}_{\mathit{i}\mathit{r}\mathit{t}}={\mathit{t}}_{\mathit{i}\mathit{r}\mathit{t}}^0·\left(1+\mathsf{\alpha }·{\left({\displaystyle \frac{{\mathit{f}}_{\mathit{i}\mathit{r}\mathit{t}}}{{\mathit{k}}_{\mathit{i}\mathit{r}}}}\right)}^{\mathit{\beta }}\right)$$

(19)

$${\mathit{f}}_{\mathit{i}\mathit{r}\mathit{t}}={\mathit{f}}_{\mathit{i}\mathit{r}\mathit{t}}^0+\sum _{\mathit{k}\in \mathcal{K}} {\mathit{u}}_{\mathit{k}}·{\mathit{z}}_{\mathit{i}\mathit{k}\mathit{r}\mathit{t}},\forall \mathit{i}\in \mathcal{I},\mathit{r}\in \mathcal{R},\mathit{t}\in \mathcal{T}$$

(20)

$${\mathit{t}}_{\mathit{i}\mathit{p}\mathit{t}}^{\mathit{I}}={\displaystyle \frac{\frac{1}{2}({\mathit{I}}_{\mathit{i}\mathit{p},\mathit{t}-1}+{\mathit{I}}_{\mathit{i}\mathit{p}\mathit{t}})}{{\sum }_{\mathit{r}\in \mathcal{R}}{\sum }_{\mathit{k}\in \mathcal{K}}{\mathit{x}}_{\mathit{i}\mathit{p}\mathit{k}\mathit{r}\mathit{t}}+\mathit{\epsilon }}}·∆\mathit{t},\forall \mathit{i}\in \mathcal{I},\mathit{r}\in \mathcal{R},\mathit{t}\in \mathcal{T},\mathit{t}\ge 2$$

(21)

$${\mathit{t}}_{\mathit{i}\mathit{p}\mathit{t}}^{\mathit{I}}={\displaystyle \frac{{\mathit{I}}_{\mathit{i}\mathit{p}1}}{{\sum }_{\mathit{r}\in \mathcal{R}}{\sum }_{\mathit{k}\in \mathcal{K}}{\mathit{x}}_{\mathit{i}\mathit{p}\mathit{k}\mathit{r}\mathit{t}}+\mathit{\epsilon }}}·∆\mathit{t},\forall \mathit{i}\in \mathcal{I},\mathit{r}\in \mathcal{R}$$

(22)

Equations (19)–(22) characterize congestion-adjusted transportation time and inventory-based equivalent residence time. Equation (19) calculates second-stage transportation time using the BPR function, where $t_{irt}^0$, $f_{irt}^0$, and $k_{ir}$ represent the free-flow travel time, baseline traffic flow, and effective capacity of the aggregated transportation link, respectively. Equation (20) incorporates logistics vehicle flows into the total traffic flow, where the additional traffic contribution is determined by vehicle-allocation decisions.

Equations (21) and (22) estimate product storage exposure based on the inventory–flow–residence-time relationship [34]:

$$\mathit{L}=\mathsf{\lambda }\mathrm{W}$$

(23)

where $L$, $\mathsf{\lambda }$, and $\mathrm{W}$ denote average inventory, average throughput rate, and average residence time, respectively. For $t\ge 2$, the average inventory is approximated by ${\displaystyle \frac{1}{2}}(I_{ip,t-1}+I_{ipt})$, while the outbound flow rate is calculated based on the delivered quantity within the operational period. For the initial period, Equation (22) uses the initial inventory level to estimate the corresponding equivalent residence time. The parameter $∆t$ represents the operational-period length used to convert throughput into the corresponding time scale.

$$\begin{array}{c}{\mathit{g}}_{\mathit{i}\mathit{k}\mathit{r}\mathit{t}}\le {\mathit{z}}_{\mathit{i}\mathit{k}\mathit{r}\mathit{t}},\forall \mathit{i}\in \mathcal{I},\mathit{r}\in \mathcal{R},\mathit{k}\in \mathcal{K},\mathit{t}\in \mathcal{T}\end{array}$$

(24)

$${\mathit{x}}_{\mathit{i}1\mathit{k}\mathit{r}\mathit{t}}\le \mathit{M}{\mathit{g}}_{\mathit{i}\mathit{k}\mathit{r}\mathit{t}},\forall \mathit{i}\in \mathcal{I},\mathit{k}\in \mathcal{K},\mathit{r}\in \mathcal{R},\mathit{t}\in \mathcal{T}$$

(25)

$${\mathit{S}}_{\mathit{i}\mathit{p}\mathit{t}}^{\mathit{F}}={\mathit{g}}_{\mathit{i}\mathit{t}}^{\mathit{F}}·\mathbf{exp}\left[-{\left({\displaystyle \frac{{\mathit{t}}_{\mathit{i}}^{\mathit{F}}}{{\mathit{d}}_{\mathit{p}\mathit{t}}^{\mathit{R}}}}\right)}^{{\mathit{\beta }}_{\mathit{p}\mathit{t}}^{\mathit{R}}}\right]+\left(1-{\mathit{g}}_{\mathit{i}\mathit{t}}^{\mathit{F}}\right)\mathbf{exp}\left[-{\left({\displaystyle \frac{{\mathit{t}}_{\mathit{i}}^{\mathit{F}}}{{\mathit{d}}_{\mathit{p}\mathit{t}}^{\mathit{O}}}}\right)}^{{\mathit{\beta }}_{\mathit{p}\mathit{t}}^{\mathit{O}}}\right]$$

(26)

$${\mathit{S}}_{\mathit{i}\mathit{p}\mathit{k}\mathit{r}\mathit{t}}^{\mathit{D}}={\mathit{g}}_{\mathit{i}\mathit{k}\mathit{r}\mathit{t}}·\mathbf{exp}\left[-{\left({\displaystyle \frac{{\mathit{t}}_{\mathit{i}\mathit{r}\mathit{t}}+{\mathit{t}}_{\mathit{i}\mathit{p}\mathit{t}}^{\mathit{I}}}{{\mathit{d}}_{\mathit{p}\mathit{t}}^{\mathit{R}}}}\right)}^{{\mathit{\beta }}_{\mathit{p}\mathit{t}}^{\mathit{R}}}\right]+\left(1-{\mathit{g}}_{\mathit{i}\mathit{k}\mathit{r}\mathit{t}}\right)\mathbf{exp}\left[-{\left({\displaystyle \frac{{\mathit{t}}_{\mathit{i}\mathit{r}\mathit{t}}+{\mathit{t}}_{\mathit{i}\mathit{p}\mathit{t}}^{\mathit{I}}}{{\mathit{d}}_{\mathit{p}\mathit{t}}^{\mathit{O}}}}\right)}^{{\mathit{\beta }}_{\mathit{p}\mathit{t}}^{\mathit{O}}}\right]$$

(27)

$${\mathit{F}}_{\mathit{i}\mathit{p}\mathit{k}\mathit{r}\mathit{t}}={\mathit{S}}_{\mathit{i}\mathit{p}\mathit{t}}^{\mathit{F}}\times {\mathit{S}}_{\mathit{i}\mathit{p}\mathit{k}\mathit{r}\mathit{t}}^{\mathit{D}},\forall \mathit{i}\in \mathcal{I},\mathit{p}\in \mathcal{P},\mathit{r}\in \mathcal{R},\mathit{k}\in \mathcal{K},\mathit{t}\in \mathcal{T}$$

(28)

$${\mathit{x}}_{\mathit{i}\mathit{p}\mathit{k}\mathit{r}\mathit{t}}·\left(1-{\mathit{F}}_{\mathit{i}\mathit{p}\mathit{k}\mathit{r}\mathit{t}}\right)\le {\mathit{x}}_{\mathit{i}\mathit{p}\mathit{k}\mathit{r}\mathit{t}}·{\mathit{\alpha }}_{\mathit{p}},\forall \mathit{i}\in \mathcal{I},\mathit{p}\in \mathcal{P},\mathit{r}\in \mathcal{R},\mathit{k}\in \mathcal{K},\mathit{t}\in \mathcal{T}$$

(29)

Equations (24)–(29) describe refrigeration-operation and freshness-preservation requirements. Equation (24) links refrigeration activation with vehicle usage, allowing refrigeration operation only when transportation activities occur. Equation (25) imposes refrigeration requirements for products requiring temperature-controlled transportation. Equation (26) calculates the freshness retention level during the first transportation stage under refrigerated and non-refrigerated conditions. Equation (27) represents freshness retention during the second transportation stage and distribution-center residence period, where the total exposure time includes transportation time and equivalent storage residence time. The Weibull-based survival formulation is applied under different temperature-control conditions. Equation (28) combines freshness retention across the two transportation stages using a multiplicative formulation. Equation (29) restricts the freshness loss of delivered products within the acceptable threshold, ${\alpha }_p$.

Overall, the proposed MINLP formulation integrates demand requirements, production and inventory decisions, transportation allocation, and freshness preservation within a unified optimization framework. Based on the order-driven supply structure and the operational assumptions described above, the formulation captures the interactions among economic performance, transportation-related carbon emissions, and product freshness in cold-chain operations. This formulation provides the foundation for the subsequent multi-objective solution approach and computational experiments.

4. Solution Procedure

The low-carbon dairy cold-chain network optimization model developed in this study integrates multiple operational decisions, including distribution-center selection, product transportation allocation, vehicle-capacity assignment, refrigeration operation, and inventory control, while simultaneously optimizing three conflicting objectives: total cost, transportation-related carbon emissions, and the quantity- and importance-weighted average freshness-loss rate. The formulation involves binary, integer, and continuous decision variables, as well as nonlinear relationships associated with transportation congestion, inventory turnover, and freshness deterioration. Combined with capacity constraints, inventory balance requirements, and freshness preservation restrictions, these characteristics result in a complex multi-objective mixed-integer nonlinear programming (MINLP) problem.

Given the combinatorial complexity and nonlinear nature of the formulation, exact optimization methods may become computationally prohibitive for medium- and large-scale instances. Therefore, the non-dominated sorting genetic algorithm II (NSGA-II) is adopted as the primary solution method to generate a diverse set of feasible Pareto solutions with satisfactory convergence performance. In addition, the multi-objective evolutionary algorithm based on decomposition (MOEA/D) is introduced as a comparative algorithm to evaluate the performance of NSGA-II under identical computational settings. After obtaining the Pareto solution set, the entropy-weighted technique for order preference by similarity to ideal solution (TOPSIS) method is employed as a posteriori decision-support approach to identify a representative compromise solution by considering the relative information contribution of the three objectives.

4.1. Principles and Mechanisms of the NSGA-II Algorithm

The non-dominated sorting genetic algorithm II (NSGA-II) is adopted as the primary optimization algorithm for solving the proposed low-carbon dairy cold-chain network model. Unlike single-objective optimization methods, NSGA-II does not require predefined preference weights among objectives and can directly obtain a set of trade-off solutions among conflicting objectives. This feature is particularly suitable for the proposed model, which simultaneously optimizes total cost, transportation-related carbon emissions, and the quantity- and importance-weighted average freshness-loss rate. In addition, the model involves binary, integer, and continuous decision variables associated with distribution-center selection, transportation allocation, vehicle-capacity assignment, refrigeration operation, and inventory control, resulting in a highly complex solution space that is difficult to efficiently explore using exact optimization methods.

The implementation of NSGA-II is customized according to the characteristics of the proposed MINLP model, as illustrated in Figure 2. The algorithm takes network parameters, demand requirements, operational costs, emission factors, and freshness-related parameters as inputs. A hierarchical hybrid encoding strategy is then developed to generate the initial population, where each chromosome represents a complete cold-chain network configuration and operational plan. After decoding, the corresponding objective values are calculated, and infeasible solutions caused by capacity limitations, inventory imbalance, or freshness constraints are repaired before evolutionary operations. Selection, crossover, and mutation operators are subsequently applied to generate new candidate solutions until the termination criterion is reached, resulting in a final set of non-dominated Pareto solutions.

The core mechanism of NSGA-II relies on non-dominated sorting, crowding-distance evaluation, and environmental selection, as illustrated in Figure 3. During the non-dominated sorting process, solutions are classified into different Pareto fronts according to their dominance relationships. A solution dominates another if it is no worse in all objectives and strictly better in at least one objective. Solutions in the first front represent the current non-dominated alternatives, while subsequent fronts contain solutions with relatively lower priority. To preserve diversity among solutions with the same Pareto rank, the crowding distance is introduced:

$${\mathit{C}\mathit{D}}_{\mathit{i}}={\sum }_{\mathit{m}=1}^{\mathit{M}}{\displaystyle \frac{{\mathit{f}}_{\mathit{i}+1}^{\mathit{m}}-{\mathit{f}}_{\mathit{i}-1}^{\mathit{m}}}{{\mathit{f}}_{\mathit{m}\mathit{a}\mathit{x}}^{\mathit{m}}-{\mathit{f}}_{\mathit{m}\mathit{i}\mathit{n}}^{\mathit{m}}}}$$

(30)

where $M$ represents the number of objectives, and $f_{i+1}^m$ and $f_{i-1}^m$ denote the adjacent objective values of solution $i$ in objective $m$. A larger crowding distance indicates that the solution is located in a less crowded region of the objective space and therefore receives a higher probability of preservation. Based on Pareto rank and crowding distance, environmental selection retains high-quality and well-distributed solutions for the next generation. Through this mechanism, NSGA-II simultaneously enhances convergence toward the Pareto frontier and maintains diverse cold-chain network alternatives representing different operational preferences.

4.2. Entropy-Weight TOPSIS-Based Compromise Solution Selection

After the NSGA-II optimization process, a set of non-dominated Pareto solutions is obtained, where each solution represents a different trade-off among total cost, transportation-related carbon emissions, and the quantity- and importance-weighted average freshness-loss rate. Since these objectives are conflicting, the Pareto set provides multiple operational alternatives rather than a unique decision. Therefore, a posteriori decision-support procedure is required to identify a representative compromise solution for subsequent analysis.

In this study, the entropy-weighted TOPSIS method is employed to rank the obtained Pareto solutions. The role of TOPSIS is limited to post-optimization decision support and does not affect the evolutionary search process or replace the Pareto optimization procedure. Specifically, NSGA-II explores the feasible solution space and generates diverse non-dominated alternatives, while TOPSIS provides a systematic mechanism for selecting one representative solution from the obtained Pareto set. The selected solution is regarded as a compromise alternative under the adopted evaluation criteria rather than a globally preferred solution.

Assume that the Pareto solution set generated by NSGA-II contains $n$ candidate solutions and $m$ evaluation criteria. In this study, m = 3, corresponding to total cost, $Z_1$; transportation-related carbon emissions, $Z_2$; and the quantity- and importance-weighted average freshness-loss rate, $Z_3$. The original decision matrix is expressed as follows:

$$\mathit{X}={\left({\mathit{x}}_{\mathit{i}\mathit{j}}\right)}_{\mathit{n}\times \mathit{m}}$$

(31)

where $x_{ij}$ denotes the value of criterion $j$ for candidate solution $i$. Since all three criteria are minimization objectives, smaller values indicate better performance. However, these criteria differ in units and numerical scales, making direct comparison inappropriate. Therefore, the reverse min–max normalization method is applied:

$${\mathit{r}}_{\mathit{i}\mathit{j}}={\displaystyle \frac{{\mathit{x}}_{\mathit{j}}^{\mathit{m}\mathit{a}\mathit{x}}-{\mathit{x}}_{\mathit{i}\mathit{j}}}{{\mathit{x}}_{\mathit{j}}^{\mathit{m}\mathit{a}\mathit{x}}-{\mathit{x}}_{\mathit{j}}^{\mathit{m}\mathit{i}\mathit{n}}}}$$

(32)

where $x_j^{\mathrm{m}\mathrm{a}\mathrm{x}}$ and $x_j^{\mathrm{m}\mathrm{i}\mathrm{n}}$ represent the maximum and minimum values of the criterion $j$ among all Pareto solutions. After normalization, larger $r_{ij}$ values indicate better relative performance.

To determine the criterion weights, the entropy-weight method is introduced. Unlike preference-based weighting approaches, entropy weighting derives criterion weights from the distribution characteristics of the obtained Pareto solutions. Specifically, a criterion with greater variation among candidate solutions provides more discriminative information and receives a higher weight, whereas a criterion with limited variation contributes less information for distinguishing alternatives. Therefore, entropy weights represent the relative information contribution of each criterion within the current Pareto solution set rather than predefined stakeholder preferences.

The proportion of candidate solution $i$ under criterion j is calculated as follows:

$${\mathit{p}}_{\mathit{i}\mathit{j}}={\displaystyle \frac{{\mathit{r}}_{\mathit{i}\mathit{j}}}{{\displaystyle {\sum }_{\mathit{i}=1}^{\mathit{n}}}{\mathit{r}}_{\mathit{i}\mathit{j}}}}$$

(33)

The information entropy of criterion $j$ is obtained as follows:

$${\mathit{e}}_{\mathit{j}}=-\mathit{k}{\displaystyle \sum _{\mathit{i}=1}^{\mathit{n}}}{\mathit{p}}_{\mathit{i}\mathit{j}}\mathbf{l}\mathbf{n}{\mathit{p}}_{\mathit{i}\mathit{j}},\mathit{k}={\displaystyle \frac{1}{\mathbf{ln}\mathit{n}}}$$

(34)

where $e_j$ represents the entropy value of criterion $j$. The divergence coefficient and entropy weight are calculated as follows:

$${\mathit{d}}_{\mathit{j}}=1-{\mathit{e}}_{\mathit{j}}$$

(35)

$${\mathit{w}}_{\mathit{j}}={\displaystyle \frac{{\mathit{d}}_{\mathit{j}}}{{\displaystyle {\sum }_{\mathit{j}=1}^{\mathit{m}}}{\mathit{d}}_{\mathit{j}}}}$$

(36)

The resulting weights are then used to construct the baseline TOPSIS ranking. It should be noted that these weights reflect the information dispersion of the obtained Pareto solutions and do not directly represent the preferences of specific stakeholders. Different preference settings may lead to different compromise selections. Therefore, the effects of equal weighting, preference-oriented weighting schemes, and normalization strategies on the selected solution are further examined in Appendix A.9.

After obtaining the criterion weights, the weighted normalized decision matrix is constructed as follows:

$${\mathit{v}}_{\mathit{i}\mathit{j}}={\mathit{w}}_{\mathit{j}}{\mathit{r}}_{\mathit{i}\mathit{j}}$$

(37)

The positive ideal solution, $A^{+}$, and negative ideal solution, $A^{-}$, are defined as follows:

$$\begin{gathered} {\mathit{A}}^{+}=\left({\mathit{v}}_1^{+},{\mathit{v}}_2^{+},\dots ,{\mathit{v}}_{\mathit{m}}^{+}\right),{\mathit{v}}_{\mathit{j}}^{+}=\underset{\mathit{i}}{\mathbf{m}\mathbf{a}\mathbf{x}}{\mathit{v}}_{\mathit{i}\mathit{j}}, \\ {\mathit{A}}^{-}=\left({\mathit{v}}_1^{-},{\mathit{v}}_2^{-},\dots ,{\mathit{v}}_{\mathit{m}}^{-}\right),{\mathit{v}}_{\mathit{j}}^{-}=\underset{\mathit{i}}{\mathbf{m}\mathbf{i}\mathbf{n}}{\mathit{v}}_{\mathit{i}\mathit{j}} \end{gathered}$$

(38)

The distances between candidate solution $i$ and the two ideal solutions are calculated using the Euclidean distance:

$$\begin{gathered} {\mathit{D}}_{\mathit{i}}^{+}=\sqrt{{\displaystyle \sum _{\mathit{j}=1}^{\mathit{m}}}{({\mathit{v}}_{\mathit{i}\mathit{j}}-{\mathit{v}}_{\mathit{j}}^{+})}^2}, \\ {\mathit{D}}_{\mathit{i}}^{-}=\sqrt{{\displaystyle \sum _{\mathit{j}=1}^{\mathit{m}}}{({\mathit{v}}_{\mathit{i}\mathit{j}}-{\mathit{v}}_{\mathit{j}}^{-})}^2} \end{gathered}$$

(39)

The relative closeness coefficient is then calculated as follows:

$${\mathit{C}}_{\mathit{i}}={\displaystyle \frac{{\mathit{D}}_{\mathit{i}}^{-}}{{\mathit{D}}_{\mathit{i}}^{+}+{\mathit{D}}_{\mathit{i}}^{-}}}$$

(40)

where $C_i$∈ [0, 1]. A larger $C_i$ indicates that candidate solution is closer to the positive ideal solution and farther from the negative ideal solution. The solution with the largest $C_i$ is selected as the representative compromise solution for subsequent operational interpretation.

The proposed solution framework integrates NSGA-II with entropy-weighted TOPSIS to address the multi-objective optimization and decision-making problem of the low-carbon dairy cold-chain network. NSGA-II generates a diverse set of non-dominated solutions representing trade-offs among total cost, transportation-related carbon emissions, and the quantity- and importance-weighted average freshness-loss rate. Entropy-weighted TOPSIS subsequently ranks these Pareto alternatives and identifies a representative compromise solution for operational interpretation. Detailed optimization results and compromise-solution analysis are presented in Section 5.

5. Numerical Experiments

All computational experiments were performed under an identical hardware and software environment to ensure consistency across algorithmic comparisons and numerical analyses. The proposed optimization model and solution algorithms were implemented in MATLAB R2023b (version 23.2.0.2365128; The MathWorks, Inc., Natick, MA, USA). Computations were performed on a Lenovo Xiaoxin Pro 16 APH8 laptop (model 83AR; Lenovo, Beijing, China) equipped with an AMD Ryzen 7 7840HS processor (3.80 GHz, 8 cores and 16 logical processors; Advanced Micro Devices, Inc., Santa Clara, CA, USA) and 32.0 GB of RAM, running the 64-bit Microsoft Windows 11 Home operating system, version 23H2 (OS Build 22631.6345; Microsoft Corporation, Redmond, WA, USA).

This section presents four groups of experiments to evaluate the proposed model and solution framework from different perspectives. First, NSGA-II is adopted as the primary optimization algorithm to investigate convergence behavior, Pareto solution characteristics, and solution quality through small-scale mixed-integer approximation tests. Second, NSGA-II is compared with MOEA/D under identical computational budgets and evaluation metrics. Third, multi-scale case studies are conducted to assess the feasibility, computational burden, and scalability of the proposed model and algorithms under different network sizes. Finally, sensitivity and scenario analyses are performed to examine the effects of key parameters on network performance.

5.1. Experimental Settings and Data Description

A numerical case study is developed to evaluate the computational performance of the proposed model and solution framework. The experimental dataset combines simulated network information, publicly available historical sales data, literature-based operational and freshness-related parameters, and public emission factors. Since complete enterprise-level spatial and operational records are unavailable, the locations of candidate distribution centers and retailer demand nodes are generated through simulation while preserving the structural characteristics of the three-echelon dairy cold-chain network. This data-generation approach is consistent with common practices in logistics network design studies when real-world spatial datasets are inaccessible [3]. Demand-related parameters are estimated from the publicly available Dairy Supply Chain Sales Dataset [31], where historical sales records are used to derive product-level demand statistics for the safe-demand transformation described in Section 3.2. Other parameters associated with transportation, refrigeration, freshness degradation, and carbon emissions are obtained or calibrated from the relevant literature and relevant public databases, including transportation characteristics, Weibull freshness parameters, and emission coefficients. Detailed parameter values, units, and data sources are provided in Appendix A.1, Appendix A.2, Appendix A.3, Appendix A.4, Appendix A.5 and Appendix A.6.

The algorithmic parameters are configured to balance solution quality and computational efficiency. The basic settings of NSGA-II and MOEA/D follow previous studies [8,9], while specific parameter values are determined through preliminary experiments based on objective performance indicators and computational requirements. To ensure a fair comparison, NSGA-II and MOEA/D adopt identical population sizes, maximum generations, termination criteria, and constraint-handling strategies. The standalone NSGA-II experiment employs a population size of 100 and 200 generations to investigate convergence behavior and Pareto solution characteristics. The algorithm comparison experiment uses the same settings and performs 20 independent runs for each algorithm. For sensitivity and scenario analyses, reduced population sizes and generation numbers are adopted to alleviate the computational burden caused by repeated optimization under multiple parameter configurations. The crossover rate, mutation rate, and random seed remain unchanged across all experiments to ensure reproducibility. Detailed algorithmic parameters are summarized in

Table 3. Algorithmic parameter settings for numerical experiments.

Symbol/Code	Value	Source
Single NSGA-II: popSize	100	Final standalone NSGA-II experiment
Single NSGA-II: maxGen	200
Algorithm comparison: popSize	100	NSGA-II and MOEA/D comparison setting
Algorithm comparison: maxGen	200
algorithmRepeatTimes	20	Independent runs for algorithm comparison
Sensitivity analysis: popSize	80	Sensitivity and scenario analysis setting
Sensitivity analysis: maxGen	150
sensitivityRepeatTimes	5	Independent runs for each sensitivity scenario
crossoverRate	0.9	General evolutionary algorithm setting
mutationRate	0.2
Rng	20260605	Fixed random seed for reproducibility

.

5.2. Analysis and Validation of NSGA-II Solutions

In the first experiment, NSGA-II is adopted as the primary optimization algorithm to solve the proposed multi-objective model. This experiment evaluates the quality of the obtained solutions from three complementary aspects: convergence performance, Pareto solution characteristics, and solution quality validation. The evaluation procedure consists of three stages. First, NSGA-II is executed using the parameter settings described in Section 5.1 to generate feasible non-dominated solutions. The convergence behavior and objective-space distribution of the resulting Pareto solutions are then analyzed to assess whether the algorithm can effectively approach the Pareto front while maintaining diverse trade-off solutions. Finally, a reduced-scale benchmark experiment based on a mixed-integer approximation model is conducted to provide an independent reference for assessing the approximation quality of the NSGA-II solutions.

The hypervolume (HV) indicator is adopted to quantitatively assess the convergence performance of NSGA-II. HV measures the volume of the objective space dominated by the obtained Pareto solution set with respect to a predefined reference point. A larger HV value generally indicates better convergence toward the Pareto front and improved diversity of the solution set. Before calculating HV, the three objectives are normalized using the min–max normalization method:

$${\mathit{f}}_{\mathit{i}}^{\prime }={\displaystyle \frac{{\mathit{f}}_{\mathit{i}}^{\prime }-{\mathit{f}}_{\mathit{i}}^{\mathit{m}\mathit{i}\mathit{n}}}{{\mathit{f}}_{\mathit{i}}^{\mathit{m}\mathit{a}\mathit{x}}-{\mathit{f}}_{\mathit{i}}^{\mathit{m}\mathit{i}\mathit{n}}}}$$

(41)

where $f_i^{min}$ and $f_i^{max}$ denote the minimum and maximum values of objective among the feasible solutions generated by NSGA-II, respectively. The same normalization bounds are maintained throughout the evolutionary process to ensure consistency and comparability across different generations. A reference point of (1.1, 1.1, 1.1) is selected for HV calculation, representing a dominated point beyond the normalized objective range and avoiding boundary effects associated with using extreme points.

As shown in Figure 4, the HV value continuously increases from 0.4540 to 1.1639 throughout the evolutionary process. The rapid improvement in the early generations indicates that NSGA-II can effectively explore promising regions of the solution space through evolutionary operators. After approximately 60 generations, the HV curve gradually stabilizes, suggesting that the population has converged toward a relatively stable Pareto region and that further iterations provide limited improvements. Meanwhile, the number of non-dominated solutions continues to increase during the optimization process, demonstrating that NSGA-II can enhance convergence performance while preserving solution diversity.

Following the convergence analysis of NSGA-II, the distribution characteristics of the final non-dominated solutions are further examined in the three-dimensional objective space. As illustrated in Figure 5, the obtained solutions exhibit evident trade-offs among total cost, transportation-related carbon emissions, and the quantity- and importance-weighted average freshness-loss rate. Improvements in one objective are generally accompanied by deteriorations in at least one of the remaining objectives, indicating that no single solution can simultaneously optimize all objectives. These results confirm the conflicting nature of economic, environmental, and freshness-related objectives and further justify the adoption of a multi-objective optimization framework.

To further analyze the operational implications of these trade-offs, three representative extreme solutions are selected from the final Pareto set. Specifically, P21, P6, and P19 correspond to the solutions achieving the minimum total cost, minimum transportation-related carbon emissions, and minimum freshness-loss rate, respectively. These solutions reflect different managerial preferences and provide representative references for examining how objective priorities affect facility selection, inventory allocation, transportation configuration, and refrigeration decisions.

Figure 6 illustrates that different objective preferences lead to distinct operational configurations. The minimum-cost solution reduces economic expenditure primarily through network consolidation and moderate inventory control. The minimum-emission solution adopts an alternative configuration by adjusting transportation allocation and inventory decisions to reduce transportation-related carbon emissions. In contrast, the minimum-freshness-loss solution prioritizes product quality preservation by reducing exposure time and increasing refrigeration intensity. These results indicate that improvements in individual objectives are achieved through different combinations of network structure, inventory exposure, and refrigeration operations rather than through a universally optimal operational configuration.

Since none of the extreme solutions simultaneously outperforms the others across all objectives, a compromise solution is required for practical decision-making. Therefore, the entropy-weighted TOPSIS method is employed as a posteriori decision-support approach to rank the obtained Pareto solutions. The entropy weights are derived from the objective-value distribution of the solution set rather than predefined managerial preferences, thereby reflecting the relative information contribution of each objective.

As shown in Figure 7, based on the entropy-weighted TOPSIS ranking, P21 is identified as the representative compromise solution with a closeness coefficient of 0.8152. The corresponding total cost, transportation-related carbon emissions, and quantity- and importance-weighted average freshness-loss rate are 4.1941 × 10⁷ CNY, 1.7828 × 10⁴ kg CO₂, and 6.60%, respectively.

Although HV provides an evaluation of the convergence and diversity performance of NSGA-II, it does not directly quantify the deviation between heuristic solutions and benchmark solutions. Therefore, an additional small-scale benchmark experiment is conducted to quantitatively assess the approximation quality of the obtained solutions.

Considering the computational complexity of the original multi-objective mixed-integer nonlinear programming model, a reduced-scale instance with three candidate distribution centers and six retailer demand nodes is constructed while preserving the main decision structure of the original problem. A mixed-integer approximation model is solved using an ε-constraint framework, in which one objective is optimized while the remaining objectives are converted into constraints with predefined bounds to generate benchmark trade-off solutions.

To reduce the impact of stochastic variation, NSGA-II is independently executed 10 times using different random seeds. The feasible non-dominated solutions obtained from all runs are aggregated and compared with the benchmark solutions. A benchmark solution is considered successfully matched if at least one NSGA-II solution satisfies the corresponding ε-constraint region. The approximation gap for each matched case is calculated as follows:

$${\mathit{G}\mathit{a}\mathit{p}}_{\mathit{i}}^{\prime }={\displaystyle \frac{|{\mathit{f}}_{\mathit{i}}^{\mathit{N}\mathit{S}\mathit{G}\mathit{A}-\mathit{I}\mathit{I}}-{\mathit{f}}_{\mathit{i}}^{\mathit{B}\mathit{e}\mathit{n}\mathit{c}\mathit{h}\mathit{m}\mathit{a}\mathit{r}\mathit{k}}|}{\left|{\mathit{f}}_{\mathit{i}}^{\mathit{B}\mathit{e}\mathit{n}\mathit{c}\mathit{h}\mathit{m}\mathit{a}\mathit{r}\mathit{k}}\right|}}\times 100\%$$

(42)

where $f_i^{NSGA-II}$ and $f_i^{Benchmark}$ denote the objective values of the matched NSGA-II solution and the corresponding benchmark solution, respectively.

For the representative 3 × 6 instance, the aggregated NSGA-II solution set successfully matches 14 out of 15 benchmark cases, corresponding to a match rate of 93.33%. The average gaps for cost-oriented and freshness-oriented cases are 1.64% and 0.53%, respectively, indicating close agreement between NSGA-II solutions and benchmark solutions in these trade-off regions. For emission-oriented cases, the average gap increases to 22.74%, which is mainly attributed to the discrete transportation decisions embedded in the emission objective and the difficulty of reproducing specific low-emission configurations through evolutionary search.

Overall, the benchmark comparison provides quantitative evidence that NSGA-II can effectively approximate high-quality regions of the multi-objective solution space. These results demonstrate the solution quality and approximation capability of NSGA-II rather than serving as evidence of exact global optimality.

5.3. Comparative Study of NSGA-II and MOEA/D

After assessing the standalone performance of NSGA-II, a comparative experiment is conducted to further examine its performance against another representative multi-objective evolutionary algorithm. MOEA/D is selected as the benchmark algorithm because it represents a decomposition-based optimization paradigm that differs fundamentally from the Pareto-dominance-based mechanism adopted by NSGA-II. Specifically, NSGA-II employs non-dominated sorting and crowding distance to balance convergence and diversity, whereas MOEA/D decomposes the multi-objective problem into a set of scalar subproblems that are optimized collaboratively through neighborhood relationships.

To ensure a fair comparison, both algorithms are applied to the same multi-objective MINLP model under identical computational settings, including population size, maximum generations, function evaluation budget, termination criteria, and number of independent runs, as specified in Section 5.1. The comparison considers three aspects: Pareto-front quality, solution distribution, and computational efficiency. Accordingly, HV, IGD, Spacing, and CPU time are adopted as performance metrics. In addition, the Wilcoxon rank-sum test and Cliff’s delta effect size are employed to examine statistical significance and practical differences between the two algorithms.

The HV values are calculated using the same normalization strategy and reference point setting described in Section 5.2. IGD is employed to measure the distance between the obtained solution set and an approximate reference Pareto set. Since the exact Pareto front of the proposed multi-objective MINLP model is unavailable, the reference set is constructed by aggregating feasible solutions obtained from multiple independent runs of NSGA-II and MOEA/D. After removing duplicate solutions and performing non-dominated sorting, the remaining solutions are considered the approximate reference Pareto set.

The IGD value is calculated as the average Euclidean distance between each solution in the reference set and its nearest solution in the obtained Pareto set. A smaller IGD value indicates that the obtained solutions are closer to the reference Pareto region and achieve better convergence performance. Spacing is used to assess the uniformity of solution distribution, where a smaller value represents a more evenly distributed Pareto solution set.

Table 4. Performance comparison between NSGA-II and MOEA/D.

Metric	NSGA-II	MOEA/D	p-Value	Effect Size (Cliff’s δ)
HV	0.8512 ± 0.0923	0.7461 ± 0.0761	0.000622	0.675
IGD	0.1372 ± 0.0392	0.1841 ± 0.0409	0.000687	−0.620
Spacing	0.0724 ± 0.0280	0.0735 ± 0.0229	0.776391	−0.025
CPU time (s)	612.26 ± 35.76	2224.27 ± 263.84	6.80 × 10−8	−1.000

summarizes the comparative results of NSGA-II and MOEA/D over 20 independent runs. Regarding Pareto-front quality, NSGA-II achieves a higher average HV value (0.8512 ± 0.0923) and a lower average IGD value (0.1372 ± 0.0392) than MOEA/D (HV: 0.7461 ± 0.0761; IGD: 0.1841 ± 0.0409). The Wilcoxon rank-sum test confirms that the differences in HV and IGD are statistically significant (p < 0.01), while the corresponding Cliff’s delta values indicate moderate-to-large practical effects. These results demonstrate that NSGA-II obtains solutions closer to the approximate Pareto region while maintaining a larger dominated objective space.

For solution distribution, NSGA-II and MOEA/D achieve similar Spacing values (0.0724 ± 0.0280 and 0.0735 ± 0.0229, respectively). The Wilcoxon test shows no significant difference (p > 0.05), with an effect size close to zero, indicating comparable diversity preservation capabilities. Therefore, the primary advantage of NSGA-II lies in convergence performance rather than distribution uniformity.

Regarding computational efficiency, NSGA-II requires substantially less CPU time than MOEA/D under the same computational budget, with average times of 612.26 ± 35.76 s and 2224.27 ± 263.84 s, respectively. The statistical test confirms a significant difference, and the large Cliff’s delta value indicates a strong practical effect. This difference is mainly attributed to the distinct search mechanisms of the two algorithms. MOEA/D introduces additional computational overhead through neighborhood-based information exchange and the optimization of multiple scalar subproblems, whereas NSGA-II directly maintains population diversity through non-dominated sorting and crowding-distance mechanisms.

Overall, the comparative results demonstrate that NSGA-II achieves superior Pareto-front quality and computational efficiency compared with MOEA/D under the same computational budget for the proposed dairy cold-chain optimization problem. Although both algorithms maintain comparable solution diversity, NSGA-II exhibits advantages in convergence performance and computational cost. Therefore, NSGA-II is selected as the primary optimization algorithm for the subsequent multi-scale experiments and sensitivity analyses.

5.4. Multi-Scale Computational Scalability Analysis

To evaluate the scalability of the proposed optimization framework under different network sizes, three multi-scale instances with increasing numbers of candidate distribution centers and retailer demand nodes were constructed. The model structure, parameter-generation rules, and algorithm settings were kept consistent across all instances to ensure comparability. The analysis focuses on the impacts of network expansion on computational efficiency, convergence performance, and optimization results.

Table 5. Computational performance across different network scales.

Scale	Network Size (I, R)	Total Safe Demand	Pareto Solutions	HV	CPU Time	${\mathit{Z}}_1$	${\mathit{Z}}_2$	${\mathit{Z}}_3(\%)$
Small	(5, 20)	1,051,728	11.80	1.2625	524.51	16,155,984	6631.13	19.23
Medium	(10, 40)	1,880,796	20.40	0.8920	716.67	25,693,408	12,789.10	10.19
Large	(15, 60)	2,843,219	29.00	1.1172	815.49	39,419,274	14,644.41	10.42

summarizes the computational performance of NSGA-II under different network scales. As the network size increases from the small case (five candidate distribution centers and 20 retailers) to the large case (15 candidate distribution centers and 60 retailers), the number of decision variables and total safe demand increase substantially, leading to higher computational requirements. Accordingly, CPU time increases from 524.51 s to 815.49 s. However, the computational burden grows moderately relative to the network expansion, indicating that the proposed solution framework maintains acceptable scalability.

Across all tested scales, NSGA-II consistently generates feasible non-dominated solution sets with satisfactory convergence performance. The increases in total cost and transportation-related carbon emissions are mainly attributed to the enlarged network structure and higher demand requirements, whereas the quantity- and importance-weighted average freshness-loss rate remains relatively stable in the medium and large cases. This result suggests that the proposed framework can maintain freshness-preservation performance under larger-scale network configurations.

Overall, the multi-scale experiments demonstrate that the proposed NSGA-II-based framework can effectively handle increasing network complexity while maintaining satisfactory solution quality and computational efficiency. The results support its applicability to tactical-level dairy cold-chain network planning problems with different scales.

5.5. Sensitivity and Scenario Analysis

To assess the robustness of the proposed model under varying operational and environmental conditions, a comprehensive sensitivity and scenario analysis was conducted by considering four critical factors: service level, transportation conditions, road capacity, and product shelf-life characteristics. All sensitivity experiments were conducted based on the benchmark solution. Only the corresponding parameters were varied, while the remaining model settings were kept unchanged to ensure a fair comparison. The results provide further insights into the robustness of the proposed framework and reveal its managerial implications under diverse supply-chain scenarios.

The service level, $S{L}_{rpt}$, represents the required demand fulfillment reliability and reflects managerial preferences regarding customer-service requirements. It directly influences the calculation of safety demand, whereas production capacity remains fixed as an independent operational constraint across all scenarios. The service level was varied within the range of [0.85, 0.9, 0.95, 0.975, 0.99], while all other model parameters were kept unchanged. The resulting impacts on objective values and key decision-performance indicators are presented in Figure 8.

As shown in Figure 8, an increase in the service level substantially expands the operational scale of the supply chain. Specifically, when the service level increases from 0.850 to 0.990, the safety demand rises from 2.392 × 10⁶ to 3.349 × 10⁶. This increase leads to higher inventory requirements, transportation activities, and resource consumption. Consequently, the total cost increases from 3.418 × 10⁷ CNY to 4.647 × 10⁷ CNY, while carbon emissions increase from 1.440 × 10⁴ to 1.809 × 10⁴. These results demonstrate that a higher service level improves demand-fulfillment reliability but inevitably incurs additional economic and environmental costs. Therefore, the service level should be regarded as a strategic decision parameter rather than a predetermined requirement. Managers should carefully balance customer-service expectations against the associated operational expenditures and environmental impacts.

The congestion intensity coefficient characterizes the impact of traffic congestion on travel-time deterioration in the BPR function and represents transportation condition uncertainty in cold-chain distribution. Unlike the service-level analysis, which focuses on demand fulfillment requirements, this experiment examines how external transportation conditions affect operational decisions and system performance. The service level was fixed at $S{L}_{rpt}=0.95$, while $\mathsf{\alpha }$ was varied within [0, 0.075, 0.15, 0.225, 0.3]. The effects of different congestion conditions on objective values, operational indicators, and overall decision performance are presented in Figure 9.

As shown in Figure 9, increasing congestion intensity mainly affects transportation-related indicators rather than the overall economic objective. As congestion increases from 0 to 0.30, the average transportation time increases from 0.534 h to 0.576 h, and refrigeration-related emissions increase from 844.6 to 961.2 due to prolonged vehicle operation and refrigeration duration. Meanwhile, total cost remains relatively stable, indicating that the proposed optimization framework can adapt to adverse transportation conditions through coordinated adjustments in distribution-center assignment, fleet configuration, shipment allocation, and inventory decisions. A non-monotonic pattern is observed in freshness loss, which increases at $\alpha =0.075$ but decreases under higher congestion levels. This phenomenon does not imply that congestion improves product freshness; instead, it reflects the adaptive response of the re-optimized supply-chain system. Under stronger congestion penalties, the model tends to avoid highly congested routes and adjust operational decisions, thereby reducing the combined storage and transportation exposure time. From the perspective of overall performance, the TOPSIS closeness coefficient reaches its highest value at $\alpha =0.225$, suggesting that moderate congestion conditions may achieve a more balanced trade-off among cost, emissions, and freshness preservation. However, excessive congestion ($\alpha =0.30$) increases transportation burdens and slightly deteriorates overall performance. These results indicate that transportation congestion should be considered an important factor in cold-chain planning, while the proposed model maintains sufficient adaptability under different congestion scenarios.

While the congestion coefficient captures the severity of congestion propagation in the BPR function, the road-capacity parameter, $k_{ir}$, determines the underlying infrastructure constraint that affects congestion formation. Therefore, a complementary sensitivity analysis was conducted to examine whether the assumption of uniform road capacity influences the robustness of the model conclusions. Since link-specific road-capacity observations are unavailable for the simulated network, the baseline value $k_{ir}$ = 1000 is treated as a representative effective-capacity parameter rather than a geographically calibrated link-level measurement. The capacity parameter was varied among $\{125,250,500,1000,1500\}$, representing severe capacity stress, constrained, baseline, and higher-capacity transportation conditions. The effective capacity and traffic flow were evaluated at the same aggregation level as Equations (19) and (20), such that ${\displaystyle \frac{f_{irt}}{k_{ir}}}$ represents a dimensionless volume-to-capacity indicator. All other model and algorithm parameters remained unchanged, and each scenario was independently solved five times. The results are summarized in

Table 6. Effects of different road-capacity scenarios on model performance.

${\mathit{k}}_{\mathit{i}\mathit{r}\mathit{t}}$	Feasible	Mean V/C	Mean Travel Time (h)	${\mathit{Z}}_1$	${\mathit{Z}}_2$	${\mathit{Z}}_3$
125	100%	0.8289	0.61409	42,938,937	15,135.25	0.06515
250	100%	0.4143	0.57026	42,744,564	15,399.39	0.05729
500	100%	0.2071	0.56513	42,744,681	15,255.48	0.05849
1000	100%	0.1035	0.56498	42,744,681	15,255.29	0.05849
1500	100%	0.0690	0.56497	42,744,681	15,255.28	0.05849

.

As shown in Table 6, all capacity scenarios maintained 100% feasibility, indicating that the proposed optimization framework can obtain feasible solutions under different transportation capacity conditions. As increased from 125 to 1500, the mean volume-to-capacity ratio decreased substantially from 0.8289 to 0.0690, confirming that the capacity parameter effectively regulates transportation congestion intensity. However, when $k_{ir}$≥ 500, the volume-to-capacity ratio remained below 0.21, and the corresponding variations in travel time and the three objectives were negligible. This indicates that the baseline network operates in a relatively uncongested region of the BPR function, where further capacity improvements provide limited performance benefits. In contrast, under severe capacity stress ($k_{ir}$ = 125), the average second-leg travel time increased from 0.56498 h to 0.61409 h (+8.69%), and the normalized freshness-loss objective, $Z_3$, increased from 0.05849 to 0.06515 (+11.39%). The non-monotonic variations in $Z_1$ and $Z_2$ are mainly attributed to adaptive adjustments in the re-optimized multi-objective solutions rather than a direct monotonic relationship between capacity and performance. Overall, the results demonstrate that the main managerial conclusions remain robust under moderate-to-high-capacity conditions, whereas severe capacity degradation may introduce nonlinear congestion effects and compromise cold-chain delivery performance.

While previous analyses focus on demand requirements and transportation conditions, the shelf-life characteristics of perishable products directly determine freshness deterioration. Therefore, a scenario-based sensitivity analysis was conducted to evaluate the impact of Weibull lifetime-scale uncertainty on model performance. Specifically, the baseline lifetime-scale parameters were adjusted as $d_{pt}^R={\lambda }_s{d}_{pt}^R$ and $d_{pt}^O={\lambda }_s{d}_{pt}^O$, where ${\lambda }_s=\{0.8, 0.9, 1.0, 1.1, 1.2\}$ represents 80–120% of the baseline shelf-life scale. The Weibull shape parameters and all other model parameters remained unchanged. This scenario analysis does not treat the deterministic Weibull function as a stochastic shelf-life distribution. Instead, it evaluates the sensitivity of the proposed framework to potential deviations in shelf-life parameter calibration. To distinguish the direct deterioration effect from the adaptive response of the optimized supply-chain system, two complementary analyses were performed. First, the baseline compromise solution was fixed under different lifetime-scale scenarios to examine the direct impact of shelf-life variation. Second, each scenario was independently re-optimized using identical random seeds to evaluate system-level adaptation. The results are presented in Figure 10.

As shown in Figure 10a, the fixed-solution analysis exhibits the expected monotonic relationship between shelf-life scale and freshness deterioration. As increases from 0.8 to 1.2, the weighted average freshness-loss rate decreases from 7.170% to 3.525%, confirming that longer effective shelf life directly reduces product deterioration under unchanged operational decisions. In contrast, the re-optimized results in Figure 10b demonstrate a more complex system-level response. Compared with the baseline scenario (${\lambda }_s=1.0$), total cost, $Z_1$, remains relatively stable (0.999–1.028), while carbon emissions ($Z_2$) and refrigeration utilization vary within limited ranges. However, the freshness-loss objective, $Z_3$, exhibits greater sensitivity (0.885–1.950), indicating that freshness-related performance is more vulnerable to shelf-life calibration errors than economic and environmental objectives. The difference between the two analyses highlights the importance of operational adaptation: changes in shelf-life parameters may alter Pareto trade-offs and trigger adjustments in transportation, inventory, and refrigeration decisions, resulting in non-monotonic system responses. Overall, the results demonstrate that accurate shelf-life calibration is particularly important for freshness-oriented decision-making, while the proposed optimization framework maintains robust economic and operational performance within the tested uncertainty range.

Overall, the sensitivity and scenario analyses demonstrate the robustness of the proposed framework under varying demand, transportation, and product-related conditions. The results reveal heterogeneous impacts of different uncertainty factors: service-level requirements mainly influence system scale and resource consumption, transportation conditions affect operational efficiency through congestion-related effects, and shelf-life parameters directly determine freshness preservation performance. Despite these variations, the optimized supply-chain system can adapt through coordinated adjustments in network configuration, inventory allocation, transportation planning, and refrigeration decisions. These findings highlight the importance of jointly considering customer-service requirements, transportation conditions, and product-deterioration characteristics in cold-chain supply-chain planning, while providing practical guidance for balancing economic, environmental, and freshness-related objectives under uncertain operating conditions.

6. Conclusions and Future Research

6.1. Conclusions

This study developed an order-driven multi-objective mixed-integer nonlinear programming (MINLP) framework for the tactical planning of a three-echelon low-carbon dairy cold-chain network. The model coordinates distribution-center selection, order-driven supply, inventory balance, transportation allocation, vehicle configuration, and refrigeration decisions while minimizing total cost, transportation-related carbon emissions, and the quantity- and importance-weighted average freshness-loss rate. Demand variability is incorporated through service-level-based safe-demand requirements, whereas product deterioration is characterized by a Weibull-based freshness formulation that links first-leg transportation, inventory-turnover-based equivalent residence time, and second-leg distribution. Congestion-adjusted travel time further captures the effects of transportation conditions on refrigeration emissions and freshness deterioration. The main contribution lies in coordinating these interrelated decisions within an internal production–distribution system rather than introducing the individual modeling components as separate methodological innovations.

The computational results reveal distinct trade-offs among the economic, environmental, and freshness objectives. Objective-oriented Pareto solutions correspond to different combinations of facility configuration, inventory exposure, transportation allocation, and refrigeration intensity, indicating that no single operational configuration performs best across all three objectives. The reduced-scale benchmark provides additional quantitative evidence of solution quality: the pooled NSGA-II solutions matched 93.33% of the valid ε-constraint benchmark cases, with close agreement in the cost- and freshness-oriented regions. Larger gaps in some emission-oriented cases, however, indicate that approximation quality varies across objectives. Under the same computational budget, NSGA-II achieved higher HV, lower IGD, and substantially shorter CPU time than MOEA/D, while no significant difference was observed in Spacing. The multi-scale experiments indicate that satisfactory Pareto-search performance is maintained as network size increases. Sensitivity and scenario analyses further reveal that service levels, Weibull shelf-life characteristics, and road capacity have distinct effects on economic, environmental, and freshness performance.

Overall, the findings support the proposed framework as a quantitative tool for examining the interactions among demand protection, inventory exposure, transportation conditions, refrigeration operation, and freshness deterioration in tactical dairy cold-chain planning. The computational results should be interpreted as evidence of model behavior and solution performance under the constructed numerical settings rather than as direct validation of industrial applicability. Within this scope, this study provides an integrated basis for examining how downstream demand requirements propagate through supply and distribution decisions and how the resulting operational choices shape trade-offs among cost, carbon emissions, and freshness.

6.2. Potential Managerial Implications

The proposed framework provides several implications for tactical planning within the internal production–distribution systems of dairy enterprises. Rather than prescribing a universally optimal strategy, the numerical results illustrate how network configuration, service requirements, inventory exposure, transportation conditions, and refrigeration decisions jointly affect economic, environmental, and freshness performance. Within the scope of the numerical case, the following implications can be drawn:

(1)

Multi-objective trade-offs should be explicitly considered in cold-chain planning. The Pareto results show that cost, carbon emissions, and freshness objectives favor different combinations of facility utilization, inventory allocation, transportation, and refrigeration decisions. A cost-oriented plan may therefore compromise environmental or freshness performance. Enterprises can use the Pareto set to compare alternative configurations according to their priorities, while the TOPSIS compromise solution serves as a reference under a specified weighting scheme rather than a universally preferred solution.

(2)

Service levels should reflect demand and service characteristics. Higher service levels increase safe-demand requirements and consequently affect planned supply, inventory, and resource utilization. Uniformly high service levels may therefore impose unnecessary operating burdens. Enterprises can differentiate service targets across products, customer categories, or operational periods according to demand variability and contractual requirements while considering the associated economic and environmental impacts.

(3)

Inventory decisions should balance demand protection and freshness exposure. Additional inventory can support downstream demand fulfillment, but prolonged storage increases freshness deterioration. Tactical planning should therefore emphasize inventory turnover rather than inventory accumulation. Coordinating planned supply, inventory balance, and outbound distribution can reduce unnecessary storage exposure while maintaining adequate demand protection, particularly for products with short or uncertain shelf lives.

(4)

Transportation capacity and congestion should be incorporated into tactical network planning. Road-capacity scenarios show that transportation conditions affect travel time and, consequently, transportation efficiency, refrigeration operation, emissions, and freshness exposure. Distribution-center utilization and transportation-capacity allocation should therefore reflect major corridor conditions rather than relying solely on nominal distance or uncongested travel time. Detailed routing, vehicle sequencing, and real-time congestion response remain operational-level decisions beyond the scope of the present framework.

(5)

Freshness preservation and low-carbon operations should be coordinated. Refrigeration mitigates freshness deterioration but increases energy consumption and associated emissions, while freshness is also influenced by transportation exposure and inventory residence time. Increasing refrigeration intensity alone may therefore be inefficient. Coordinating inventory turnover, transportation conditions, and refrigeration allocation can provide a more balanced means of maintaining freshness while controlling economic and environmental burdens.

6.3. Limitations and Future Research

This study has several limitations that define the scope of the proposed framework and suggest directions for future research. First, the numerical case combines publicly available sales data, simulated spatial nodes, and literature-based parameters rather than a complete operational dataset from a specific dairy enterprise. The experiments therefore assess model behavior and computational performance under controlled settings but do not establish external validity against observed facility configurations, operating costs, delivery performance, or freshness outcomes. Demand variability is represented using marginal service-level quantiles, with exogenously specified service levels and full satisfaction of the resulting safe-demand requirements. As examined in Appendix A.8, this representation is primarily applicable to independent or weakly correlated demand and does not explicitly capture stronger cross-retailer, cross-product, or inter-period dependencies. Future research could use enterprise-level longitudinal data to estimate joint demand structures and develop stochastic, robust, or distributionally robust formulations for more strongly correlated demand. Service levels could also be endogenized through differentiated minimum-service requirements or shortage penalties to capture service–resource trade-offs more flexibly.

Second, several physical and operational processes are represented at an aggregate level to preserve the tractability of the tactical MINLP formulation. The three 120-day periods represent aggregated operational conditions rather than continuous daily operations, while freshness exposure is evaluated on a daily time scale using transportation time and an inventory turnover-based equivalent residence time. Accordingly, the inventory representation is most appropriate under relatively low-to-moderate inventory-age heterogeneity, as evaluated in Appendix A.7, rather than in systems requiring detailed batch-level age tracking. Similarly, fixed Weibull parameters provide a tractable representation of shelf-life reliability but do not endogenously capture temperature-dependent deterioration or product-specific parameter variation. The environmental boundary includes transportation fuel consumption and vehicle-refrigeration electricity emissions but excludes emissions from facility operations, stationary cold storage, production, and waste treatment. Fuel consumption and refrigeration power are represented by average parameters. Although Appendix A.4 evaluates the sensitivity of the fuel-consumption assumption to vehicle loading, load- and speed-dependent energy consumption and dynamic refrigeration loads remain outside the endogenous formulation. Transportation is likewise represented at an aggregate network level: first-leg travel is approximated using a representative average speed, while second-leg congestion is modeled using the BPR function over aggregated DC–retailer connections rather than detailed physical road links. Future research could incorporate age-indexed inventory and shorter rolling horizons; data-driven and temperature-dependent deterioration and energy models; broader life-cycle emission boundaries; and GIS-based time-dependent routing and scheduling supported by enterprise-level operational and traffic data.