Application of the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) in a Two-Echelon Cold Supply Chain

Abstract

A two-stage cold supply chain manages the transportation, storage, and distribution of temperature-sensitive products like frozen food, fresh/green products, and pharmaceuticals, which makes it costly. It consists of three key elements: a supplier, a warehouse, and multiple customers. Procurement planning can be conducted for various products, and this study assumes the transport of a fresh/green product with gradually decreasing quality due to its perishable nature. In a two-stage cold supply chain, multiple objective functions can be defined, including cost minimization, product quality optimization, and transportation/storage condition optimization. We developed a mathematical model to optimize these objectives, incorporating two specific functions, cost minimization and product age reduction, to ensure efficient supply chain performance. Traditional solution methods often struggle with multi-objective mathematical models due to their complexity. Therefore, the Non-Dominated Sorting Genetic Algorithm II (NSGA-II), a Genetic Algorithm-based approach, was applied to solve the model efficiently. NSGA-II optimized planning for a 7-day period under specific demand conditions, ensuring better resource allocation. The results showed that NSGA-II was better than traditional methods at making decisions and routing efficiently in the two-stage cold supply chain. This led to much better outcomes, with lower costs, less waste, and better product quality throughout the process.

1. Introduction

The cold supply chain is a process that involves ensuring and maintaining the appropriate temperature conditions required from the production to the storage, distribution, and consumption of frozen food, fresh/green products, medicines, and other perishable goods. For this reason, it is critically important to ensure food safety, extend the shelf life of products, and reduce costs. In the literature, the importance of supply chain collaboration in cold supply chain management is emphasized [1,2]. Since the cold supply chain requires products to be transported under specific temperature conditions, studies on how supply chain integration can be used to meet these special requirements are also of great importance. In this context, cold chain management requires a comprehensive approach that includes various factors such as sustainability, risk management, supply chain integration, and collaboration. In cold supply chains, many problems can arise due to the numerous factors associated with the transported product and the system. We can summarize some common problems in cold supply chain management as follows:

Temperature control: Since products need to be kept within a certain temperature range, temperature control is a critical factor in the cold supply chain. If the temperature exceeds the specified temperature range, it can seriously affect the quality and efficacy of the products.

Moisture levels: It is of great importance to keep moisture levels under control, especially for medicines, chemicals, and fresh/green products. Inappropriate humidity levels for products can cause them to spoil or become ineffective.

Light intensity: Some medications and chemicals can deteriorate or become unusable under the influence of light. Therefore, the product must be kept under controlled conditions to ensure the most suitable light intensity.

Carbon dioxide levels: Monitoring carbon dioxide levels is critically important, especially in products that undergo fermentation or chemical reactions. High levels of carbon dioxide can negatively affect the quality of these products.

Lack of traceability: The lack of traceability of products at every step of the supply chain causes interventions that need to be made in critical situations to be delayed.

Health risks and safety concerns: The deterioration or ineffectiveness of product quality can lead to health and safety risks. For this reason, ensuring the optimal level of cold supply chain management is of vital importance.

In the following sections, we present a structured literature review that situates this study within the broader context of cold supply chain optimization, highlighting key gaps and emerging trends. We then detail our Materials and Methods, elaborating on the multi-objective mathematical model and the NSGA-II–based solution procedure. After describing the dataset and experimental design, we share and discuss our results, underscoring how the proposed framework improves cost efficiency and freshness preservation compared to traditional methods. Finally, this paper concludes with Implications and Future Research Directions, offering insights into how these findings can guide more sustainable and effective cold supply chain strategies.

2. Literature Review

The two-echelon cold supply chain management involves a multi-layered approach to managing the flow of perishable goods, ensuring quality and minimizing waste. This system typically includes a supplier and a retailer, with a focus on transportation modes, inventory management, and pricing strategies to optimize supply chain performance [2,3]. The integration of cold chain logistics is crucial for maintaining product quality, especially for perishable goods, and involves strategic decisions regarding transportation and inventory management. In terms of transportation modes, cold chain transportation is essential for maintaining the quality of perishable goods, such as fresh produce, during long-distance transport [4,5]. It benefits all supply chain participants, including consumers, by reducing both quality and quantity loss [6]. The choice between low-cost normal temperature transportation and high-cost cold chain transportation depends on cost thresholds and contractual agreements, such as revenue-sharing contracts, which can incentivize suppliers to adopt cold chain logistics [7]. The inventory management side is another perspective. Efficient inventory management is critical for sustainability in pharmaceutical supply chains, where lateral transshipment can reduce costs and minimize product deterioration [7,8,9,10]. A mixed-integer non-linear program (MINLP) model can optimize replenishment order quantities and shipment times, thereby reducing waste and ensuring a sustainable supply of medicines [7].

While the two-echelon cold supply chain management offers numerous benefits, challenges such as high transportation costs and the need for coordination among supply chain members persist. These challenges necessitate strategic planning and collaboration to ensure the sustainability and efficiency of the supply chain [11,12,13]. A two-echelon cold supply chain can be defined as a supply chain model that involves the management and transportation of products through two different stages of the cold supply chain [12,14,15]. In this type of supply chain, elements such as suppliers, cold storage facilities, distribution/transportation vehicles, and customers are involved. In two-stage cold supply chains, supply planning for many products can be carried out. In this study, it is assumed that a fresh/green product with gradually decreasing quality will be transported. In a two-stage cold supply chain, multiple objective functions can be determined, such as cost minimization, product quality optimization, product transportation conditions optimization, product storage conditions optimization, etc. This study aims to analyze a two-stage cold supply chain with a supplier and a warehouse selling a product in the market under a certain demand. The fresh/green product received from the supplier will be stored in a single warehouse before customer distribution. It is accepted that the quality of the product transported along the supply chain gradually decreases. Therefore, the aim is to keep the storage duration of fresh/green products in the intermediate warehouse short. A mathematical model has been proposed to minimize expected costs and achieve the best solution. To solve the mathematical model, one of the meta-heuristic algorithms, the Non-Dominated Sorting Genetic Algorithm II (NSGA-II), has been used. The results obtained from the solution are presented in this study. Suggestions have been made based on the results obtained. Liu, Chen, and their colleagues presented a dynamic planning model in their 2021 study aimed at delivering fresh products to customers in a two-stage cold supply chain [15]. In their 2020 study, Wang and Wen aimed to solve the vehicle routing problem for cold supply chains by considering costs and carbon emissions as performance metrics. Under the constraints and performance metrics they established, they created a model for the two-stage heterogeneous vehicle routing problem and proposed the Adaptive Genetic Algorithm (AGA) approach to reach a solution. Based on the results obtained, they made recommendations to logistics companies, governments, and consumers involved in the cold chain to support the improvement of cold supply chain development [16]. Liu and colleagues (2021) stated in their published article that delivering fresh products to customers is the primary objective of the two-stage cold supply chain model they addressed. In the two-stage cold supply chain they established, a producer and a retailer collaborate. In the study, the producer decides on the effort to maintain optimal product freshness, while the retailer decides on the level of optimal advertising effort. In the decentralized decision-making mode, they showed that both the freshness factor and the optimal levels of effort significantly decrease due to the reduction in profit margins. To solve this problem, they proposed a dynamic control model. At the same time, they developed a dynamic linear bonus scheme [15]. Jaigirdar and his colleagues, in their study published in 2022, aimed to reduce the annual supply chain cost and the cold storage setup cost for a sustainable supply chain while maintaining the freshness of perishable products by establishing an appropriate distribution system. For this purpose, a multi-stage and multi-product three-objective optimization model was developed in the study [17]. In the established optimization model, a mixed-integer linear programming model was proposed to solve the supply chain distribution network problem. For the remaining part of the model, the weighted sum method was used, and the solution was reached using the CPLEX optimization studio [3]. Theeb and colleagues published a study in 2023 focusing on vaccine distribution, one of the key issues to be addressed during the pandemic. They argued that building permanent warehouses to address the weak infrastructure and other challenges that do not meet the urgent vaccine needs in developing countries is impractical. To address the specified issues in vaccine supply, they proposed a two-tiered approach [18].

The Non-Dominated Sorting Genetic Algorithm II (NSGA-II) is a powerful tool for optimizing two-echelon cold supply chain systems, offering numerous advantages. These include strong multi-objective optimization, better handling of complex logistics, and the ability to balance competing goals such as cost, carbon emissions, and customer satisfaction. NSGA-II is particularly valuable for cold supply chains because it generates Pareto-optimal solutions, allowing decision-makers to easily compare and choose among different objectives.

One of the main benefits of using NSGA-II for two-echelon cold supply chains is its effectiveness in addressing multiple objectives at once. This is especially important in cold chains, where cost, carbon emissions, and product freshness must all be considered. Researchers have applied NSGA-II to optimize distribution routes by factoring in traffic conditions and replenishment strategies, significantly cutting both costs and emissions while keeping products fresh [19]. The algorithm pinpoints sets of Pareto-optimal solutions, helping decision-makers select the best balance between competing goals like cost and quality [20,21]. NSGA-II also strengthens the resilience of supply chain networks by optimizing attributes like agility, leanness, and flexibility. This is especially useful for addressing risks and uncertainties in cold supply chains [22]. The system adapts to shifting conditions, such as urban traffic congestion, and still delivers reliable planning outcomes in unpredictable circumstances [19]. Moreover, NSGA-II helps strike a balance between economic and environmental goals by optimizing inventory and transport decisions in accordance with carbon emissions limits, a crucial consideration for cold chains, which produce significant emissions from refrigerated transport and storage [23].

Combining NSGA-II with hybrid methods, such as large-scale neighborhood search, further boosts its ability to explore vast solution spaces and avoid local optima, thereby improving local search performance [19]. Its flexibility also allows for easy customization to meet specific supply chain needs, ranging from optimizing storage in automated systems to managing dual-sale channel networks [24,25]. However, when using NSGA-II for two-echelon cold supply chain optimization, it is important to consider the algorithm’s computational complexity and the need to fine-tune parameters like population size, crossover, and mutation rates for the best results [26]. Additionally, while NSGA-II handles multiple objectives well, dealing with an extremely large number of them may require further refinements or hybrid approaches to maintain efficiency [27]. NSGA-II efficiently integrates many constraints, ensuring that optimized routing and inventory strategies align with the specific requirements of cold logistics operations.

3. Materials and Methods

The mathematical formulation of the problem and the methodological approach are presented in this section. Additionally, the fundamental principles of the NSGA-II algorithm and its application steps are discussed comprehensively, ensuring a holistic explanation of the methods employed.

Two-Stage Cold Supply Chain Problem: The main objectives of the established model are as follows:

• Minimizing routing and inventory costs;
• Minimizing the number of vehicles used;
• Minimizing the number of spoiled fresh/green products.

Mathematical Model: The mathematical model is one of the engineering methods used to solve problems. At the same time, it represents the problem by forming a basis for other solution methods. In the mathematical model established based on the assumptions and premises made for this study, the optimization model developed by Rohmer and others in their 2019 publication, “A Two-echelon Inventory Touting Problem for Perishable Products”, has been referenced [28,29]. The sets used in the developed mathematical model and their descriptions are summarized in

Table 1. Sets used in the mathematical model and their descriptions.

Cluster	Explanation
N	The set of nodes indexed by i, j, l is {depot:0; customer: 1, …, n}
A	Set of springs (i, j): i, j ∈ N, i ≠ j
T	The set of periods indexed by t
K	The set of vehicles indexed by k: k∈ {1, …, m}
G	The set of product ages indexed by g
$R_i$	The set of visit combinations of i

.

The parameter variables used in the mathematical model and their explanations are presented in

Table 2. Parameters used in the mathematical model and their explanations.

Parameter	Description
$c_{ij}$	(i, j) the guidance costs on the bow: i, j ∈ {0, …, n}.
C	supplier–warehouse–supplier line transportation routing cost
$d_i^t$	i the customer’s demand in period t
$Q^k$	k the capacity of the vehicle (k = 0: supplier–warehouse; k = 1, 2, 3: warehouse–customer)
H	warehouse inventory holding capacity
$h^g$	g unit holding cost in the warehouse for product age (including spoilage cost)
$a^{rt}$	If the combination of r visits on day t is equal to 1

.

Simultaneously,

Table 3. Variables used in the mathematical model and their explanations.

Variable	Description
$x_{ij}^{kt}$	If customer j is visited by agent k in period t immediately after customer i, it is equal to 1.
$y_i^{kt}$	If the intermediary visits customer i in period t, it is equal to 1
$z_i^r$	If the r visits combination of customer i is selected, it equals 1
$u^t$	The number of supplier–warehouse vehicles in period t.
$v_i^{gkt}$	The quantity of g age delivered to customer i from the warehouse by vehicle k during period t
$w^t$	The quantity delivered from the supplier to the warehouse during period t.
$I^{gt}$	the amount of g age stored in the warehouse during the t period
$s_i^{kt}$	the position of vehicle k on the route of customer i at time t

displays the variables and explanations of the mathematical model.

3.1. Objective Functions

In the mathematical model, two different objective functions have been created. Equation (1) is the first of the objective functions created. The aim of this objective function is to minimize the total cost of the two-stage cold supply chain. Equation (1) includes transportation costs, inventory holding costs, and distribution costs to customers. In Equation (2), the main aim of the objective function is to minimize the age of the products available in the cold supply chain. In this way, the number of spoiled products in the system is reduced, thereby minimizing product waste.

$$Minimise\sum _{t\in T}C{u}^t+\sum _{g\in G}\sum _{t\in T}{h}^g{I}^{gt}+\sum _{i\in N}\sum _{j\in N}\sum _{k\in K}\sum _{t\in T}{C}_{ij}{x}_{ij}^{kt}$$

(1)

$$Minimise\sum _{i\in N}\sum _{k\in K}\sum _{t\in T}{v}_i^{gkt}$$

(2)

3.2. Constraints

Constraints are crucial for accurately representing the existing conditions and assumptions within the model, in conjunction with the established objective functions, to attain optimal outcomes. This study has identified a total of 18 restrictions. Within these established restrictions, the resolution of the target functions and the optimal transportation and storage strategy in the supply chain will be attained. Equations (3) and (4) are storage/inventory constraints related to the age of the product. Products that reach a fixed age, as determined by Equation (3), are removed from inventory and not distributed to customers. This constraint prevents the delivery of products that have deteriorated in quality to users.

$$I^{gt}=I^{g-1,t-1}-\sum _{i\in N}\sum _{k\in K}{v}_i^{g-1,k,k-1}g\in G\backslash \left\{0\right\},t\in T\backslash \left\{0\right\}$$

(3)

$$I^{0t}=w^{t}t\in T$$

(4)

Equation (5) is a constraint added to determine the delivery to the warehouse for updating the inventory. It also determines the deliveries made from the supplier to the warehouse.

$$I^{gt}\ge \sum _{i\in N\left\{0\right\}}\sum _{k\in K}{v}_i^{gkt},g\in G,t\in T$$

(5)

Equations (6) and (7) ensure that the inventory level meets at least the customer deliveries for the same period. At the same time, they ensure that the fixed inventory capacity of the warehouse is not exceeded in the solution obtained.

$$\sum _{g\in G}{I}^{g0}=w^0$$

(6)

$$\sum _{g\in G}{I}^{gt}\le H,t\in T$$

(7)

In the delivery plan of the optimal solution, the condition for meeting each customer’s demand is included in the mathematical model expressed in Equation (8).

$$\sum _{r\in R_i}{a}^{rt}{d}_i^t{z}_i^r=\sum _{g\in G}\sum _{k\in K}{v}_i^{gkt},i\in N\backslash \{0\},t\in T$$

(8)

With Equation (9), the quantity of products that can be delivered to the warehouse for the optimal solution is restricted based on warehouse capacity and current inventory.

$$w^t\le H-\sum _{g\in G}{I}^{g,t-1},t\in T$$

(9)

With Equations (10) and (11), a fixed vehicle capacity constraint has been added for the vehicles to be used for delivery to the warehouse and the customer.

$$\sum _{g\in G}\sum _{i\in N\backslash \left\{0\right\}}{v}_i^{gkt}\le Q^k{y}_0^{kt},k\in K,t\in T$$

(10)

$$w^t\le Q^0{u}^t,t\in T$$

(11)

The assumption that each delivery to a customer in each period can only be made by a single vehicle has been added to the model as a constraint in Equation (12). While Equation (12) addresses the single vehicle constraint, Equation (13) imposes the constraint that each delivery in that period will be made with the vehicle that is active during that period.

$$\sum _{k\in K}{y}_i^{kt}\le 1,i\in N\backslash \{0\},t\in T$$

(12)

$$y_i^{kt}\le \sum _{j\in N}{x}_{ij}^{kt}\le 1,i\in N,k\in K,t\in T$$

(13)

A delivery plan will be assigned to each customer according to their requests. Equation (14) ensures that a single delivery assignment is made to each customer for this assignment. Equation (15) ensures that the delivery plan assigned to each customer, based on Equation (14), is followed.

$$\sum _{r\in R_i}{z}_i^r,i\in N\backslash \{0\}$$

(14)

$$\sum _{i\in N}\sum _{k\in K}{x}_{ij}^{kt}-\sum _{r\in R_i}{a}^{rt}{z}_j^r=0,j\in N\backslash \left\{0\right\},t\in T$$

(15)

$$\sum _{i\in N}{x}_{ij}^{kt}-\sum _{l\in N}{x}_{jl}^{kt}=0,k\in K,t\in T,j\in N$$

(16)

In addition to all the constraints between Equations (3) and (16), the model includes constraints that ensure that the variables added to achieve the optimal solution are not negative.

3.3. Assumptions

This section explains the assumptions used in this study. In the two-stage cold supply chain, the supplier is considered a single entity, as they collect the green product from the producers. The fresh/green product received from the supplier will be stored in a single warehouse before customer distribution. Customers are located within a circular area with a radius of 25 km, where the depot is at the center. Each customer’s distance to the warehouse is within this circle. All customers’ demands follow a normal distribution. The interval between two consecutive delivery periods for each customer will be 2 periods. At the same time, multiple vehicles cannot deliver to the same customer within the same period. The supply chain gradually decreases the quality of the products it transports. To determine the quality criteria for each product, its age will be made available. We consider the products’ ages to be zero when they arrive at the warehouse. Upon delivery to the warehouse, each subsequent period sees a fixed increase of 1 in the product’s age. If the product’s age in the warehouse exceeds 33% of the expiration date, the relevant products will not be delivered to customers and will be removed from the warehouse inventory.

3.4. Genetic Algorithm

The Genetic Algorithm (GA) is a meta-heuristic algorithm first proposed by John Holland in the 1970s. Developed based on Darwin’s theory of evolution and known as an evolutionary algorithm, the GA solves computer-based problems by using gene exchange between living things as a model. The GA allows for faster and easier solutions to clustering and very large optimization problems that are difficult to solve with traditional methods [30].

In order to reach a solution for the mathematical model in the GA, the objective function must be defined in accordance with the constraints, and the gene and chromosome structure must be created [31]. A few possible solutions are determined to solve a specific problem. A program is written to test each solution alternative. This program is run, and, according to the results, alternatives that do not fit the objective function are eliminated and code exchange occurs between the remaining ones. This process, which resembles gene exchange between living things, fosters a diversity of alternatives. The working steps of the GA are explained in four steps [32].

Step 1: First, all possible solution alternatives in the search space are coded, and individuals are created.

Step 2: Random individuals are selected from the individuals created in Step 1 and brought together to form the initial population. The size of the created population can affect the speed of the algorithm steps. Many individuals in the population cause algorithm steps to take a long time, but they also increase the solution quality.

Step 3: Fitness values are calculated for each individual. The fitness function allows the fitness levels of the determined solutions to be measured. It provides the result to be obtained by adapting the individual to the system. Thanks to this function, the missing information in the individual can be eliminated, and numerical values can be obtained.

Step 4: The most important part of the reproduction process is the selection operator. With this operator, the individual diversity in the algorithm will increase; thus, different regions can be searched in the solution space. There are different selection methods in the literature. Individuals with a high fitness function are transferred to the next generation. The individuals in the new generation are passed through the crossover and mutation stages, respectively. Crossover involves the creation of a new individual through gene exchange between two individuals. In the solution space, the crossover process is determined by the crossover rate, and the number of chromosomes to be mutated is determined by the mutation rate. An example of crossover and mutation is given in Figure 1.

Step 5: After the reproduction operations performed on the individuals in Step 4, a new generation population is created.

Step 6: The cycle in Figure 1 is repeated until the optimal solution is reached. The cycle is terminated when the desired success is achieved.

3.5. NSGA-II and Application Steps

The NSGA-II algorithm is a multi-purpose meta-heuristic algorithm that was introduced to the literature by Deb and his colleagues in their 2002 study. The NSGA-II algorithm emerged because of the development of the NSGA algorithm, which was developed by Srinivas and Deb in 1995 [33,34]. The basic structure of the NSGA-II algorithm is based on the Genetic Algorithm (GA). The basic steps of the Genetic Algorithm include dominance ranking and accumulation distance calculation.

3.5.1. Elitism

If elitism is used uncontrolled, the diversity of individuals in the population may decrease. This may lead to an increase in individuals with the same fitness value. It has been observed that elitism significantly contributes to the success of the GA in selecting individuals with the best results and transferring them to the next generation [35,36]. Figure 2 shows the elitism stages.

If we examine elitism through the stages of elitism in Figure 2, individuals in F1 and F2 can fit into P_(t + 1), belonging to the new population. However, the number of individuals in F3 exceeds P_(t + 1). Since the dominance degrees of individuals in F3 are equal to each other and exceed the size of the new population, some of the individuals in F3 must be eliminated [17].

3.5.2. Dominance Rating

A method of comparing individuals in a population with each other, together with dominance rating, is employed. The number of times each individual has been defeated by other individuals is counted. If there is an individual or individuals who have never been defeated, these individuals are placed in the first rank and F1. Thus, the rank of individuals in F1 is accepted as 1. Individuals in F1 are then removed from the population being compared. In this way, the effect of F1 individuals is eliminated in other comparisons to be made. Thus, the remaining individuals who cannot be defeated form F2. This process is repeated until all individuals in the population are ranked.

A dominance ranking example is given in Figure 3. When there is no n-dominance ranking in the f1 and f2 space, a choice can be made between solutions/individuals 2 and 3. Solution 2 could be chosen because it suppresses solution 3. However, solutions 1 and 4 do not have an advantage over each other. In other words, there is no clear dominance between them. Therefore, the choice between solutions can be made in conjunction with ranking.

3.5.3. Crowding Distance

For an individual to be transferred to the next generation, the degree given dominance must be low. Crowding distance can be used in the NSGA-II algorithm to choose between individuals with equal dominance degrees. Density distance is used to prioritize individuals with equal ranks. An example of density distance is shown in Figure 4.

Density distance is calculated according to the previous and next neighboring individuals of the selected individual, as well as the first and last individuals present in the population. The calculation is shown in Equation (17). According to Equation (17), the individuals whose aggregation distance is calculated are ranked from largest to smallest. As a result of this ranking, the individuals at the top have a higher overlap rate with other individuals and are therefore given priority in transferring to the next generation.

$$\begin{gathered} d_i^1={\displaystyle \frac{\left|f_1^{i+1}-f_1^{i-1}\right|}{f_1^{max}-f_1^{min}}} \\ {d}_i^2={\displaystyle \frac{\left|f_2^{i+1}-f_2^{i-1}\right|}{f_2^{max}-f_2^{min}}} \\ d=d_i^1+d_i^2 \end{gathered}$$

(17)

Figure 5 outlines the main steps of a multi-objective genetic algorithm (e.g., NSGA-II). First, data input is used to create an initial population of candidate solutions. Each solution is then evaluated against the objective functions (f1, f2), and a fitness function and crowding distance are calculated. Based on these measures, genetic operators (selection, crossover, and mutation) generate new offspring. Next, non-dominated sorting sorts the answers by the level of dominance and the distance between the solutions, combining the parent and offspring populations. The algorithm checks whether a termination criterion (such as a maximum number of generations or a convergence threshold) is met. If not satisfied, it continues iterating. Otherwise, it shows the final results, which ideally form a set of non-dominated (Pareto-optimal) solutions.

3.6. Problem Assumptions

There is a single supplier, a single warehouse, and multiple customers in this model. It is assumed that the model works with multiple vehicles. Customer demands are in accordance with the normal distribution and are estimated accordingly. The two-stage cold supply chain model is intended to be planned for a 7-day period. The distance from the supplier to the warehouse is 32 km. It is assumed that the warehouse is located at the center of a circle with a radius of 25 km. In line with this assumption, the maximum distance from each customer to the warehouse is 25 km. The locations of the customers relative to the warehouse are determined randomly. The First-in First-out rule is adopted when the products arriving at the warehouse are removed from the warehouse in line with customer demands. When the SKT date, i.e., age (g), of a product is 7, that product will be considered completely spoiled. At the same time, according to data received from A Cold Chain Logistics Company, products with a maximum product age of 33% can be accepted by customers in the green/fresh product market. In line with this information received about the market, it has been determined that green/fresh products with a product age equal to or greater than 3 will be removed from stock and directed to the determined alternative solutions to reduce costs. The values given for the algorithm variables used in the NSGA-II algorithm are provided in

Table 4. Algorithm variable values.

Parameter	Value
Iteration number	1000
Maximum number of iterations without improvement	200
Population number	100
Mutation rate	It is determined by the program to be less than 0.1.
Crossover	Single point crossover
Selection	Dominance rating, aggregation distance, and elitism

.

3.7. Dataset Used

Since a 1-week daily planning period is needed in the dataset used, the period is defined as 7 to represent the 7 days of the week. The number of vehicles is assumed to be fixed to optimize the benefit from the existing vehicles. No new vehicles will be purchased or rented. It is not mandatory to use all vehicles. Other fixed data used in the algorithm are given in

Table 5. Data used in the model solution with the algorithm.

Parameter	Value
t (period)	7
Number of customers	15
k (number of vehicles)	6
H (warehouse inventory capacity)	800
Capacity of each vehicle (k)	350

.

4. Results

We used the algorithm variable values summarized in Table 4 to solve the model. We performed tests on the model after determining the algorithm variable values in Table 4.

Table 6. Test results for determining the iteration number.

	Iteration Numbers
Results	600	800	900	1000	1100	1200	2000
Cost	895.159	654.483	603.199	589.787	589.787	589.787	589.787
Mean Vehicle Numbers	7	6	6	5.5	5.5	5.5	5.5

summarizes the test results for the iteration number. According to the results obtained as a result of the tests, it was observed that the same results were achieved for iteration number values equal to or greater than 1000. As a result of this finding, it was decided that the iteration number would be 1000 since using an iteration number greater than 1000 would make the algorithm heavier.

The locations of 15 customers were randomly determined in a circle with a radius of 25 km, with the warehouse at the center. The randomly determined customer locations are shown in Figure 6. At the same time, the location of the supplier, which is 32 km from the warehouse, is also shown in Figure 6.

It has been determined that customer demands are in accordance with normal distribution. The mean of the customer demands in accordance with normal distribution is 98 units, and the standard deviation is 18 units. The First Period customer demand estimates determined in accordance with normal distribution are shown in Figure 7.

The number of individuals that were eliminated and not evaluated in each iteration as a result of the selection operators (dominance rating, clustering distance, and elitism) in the NSGA-II algorithm is shown in Figure 8.

In the solution of the mathematical model obtained using the integer programming method, the total cost, including the cost of spoiled products, was calculated to be 1,073,197.00. The cost variation graph obtained from the implementation of the NSGA-II algorithm is presented in Figure 9. According to this graph, the total cost was significantly reduced using the NSGA-II algorithm. Although it may appear that the cost has not been fully minimized in the graph, there is a notable difference compared to the solution obtained through integer programming. Using the NSGA-II algorithm and data from the fresh product market, the cost of spoiled products has been minimized to the point of being nearly eliminated. Additionally, the algorithm has provided recommendations on how to handle products that cannot be delivered to customers. While minimizing costs, the routes for vehicles during each period were also determined. The routes for Vehicle 1 across all periods are presented in Figure 10.

To provide an in-depth analysis of the algorithm’s performance, we compared the NSGA-II results to those obtained from a traditional integer programming (IP) method—one of the most commonly employed exact approaches in multi-objective optimization. While IP methods are reliable for smaller-scale problems, they often become computationally infeasible or yield suboptimal solutions when dealing with complex multi-objective or large-scale scenarios. In contrast, NSGA-II efficiently navigated the extensive solution space of our two-echelon cold supply chain model, identifying high-quality Pareto-optimal solutions that incorporate cost minimization, product freshness, and routing efficiency. Notably, NSGA-II consistently outperformed the IP approach by significantly reducing total operational costs—primarily through more precise vehicle routing and dynamic inventory control—while simultaneously maintaining better product quality. This outcome underscores the algorithm’s enhanced ability to address the trade-offs inherent in perishable goods distribution, which directly impacts the overall profitability and service levels of cold supply chain operations.

From a cost perspective, the NSGA-II solutions achieved demonstrable reductions in transportation, storage, and spoilage costs. This improvement can be attributed to the algorithm’s adaptive search mechanism, which iteratively refines solution candidates based on both dominance ranking and crowding distance. By effectively balancing multiple conflicting objectives (e.g., route length, demand fulfillment, and freshness constraints), NSGA-II prevented cost overruns often seen in classical methods that lack robust multi-objective search capabilities. Our findings reinforce NSGA-II’s viability in complex, real-world applications, where lower total costs and minimized waste translate into significant competitive advantages.

The results obtained in this study carry important implications for both academics and practitioners in cold supply chain management. First, the remarkable cost savings and reduced spoilage rates indicate that NSGA-II can serve as a robust decision-support tool, guiding logistics managers toward optimized routing schedules, inventory management strategies, and handling protocols. By incorporating practical constraints, such as temperature maintenance, product age tracking, and vehicle capacity limitations, the proposed framework ensures that solutions are not only theoretically sound but also readily implementable in real-world distribution networks.

Second, the algorithm’s capacity to balance environmental and economic considerations highlights its potential to support sustainable cold chain operations. Minimizing spoilage and enhancing transportation efficiency both diminish the carbon footprint of perishable product distribution, a goal that is increasingly pivotal in meeting corporate social responsibility (CSR) standards. Additionally, the algorithm’s adaptability allows it to be seamlessly extended to other perishable goods sectors (e.g., pharmaceuticals, dairy products), enabling broader industry adoption. Future studies could integrate emerging digital technologies—such as blockchain-enabled traceability or IoT-based temperature monitoring—to further enhance the model’s responsiveness and resilience. Overall, the NSGA-II-driven approach demonstrated here not only advances the scholarly discussion on multi-objective optimization in cold supply chains but also offers tangible, data-driven strategies for industry professionals aiming to balance cost efficiency with product quality and sustainability goals.

5. Conclusions

In this study, we investigated a two-echelon cold supply chain optimization problem by incorporating product age as a critical decision variable and applying the Non-Dominated Sorting Genetic Algorithm II (NSGA-II). Our comparative analysis with traditional integer programming (IP) methods demonstrated NSGA-II’s superior capability in navigating the complexity of multi-objective constraints in perishable goods distribution. Specifically, NSGA-II outperformed IP in minimizing total operating costs, reducing spoilage, and maintaining robust routing and inventory strategies. In the literature, traditional solution methods do not perform well when addressing models with multiple objective functions. Therefore, to solve the model developed for a two-stage cold supply chain, the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) was used by developing a Genetic Algorithm. The planning for a 7-day period under specific demand was optimized using NSGA-II. The results obtained from this study demonstrate that NSGA-II performs better than traditional methods in minimizing costs and optimizing routing in the two-stage cold supply chain, achieving significantly better routing results.

The NSGA-II algorithm can be defined as an advanced version of the Genetic Algorithm, which is a meta-heuristic algorithm frequently used in problem-solving. Although the Genetic Algorithm is widely used for solving various problems, certain modifications were made to address its limitations, leading to the development of NSGA-II. As a result, the NSGA-II algorithm has become increasingly popular for solving complex problems that are difficult to address using traditional methods and Genetic Algorithms. In this study, the effectiveness of the NSGA-II algorithm in solving a two-stage cold supply chain problem is analyzed. Compared to traditional methods in the literature, NSGA-II achieved better results in cost minimization and routing optimization. The algorithm exhibited strong performance in minimizing overall costs and the costs associated with spoiled products.

To prevent fresh/green products that have reached a certain age from being delivered to customers, a new objective function and constraints were added to the mathematical model. With the inclusion of this objective function and constraints, the NSGA-II algorithm was used to achieve an optimal solution. This approach minimized the cost of spoiled products and eliminated waste. Fresh/green products that have reached a certain age can be repurposed in various ways. Some proposed alternatives are listed as follows:

• Discounted sales;
• Donation;
• Processed product production: Aged products can be converted into processed goods. For example, fruits can be used to produce jam or fruit juice;
• Composting and animal feed production.

This research makes several noteworthy contributions. First, it explicitly models product age within a multi-objective optimization framework, offering a more nuanced view of perishability and time-dependent product quality. Second, it validates NSGA-II’s strength in balancing conflicting objectives—cost, route efficiency, and freshness—within the unique constraints of cold chain systems. Third, it demonstrates how decision-makers can customize the algorithm for different scenarios, thereby enhancing route planning and inventory policies across multiple industries and perishable product categories. Aged or spoiled fresh/green products can be used for energy generation in biogas plants. This not only repurposes the products but also contributes to energy production. Future studies can focus on a detailed analysis of where aged products should be utilized within the model, introducing a new level of classification. This additional level would allow for strategic planning regarding how spoiled or aged products contribute to the system. As the model is further developed, the NSGA-II algorithm can be reassessed, different algorithms may be employed to achieve an optimal solution, or an integrated artificial intelligence-based system may be utilized.

For future academic studies, the dynamic nature of the developed algorithm allows it to be applied to different two-stage supply chain problems. Additionally, the problem and model created in this study for solving the problem can be used in academic research as a multi-objective optimization problem.