Optimizing Dual Distribution Scheme in Pharmaceutical Cold Chain for Cost and Carbon Emissions Reduction

Abstract

The reduction in comprehensive costs and carbon emissions in the process of trans-regional and multi-seasonal pharmaceutical cold chain logistics is a highly debated topic. To simultaneously address these issues, we propose a dual distribution method based on the vehicle routing problem (VRP), wherein either a refrigerated truck or van equipped with a refrigerated container is selected for drug delivery. To verify the effectiveness of this method, we design a comparison function to calculate the depletion of consumables and create a better distribution scheme for customer nodes in multiple temperature zones with various demands. To further optimize the model, we introduce the Grey Wolf Optimizer (GWO) to establish a large-scale neighborhood search vehicle routing model. Additionally, we utilize the greedy algorithm and an upgraded nonlinear factor to accelerate convergence speed and enhance accuracy. Ultimately, our experiments demonstrate significant advantages over existing algorithms, while providing valuable insights into economical and environmentally friendly pharmaceutical cold chain logistics.

1. Introduction

The sudden outbreak of the novel coronavirus thrust the issue of pharmaceutical cold chain logistics into the spotlight. The pharmaceutical cold chain is a crucial link in drug transport, with tremendous potential for industrial development [1]. Meanwhile, the emergence of “two votes” and the new GSP certification specification, accompanied by stricter supervision over procedures such as procurement, storage, and sale, make the direct distribution from pharmaceutical plants to dealers or hospitals seemingly more profitable and feasible. However, many logistics enterprises cannot meet the requirements of the latest pharmaceutical cold chain logistics; as a result, drug distribution tasks are performed independently by pharmaceutical manufacturers. In addition, since the pharmaceutical cold chain logistics is still in its early stage [2], the comprehensive cost and carbon emissions are not ideal. Pertaining to this condition, studies showed that the logistics activities of carrier vehicles, especially those involved in cold chain logistics, are a main factor contributing to global warming (see [3,4]). Moreover, this goes against the prevalent “low-carbon environmental protection” concept, while being unfavorable toward the battle of pollution prevention and control. Therefore, optimizing the pharmaceutical cold chain logistics path and reducing carbon emissions in logistics management became the focus of research [3]. Currently, there are three main research topics for cold chain logistics optimization. The first one is the optimization of cold chain logistics efficiency, which involves improving properties, infrastructure maintenance, and management practices [5]. The second one is the optimization of logistics decision making, such as route optimization and energy optimization [6]. The last one is about environmentally friendly logistics that reduce carbon emissions and energy consumption [7].

The contributions of this study are as follows:

(1)

Proposal of a new medicine cold chain distribution model, which is different from the existing single transport vehicle scheme.

(2)

Considerations of the influence of spatial and seasonal temperature variations on customers from distinct areas and a further refinement of transportation costs, especially for the issues of consumables in cold zones.

(3)

Acceleration of the convergence speed, while avoiding local optimal solutions via a nonlinear convergence factor, and further improvements of the traditional Grey Wolf Optimizer algorithm through the introduction of the greedy algorithm principle and the damage repair idea in the large-scale neighborhood search algorithm.

The specific research organization of this paper is as follows: Section 2 reviews the progress of the related research through a literature review; Section 3 analyzes the existing problems of the pharmaceutical cold chain and puts forward a cold chain transportation scheme with the dual distribution method; Section 4 designs the comparison function to determine the distribution mode and establishes a path optimization model; Section 5 introduces a large-scale neighborhood search algorithm, the Grey Wolf Optimizer (GWO), and our improvements of this algorithm; Section 6 demonstrates the superiority of our proposed method through an example analysis and an efficiency comparison with existing algorithms; Section 7 summarizes the research results and plans for future research.

2. Literature Review

Traditionally, cold chain logistics refers to the continuous processes employed to ensure the refrigerated preservation of perishable foodstuffs from production to consumption stages (see Zhao et al. [1]). This involves various processes such as initial refrigeration, refrigeration during post-harvest, transportation, retail distribution, and home storage. However, pharmaceutical cold chain logistics is significantly different from traditional food cold chain logistics, as improper operations could greatly impact medicine quality, making it much more vulnerable. As a result, the comprehensive cost of pharmaceuticals is considerably high (see Han et al. [3]), and optimizing pharmaceutical cold chain logistics has become a pressing topic.

The studies on traditional cold chain optimization models are as follows: Wang et al. [8] took the urban cold chain delivery problem in the case of a time-varying road network as their research object and designed a uniparental genetic algorithm to solve a multi-objective optimization model with multiple factors such as total delivery cost, customer satisfaction, and customer value to provide a reference for urban cold chain delivery decisions. Liu et al. [9] built a joint distribution–green vehicle path problem (JD–GVRP) model to deliver cold chain products by considering carbon tax policy mutual collaboration. Ma et al. [10] introduced a safety factor and set up constraints based on the equilibrium of the increase and decrease in customer demand. They established a mathematical model intended to minimize the comprehensive cost to solve the optimization problem of a logistics vehicle path with random demand. In addition, they used an adaptive genetic algorithm to solve the mathematical model to verify the practicality and innovation of the optimization model. For external factors such as road conditions, Ren et al. [11] incorporated the forbidden search operator and dynamic probabilistic selection of just models under knowledge-based business strategies into the ant colony algorithm to solve the path optimization problem of urban cold chain logistics distribution. Theophilus et al. [12] introduced a new mixed integer formula, to determine the product decay throughout the service process when the truck arrives, to improve fresh produce distribution efficiency. The situation and a temperature-controlled zone dedicated to fresh products were considered to ensure the lowest total cost of the truck delivery process.

Unlike the general cold chain industry, the whole circulation process of the cold pharmaceutical chain from production to pre-use must be within a specified temperature range; since exceeding this temperature range leads to drug failure, there is no tolerance for cargo loss rate. For this reason, Chen et al. [13] developed a combined contract coordination model with a quantity discount and revenue-sharing contract under decentralized decision making that was applied to the coordination and optimization of the pharmaceutical supply chain. Xiong et al. [14] established a mathematical model with the objective function of minimizing the distribution cost based on the new version of the product supply specification standard with the constraints of supply–demand balance and transportation volume for each cycle. They demonstrated the effectiveness of the proposed method through case studies. Setak et al. [15] proposed a three-level pharmaceutical supply chain model considering competition and integrity under uncertainty and used a two-stage stochastic approach to model the problem. A hybrid genetic algorithm was applied to solve the transportation problem of pharmaceutical companies in the Erbouche Mountains to verify the model’s validity. Meisam et al. [16] considered a four-level multi-phase supply chain including manufacturers, distribution centers, hospitals, and patients for the multi-objective optimization problem of the cold pharmaceutical chain. They proposed a multi-objective non-dominated ranking genetic algorithm to solve a multi-objective model and used the multi-objective non-dominated ranking genetic algorithm to solve a supply chain problem in the medical sector in Iran, proving that the proposed method can provide a more reliable solution in a real pharmaceutical supply chain. Zahir et al. [17] proposed a pharmaceutical supply chain network design mathematical model. They discussed a new robust likelihood optimization method to validate the effectiveness of the proposed mathematical model. Wen [18] constructed a time-reliability calculation method for pharmaceutical cold chain logistics without intersecting the minimum set of paths, established a time-reliability-based optimization model for pharmaceutical cold chain logistics networks, and designed a stochastic simulation genetic algorithm to solve the model to solve the pharmaceutical cold chain logistics distribution problem that satisfies node reliability. Zhang et al. [19] determined the number of shipments and vessel deployments in a multi-period planning environment with independent perishable product demand to reduce the cost of transporting perishable products such as in the cold pharmaceutical chain. Goodarzian et al. [20] developed two new meta-heuristic algorithms for the pharmaceutical supply chain network (PSCN) design problem and improved the social engineering optimization (ISEO) and the hybrid firefly and simulated annealing algorithm (HFFA-SA) to solve a new mixed-integer nonlinear programming model. This new mixed-integer nonlinear programming model is solved to minimize the total cost and delivery time of pharmaceutical products to hospitals and pharmacies, while maximizing the reliability of the transportation system.

In addition to the optimization of the path optimization in the pharmaceutical cold chain, many researchers simultaneously studied the refrigeration materials and new devices used in the pharmaceutical cold chain. Cui et al. [21] proposed a hybrid method based on an improved bidirectional gated cyclic unit (IBGRU) network and unscented Kalman filter (UFK), which achieves battery stability at different temperatures and has good estimation accuracy and robustness. Li et al. [22] proposed two kinds of Li-ion battery estimation models and a variety of equivalent circuit models for different practical applications and designed an improved particle swarm optimization algorithm (IPSO) to optimize the model. The experimental results show that the proposed method is more flexible for Li-ion battery health state estimation. Cui et al. [23] proposed a hybrid method based on the CNN-BWGRU network to solve the problem of the accurate estimation of the battery State of Charge (SOC), which provides useful guidance for the safe and stable operation of batteries in the natural environment. Liu et al. [24] designed a time convolution model of a supercapacitor to more accurately predict its remaining service life. At the same time, they used the Adam convolution algorithm to optimize the time network parameter adjustment process; through the experiment, it is concluded that the time convolution network model has stronger robustness and higher accuracy for the prediction of the residual service life of the supercapacitor. Taher et al. [25] assessed refrigerators in the air thermal behavior of two kinds of situations; throughout the operation cycle (charging and discharging) used in the different phase change materials, the results show that their study system of the integrated PCM can significantly reduce the consumption of cooling needed for when the door opened, thereby reducing greenhouse gas emissions and providing significant energy savings. Leungtongkum et al. [26] analyzed the effects of the melting point and position of phase change materials (PCM), the quality and load of PCM, insulation materials, external temperature, and other conditions on product temperature and energy consumption and applied PCM in refrigeration equipment. Zhou et al. [27] built a high- and low-temperature test chamber test bench, studied the performance comparison of the R404A with or without a heat recovery cycle, studied the refrigeration cycle of the new refrigerant R4484, and proved through experiments that the stability of the R4484 refrigeration system under high-temperature conditions was due to the R404A refrigeration system.

In summary, the domestic and international research on pharmaceutical cold chain distribution still has the following shortcomings:

(1)

The choice of distribution mode is single, and only refrigerated trucks with refrigeration functions are considered for transportation.

(2)

The existing research mostly limits the transportation season to spring or autumn, i.e., the external temperature of the truck during the transportation is determined to be in the range of 10–20 °C, without considering seasonal temperature variation.

(3)

The truck transportation area is located in the same temperature zone, without considering the distribution environment across multiple temperature zones, so it is more instructive to consider the new distribution model in pharmaceutical cold chain logistics and the spatial and seasonal temperatures variations according to the actual customer demand.

3. Problem Analysis

3.1. Problem Description

Pharmaceutical cold chain logistics require more stringent preservation conditions during transportation compared to traditional cold chain logistics. To effectively reduce the total cost and carbon emissions of these logistics, this paper takes into consideration both seasonal and spatial temperature variations. At the beginning of our logistics process, hospitals and first-tier suppliers issue orders. The supply chain department of the pharmaceutical company then determines the mode of transportation according to the order, taking into account the temperature zone and current season of the customer node. Once the decision is made, the goods are transported either by a refrigerated truck or van equipped with a refrigerated box from the pharmaceutical factory to each customer’s location. The specific flowchart is shown in Figure 1.

3.2. Model Assumptions

To ensure that the problem is solved with the minimum comprehensive cold chain logistics cost under the known customer demand and to maintain the maximum vehicle capacity with fixed cost and carbon emissions in each region, for the actual problem, the following assumptions are made.

(1)

The information of all customers is known. The locations of the pharmaceutical plant and customers are fixed, and the delivery demand of each customer is provided.

(2)

Each customer’s order cannot be divided into several delivery missions, and each customer can only be served for once.

(3)

Road congestion or traffic jams are left out, so the transport vehicle can deliver at a constant speed and reach the area specified by the customer.

(4)

Excluding the carbon emissions caused by the operation of the pharmaceutical plant, only the carbon emissions generated during transportation are taken into account.

(5)

The maximum weight of each customer’s demand cannot exceed the loading capacity of the carrier vehicle.

(6)

Every carrier vehicle departs from the pharmaceutical factory and eventually returns to it after passing through each customer point.

(7)

The pharmaceutical company has sufficient stock to cope with the demand of every customer.

(8)

There is no malfunction of the Probability Density Function (PDF) thermometer and no deterioration of the drugs during transportation.

3.3. Model Parameters

For a more intuitive representation of the model, the tables of the setup parameters and variables are shown in

Table 1. Parameters and variable symbol description table.

Symbols	Meaning	Symbols	Meaning
I	Set of customers served by the pharmacy, I = {1, 2,…, I}	$f_{ce}$	Carbon costs
$N_i$	Set of customers in region i, Z = {1, 2,…, N}	$s{0}_{K_R}$	Gasoline emission rate of refrigerated trucks at no load
$K_R$	Reefer collection, K1 = {1, 2,…, R}	$s{f}_{K_R}$	Gasoline emission rate when refrigerated trucks are fully loaded
$K_V$	Van collection, K2 = {1, 2,…, V}	$f_C$	Cost of using a single disposable PDF thermometer
$K_t$	Reefer collection, K2 = {1, 2,…, T}	$c_{ij}$	Distance between clients i, j
$q_{K_r}$	Maximum loading capacity of refrigerated trucks	cer	Carbon emission rate
$q_{K_V}$	Maximum loading capacity of vans	$s{0}_{K_V}$	Gasoline emission rate when van is empty
$f_{K_R}$	Fixed usage costs of refrigerated trucks	$s{f}_{K_V}$	Gasoline emission rate for vans at full load
$f_{K_R}^n$	Depreciation cost of refrigerated trucks	$T_i$	Total demand in region i
$c_{K_R}$	Reefer truck transportation cost per distance	$q_i$	i-th customer demand
$f_{K_V}$	Fixed costs of vans	$u_{K_R}$	Real-time load of refrigerated trucks
$f_{K_V}^n$	Depreciation cost of vans	$u_{K_V}$	Real-time load of vans
$c_{K_V}$	Van transportation cost per unit distance	$Q_1$	Set refrigerated truck transport costs within the area
$x_q$	Customer demand	$Q_2$	Cost of refrigerated container transport within the specified area by van
$f_{Kj}$	Fixed cost of reefer	S	Branch eliminates the constraint set

and

Table 2. Decision variables table.

Symbols	Meaning
$x_t$	Transport mode decision variables
$x_c$	Decision variable for coolant storage amount
$x_i$	Distribution mode selection coefficient: $T_i$ = 1 if use refrigerated trucks for distribution; $T_i$ = 0 if use refrigerated boxes carried by vans for distribution
$x_{ij}^{K_R}$	Binary variable: equal to 1 if the refrigerated truck was driven from customer i to customer j; otherwise 0
$x_{ij}^{K_V}$	Binary variable: equal to 1 if the van is driven from customer i to customer j; otherwise 0
$y_i^{K_R}$	Binary variable: equal to 1 if the requirement of customer i is fulfilled by refrigerated car $K_R$; otherwise 0
$y_i^{K_V}$	Binary variable: equal to 1 if the requirement of customer I is fulfilled by van $K_V$; otherwise 0

, respectively.

3.4. Analysis of Decision Variables

In order to reduce the costs and carbon emissions in the distribution process, we propose a dual distribution model that uses both refrigerated trucks and van-loaded refrigerated containers based on the specific demands of each regional customer point. To determine the most cost-efficient mode of distribution for each customer point, we calculate the costs associated with each option. For example, the total transportation cost for a 1013 km route using a refrigerated truck is RMB 3054, which includes the fixed costs as well as the fuel costs and carbon taxes. The additional costs include the use of reefer boxes, disposable PDF thermometers, refrigerant storage, and vehicle transportation, which total RMB 54.6 per piece of a drug. Based on this calculation, we select refrigerated trucks for transportation when the demand exceeds 56 pieces and vans with refrigerated boxes when the demand is less than or equal to 56 pieces.

We set the cost of transporting refrigerated trucks in the region as $Q_1$ and the cost of transporting refrigerated containers in vans as $Q_2$; see Equations (1) and (2), respectively.

$$Q_1={\displaystyle \sum _{N_i=1}^N(f_{K_R}^n}+f_{K_R})+c_{zi}{c}_{K_R}$$

(1)

$$Q_2={\displaystyle \sum _{N_i=1}^N(f_{K_V}^n}+f_{K_V})+c_{zi}{f}_{K_V}+x_{zq}(f_C+f_P+f_{K_f})$$

(2)

The choice of distribution method is made after calculating the demand cost in the region, and the decision variable $x_i$ is expressed as follows:

$$T_i=\left\{\begin{array}{c}1\hspace{1em}{Q}_1-Q_2\ge 0\\ 0\hspace{1em}{Q}_1-Q_2<0\end{array}\right.$$

$$x_i=\left\{\begin{array}{c}{\mathrm{T}}_{\mathrm{i}}=1\hspace{1em}\mathrm{Use} \mathrm{of} \mathrm{refrigerated} \mathrm{trucks} \mathrm{for} \mathrm{distribution}\\ {\mathrm{T}}_{\mathrm{i}}=0\hspace{1em}\mathrm{Use} \mathrm{of} \mathrm{vans} \mathrm{with} \mathrm{refrigerated} \mathrm{containers} \end{array}\right.$$

The pharmacy number is set to 0, and each customer node is denoted by i, j (i, j = 1, 2, 3, …, N). The decision variables $x_{ij}^{K_R}$, $x_{ij}^{K_V}$, $y_i^{K_R}$, and $y_i^{K_V}$ are denoted as follows:

$$x_{ij}^{K_R}=\left\{\begin{array}{l}1\hspace{1em}\mathrm{Reefer} \mathrm{truck} \mathrm{travels} \mathrm{from} \mathrm{customer} \mathrm{i} \mathrm{to} \mathrm{customer} \mathrm{j}\\ 0\hspace{1em}\mathrm{otherwise}\end{array}\right.$$

$$x_{ij}^{K_V}=\left\{\begin{array}{l}1\hspace{1em}\mathrm{The} \mathrm{van} \mathrm{travels} \mathrm{from} \mathrm{customer} \mathrm{i} \mathrm{to} \mathrm{customer} \mathrm{j}\\ 0\hspace{1em}\mathrm{otherwise}\end{array}\right.$$

$$y_i^{K_R}=\left\{\begin{array}{l}1\hspace{1em}\mathrm{The} \mathrm{need} \mathrm{of} \mathrm{customer} \mathrm{i} \mathrm{is} \mathrm{fulfilled} \mathrm{by} \mathrm{reefer} {\mathrm{K}}_R \\ 0\hspace{1em}\mathrm{otherwise}\end{array}\right.$$

$$y_i^{K_V}=\left\{\begin{array}{l}1\hspace{1em}\mathrm{The} \mathrm{need} \mathrm{of} \mathrm{customer} \mathrm{i} \mathrm{is} \mathrm{fulfilled} \mathrm{by} \mathrm{van} {\mathrm{K}}_V \\ 0\hspace{1em}\mathrm{otherwise} \end{array}\right.$$

Considering the regional and seasonal temperature variance, carrier vans need to place a corresponding quantity of refrigerants. The quantity of refrigerants required is influenced by the annual average and seasonal temperature variations. To clarify this concept, the domestic annual average isotherm map [28] (Figure 2) and the refrigerant storage use table (

Table 3. Refrigerant storage use table.

	Season/Month	December of the Previous Year to February of the Current Year	March to April of This Year or October to November of the Current Year	May to September of the Current Year
Average Annual Temperature
19–24		7	8	10
13~18		7	8	9
9–12		6	7	9
5–8		6	6	8
0–4		6	6	8

) are presented below. The temperature data are collected from the China Meteorological Center Yearbook [29], and the isotherm is generated by ArcMap.

Taking the values of 0, 1, 2, 3, 4, and 5 for each season of refrigerant storage use in each region in the table, the decision variable $X_c$ is expressed as follows:

$$X_c=\left\{\begin{array}{cc}0& \mathrm{Use} \mathrm{of} 6 \mathrm{blocks} \mathrm{of} \mathrm{refrigerant} \mathrm{accumulator} \\ 1& \mathrm{Use} \mathrm{of} 7 \mathrm{blocks} \mathrm{of} \mathrm{refrigerant} \mathrm{accumulator} \\ 2& \mathrm{Use} \mathrm{of} 7 \mathrm{blocks} \mathrm{of} \mathrm{refrigerant} \mathrm{accumulator} \\ 3& \mathrm{Use} \mathrm{of} 9 \mathrm{blocks} \mathrm{of} \mathrm{refrigerant} \mathrm{accumulator} \\ 4& \mathrm{Use} \mathrm{of} 10 \mathrm{blocks} \mathrm{of} \mathrm{refrigerant} \mathrm{accumulator} \end{array}\right.$$

If we use six blocks of refrigerant accumulators, we cover the inner walls of the refrigerator and the upper part and lower part of the medicine. For seven blocks, we add one block to the topside of the refrigerator. For eight blocks, we put two accumulators in the slots of the box and around the whole circumference of the medicine. For nine blocks, we add one block to the topside of the refrigerator. Furthermore, for 10 blocks, we place accumulators in all the slots around the refrigerator and around the whole circumference of the medicine.

As mentioned previously, the number of refrigerant accumulators must be adjusted to accommodate temperature fluctuations. If customers are located in multiple temperature zones, changes in refrigerant storage must be taken into account. Specifically, when a van with a refrigerated box travels from a low-temperature zone to a high-temperature zone, an additional refrigerant accumulator must be added. On the other hand, if the van travels from a high-temperature zone to a low-temperature zone, no additional accumulators are necessary.

3.5. Cost Analysis

3.5.1. Fixed Costs ($Z_1$)

To provide pharmaceutical plant services to each customer point, either refrigerated trucks or vans with refrigerated boxes are used. This method incurs comprehensive costs, including fixed and depreciation costs. If vans with refrigerated boxes are utilized, then additional expenses are incurred for the refrigerated box, including depreciation costs, fixed refrigerant usage costs, and disposable PDF temperature agent usage costs. (The new version of GSP implementation requires thermometer tracking for each box). These costs are shown in Equation (3).

$$Z_1={\displaystyle \sum _{K_1=1}^R{\displaystyle \sum _{i=0}^I{x}_i{x}_{oi}^{K_R}\left(f_{K_R}^n+f_{K_R}\right)}}+{\displaystyle \sum _{K_2=1}^V{\displaystyle \sum _{i=0}^I{\displaystyle \sum _{p=1}^I{\displaystyle \sum _{f=1}^{}{x}_i\left[x_{0i}^{K_V}\left(f_{K_V}^N+f_{K_V}\right)+\left(x_c{f}_c\right)+f_f+f_{KF}+f_P\right]}}}}$$

(3)

where $x_{0i}^{K_R}$, $x_{0i}^{K_V}$ indicates 1 when the pharmacy uses refrigerated trucks $K_R$ or vans $K_V$, otherwise it indicates 0; $x_i$ is the choice of refrigerated trucks or vans for distribution, and $x_c$ indicates the current season and the amount of refrigerant storage used.

3.5.2. Transportation and Carbon Emissions Costs ($Z_2$)

The cost of this segment is divided into two stages: the transportation vehicle transportation cost $z_1$ and the carbon emissions cost $z_2$, where $z_1$ is the product of the distance traveled by the transportation vehicle and the unit transportation cost of the transportation vehicle, as shown in Equation (4).

$$z_1=c_{ij}\times c_K$$

(4)

The calculation of carbon emissions cost $z_2$ in this paper is based on the fuel conversion factor in the Greenhouse Gas Inventory Protocol [30], which is 2.676 kg $C{O}_2$/L for analysis, and we set the carbon dioxide emissions as the product of the fuel consumption and fuel conversion factor. Meanwhile, the fuel consumption rate is determined by the distanced traveled and the loading condition of the carrier vehicle, as shown in Equation (5).

$$\mathrm{Fuc}=s{0}_K+\left(\frac{s{f}_K-s{0}_K}{q_K}\right)c_{ij}$$

(5)

Therefore, the cost of the carbon emissions is

$$\begin{array}{l}{Z}_2=\left[{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{K_1=1}^{R}cer\left.\left(s{0}_{K_R}+\left.\frac{s{f}_{K_R}-s{0}_{K_R}}{q_{K_R}}{u}_{K_R}\right)x_{0i}^{K_R}{c}_{01}\right.\right]}{f}_{ce}{x}_i}\right.+{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{K_1=1}^R{c}_{0i}{c}_{K_R}}}\\ +\left[{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{K_2=1}^{V}cer\left.\left(s{0}_{K_V}+\left.\frac{s{f}_{K_V}-s{0}_{K_V}}{q_{K_V}}{u}_{K_V}\right)x_{0i}^{K_V}{c}_{01}\right.\right]}{f}_{ce}{x}_i}\right.+{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{K_2=1}^V{c}_{0i}{c}_{K_V}}}\end{array}$$

(6)

4. Building a Low-Carbon Drug Cold Chain Logistics Path Optimization Model

Based on the above analysis, a mathematical model is constructed to optimize the pharmaceutical cold chain logistics path with a minimum comprehensive cost including the fixed cost, transportation, and the carbon emissions cost.

$$\begin{array}{l}\min Z=Z_1+Z_2=\\ {\displaystyle \sum _{k=1}^K{\displaystyle \sum _{i=0}^N{x}_i{x}_{oi}^{K_R}\left(f_{K_R}^n+f_{K_R}\right)}}+{\displaystyle \sum _{k=1}^K{\displaystyle \sum _{i=0}^I{\displaystyle \sum _{p=1}^I{\displaystyle \sum _{f=1}^{}{x}_i\left[x_{0i}^{K_V}\left(f_{K_V}^N+f_{K_V}\right)+\left(x_c{f}_c\right)+f_f+f_{KF}+f_P\right]}}}}+\\ \left[{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{K_1=1}^{R}cer\left.\left(s{0}_{K_R}+\left.\frac{s{f}_{K_R}-s{0}_{K_R}}{q_{K_R}}{u}_{K_R}\right)x_{0i}^{K_R}{c}_{01}\right.\right]}{f}_{ce}{x}_i}\right.+{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{K_1=1}^R{c}_{0i}{c}_{K_R}}}+{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{K_2=1}^V{c}_{0i}{c}_{K_V}}+}\\ \left[{\displaystyle \sum _{i=1}^I{\displaystyle \sum _{K_2=1}^{V}cer\left.\left(s{0}_{K_V}+\left.\frac{s{f}_{K_V}-s{0}_{K_V}}{q_{K_V}}{u}_{K_V}\right)x_{0i}^{K_V}{c}_{01}\right.\right]}{f}_{ce}{x}_i}\right.\end{array}$$

(7)

$$\mathrm{s}.\mathrm{t}. {\displaystyle \sum _{i=1}^I{\displaystyle \sum _{k=1}^R{x}_{ij}^k=1}}\hspace{1em}\mathrm{i}, \mathrm{j}\in \left\{1,2,\dots ,I\right\}$$

(8)

$${\displaystyle \sum _{i=1}^I{\displaystyle \sum _{k=1}^K{x}_i^k=1}}\hspace{1em}\mathrm{i}\in \left\{1,2,\dots ,I\right\}$$

(9)

$${\displaystyle \sum _{i=1}^I{\displaystyle \sum _{k=1}^K{q}_i{x}_{i0}^{K_R}\le q_{K_R}}}$$

(10)

$${\displaystyle \sum _{i=1}^I{\displaystyle \sum _{k=1}^K{q}_i{x}_{i0}^{K_V}\le q_{K_V}}}$$

(11)

$$y_0^{\alpha }{\displaystyle \sum _{i=1}^I{q}_i}\le Q_k$$

(12)

$${\displaystyle \sum _{i=1}^I{\displaystyle \sum _{k=1}^K{x}_{0i}^{K_R}=1}}\hspace{1em}\mathrm{i}\in \left\{1,2,\dots ,I\right\}$$

(13)

$${\displaystyle \sum _{i=1}^I{\displaystyle \sum _{k=1}^K{x}_{0i}^{K_V}=1}}\hspace{1em}\mathrm{i}\in \left\{1,2,\dots ,I\right\}$$

(14)

$$x_{0i}^{K_R}\left(x_{0i}^{K_R}-1\right)=0$$

(15)

$$x_{0i}^{K_V}\left(x_{0i}^{K_V}-1\right)=0$$

(16)

$${\displaystyle \sum _{i\in S}{\displaystyle \sum _{j\in s}{x}_{ij}^{K_R}\le \left|S\right|-1\hspace{1em}S\subseteq I,K_R\in R}}$$

(17)

$${\displaystyle \sum _{i\in S}{\displaystyle \sum _{j\in s}{x}_{ij}^{K_V}\le \left|S\right|-1\hspace{1em}S\subseteq I,K_V\in V}}$$

(18)

Equation (7) is the objective function of this paper, including the fixed cost, transportation, and the carbon emissions cost; Equation (8) indicates that each customer can only be served once by the carrier vehicle sent from the pharmaceutical plant; Equation (9) indicates that each customer can only be served by one carrier vehicle; Equations (10) and (11) indicate the limits of the vehicle load capacity of the two different distribution methods; Equation (12) indicates that the total demand of customers on any distribution route cannot exceed the maximum load capacity of the carrier vehicles; Equations (13) and (14) indicate that both distribution methods are guaranteed to start from the pharmaceutical plant and eventually return to it; Equations (15) and (16) indicate the 0–1 variable constraints; Equations (17) and (18) indicate the elimination of distribution routes that constitute incomplete routes.

5. Grey Wolf Algorithm

5.1. Principle of Traditional Grey Wolf Algorithm

The vehicle path problem [31] is already proven to be an NP-hard problem, which is significantly enormous in size and almost impossible to be implemented by exact algorithms. In this paper, we attempt to solve this problem by optimizing an existing algorithm, the grey wolf algorithm. The Grey Wolf Optimizer (GWO) is a pack optimization algorithm, proposed by Mirjalili et al. in 2014 [32], which performs search optimization by simulating the predatory behavior of wolf packs. The algorithm population was divided into α, β, δ, and ω, and the priority was set. Alpha has the highest priority and is responsible for guiding the search direction of the algorithm. The priority of β, δ, and ω successively decreases, and the population hierarchy diagram is shown in Figure 3.

To simulate the social hierarchy of a grey wolf pack, we set a unit α, a unit β, and a unit δ, and they will guide the other units to search for the target. During the search process, the position of unit α is $X_{\alpha }$ ($X_{\alpha 1}, X_{\alpha 2}, \cdots X_{\alpha n}$). The position of unit β is $X_{\beta }$ ($X_{\beta 1}, X_{\beta 2}, \cdots X_{\beta n}$). The location of unit δ is $X_{\delta }$ ($X_{\delta 1}, X_{\delta 2}, \cdots X_{\delta n}$). Unit i could be an individual in α, β, δ, and ω with a current position that is $X_i$ ($X_{i1}, X_{i2}, \cdots X_{in}$). Unit α directs unit i to move to the next position, which is $X_{\alpha i}$ ($X_{\alpha i1}, X_{\alpha i2}, \cdots X_{\alpha in}$). The formula of the calculation is as follows:

$$D_{\alpha N}=\left|C_1{X}_{\alpha N}-X_{iN}\right|$$

(19)

$$X_{\alpha iN}=X_{\alpha N}-A_1{D}_{\alpha N}$$

(20)

Equation (19) represents the distance between units and their target, and Equation (20) indicates the location of the command given by the alpha to unit i at the Nth iteration. A and C are coefficients, and the formula is calculated as follows:

$$A_1=2a{\mathrm{r}}_1-a$$

(21)

$$C_1=2r_2$$

(22)

where a is the convergence factor, which decreases linearly from 2 to 0 as the number of iterations increases, and $r_1$, $r_2$ are random numbers from 0 to 1.

In the process of iteration, units β and δ lead the population to surround the target under the leadership of unit α. Assuming that units α, β, and δ know the potential location of the target and direct the other units to update their positions according to the positions of α, β, and δ, to gradually approach the target, the position update diagram is shown in Figure 4.

The way a unit approaches the target is as follows:

$$D_{\beta N}=\left|C_2{X}_{\beta N}-X_{iN}\right|$$

(23)

$$D_{\delta N}=\left|C_3{X}_{\delta N}-X_{iN}\right|$$

(24)

where $C_1=C_2=C_3=2r_2$, and the next potential position of the population guided by the unit α is

$$X_{\beta iN}=X_{\beta N}-A_2{D}_{\beta N}$$

(25)

$$X_{\delta iN}=X_{\delta N}-A_3{D}_{\delta N}$$

(26)

Equation (25) represents the next potential position of the unit i guided by unit β, and Equation (26) represents the next potential position of unit i guided by unit δ, where $A_1=A_2=A_3=2a{r}_1-a$; then, the next position of the unit i guided by the unit α, β and δ $X_i$ is calculated as

$$X_{iN}=\frac{X_{\alpha iN}+X_{\beta iN}+X_{\delta iN}}{3}$$

(27)

5.2. Improved Design of the Grey Wolf Algorithm for Large-Scale Neighborhood Search

The traditional Grey Wolf Optimizer (GWO) suffers from several issues such as a slow convergence speed in later stages of the iteration, being prone to converge for optimal local solutions, and poor accuracy. In order to address these issues, we introduce a nonlinear convergence factor to accelerate the convergence speed during the later stages of the iteration. Additionally, we incorporate a large-scale neighborhood search algorithm along with destruction and recovery operators in the local search process. This allows the population to adjust their positions toward more optimized directions, ultimately leading to improved accuracy but avoiding local convergence. The flow chart of the improved grey wolf algorithm is shown in Figure 5.

5.2.1. Nonlinear Convergence Factor

The convergence factor in the traditional GWO algorithm decreases linearly from 2 to 0 with each iteration. However, since the search process in the GWO algorithm is non-linear, the linear convergence factor is unable to fully reflect the change process of the grey wolf position. This can result in slow convergence and an increased likelihood of falling into local optima. Therefore, we propose a nonlinear convergence factor that varies according to the following formula.

$$\left\{\begin{array}{l}a=1+{\left[\frac{\cos (t-1)\pi }{T-1}\right]}^n\hspace{1em}t\le \frac{1}{2}T\\ a=1-{\left[\frac{\cos (t-1)\pi }{T-1}\right]}^n\hspace{1em}\frac{1}{2}T<t\le T \end{array}\right.$$

(28)

where t is the current number of iterations, T is the maximum number of iterations, and n is the decreasing exponent.

According to Mirgalili’s proposed theory [34], the grey wolf population performs a global search when $\left|\mathrm{A}\right|>1$ and a local search when $\left|\mathrm{A}\right|<1$. The improved nonlinear convergence factor decreases slowly at the beginning of the iteration, and it maintains population diversity and improves search efficiency while preventing the algorithm from falling into local optimal solutions. In the later iterations, the reduction speed is rapid, thus improving the accuracy of the exact local search and accelerating the global convergence of the algorithm.

5.2.2. Encoding and Decoding

In the process of encoding and decoding, if the number of customer nodes is m, and a total of n regions are assigned for distribution, then the length of the individual grey wolf code is m + n. Assuming that the number of customers is seven and a total of three regions are assigned, then the number of customer points to be distributed in each region is 3, 2, and 2. The IGWO algorithm encoding diagram is shown in Figure 6.

The six digits on the left side of the dotted line are the six customer nodes arrangement, and the three digits on the right side indicate the number of customer points to be distributed in each region. The decoding is the order of the customer node arrangement on the left side of the dashed line, and the distribution is performed sequentially for the three regions, i.e., region 1: 0 1 3 2 0, region 2: 0 7 6 0, and region 3: 0 4 5 0.

5.2.3. Population Initialization

In order to improve the accuracy of the grey wolf algorithm, the principle of the greedy algorithm is used to construct the initial solution to avoid the deviation of the position where the α wolf has located too large an algorithm into the local optimum, and all the customer nodes except the pharmacy are sorted to find the nearest node j′ from customer node j, until all the customer nodes are traversed to complete the initialization operation.

5.2.4. Grey Wolf Location Update

Mimicking the crossover operation in the genetic algorithm, the location update of the grey wolf individuals is optimized, assuming that the current grey wolf individual is ${\mathrm{X}}_G$, the grey wolf individual α is ${\mathrm{X}}_{\alpha G}$, the grey wolf individual β is ${\mathrm{X}}_{\beta G}$, and the grey wolf individual δ is ${\mathrm{X}}_{\delta G}$, then the grey wolf individual ${\mathrm{X}}_G$ is updated according to the following equation.

$${\mathrm{X}}_{G+1}=\left\{\begin{array}{l}cross\left[{\mathrm{X}}_G,{\mathrm{X}}_{\alpha G}\right], \mathrm{r}\le \frac{1}{3}\\ cross\left[{\mathrm{X}}_G,{\mathrm{X}}_{\beta G}\right],\frac{1}{3}<r\le \frac{2}{3}\\ cross\left[{\mathrm{X}}_G,{\mathrm{X}}_{\delta G}\right], \mathrm{r}>\frac{2}{3}\end{array}\right.$$

(29)

where the cross is the crossover operation of two grey wolf individuals, and r is a random number between [0, 1].

Assuming that the number of customers is 7, the pharmacy number is 0, the distribution area is divided into 3, and the deliveries start from the pharmacy and eventually return to the pharmacy, there are two grey wolf individuals, ${\mathrm{X}}_1$ and ${\mathrm{X}}_2$; ${\mathrm{X}}_1$: 1, 3, 2, 7, 6, 4, 5 and ${\mathrm{X}}_2$: 2, 7, 5, 4, 3, 1, 6. At the same position in ${\mathrm{X}}_1$ and ${\mathrm{X}}_2$, a gene string is selected and crossed to form a new grey wolf individual, the problematic (genetically identical) gene in the new individual is repaired, and the operation diagram for the crossover of these two grey wolf individuals is shown in Figure 7.

5.2.5. Local Search Operations

Letting the population of the grey wolf update their location globally, take grey wolf α, β, δ as the object of the local search operation, apply the idea of damage repair in a large-scale neighborhood search algorithm for local search, and assume that the number of customer points is m, divided into n regions; then, the damage operator removes k ($1\le \mathrm{k}\le k_{\max }$) customer nodes from each of s ($2\le s\le m$) adjacent paths of the current solution to obtain the set R of removed customer nodes, forming the destructive solution $S_d$, where $k_{\max }$ is formulated as

$$k_{\max }=\min \left(\left|\overline{p_n}\right|,\left|p_c\right|\right)$$

(30)

where $\left|\overline{p_n}\right|$ is the average value of the regional distribution customer nodes of the current solution, and $\left|p_c\right|$ is the number of the regional distribution customer nodes of the current solution.

Inserting a client node in R back to a location in $S_d$, the insertion cost of inserting the client node back to $C_I$ is obtained by subtracting the maximum of the path distance of the client node from all the path distances of $S_d$. The resulting insertion cost is sorted from smallest to largest ($C_{I1},C_{I2},C_{I3},\cdots ,C_{In}$), and the regret value $V_S$ is $C_{I2}-C_{I1}$.

The specific steps to repair the operator are

(1)

Initialize the repaired solution $S_d$.

(2)

If R is non-empty, advance to step (3); otherwise, advance to step (6).

(3)

Calculate the number of client nodes in R, and calculate the regret value of each interpolated client node $V_S$.

(4)

Identify the maximum $V_S$ and insert the unsettled client point back into the $S_d$ insertion cost minimum position.

(5)

Update the set R and return to step (2).

(6)

End the repair process and output.

The pseudo-code for local search operation [35] and the pseudo-code for IGWO are shown in Algorithms 1 and 2 respectively.

Algorithm 1 Local Search Operation

Input: Current solution S, Number of customer nodes n, Number of regions m

Output: The removed customer node set R and the broken solution S*

1 $R\leftarrow \varnothing$ The set of customer nodes that are removed, initially empty

2 $T\leftarrow \varnothing$ The set of broken paths, initially empty

3 Determine the number of routes to remove $k\leftarrow Random[2,i]$

4 Select a customer node $i_{seed}$ at random from the current solution S

5 $\mathrm{for} i\in adj(i_{seed}) \mathrm{and} \left|T\right|<k \mathrm{do}$

6 $\mathrm{if} tour(i)\notin T \mathrm{then}$

7 $l\leftarrow Random\left[1,\min \left(\left|\overline{t_1}\right|,\left|t_2\right|\right)\right]$

8 $l\leftarrow I\cup \mathrm{removeSelected}(i,l)$

9 $T\leftarrow T\cup tour(i)$

10 end if

11 end for

12 Remove the form in set R from the solution S to form the failure solution S*

13 Return R and S*

Where $adj(i_{seed})$ represents the city sequence with the distance from customer node $i_{seed}$ sorted from large to small; $i_{seed}$ is the first element in the sequence; $tour(i)$ indicates the path where customer node i resides; And $\mathrm{removeSelected}(i,l)$ means to randomly remove l consecutive customer nodes from $tour(i)$, including customer node i.

Algorithm 2 IGWO

Input: Population size I, Dimension D, The maximum number of iterations $T_{\max }$, The objective function $f\left(x\right)$

Output: Optimal solution

1 The greedy algorithm principle is used to initialize the population, The population individual is $X_i=\left(X_i^1,X_i^2,\dots ,X_i^I\right)$

2 Calculate all individual fitness values

3 In order of fitness value from largest to smallest

4 Individuals α, β and δ corresponding to the first three fitness values were selected. Their location information was $X_{\alpha },X_{\beta },X_{\delta }$

5 Calculate $a,A_1,C_1$ by Equations (21), (22), and (28)

6 for $i=1:I$

7 Formula (27) was used to obtain the position difference between individual grey wolves and each leader Wolf

8 The new position updating Equation (29) was used to update the individual position of grey wolves and update the value of α, β and δ

9 end

10 Calculate the fitness value of each individual grey Wolf and update the historical optimal position X’ of $X_{\alpha },X_{\beta } \mathrm{and} X_{\delta }$

11 end

6. Example Solution and Analysis

6.1. Numerical Experiment Settings

The program was written in Matlab 2016a 9.0 and performed on a Windows 10 computer with an AMD Ryzen 7 4800H CPU @2.90 GHZ processor, 16 G RAM, and 64-bit OS. The simulation is based on the supply chain in December 2020 of a pharmaceutical plant in North China. There are 46 customer nodes in the supply chain, and it passes by nine provinces covering three temperature zones. The distribution of the customer nodes is demonstrated in a square area of 100 × 100 under the conversion ratio of 1:10 km, as shown in Figure 8. The temperature data and customer nodes data are collected from the China Meteorological Center Yearbook and the China Cold Chain Logistics Development Report (2021) [36].

In the simulation area, the coordinates of the pharmaceutical factory are (58.3, 69.4), and the logistics process is divided into two distribution modes: cold chain vehicle transportation and van-loaded reefer transportation. The diesel consumption of cold chain trucks is 0.12 L/km when empty and 0.43 L/km when fully loaded. The diesel consumption of vans is 0.11 L/km when empty and 0.18 L/km when fully loaded. The unit price of diesel is 7.18 RMB/liter; the carbon emissions are in accordance with the Greenhouse Gas Inventory Protocol fuel conversion factor: 2.676 kg/L; the carbon tax rate is 30 RMB/t; the fixed cost of a van is RMB 50, the fixed cost of a reefer truck is RMB 70; the depreciation cost of reefer box use is 1.5 RMB/use; the PDF thermometer unit price is 24.8 RMB/use, according to each customer’s location in the temperature zone and simulation when in the season; and the single refrigerant storage use cost is RMB 2. The customer nodes and demand table is shown in

Table 4. Customer nodes and demand table.

Customer Number	Coordinate Location	Demand/Piece	Customer Number	Coordinate Location	Demand/Piece	Customer Number	Coordinate Location	Demand/Piece
1	(91.5, 94.7)	7	17	(22.9, 66.4)	6	33	(66.3, 34.7)	21
2	(85.9, 88.2)	5	18	(21.9, 72.3)	5	34	(46.9, 34.6)	3
3	(85.9, 88.2)	13	19	(78.1, 84.7)	8	35	(43, 34.1)	5
4	(83.2, 81)	3	20	(37.5, 52.3)	6	36	(49.5, 69.7)	10
5	(88.2, 82.2)	9	21	(49.4, 55.1)	8	37	(67.7, 29.3)	11
6	(81.4, 75.5)	5	22	(49.2, 52.3)	12	38	(65.2, 25.8)	9
7	(77.3, 71.2)	8	23	(66.4, 45.6)	6	39	(57.6, 29.5)	8
8	(68.7, 67.9)	5	24	(70.3, 47.1)	3	40	(46.4, 29.5)	16
9	(66.3, 77.9)	3	25	(76.4, 47.7)	14	41	(38.7, 29.4)	23
10	(49.6, 75.2)	9	26	(49.7, 44.4)	8	42	(43.9, 23.6)	11
11	(60.4, 66.1)	7	27	(84.4, 44.8)	15	43	(29.7, 41.3)	27
12	(64.4, 64)	11	28	(65.5, 42.4)	13	44	(50.7, 39.5)	4
13	(58.5, 61.9)	3	29	(49.4, 40.5)	9	45	(53.1, 35.6)	6
14	(54.3, 61.5)	6	30	(41.9, 41)	13	46	(40, 76.5)	8
15	(40.8, 69.8)	7	31	(62.8, 36.8)	17
16	(33.3, 73.9)	11	32	(73.8, 35.8)	7

.

6.2. Distribution Route Optimization

This study focused on enhancing the grey wolf algorithm to partition customer nodes and optimize transportation routes. The resulting optimization divided 46 customer nodes into seven regions for more efficient distribution. Visualizations of the route optimization for each region and the iterative optimization process were created using the algorithm. Furthermore, a set of examples consisting of 202 samples was added to provide evidence for the effectiveness of the algorithm. The figures illustrating the route optimization and iteration diagrams for each region can be found in Figure 9.

The specific distribution van or refrigerated truck travels to all the customer nodes and finally heads back to the pharmaceutical factory. For example, the distribution order of region 1 is as follows.

Region 1: 0-7-6-4 5-3-1-2-19-9-0;

Thus, the carbon emissions and the comprehensive cost of the logistics process are the lowest. Moreover, according to the total demand of each region, different distribution methods are selected, and the specific delivery methods are shown in

Table 5. Regional demand table.

Region	Total Demand/Piece	Delivery Method
Region 1	58	Vans
Region 2	48	Vans
Region 3	72	Refrigerated trucks
Region 4	61	Refrigerated trucks
Region 5	79	Refrigerated trucks
Region 6	50	Vans
Region 7	62	Vans

. The comparison of the new and old transportation methods is shown in

Table 6. Comparison table of new and old transportation methods.

Region	IGWO Logistics Cost/RMB	Traditional Distribution Method Logistics Cost/RMB	IGWO Carbon Emissions/t	Traditional Distribution Method Carbon Emissions/t
Region 1	3443.01	3720.26	0.574	1.34
Region 2	3170.29	3783.01	0.584	1.362
Region 3	3011.86	3011.86	1.11	1.11
Region 4	2735.10	2735.10	1.008	1.008
Region 5	2960.33	2960.33	1.091	1.091
Region 6	2787.20	3035.86	0.480	1.119
Region 7	3815.46	4514.45	0.714	1.507
Total	21,923.25	23,760.87	5.561	8.537

.

By comparing our dual-method distribution scheme with the traditional refrigerated truck distribution, there is a significant reduction in both comprehensive and carbon emissions, which decreased by 7.73% and 34.86%, respectively.

6.3. Sensitivity Analysis

6.3.1. Effectiveness of Improved Grey Wolf Algorithm

To verify the superiority of our algorithm, we compared it with three other traditional search optimization algorithms: the traditional grey wolf algorithm, ant colony algorithm, and large neighborhood search algorithm. Furthermore, to ensure that the other three algorithms would not fall into the local optimal solution as a result of logical defects, we also improved those three comparison algorithms. The algorithm parameters and improvements are as follows.

First, for the ant colony algorithm, an improved ant colony algorithm strategy, with the adaptive-adjusting pheromone volatile factor, is introduced, and the global performance of the algorithm is improved by adaptively changing pheromone volatile factor $\rho$. Letting $\rho$ start with $\rho \left(t_0\right)=1$, $\rho$ performs an adaptive adjustment according to the following formula:

$${\rho }_{\left(t\right)}=\left\{\begin{array}{l}0.95\rho \left(t-1\right) , if 0.95\rho \left(t-1\right)\ge {\rho }_{\min }\\ {\rho }_{\min } ,\hspace{1em}\mathrm{otherwise} \end{array}\right.$$

After conducting multiple calculations, we established the following parameters for the adaptive ant colony algorithm to ensure the optimal results. The concentration weight of the pheromone (alpha) and the heuristic information weight (beta) are 1 and 2, respectively. These values balance the effects of the pheromone and heuristic information during the search process. Meanwhile, the evaporation rate of the pheromones (rho) is set to 0.5, regulating the persistence and update speed of the pheromones. We also set the exploration probability (q0) to 0.9, encouraging the ants to choose the optimal solution. To determine when the algorithm should stop searching, we used a convergence criterion of 0.001. The search process terminates when the difference between a newly generated solution, and the current best solution is smaller than this value.

Moreover, we introduced the static weighted average strategy [37] to the grey wolf algorithm, assigning weights to the three head wolves to show the hierarchical structure of the pyramid. The alpha wolves were assigned a weight of 0.5, beta wolves 0.3, and delta wolves 0.2.

Lastly, to expedite the convergence of the large-scale neighborhood search algorithm, we used the C-W saving method to construct an initial solution. The number of iterations, numIterations, was set to 300, while the neighborhood size was defined as 100, indicating the number of paths that require modification during each search. Additionally, we set the tabu length to 13, dictating the duration that a path cannot be revisited.

A comparison between the initial solution by GWO and its three variant algorithm is shown below in

Table 7. Comparison table of four algorithms.

Results	Initial Solution (GWO)	IGWO	AGO	LNS
Region 1 path length/km	1223.09	1191.965	1632.375	1771.365
Region 2 path length/km	910.7994	1212.071	911.97	995.585
Region 3 path length/km	1024.642	987.709	1197.331	1172.693
Region 4 path length/km	1554.866	896.957	1621.225	1193.375
Region 5 path length/km	940.924	970.818	1092.455	973.78
Region 6 path length/km	1306.518	995.585	966.274	1371.156
Region 7 path length/km	1459.393	1482.041	1233.797	1218.066
Algorithm running time/s	11.8979	9.889	10.34	12.18318
Total delivery distance/km	8420.227	7737.146	8655.427	8696.02

.

Examining the upper right iteration diagram of Figure 9, we find that the large-scale neighborhood search algorithm falls into the local optimal solution three times in the process of solving, and the third stagnation time is very long, which affects the solving speed of the algorithm. The ant colony algorithm is slow in the early iteration, and the grey wolf algorithm is obviously better than the other two algorithms. However, our improved grey wolf algorithm still has an advantage over all the other algorithms: compared with them, the calculation time is reduced by 16.88%, 3.39%, and 18.83%, respectively. In terms of the optimization results, the improved grey wolf algorithm also has the highest solution quality. Compared with the other three algorithms, the total delivery route length is reduced by 8.11%, 10.61%, and 11.03%, respectively. In addition, to verify the superiority of our algorithm in more general cases, we used a new set of artificial data with 202 customer nodes and compared the convergence speed between the traditional GWO and our algorithm (see the bottom charts in Figure 9). This distribution pattern is four times the size of the previous one, and it is larger than any current pharmaceutical cold chain logistics in China [3]. Owing to the sharp increase in customer nodes, it took more than 50 iterations for our algorithm to reach the optimal solution, while the traditional GWO took more than 150 iterations.

Furthermore, we compared the GAP values of the four algorithms by continuously increasing the number of client nodes.

Table 8. Comparison table of four algorithms.

Number of Nodes	LNS	AGO	GWO	IGWO
80	0.01	0.07	0.22	0.00
120	0.00	0.10	0.52	0.00
160	0.04	0.13	1.17	0.00
200	0.01	0.09	2.59	0.00
240	0.12	0.70	3.93	0.00
280	1.33	2.54	4.59	0.00
320	2.02	2.97	4.99	0.00
360	2.84	3.66	5.76	0.00

compares the GAP values, defined as (mean value of 20 runs of a certain algorithm—optimal solution obtained by all algorithms)/optimal solution obtained by all algorithms * 100%. The results indicate that the IGWO algorithm has a smaller dispersion, signifying its superior stability and performance when compared to the other algorithms. When the number of target nodes is less than 160, there is little difference in the optimization time among the four algorithms, with the IGWO algorithm exhibiting a slight advantage. However, when the number of target nodes exceeds 160, the GWO algorithm’s slope increases, while the slopes of the LNS, AGO, and GWO algorithms also increase when the number of target nodes surpasses 240. This is because the three algorithms converge too early during the later iterations, resulting in local optimal solutions instead of the global optimal solution.

The comparison reveals that the IGWO algorithm outperforms the others when optimizing more target nodes, demonstrating higher efficiency and a better path planning ability in complex environments. It provides superior path optimization compared to the other comparative algorithms, verifying the feasibility and effectiveness of the proposed improvement strategy. Figure 10 presents a comparison of the running times for the four algorithms across various numbers of nodes.

6.3.2. Effectiveness of Comprehensive Cost Reduction

In our numerical experiments, the specific parameters were kept constant. To demonstrate the effectiveness at reducing carbon emissions and logistics cost, we introduced the traditional distribution method as a control group. The comparison between the five distribution methods is shown below in

Table 9. Comparison table of cost and carbon emissions of five methods.

Region	Transportation Consumption	IGWO	Traditional Distribution Method	GWO	AG	LNS
1	Delivery method	Vans	Refrigerated trucks	Refrigerated trucks	Refrigerated trucks	Refrigerated trucks
	Logistics cost/RMB	3443.01	3720.26	3729.60	4977.64	5401.46
	Carbon emissions/t	0.574	1.34	1.375	1.835	1.991
2	Delivery method	Vans	Refrigerated trucks	Vans	Vans	Vans
	Logistics cost/RMB	3190.29	3783.01	1483.01	2169.12	2252.396
	Carbon emissions/t	0.584	1.362	0.439	0.439	0.480
3	Delivery method	Refrigerated trucks	Refrigerated trucks	Vans	Vans	Vans
	Logistics cost/RMB	3011.86	3011.86	3059.15	2625.17	2603.64
	Carbon emissions/t	1.11	1.11	0.494	0.575	0.565
4	Delivery method	Refrigerated trucks	Refrigerated trucks	Vans	Vans	Vans
	Logistics cost/RMB	2735.10	2735.10	4062.44	3759.59	2768.95
	Carbon emissions/t	1.008	1.008	0.749	0.781	0.575
5	Delivery method	Refrigerated trucks	Refrigerated trucks	Refrigerated trucks	Refrigerated trucks	Refrigerated trucks
	Logistics cost/RMB	2960.33	2960.33	2869.19	3331.25	2969.35
	Carbon emissions/t	1.091	1.091	1.058	1.228	1.094
6	Delivery method	Vans	Refrigerated trucks	Refrigerated trucks	Refrigerated trucks	Refrigerated trucks
	Logistics cost/RMB	2797.20	3035.86	3983.98	2946.48	4181.09
	Carbon emissions/t	0.480	1.119	1.468	1.086	1.541
7	Delivery method	Vans	Refrigerated trucks	Vans	Refrigerated trucks	Refrigerated trucks
	Logistics cost/RMB	3815.46	4514.45	3647.303	3762.25	3714.27
	Carbon emissions/t	0.714	1.507	0.703	1.387	1.369
Total	Logistics cost/RMB	21,923.26	23,760.87	22,834.67	23,571.49	23,891.17
	Carbon emissions/t	5.561	8.599	6.871	7.331	7.615

. The comparison chart of the five approaches in Example 1 is shown in Figure 11.

Given the results in Table 7, it is obvious that our proposed method in this paper has a significant improvement in the context of comprehensive cost and carbon emissions. Compared with the traditional drug cold chain logistics model, grey wolf algorithm, ant colony algorithm, and large neighborhood search algorithm, the comprehensive cost decreased by 7.73%, 3.99%, 6.99%, and 8.24%, respectively, and the carbon emissions decreased by 35.33%, 19.07%, 24.14%, and 26.97%, respectively. Moreover, in order to figure out to what extent this optimization improves the dual distribution scheme in each season, we took the refrigerant-related costs into consideration. The above parameters were recalculated using the refrigerant storage consumption date from May to August, and the comparison between the new results is shown in

Table 10. Comparison table of cost and carbon emissions of five methods from May to August.

Region	Transportation Consumption	IGWO	Traditional Distribution Method	GWO	AG	LNS
1	Delivery method	Vans	Refrigerated trucks	Refrigerated trucks	Refrigerated trucks	Refrigerated trucks
	Logistics cost/RMB	3613.6170	3720.26	3729.60	4977.64	5401.46
	Carbon emissions/t	0.574	1.34	1.375	1.835	1.991
2	Delivery method	Vans	Refrigerated trucks	Vans	Vans	Vans
	Logistics cost/RMB	3362.29	3783.01	1668.777	2308.12	2378.40
	Carbon emissions/t	0.584	1.362	0.439	0.439	0.480
3	Delivery method	Refrigerated trucks	Refrigerated trucks	Refrigerated trucks	Vans	Vans
	Logistics cost/RMB	3011.86	3011.86	3124.47	2865.17	2761.64
	Carbon emissions/t	1.11	1.11	1.152	0.575	0.565
4	Delivery method	Refrigerated trucks	Refrigerated trucks	Vans	Vans	Vans
	Logistics cost/RMB	2735.10	2735.10	4531.0513	3979.59	2993.99
	Carbon emissions/t	1.008	1.008	0.749	0.781	0.575
5	Delivery method	Refrigerated trucks	Refrigerated trucks	Refrigerated trucks	Refrigerated trucks	Refrigerated trucks
	Logistics cost/RMB	2960.33	2960.33	2837.450	3331.25	2969.35
	Carbon emissions/t	1.091	1.091	1.058	1.228	1.094
6	Delivery method	Refrigerated trucks	Refrigerated trucks	Refrigerated trucks	Refrigerated trucks	Refrigerated trucks
	Logistics cost/RMB	3035.86	3035.86	3983.98	2946.48	4181.09
	Carbon emissions/t	1.119	1.119	1.468	1.086	1.541
7	Distribution method	Vans	Refrigerated trucks	Vans	Refrigerated trucks	Refrigerated trucks
	Logistics cost/RMB	4044.95	4514.45	3759.303	3762.25	3714.27
	Carbon emissions/t	0.714	1.507	0.703	1.387	1.369
Total	Logistics cost/RMB	23,124.01	23,760.87	23,634.63	24,170.49	24,400.20
	Carbon emissions/t	6.2	8.599	6.944	7.331	7.615

. The graph comparing the five methods is shown in Figure 12.

R1, R2, R3, R4, R5, R6, and R7 represent the seven regions, COST represents the comprehensive logistics cost, and CE represents the carbon emissions.

It is obvious that the refrigerant-related costs increase due to seasonal temperature variations (Table 9). However, our improved algorithm still effectively reduced the comprehensive cost and carbon emissions. Compared with the traditional distribution scheme, the traditional grey wolf algorithm, the ant colony algorithm, and the large neighborhood search algorithm, the comprehensive logistics cost is reduced by 2.68%, 2.16%, 4.33%, and 5.23%, respectively, and the carbon emissions are reduced by 27.90%, 10.71%, 15.43%, and 18.58%, respectively.

7. Conclusions

In this study, we developed an improved mathematical model for pharmaceutical cold chain logistics path optimization that considers both comprehensive cost and carbon emissions reduction. To effectively classify delivery methods, we constructed comparison functions. We optimized the grey wolf algorithm by introducing a nonlinear factor and accounting for seasonal and spatial temperature variations to specify the use of refrigerant accumulators.

Our numerical experiment using full-scale example data verified that our improvements resulted in increased accuracy and efficiency for cold chain logistics. The decision time and route length were algorithmically reduced by 16.88% and 8.11%, respectively. Practically, we saved 7.73% of the conventional comprehensive cost and, remarkably, reduced over one-third of the overall carbon emissions. Therefore, our innovative dual distribution method application was found to be more effective in improving cold chain logistics capacity than the mathematical method optimization.

From a management perspective, the findings of this study hold significant implications for the pharmaceutical industry. The 2030 Agenda for Sustainable Development emphasizes the need to guarantee affordable and high-quality medicines for all. Ensuring the safety and effectiveness of these medicines is crucial, particularly in developing nations where the infrastructure may be inadequate. Therefore, optimizing cold chain logistics using innovative methods, such as the dual distribution approach suggested in this study, can make a substantial contribution toward achieving the Sustainable Development Goals.

However, our research identified several neglected issues and operational limitations in pharmaceutical cold chain logistics. Based on our findings, there are managerial insights for future research within pharmaceutical cold chain logistics:

• While the dual distribution method proposed in this study mitigates temperature variation in cold chain logistics, it overlooks other factors that impact logistics quality, such as the human costs associated with loading and unloading longer routes. Future efforts should undertake a more comprehensive analysis that simultaneously evaluates all factors.
• Our example data analysis illustrates the effectiveness of our algorithm on a larger scale. However, our distribution method remained unchanged from the original scenario. Therefore, further consideration of the transportation modes is necessary. For instance, we can utilize different types of carrier vehicles in a single long route to dynamically adjust our vehicle type according to customer demand variations.
• Our study, along with other current research, assumes that there will be no emergencies during cold chain logistics. However, customer nodes along longer routes can significantly change, leading to massive node variations that can cause systematic inefficiencies in logistics optimization. As such, further optimization of the self-repairing mechanisms is necessary. Dynamic route optimization can be introduced into our model to address possible dynamic customer nodes and demand variations.

In conclusion, this study provides valuable insights into optimizing pharmaceutical cold chain logistics from both cost and carbon emissions reduction perspectives. Nonetheless, additional research is required to create more comprehensive models that consider all relevant variables, including those associated with human factors and emergency situations. Adopting such models will enable the industry to achieve greater efficiency and sustainability and contribute toward achieving the objectives of the 2030 Agenda for Sustainable Development.