Cold Chain Low-carbon Logistics Path Optimisation Method Based on Improved Hummingbird Optimisation Algorithm

INTRODUCTION:The research of scientific and reasonable logistics and distribution programme time the pursuit of each logistics enterprise, not only can improve customer satisfaction and corporate image, but also help to reduce distribution costs. OBJECTIVES: For the current cold chain low-carbon logistics distribution path optimisation methods there are problems such as easy to fall into the local optimum, optimisation time-consuming. METHODS: This paper proposes a cold chain low-carbon logistics distribution path optimisation method based on the improved Hummingbird optimisation algorithm. Firstly, by analyzing the characteristics of the cold chain low-carbon logistics distribution path optimization problem, designing the cold chain low-carbon logistics path optimization objective function and constraints, and constructing a cold chain low-carbon logistics distribution path optimization model based on a soft time window; then, the hummingbird optimization algorithm is improved by using the initialization strategy of the set of good points and the cardinality leap strategy, to overcome the defects of the hummingbird optimization algorithm; secondly, a method based on intelligent optimization algorithm is proposed by designing the double-layer array coding and the adaptive function, combined with the improved hummingbird optimization algorithm. A cold chain low-carbon logistics path optimization method based on intelligent optimization algorithm is proposed; finally, the superiority and robustness of the proposed method are verified by simulation experimental analysis. RESULTS: The results show that the proposed method not only improves the optimisation time, but also increases the optimisation fitness value. CONCLUSION: This paper solves the problem that the optimisation of the green low-carbon logistics path optimisation problem is time-consuming and prone to falling into local optimum.


Introduction
As the consumption capacity of China's residents continues to climb, more and more large-scale e-commerce platforms layout fresh food retail [1].Fresh products are easy to deteriorate in the transport process, it is necessary to take refrigeration and thermal insulation measures to transport, but also to meet the customer's time needs [2].The research of scientific and reasonable logistics and distribution programme time the pursuit of each logistics enterprise, not only can improve customer satisfaction and corporate image, but also conducive to reducing distribution costs [3].Therefore, it is very necessary to study the optimisation of cold chain low-carbon logistics path [4].
Cold chain low-carbon logistics and distribution path optimisation is essentially a Vehicle Routing Problem (VRP) [5].The cold chain low-carbon logistics distribution path optimisation technique considers all the costs in the cold chain low-carbon logistics distribution process on the basis of meeting the maximum vehicle load, and solves the cold chain low-carbon logistics distribution path optimisation model with the minimum total cost based on the constraints of customer demand, vehicle load and time window [6].Commonly used logistics path optimisation methods include exact optimisation algorithms and heuristic algorithms [7].Logistics distribution methods based on exact optimisation algorithms can fall into dimensional explosion, making it difficult for the algorithm to be met with a satisfactory solution in a short period of time [8].Logistics path optimisation methods based on heuristic algorithms converge quickly and are easy to implement, but they are also prone to the local optimum problem [9].Literature [10] combines cuckoo algorithm and intelligent water droplet algorithm to improve the ability to solve the logistics vehicle path planning problem; Literature [11] proposes an improved ant colony algorithm for the multiwarehouse green vehicle planning problem by comprehensively considering the cost of economy and environmental pollution; Literature [12] introduces a pheromone oscillation process to transform the firefly algorithm and applies it to the logistics vehicle planning problem; Literature [13] proposes a logistics vehicle planning problem with time windows based on improved particle swarm algorithm considering distribution and recovery costs; literature [14] designed a joint adaptive large-scale optimization algorithm and solved the timedependent logistics vehicle problem under fuzzy demand; literature [15] proposed a stochastic logistics and distribution vehicle method for multi-centre demand based on neighbourhood searcher strategy to improve the cultural gene algorithm; literature [16] combined the scanning algorithm and improved particle swarm algorithm to propose a time-uncertain logistics distribution vehicle path optimisation method.Review and collation of related literature, although logistics distribution research has achieved relatively fruitful results, but there are still shortcomings: 1) traditional intelligent optimisation algorithms are simple in structure, and can easily fall into the local optimum; 2) the current logistics distribution model can not comprehensively consider the constraints and costs [17].
Aiming at the shortcomings of the current logistics path optimisation method, this paper proposes a logistics path optimisation method based on the improved Hummingbird optimisation algorithm considering cold chain features and low-carbon features.Firstly, we analyse the constraints and costs of low-carbon logistics path optimization for cold chain, and construct a logistics path optimization model considering cold chain and low-carbon factors; then we improve the performance of the hummingbird optimization algorithm through two effective strategies, and put forward a low-carbon logistics path optimization problem for cold chain based on the improved hummingbird optimization algorithm.Compared with other optimisation algorithms, the proposed improved Hummingbird optimisation algorithm shows better convergence speed and accuracy in the cold chain lowcarbon logistics path optimisation problem.

Optimisation model for cold chain lowcarbon logistics and distribution routes
According to the characteristics of the cold chain lowcarbon logistics distribution path optimisation model, the soft time window-based cold chain low-carbon logistics distribution path optimisation model was constructed by comprehensively considering the fixed cost of the distribution vehicle, the transportation cost, the cost of cargo damage arising from the delivery of products to consumers, the refrigeration cost of ensuring that the items in the vehicle are in a low-temperature environment, the carbon emission cost arising from the distribution, and the penalty cost [18].

Fixed costs
Fixed costs include salaries of distribution vehicle drivers, vehicle depreciation costs, and maintenance and upkeep costs [19].The specific calculations are as follows: where k f denotes the fixed cost of the kth vehicle and m is the number of vehicles required for distribution.

Transport costs
Transportation cost is the cost incurred by the amount of fuel consumed by the distribution vehicle to complete the distribution, which is proportional to the distance travelled for transportation, and is calculated by the following formula:

Cargo damage costs
Cargo damage costs mainly include the logistics and distribution process with the increase in time decay and deterioration of the cost of cargo damage and loading and unloading cost of cargo damage, its cost and time showing exponential changes in the relationship [20], the specific calculations are as follows: ( ) ( ) p denotes the cost of refrigeration per unit of time in the loading and unloading process.

Penalty costs
Penalty costs are additional costs incurred during the delivery process, which are incurred if the delivery vehicle does not deliver within the specified time period, including waiting costs earlier than the time window and late costs later than the time window.Consumer j penalty cost is calculated as follows: where k j t denotes the time for delivery vehicle k to reach demand point j,  ) ( ) ( )

Cost of carbon emissions
Carbon emission cost refers to the cold chain lowcarbon logistics and distribution process, vehicle driving fuel consumption and consumption of refrigerant will produce carbon dioxide, and the fuel consumption is related to the vehicle weight and travelling distance [22].The formula for calculating the carbon emission cost of vehicle travelling fuel consumption is as follows: ( ) where c p denotes the price of the carbon tax, 1 E denotes the carbon emissions per kilometre, e denotes the CO2 gas emission factor, and ij q denotes the amount of freight transported from vehicle i to j.
The cost of carbon emissions from refrigeration equipment is: where 2 E denotes unit consumption.
The total carbon cost is:

Modelling
In summary, the total cost model for cold chain distribution path optimisation includes objective function and conditional constraints.The cold chain distribution path optimisation objective function is designed as follows: Cold chain logistics and distribution vehicle constraints: Vehicle constraints required for distribution: Where m indicates the number of vehicles required for distribution.
Consumer constraints on distribution needs: where n denotes the number of consumers.
Maximum load capacity constraints for cold chain logistics and distribution vehicles: Where, M Q denotes the maximum load capacity of the delivery vehicle, and j Q denotes the delivery weight of the jth consumer.
Cold chain logistics and distribution vehicle constraints:

Hummingbird Optimisation Algorithm
Hummingbird optimisation algorithm (HOA) algorithm is a stochastic optimisation algorithm that simulates the process of honey harvesting by hummingbirds [23].Initially, the algorithm creates multiple hummingbird individuals randomly in the search space.The position of each individual corresponds to a feasible solution of the optimisation problem.The quality of the food source is the objective function value, and the best food source is the global optimal solution.The whole optimisation process of HOA can be divided into two phases: self search and guided search.The specific form of the algorithm is as follows: (1) Self-search phase Assume that the size of the hummingbird population is NP and the dimension in the search space is , , , , It is the i th hummingbird individual in the t th moment.In the self-searching stage, hummingbirds can find food sources based on their previous experience.When the hummingbird can continuously find better food sources ( indicates the correctness of previous experience.Therefore, the position of each hummingbird is updated based on the previous gradient information: 11 () Where When a hummingbird keeps searching but cannot find better results ( ), it means that the hummingbird's previous experience is no longer applicable.
In this case, the hummingbird randomly changes the direction of its search.This process is simulated using Levy flights, which are important non-Gaussian random walks whose random steps obey a large-tailed probability distribution.Due to the infinite and rapid growth of the variance, the most important feature of this flight pattern is the ability to maximise the exploration of space in uncertain environments.Levy flight is able to search more efficiently than conventional random walks such as Brownian motion.
Figure 1 shows the two-dimensional motion trajectories of Levy flight and Brownian motion over 1000 time steps.As shown in Figure 1, the Levy flight is able to produce larger jumps than the Brownian motion, thus exploring space more extensively.Therefore, it is more suitable for large-scale search.The search process based on Levy flight is represented as follows: Where Levy  is calculated as follows: 1/ () where  and denote two random numbers that obey Gaussian distributions 2 (0, ) where (z)  denotes the gamma function and the size of  is set to 1.5 .
2) Guided search phase At this stage, the best current individual hummingbird is called the territorial bird and the others are called followers.This territorial bird patrols up and around its territory to prevent other companions from approaching.This behaviour can be expressed as: Where , Tt X denotes the position of the territorial bird at the time of t , , Tt X is a random number with a value in the range of [ 1,1] − , and  is a step factor, defined as follows: where ub and lb denote the upper and lower bounds of the search space, respectively.There are two scenarios for following hummingbird movement patterns.Scenario 1: When the territorial bird does not detect a threat, the following bird will quickly approach its territory: , 1 , , , ( ) where , Ft j X is the position of the first - j th following bird at the moment T, and MF randomly takes the value 1 or 2. Scenario 2: When the territorial bird spots the following bird, the following bird is driven away and flies away in a peripheral direction.During this process, the follower bird j will randomly pick a companion ( k kj  ) to follow.If the position of k is better, j will move towards it, otherwise, it will move away.The above can be expressed by the following equation: , Where, , , , {1, indicates that the following bird J is ranked from smallest to largest among all peers according to the fitness value.
In addition, a boundary control strategy for preventing invalid searches can be described as follows: , , , ( ) Finally, the HOA uses a greedy strategy for population updating, i.e., retains 1 t i X + only if the fitness value of 1 t i X + is better than that of t i X , otherwise the individual is not updated.This scheme is described as follows: The pseudo-code of the HOA algorithm is shown in Figure 2:

Improvement strategies
In order to enhance the algorithm's full-domain exploration capability and avoid the algorithm from falling into local optimum, this paper adopts the good point set initialisation strategy [24] and the cardinal leap strategy [25] to improve the Hummingbird optimisation algorithm.

Good point set initialisation strategy
The quality of the initialised population of HOA algorithm affects the solution optimisation speed of the algorithm, and an excellent population initialisation strategy can make the individuals of the population traverse the whole search space more evenly, increase the population diversity and improve the convergence speed of the algorithm.In order to improve the diversity of the population search and make the population evenly distributed in the search space, this paper proposes a good point set initialisation strategy to improve the initialisation of the HOA algorithm.Suppose s G is a unit cube in s- dimensional Euclidean space, if s rG  , for: , , , ,1 Its deviation is satisfied: Then ( ) n Pk is called the set of good points and r is the good point.
where p is the smallest prime number satisfying ( ) 32 ps − .The initialised population distribution graph using the set of good points is shown in Figure 3.

Cardinality leapfrog strategy
In the HOA algorithm, hummingbird individuals can increase the diversity of the algorithm by using the Levy flight strategy during the population iteration process, but with the increase of iteration number, the population variability decreases, and the distribution of algorithm diversity is limited.To address the above problems, this paper proposes the cardinal leap strategy, which makes the hummingbird jump out of the local optimum in the selfsearch phase and improves the algorithm's ability to find the optimum.The mathematical model of the cardinal leap strategy is as follows: ( ) where 1 t id X + denotes the value of the i th hummingbird in the d th dimension in the first 1 t + iteration,  is the step control coefficient(Figure 4), and ( ) L  is the Lévy flight random search path.
In order to manipulate the step size control of the leaps, the cardinality adaptive step size is proposed based on the cardinality distribution function,  solving Eq: Where, n denotes the degree of freedom, t is the current number of iterations, max t is the maximum number of iterations, and ( ) n f  is the probability density function of the chi-square distribution, Eq: ( ) ( ) where ( )

Improvement of algorithm flow
According to the HOA algorithm with improved optimisation strategy, the flowchart of HOA algorithm (HOA based on good point set initialisation strategy and chi-square transition, GChiHOA) is shown in Figure 5 with the following steps: Step 2: Calculate the fitness value and record the current optimal individual; Step 3: Improve the Levy flight operator using the cardinality leapfrog strategy to perform the self-search phase; Step 4: Execute the search phase for territorial and following birds; Step 5: Calculate the fitness value and update the optimal individual; Step 6: Determine whether the number of iterations reaches the maximum number of iterations.If the maximum number of iterations is reached, carry out the output of the optimal solution and optimal value; otherwise, go to step 3.

Optimising solutions
In the cold chain low carbon logistics path optimization method based on GChiHOA algorithm, the population individuals represent the feasible solution of the cold chain low carbon logistics path optimization problem, i.e., the feasible path traversing the target distribution nodes; the optimal individuals represent the optimal solution obtained from the cold chain low carbon logistics path optimization problem, i.e., the optimal logistics and distribution paths; the change of the position of the population individuals represents the path passing through the changes of the distribution nodes or the optimization planning process, and the fitness value corresponds to the value of the objective function.The change of individual position of the population represents the change of the path through the distribution node or the planning optimisation process, and the fitness value corresponds to the value of the objective function, and the specific relationship is shown in Figure 6.

Individual information design
According to the correspondence between the GChiHOA algorithm and the problem in Fig. 6, it can be seen that in this paper, the population individual position indicates the feasible path traversing the target distribution node, and the array coding method is used to explain the structure of the population individual position, i.e.,

dim 
, in which the dim dimension denotes the number of the distribution node, and the structure of the solution is shown in Figure 7. From Figure 7, it can be seen that the distribution centre and customer points are discrete points, and a two-layer array coding approach is introduced to encode the GChiHOA algorithm to solve the cold chain low-carbon logistics path optimization problem.The position-order coding expression formula is as follows:   According to the constraints such as the loading capacity of each vehicle, the structure of two groups of solutions is given in Figure 7, the first group of solution location information is 0.3, 1.9, 3.9, 6.3, 4.1, 5.7, and the distribution order information is 1.3, 1.9, 2.0, 3.9, 4.3, 4.8, and the distribution node information after the solution is discretised is 1, 2, 4, 8, 5, 6, and the distribution order serial no, 4, 5, 6, which means that the first car starts from the distribution centre and returns to the distribution centre after serving the customer nodes 1, 2, 4, 8, 5, 6, and the distribution services are carried out by different cars without repetition.

Adaptation function design
The individual fitness function is used to analyse the constraints of the cold chain low-carbon logistics path optimisation problem, and to generate fitness values for evaluating the individual strengths and weaknesses of the distribution scheme by combining the fixed costs, transportation costs, the cost of damage to the product Xuan Long 9 delivered to the consumer, the cost of refrigeration to ensure that the items in the vehicle are in a low-temperature environment, the cost of carbon emissions generated by the distribution, and the cost of penalties.The individual fitness function in this paper is specified as:

Methodological steps
The essence of the cold chain low-carbon logistics path optimisation problem is to solve the optimal solution problem with multiple constraints and multiple objectives, i.e., the cold chain low-carbon logistics and distribution path optimisation problem based on the soft time window.The application process of the improved Hummingbird optimisation algorithm in cold chain low-carbon logistics path optimisation is shown in Figure 8, and the specific steps are as follows: Step 2: Select the search population randomly and uniformly using the given boundary constraints.
Step 3: The mapping between population individuals to the problem solution is done using the position-order array coding approach.
Step 4: Calculate the fitness value of the individual.
Step 5: Optimally updating the logistics delivery solution individual based on the improved Hummingbird optimisation algorithm self search strategy and search strategy.
Step 6: Determine whether the algorithm termination conditions are satisfied.If the number of search iterations is greater than the maximum number of iterations, the search is terminated and the optimal cold chain low-carbon logistics path distribution scheme is output; otherwise, continue with steps 4 to 6.

Results and analysis
In order to verify the effectiveness of the cold chain low carbon path optimisation method based on the improved Hummingbird optimisation algorithm, using the distribution logistics data of an enterprise, this paper selected five analysis algorithms for comparison.MATLAB 2021a was used to write the programme, and the test environment was a Windows 10 system, the processor was AMD Ryzen 9 5900HX with Radeon Graphics, and the memory was 16.0 GB.The parameters of each algorithm are shown in Table 1.Distribution logistics data for 16 consumers to deliver goods, the distribution centre is (0,0), the delivery service departs from the distribution centre at 5:30, and the consumers are shown in Table 2.

Conclusion
Aiming at the current cold chain low-carbon logistics and distribution path optimisation method, which is prone to fall into local optimum and takes a long time to optimise, this paper proposes a cold chain low-carbon logistics and distribution path optimisation method based on the improved Hummingbird optimisation algorithm.By analysing the characteristics, costs and constraints of the cold chain low-carbon logistics distribution path optimisation problem, a soft time window-based cold chain low-carbon logistics distribution path optimisation model is constructed; combining the double-layer array coding method and the fitness function, the hummingbird optimisation algorithm is improved by using the good point set initialisation strategy and the cardinality leap strategy to solve the cold chain low-carbon logistics distribution path optimisation problem.The results show that the cold chain low-carbon logistics path optimisation method proposed in this paper can find the optimal scheme of logistics and distribution paths in a shorter time, with uniform and reasonable node distribution and stable performance.
vehicle that did not travel from i to j, ij d denotes the distance between i and j, and k c denotes the cost per kilometre of the vehicle.

ep
denotes the cost of waiting per unit of time, l p denotes the cost of tardiness per unit of time, and the acceptable time window for demand node j is ,

X
− denote the position of the i hummingbird at t and 1 t − respectively, and rand denotes a random number in the range [0,1] .

Figure 1 .
Comparison of two random motions

Figure 2 .
Figure 2. Pseudo-code of HOA algorithm the fractional part,  is any positive number, ( ) , Cr is a constant related only to , r  , n denotes the number of points, and r is:

Figure 3 .
Figure 3. Distribution of initialised populations in the good point set

Figure 5 .
Figure 5. Flowchart of GChiHOA algorithmStep 1: Initialise the population position using the good point set strategy, set the maximum number of iterations and other parameters;Step 2: Calculate the fitness value and record the current optimal individual;

Figure 6 .
Figure 6.Schematic diagram of the optimised solution sorted in ascending order, i.e., the distribution vehicle path code.

Figure 7 .
Figure 7.The solution structure From Figure 7, it can be seen that the distribution centre number dimension 0 and the customer point is 1, 2, , N , and the vehicle returns to the distribution centre after serving a certain number of customers.According to the constraints such as the loading capacity of each vehicle, the structure of two groups of solutions is given in Figure7, the first group of solution location information is 0.3, 1.9, 3.9, 6.3, 4.1, 5.7, and the distribution order information is 1.3, 1.9, 2.0,3.9,4.3, 4.8, and the distribution node information after the solution is discretised is 1, 2, 4, 8, 5, 6, and the distribution order serial no, 4, 5, 6, which means that the first car starts from the distribution centre and returns to the distribution centre after

Figure 8 . 1 :
Figure 8. Optimisation of low-carbon logistics paths in the cold chain Step 1: Initialise the cold chain low carbon logistics path optimisation parameters, including the parameters of the mathematical model of the problem and the parameters of the improved Hummingbird optimisation algorithm.The

Table 1 14 .
Parameter settings of green low-carbon logistics path optimisation algorithm arithmetic parameterisation LSHADE Memory size H=5 and archive rate H=1.4 LSA The channel time is set to 10 NRO PFi = 0.75, Pβ = 0.1, freq = 0.05 HOA Parameter-free optimisation GChiHOA Parameter-free optimisation Cold Chain Low-carbon Logistics Path Optimisation Method Based … (a) Optimisation results of the GChiHOA algorithm (b) Optimisation results of the HOA algorithm Figure Schematic diagram of cold chain low carbon logistics distribution path based on GChiHOA algorithm and HOA algorithm