Economic optimization of fresh logistics pick-up routing problems with time windows based on gray prediction

Abstract

The rapid development of modern cold chain logistics technology has greatly expanded the sales market of agricultural products in rural areas. However, due to the uncertainty of agricultural product harvesting, relying on the experience values provided by farmers for vehicle scheduling can easily lead to low utilization of vehicle capacity during the pickup process and generate more transportation cost. Therefore, this article adopts a non-linear improved grey prediction method based on data transformation to estimate the pickup demand of fresh agricultural products, and then establishes a mathematical model that considers the fixed vehicle usage cost, the damage cost caused by non-linear fresh fruit and vegetable transportation damage and decay rate, the cooling cost generated by refrigerated transportation, and the time window penalty cost. In order to solve the model, a hybrid simulated annealing algorithm integrating genetic operators was designed to solve this problem. This hybrid algorithm combines local search strategies such as the selection operator without repeated strings and the crossover operator that preserves the best substring to improve the algorithm’s solving performance. Numerical experiments were conducted through a set of benchmark examples, and the results showed that the proposed algorithm can adapt to problem instances of different scales. In 50 customer examples, the difference between the algorithm and the standard value in this paper is 2.30%, which is 7.29% higher than C&S. Finally, the effectiveness of the grey prediction freight path optimization model was verified through a practical case simulation analysis, achieving a logistics cost savings of 9.73%.

Keywords

Pick-up routing problems fresh logistics gray prediction hybrid simulated annealing

1 Introduction

With the development of the economy and the continuous improvement of people’s living standards, meeting people’s demand for high-quality fresh products has become an important driving force for the development of cold chain logistics. The development of cold chain logistics provides quality and safety assurance for the cross regional circulation of a large number of fresh agricultural products, meeting the diverse needs of individual consumers. However, the production and supply sources of agricultural products directly determine the quality and value of fresh supply, and the quality of fresh products is an important factor affecting consumer satisfaction, while also affecting the sustainable development of cold chain enterprises. This has attracted increasing attention from the government and cold chain enterprises [1, 2]. According to the “China Agriculture Outlook Report (2023–2032)”, in 2023, China produced 799 million tons of fresh vegetables, with a consumption of 609 million tons. Nearly 190 million tons of fresh vegetables were wasted or turned into substandard products in the circulation process and abandoned by people. Therefore, for cold chain enterprises, how to timely transport fresh and newly mature high-quality agricultural products to consumers while minimizing investment is a major challenge.

The logistic development of fresh agricultural products has always been one of the key factors affecting the realization of rapid and high-quality circulation. The traditional vehicle routing problem focuses on the transportation cost and timeliness of response to customer demand. The fresh logistics pick-up routing problem has many features, e.g., the loss of agricultural products is higher than the traditional counterpart. According to a survey, in China, the loss rate of the fruit and vegetable products in the “first kilometer” is as high as 15–25%. The main reasons for this phenomenon are as follows: Firstly, due to the inadequacy of facilities such as “first-mile” origin warehouses and refrigerated storage facilities, a large number of fresh fruits and vegetables cannot be pre-cooled immediately after collection, resulting in cargo damage. This is particularly prominent in rural areas. Secondly, farmers are unable to collect fruits and vegetables according to market demand, and the waste of surplus fresh fruits and vegetables that do not reach the market increases further during long-term storage. Thirdly, due to the perishability, temperature sensitivity, and fragility of fresh produce, some products will gradually rot and deteriorate during transportation from the production base to the target market [3]. The loss rate is directly related to the cooling measures taken and is dynamically changing. Due to the long transportation time, the corresponding damage rate of goods has decreased the freshness of agricultural products and the satisfaction of buyers, causing more and more profit losses for enterprises. Therefore, the problem of picking up fresh agricultural products has been paid more and more attention [4, 5]. In the context of the rapid development of agricultural e-commerce, which brings higher time limit requirements for consumers and limited cold chain facility resources, how to coordinate the scientific formulation of fresh collection plans with reasonable planning of fresh transportation routes for decision-making, in order to reduce agricultural product damage, improve enterprise profits and customer satisfaction, is an urgent problem that fresh food enterprises need to solve.

Generally speaking, the collection volume of fresh produce is directly or indirectly affected by the maturity time of fresh produce, consumer demand, price fluctuations, cold storage resources, climate change, and other factors, exhibiting seasonal, cyclical, and random fluctuations, posing great challenges to the prediction of fresh produce collection volume. There is no universally accepted effective prediction method. The transportation of fresh goods requires high timeliness, which leads to high logistics costs and low customer satisfaction. Therefore, optimizing the pickup route for fresh agricultural products can not only reduce transportation time, but also reduce the risk of product quality deterioration. The cost and efficiency of logistics operations depend on the optimized route plan.

At present, the partner of fresh logistics is still in the exploration stage, and the degree of intellectualization is also low, especially in the cold chain logistics of agricultural products. Meanwhile, the inadequate ability to employ history data resources and low data utilization rate are frequently presented in the real world [6]. On this basis, logistics enterprises need to respond to customers’ demands quickly and accurately. Therefore, to improve the data utilization rate, pre-scheduling of vehicles can provide better transportation services for farmers based on data-driven methods. Most output and sales of fresh agricultural products is not fixed, this leads to great uncertainty of pick-up demand. A simple example, as shown in the Fig. 1, can be used for illustrate it, due to the influence of soil moisture, sunshine, manure amount, cultivation methods, diversified fresh varieties and other factors, resulting in different ripening times of fresh products, so the harvest amount of each harvest period is uncertain. For market, shows a dynamic change law for the amount of fresh products ordered each time due to the uncertainty of the sales volume of fresh products. However, the demand for picking up goods is determined by the output of fresh products and the order demand of the market, so it is also uncertain. If the uncertainty is ignored by the decision-makers of logistics sectors, it may lead to the increase of the pick-up cost, even failure to meet transportation demands. As a central part of the research on the fresh goods collection path problem, decision-making highly depends on the collection volume.

Fig. 1

An illustration of the pick-up routing problem with time windows.

Therefore, based on the characteristics of “small samples” and “partially known information” of the existing historical collection data of fresh farmers, this paper proposes an improved grey prediction method to predict the picking volume, and establishes an optimization model for the agricultural product cold chain logistics collection route problem that takes into account vehicle costs, cargo damage costs caused by fresh fruit and vegetable transportation damage and decay rates, refrigeration costs generated by refrigerated transportation, and time penalty costs, Optimize the first mile pickup path for fresh products. To solve the nonlinear cost of refrigerated transportation damage and the penalty function for violating time windows, the author designed a two-stage algorithm. In the first stage, an improved grey prediction model based on data transformation is adopted to predict the fresh picking quantity and form an initial solution. In the second stage, a hybrid heuristic algorithm combining the SA algorithm framework and GA operator is designed to schedule the pickup route, using local search strategies such as non 051repeated string selection operator and cross operator that preserves the best substring to avoid local optimal solutions. Numerous standard examples and examples.

The scientific contribution of this paper can be summarized as follows:

The design of an improved gray prediction model based on data transformation.

The establishment of a mixed integer linear programming model of fresh logistics pick-up routing problems.

The development of a hybrid simulated annealing algorithm with genetic operators, which can solve the model efficiently.

The residual parts of this paper are organized as follows: Section 2 presents a review focusing on the vehicle routing problem. In Section 3, we introduce the problem studied in this paper and present the grey forecasting method before modeling. Section 4 establishes a mathematical model for the investigated problem. Meanwhile, a hybrid heuristic algorithm is described in Section 5. Section 6 includes a case study to demonstrate the model and solution algorithm. Finally, this study has been concluded in Section 7.

2 Literature review

2.1 Demand predicting

With the increase in income of urban and rural residents, the upgrading of food consumption, and the acceleration of urbanization process, people’s demand for high-quality fresh vegetables and fruits continues to increase. At the same time, people’s response time requirements for fresh orders are getting higher and higher, they even need the products to be pre-delivered to the front warehouse near consumers to wait for orders, thus improving customer satisfaction [7]. The forecasting of fresh harvest volume belongs to time series prediction.

Many efforts have been conducted to explore the vehicle routing problem (VRP) with uncertain demand. Stewart and Golden [8] took the logistics demand of some pick-up points as a random variable and presented a specific model and solution method. Shen et al. [9] scheduled vehicle routes considering uncertain demand and time-dependent travel time. Sungur et al. [10] introduced a robust optimization method to solve the vehicle routing problem with demand uncertainty. Goel et al. [11] introduced a vehicle routing problem with random demand and service time in a time window. The earlier studies can be founded by the work of Shen et al. [9]. Incorporating the uncertainty demand into the vehicle routing problem model can increases the difficulty of solving the problem. Therefore, some scholars make demand predictions for the uncertainty in advance.

Traditional prediction models include regression models based on various types of time series data features, such as Bayesian models [12] and Autoregressive Integrated Moving Average (ARIMA) [13], Support Vector Machine (SVM) [14]. These methods perform well in certain aspects, they are still effective in predicting fresh collection volume, but they are not good at handling non-linear data.

The common machine learning methods mainly rely on deep neural networks. Yao et al. [15] established a mixed integer programming model for network distribution problems with stochastic demand based on machine learning algorithms. Based on the application of machine learning methods and Petri nets, Niu et al. [16] designed an adaptive demand predictor to estimate the customers’ demand. Similarly, Latorre-Biel et al. [17] employed the real-time prediction method based on machine learning to predict the customer de-mand on logistics network nodes. Chu et al. [18] introduced a data-driven machine learning method to predict customer order quantity changing over time, and the predicted results were used for subsequent routing optimization. Algarni A [19] This paper explores the application of deep learning in open parking lot vehicle allocation by collecting and displaying real-time parking occupancy information to guide drivers in finding available parking spaces. To achieve this goal, the authors considered the compatibility and cost-effectiveness of sensors and technology. Peng et al. [20] designed the Monte Carlo simulation algorithm to deal with the uncertainty of transportation demand, and proposed an efficient data-driven conversion method to against the time consuming mass customization. Machine learning-based algorithms are used in above literature to deal with the demand uncertainty; however, it requires a large amount of historical data. Grey prediction is a simple and effective method for small sample data prediction. The method is widely used in various scenarios with uncertain factors, especially in VRPs. Ye et al. [21] studied a new gray prediction model composed of function transformation and interval gray numbers, which was used to predict four typical interval gray sequences, respectively. Different from the commonly used linear grey model, Ma and Liu [22] introduced a kind of nonlinear multivariate grey model based on the kernel methods, namely kernel based GM(1, N). Javed et al. [23] proposed an EGM (1, 1, α, θ) to predict carbon dioxide emissions from biofuel production. Liu and Wu [24] proposed a new adjacent non-homogeneous grey model to predict renewable energy consumption in Europe, which emphasized the weight relationship between the latest values and historical data. Although the grey prediction method has been widely used in many fields, it is seldom ap-plied in the study of vehicle routing problems with uncertain demand. Considering the feature of insufficient historical data, this paper introduces an improved grey model to predict the customers’ demand.

2.2 Pick-up routing problems

The vehicle routing optimization for fresh products can effectively reduce the total logistics operation cost and improve customer satisfaction. Aman M et al. [25] explored the issue of determining the optimal production factory location and vehicle route in the jujube sugar processing industry to optimize the supply chain network. The researchers established a mixed integer linear programming (MILP) model, which minimizes the total supply time while satisfying constraints such as routing, timing, capacity and supply, as well as transportation costs, to find an ideal factory location and determine the number of vehicles and coverage areas needed. Chen and Fan [26] constructed a deterioration model considering the time window and the characteristics of fresh products, whose decay loss degree is related to the number of fresh products, the decay rate and perishability in different environments. Liu et al. [27] analyzed the economic cost, carbon emission level and freshness of perishable products. Wu and Wu [28] built a time-varying green vehicle routing problem model with soft time window considering customer’s time windows, vehicle time-varying speed, fuel consumption, carbon emission, minimum freshness and other factors, and adopted a freshness measure function of agricultural products and a measure function of carbon emission rate to evaluate costs more accurately. Incorporated the refrigeration cost into the fresh-keeping cost of fresh agricultural products, Zhu et al. [29] constructed a mathematical model to minimize distribution cost, and designed a hybrid ant colony algorithm to solve the model. Ghasemkhani et al. [30] established a fuzzy chanceconstrained programming model to deal with fuzzy parameters when evaluating multi-category perishable products, multi-period and heterogeneous fleet with time window in the distribution network. In the process of smart city distribution of fresh products, Wang et al. [31] proposed a mathematical model with the intelligent optimization objective of minimizing distribution cost and maximizing customer satisfaction. An improved quantum-pour particle swarm optimization algorithm was designed to solve the problem. Aiming at effectively optimizing the distribution path and reducing the comprehensive cost, Ning et al. [32] proposed a cold chain distribution routing optimization method for fresh agricultural products under the carbon tax mechanism. In the existing studies, many scholars focus on the “last mile” delivery of fresh logistics, and few studies forcing on the “first mile” pick-up routing optimization of fresh products.

Many scholars have adopted sophisticated cost functions in modeling the vehicle routing problem and its variations [33]. In response to the Chinese government’s carbon trading plan, Shen et al. [34] constructed a mixed integer linear programming model with minimizing the total driver’s salary, penalty cost, fuel cost and carbon emission cost. Wang et al. [35] established a mathematical model considering the time-dependent speed and the segmented penalty costs of delivery in order to reduce waiting costs and improve customer satisfaction. Zhou et al. [36] constructed a mixed integer programming model to minimize the driving cost and time window penalty cost for the collaborative delivery problem of electric vehicles with multiple warehouses and designed mixed time windows, and an extensible non dominated genetic algorithm II for solving the problem. In the practical application of garbage collection service, Chevroton et al. [37] proposed the arc-based problem with time-dependent penalty cost and con-structed an objective function to minimize the total service cost, distance cost and penalty cost. Londono et al. [38] introduced a novel objective function with penalty cost, which was applied to the vehicle routing problem with backhauls considering the transportation cost and penalty related to the collection decision. Pan [39] established a mathematical model of cold chain logistics routing optimization of agricultural products Internet of Things, the objective function of the model included fixed cost, transportation cost, dam-age cost, refrigeration cost and time penalty cost of cold chain distribution service. Hou et al. [40] proposed a multi-depot heterogeneous vehicle routing problem under the time-dependent road network and soft time window, whose objective function is a linear combination of vehicle fixed cost, time window penalty cost and vehicle transportation cost. Archetti et al. [41] investigated the vehicle routing problem occasional drivers, and each temporary driver would incur punishment costs if he/she violated the designated delivery time window. The penalty cost function defined in the literature mentioned above is mostly linear or piecewise linear functions, without the difference of penalty level. It may differ greatly from the actual situation in some cases, resulting in excessive cost during the routing optimization process. Therefore, this paper designs a nonlinear penalty function to calculate the penalty cost more accurately.

Currently, the above literature is mainly focused on the fresh products from the distribution center through the cold chain logistics transport to their customers, from the “The last mile” distribution perspective on how to reduce the total cost of distribution, and to improve the freshness of products. However, there are few studies on how to reduce logistics costs for enterprises from the perspective of “The first mile” of fresh goods. Although there are some studies investigating the fresh logistics pick-up routing problems with time windows as well as constructing optimization models and algorithms, research on the fresh logistics pick-up routing problems considering stochastic demand is still not enough, especially by using grey forecasting method.

2.3 Hybrid heuristic algorithm

Genetic algorithm (GA) and simulated annealing (SA) algorithm are popular with solving the vehicle routing problem and its variations. Ali et al. [42] compared four wind energy optimization algorithms, namely Ant Lion Algorithm, Whale Algorithm, Particle Swarm Optimization Algorithm, and Crow Search Algorithm, and evaluated their performance through a mixed decision model. The research results showed that Ant Lion Optimization Algorithm had the best optimization effect. Jin et al. [43] designed a hybrid heuristic algorithm combining taboo search algorithm and artificial immune algorithm to solve vehicle routing problems considering product classification and time window constraints. The algorithm incorporates variable neighborhood search and large neighborhood search programs to fully perturb the initial solution. Ali et al. [44] explored the optimization scheduling problem of medical appointments based on fair service quality and designed a hybrid optimization algorithm that combines Whale Optimization Algorithm (WOA) and NSGA-II for solution.

Tasan and Gen [4] designed a genetic algorithm to solve the vehicle routing problem with pick-up and deliver simultaneously. Hosseinabadi et al. [45] proposed a hybrid heuristic algorithm based on gravitational emulation local search and GA to solve capacitated vehicle routing problem (CVRP). In order to reduce distribution cost and achieve green distribution of fresh products, Li et al. [46] established an optimization model for fresh product transportation with heterogeneous fleet and designed genetic algorithm with adaptive simulated annealing mutation. Cosma et al. [47] proposed a two-level genetic algorithm for the clustered vehicle routing problem. Wei et al. [48] studied the capacitated Vehicle routing problem with two-dimensional loading constraints. A mechanism of repeatedly cooling and rising the temperature was designed to solve the problem. Misni et al. [49] proposed a hybrid harmony search-simulated annealing algorithm, which incorporated the dynamic values of harmony memory considering rate and pitch adjustment rate with the local optimization techniques to hybridize with the idea of probabilistic acceptance rule from simulated annealing, to avoid the local extreme points. Ilhan [50] designed an improved simulated algorithm with crossover operators to solve CVRPs, which can speed up the convergence of solutions. Yagmur and Kesen [51] proposed a memetic algorithm with the use of a new splitting procedure for considering both total tardiness and total tour time and simulated annealing in order to offer favorable solutions in a reasonable time. Yu et al. [52] developed two SA algorithms for the vehicle routing problem with simultaneous pickup and delivery and parcel locker.

Different from the above studies, the basis of Chen et al. [53] combined with the global search capability of GA and the excellent local search capability of SA, a hybrid heuristic algorithm that combines SA algorithm framework and GA operators is proposed to schedule the pick-up routes, which may help to improve the quality of solution. The hybrid heuristic algorithm adopts local search strategies such as a non repeated string selection operator and a crossover operator that preserves the best substring. The mutation operator uses a neighbor search method similar to a simulated annealing solution acceptance mechanism to avoid local optima.

3 Problem description and forecasting method

3.1 Problem description

The fresh logistics pick-up routing problems is described as: A homogeneous fleet vis-its the farmers distributed in a region for the pick-up of fresh produce services. Vehicles of the fleet depart from and return to the distribution center and each farmer is allowed to vis-it only once. Similar to the traditional vehicle routing problem (see Fig. 2 (a)), the vehicles serve their customers one by one according to the pick-up route until the service is completed. However, the utilization rate of vehicle capacity is low due to the different amount of agricultural products picked every day in the real process of pick-up. Therefore, the problem solved by our work is that the quantity of products collected by each farmer can be predicted by using the grey prediction method based on the historical data of the last five years. With the goal of reducing the total operating cost, the optimized pickup route is scheduled based on the forecast result, as shown in Fig. 2 (b).

Fig. 2

An illustration of the pick-up routing problem with time windows.

Obviously, the fixed routes require three vehicles to complete the collection tasks, while only two trucks are needed given farmers collecting demand in Fig. 2 (b). That is, the routing plan can significantly save costs.

3.2 Assumptions

The problem investigated in this paper is single depot, pick-up routing optimization in one horizon. To further illustrate this problem, the following assumptions are made:

There is a depot with a fixed number of vehicles center. The depot provides storing service for e fresh fruits and vegetables. The capacity of the depot is greater than the sum of the collection goods in the service area.

Locations of the logistics center and the customer are known in advance,

The soft time window of pick-up goods is known for each farmer, and the penalty coefficient of time window is the same.

Only one distribution center is considered in the system and each route is closed [53].

The logistics center only provides pickup service for a single type of fresh agricultural products.

A route is only serviced by a vehicle, and each customer can be served only once.

The speed of the vehicle is a constant.

3.3 Grey forecasting model

There are many factors affecting the quantity of agricultural products, so it is difficult to design the forecasting model. Therefore, various forecasting methods are published in scientific journals. For example, regression analysis and prediction, time series prediction, BP neural network, etc. which are based on large samples. However, the grey prediction is a kind of uncertain system with “small data” as the research object, which satisfies to the characteristics of revealing unknown grey dynamic system with known information in demand prediction of agricultural product. The grey prediction model has several superiorities compared to other prediction methods: firstly, it is different from other prediction models’ direct data modeling, but instead provides intermediate information for differential fitting modeling through the mapping processing of data sequences; secondly, it weakens the randomness of the original data sequence, especially for non-stationary data sequences, through the generation of data sequences; thirdly, it proposes module prediction and accumulation generation ideas for modeling [55]. Considering the randomness of series and grey GM (1, 1) as a classical model of grey forecasting system, it has been employed by scholars in agricultural demand forecasting. Due to the influence of weather conditions, the number of labor forces participating in harvesting and other factors, the quantity of agricultural products has a certain fluctuation. As shown in Fig. 1, predicting the amount of agricultural products to be picked (picked up) before the vehicle departure can improve the utilization rate of capacity and reduce the transportation cost.

The GM (1, 1) model is the most widely used grey model which has gained popularity because of its numerous applications [54]. The common grey prediction models are generated according to the model form, which is continuous type, discrete type, and the number of variables, which is single variable, multiple variables. These are respectively called continuous single variable grey prediction model, discrete single variable grey prediction model, continuous multiple variable grey prediction model, and discrete multiple variable grey prediction model [55]. The continuous single variable grey prediction model originates from the GM (1,1) model, which only considers a single variable data sequence {x (t₁) , x (t₂) , x (t₃) , . . . , x (t_n)}. It discards the differential equation form dy (t)/dt + ay (t) = b of GM (1,1) model and directly constructs a discrete form y (t_k) = β₁y (t_k-1) + β₂. The form is also similar to the linear relationship of polynomials. The continuous multi-variable grey prediction model originates from the GM (1,N) model, considering the impact of time delay effect, which means that the influencing variable does not necessarily change synchronously with the main variable of the system, but there is a certain time difference, i.e., the lag effect. There is a nonlinear relationship among all influencing factors in the traditional GM (1, 1) model [56], so the first-order differential equation is used to establish a prediction model for a variable to obtain the predicted value. Similarly, in the scene of fresh logistics, there is no exact linear relationship between the influencing factors of the aggregate demand, and the relationship between the influencing factors is vague. In addition, the sample of historical data is small, thus this paper chooses GM (1, 1) model to predict the pick-up demand of farmers. In addition, the sample size of historical data collected, and the sample size of historical data collected for fresh products is relatively small. Therefore, compared to the four derived models: continuous single variable grey prediction model, discrete single variable grey prediction model, continuous multiple variable grey prediction model, and discrete multiple variable grey prediction model, the traditional GM (1, 1) model is more suitable for predicting the amount of fresh logistics collection.

Therefore, compared to the four derivative models of continuous univariate grey prediction model, discrete univariate grey prediction model, continuous multivariate grey prediction model, and discrete multivariate grey prediction model, the traditional GM (1,1) model is more suitable for predicting the collection volume of fresh logistics. This paper proposes an improved gray forecasting model based on data transformation. The improved model first takes logarithms of the original data series to reduce the volatility, and then adds constants before the generated data series. Therefore, the improved model can fully extract information from the data more than the traditional model. Finally, the index of the generated data series is taken. The steps of the improved model are as follows.

–Data preprocess:

* Step 1: The logarithm of the original data is performed to weaken the volatility. Assuming that the original number is: $x_{1}^{(0)} = x_{1}^{(0)} 1 + x x_{1}^{(0)} 2 +, \dots x_{1}^{(0)} n$

and $x_{2}^{(0)} (k) = ln (x_{1}^{(0)} (k))$

* Step 2: Inserting a constant c to the transformed data sequence: $x_{3}^{(0)} (k) = {c, x_{2}^{(0)} (1), x_{2}^{(0)} (2), . . ., x_{2}^{(0)} (n)}, c ⩾ 0$

* Step 3: Performing the exponent on the data sequence.

–Modeling of Gray forecasting [58].

* Step 1: Aggregating to create a new sequence. Suppose the preprocessed data is $x^{(0)} = (x^{(0)} (1) + x^{(0)} (2) +, . . . x^{(0)} (n)),$

a new sequence can be generated by Equation (1): $x^{(1)} = (x^{(1)} (1) + x^{(1)} (2) +, . . . x^{(1)} (n))$ (1)

where, $x^{(1)} (k) = \sum_{i = 1}^{k} x^{(0)} (i), (k = 1, 2, . . ., n) .$

* Step 2: Construct data matrix B and data vector Y. $B = [\begin{matrix} - \frac{1}{2} (x^{(1)} (1) + x^{(1)} (2)) 1 \\ - \frac{1}{2} (x^{(1)} (2) + x^{(1)} (3)) 1 \\ ⋮ ⋮ \\ - \frac{1}{2} (x^{(1)} (n - 1) + x^{(1)} (n)) 1 \end{matrix}],$

$Y = [\begin{matrix} x^{(0)} (2) \\ x^{(0)} (3) \\ ⋮ \\ x^{(0)} (n) \end{matrix}]$

* Step 3: The least square method is used to solve the grey parameters by Equation (2): $u^{\partial} = {(B^{T} \times B)}^{- 1} B^{T} Y = (\begin{matrix} a \\ b \end{matrix})$ (2)

* Step 4: Constructing grey differential equations by Equation (2): $d (k) + {az}^{(1)} (k) = b,$ (3) where, a is the development coefficient, and b is the gray action.

* Step 5: Substituting parameters into Equation (3) and solving the equation by Equation (3): $\begin{matrix} {\hat{x}}^{(1)} (k + 1) = (x^{(0)} (1) - \frac{b}{a}) e^{- ak} \\ + \frac{b}{a}, (k = 1, 2, . . ., n - 1) \end{matrix}$ (4)

* Step 6: Obtaining the predicted values in the original domain by Equation (4): $\begin{matrix} {\hat{x}}^{(0)} (k + 1) = {\hat{x}}^{(1)} (k + 1) \\ \begin{matrix} \end{matrix} - {\hat{x}}^{(k)}, (k = 1, 2, . . ., n - 1) \end{matrix}$ (5)

–Model checking methods:

* Residual test: Residual

$ɛ (k) = x^{(0)} (k) - {\hat{x}}^{(0)} (k), (k = 1, 2, . . ., n)$ , Relative residual Δɛ (k) = ɛ (k)/x⁽⁰⁾ (k), and Average relative residual $\bar{Δ ɛ (k)} = \sum_{k = 1}^{n} Δ ɛ (k) / n, (k = 1, 2, . . ., n) .$

* Posterior ratio test by Equation (5): $C = \sqrt{\frac{S_{1}^{2}}{S_{0}^{2}}},$ (6)

where $S_{0}^{2} = \frac{\sum_{k = 1}^{n} {(x^{(0)} (k) - \bar{x^{(0)}})}^{2}}{n - 1}, (k = 1, 2, . . ., n)$ $S_{1}^{2} = \frac{n \sum_{k = 1} {(ɛ (k) - \bar{ɛ (k)})}^{2}}{n - 1}, (k = 1, 2, . . ., n)$ $\bar{x^{(0)}} = \frac{1}{n} \sum_{k = 1}^{n} x^{(0)} (k), (k = 1, 2, . . ., n)$ $\bar{ɛ (k)} = \frac{1}{n} \sum_{k = 1}^{n} ɛ (k), (k = 1, 2, . . ., n)$

* Probability of small error by Equation (6): $P = p {| ɛ (k) - \bar{ɛ (k)} | ⩽ 0.6745 S_{0}}$ (7)

The judgment criteria of model accuracy are summarized in Table 1.

Table 1

The performance of model accuracy

Performance	$\bar{Δ ɛ (k)}$	C	P
Outstanding	<0.1	<0.35	>0.95
Very good	0.1–0.2	0.35–0.5	0.8–0.95
Creditable	0.2–0.5	0.5–0.65	0.7–0.8
Bad	>0.5	>0.65	<0.7

4 Model

4.1 Nomenclature

To facilitate the subsequent modeling, the nomenclatures used in this section are classified and summarized as follows.

Sets:

A = arc (i, j), i, j ∈ V, i ≠ j: the set of arcs between any nodes, and an arc of any order even is used as the index, i.e., (i, j).

N ={ 1, . . . , n }: the set of customer (farmer) nodes. Nodes use natural numbers as index subscripts, and a customer node i can be represented as i ∈ N.

G = (A, N): the whole collection network of farmers and depot.

K ={ 1, . . . , m }: a homogeneous set of vehicles. Using natural numbers as index subscripts, a vehicle k can be expressed as k ∈ K.

Constants and variables:

q_i: the uncertain demand of farmer node i within a distribution horizon, i ∈ N.

$t_{i}^{s}$ : the time required for the collection vehicle to serve the farmer node i, i ∈ N.

d_ij: the travel distance between farmer node i and farmer node j, i, j ∈ N. The Euclidean distance in two-dimensional coordinates is used for the test of benchmark instances. In the case study section, the distance between nodes is obtained from the online map platform in a free-flow state according to the coordinates of the starting and ending nodes.

$t_{ij}^{k}$ : the travel time between farmer node i and farmer node j.

$a_{i}^{k}$ : the arrival time when vehicle k visits farmer node i.

$b_{i}^{k}$ : the depart time when vehicle k visits farmer node i.

$w_{i}^{k}$ : the service time when vehicle k visits farmer node i.

(l_i, u_i): the time windows of farmer node i.

Q: the capacity of vehicle.

$q_{i}^{k}$ : the vehicle load when vehicle k leaves farmer node i.

$y_{ij}^{k}$ : the load when vehicle k on arc (i, j).

Decision variables:

$x_{ij}^{k}$ : $x_{ij}^{k} = 1$ , if vehicle k flow on arc (i, j), otherwise, $x_{ij}^{k} = 0$ .

r^k: r^k = 1, if vehicle k is used, otherwise, r^k = 0.

$z_{i}^{k}$ : $z_{i}^{k} = 1$ , if the farmer i is served by vehicle k, otherwise, $z_{i}^{k} = 0$ .

The text should be in a double column format with a space of 10 mm between the two 76 mm wide columns.

4.2 Mathematical model

According to presentation above, the mixed integer linear programming is established. The cost considered in the model includes vehicle usage cost, cargo damage cost caused by transport process, refrigeration cost caused by refrigerated transport, and time cost (waiting cost for vehicles to arrive at the farmer’s location earlier, and punishment cost for vehicles exceeding the latest collection time).

The vehicle usage cost is composed of initial cost and transportation cost. The initial cost is set as a fixed value, which is positively correlated with the number of vehicles used. Then, the number of vehicles used can be saved as much as possible by incorporating fixed costs into the objective function. The transportation cost is related to fuel consumption, which is positively correlated with the distance traveled. Therefore, the vehicle usage cost is calculated by Equation (7): $C_{1} = p_{1} \sum_{k = 1}^{K} \sum_{i = 1}^{N} z_{i}^{k} r^{k} + p_{2} \sum_{i, j = 1}^{N} \sum_{k = 1}^{K} d_{ij} x_{ij}^{k}$ (8)

where, p₁ is the initial cost per vehicle, and p₂ is the transportation cost per kilometer.

The variable function of fresh product quality change due to a certain decay rate is M (t) = M₀e^-∂t, where M( t) is the quality of fresh product at the time t, t is the transportation time; M₀ is the product quality starting from the logistics center; ∂is the spoilage rate of fresh products that related to storage temperature. It is assumed that the storage temperature of fresh products in transit is constant, so the spoilage rate of fresh products at constant temperature is also constant. Due to the multiple opening and closing of the vehicle door will cause changes in the temperature of the cabin, thus it can affect the quality of fresh products. Therefore, the goods damage costs can be defined as by Equation (8): $\begin{matrix} C_{2} = \sum_{k = 1}^{K} \sum_{i = 1}^{N} z_{i}^{k} p_{f} [q_{k} (1 - e^{- \partial_{1} (a_{j}^{k} - b_{i}^{k})}) \\ + Q_{i}^{k} (1 - e^{- \partial_{2} t_{i}^{s}})] \end{matrix}$ (9)

where, p_f is the unit cost of fresh products, ∂₁ is the decay rate of fruits and vegetables at constant temperature in the process of cargo transportation, ∂₂ (∂₂ > ∂₁) is the decay rate caused when the door is closed and closed, and $Q_{i}^{k}$ is the weight of fresh products carried by refrigerated vehicles when vehicle k leaves farmer i.

Fresh products are generally transported by refrigerated trucks, so there will be refrigeration energy costs during the transportation process. The cost includes the cooling temperature in compartment and the additional cooling cost against temperature changes caused by opening and closing door operation. Therefore, the total cooling cost of the vehicle is defined as by Equation (9): $C_{3} = p_{3} \sum_{k = 1}^{K} \sum_{i, j = 1}^{N} x_{ij}^{k} t_{ij}^{k} + p_{33}^{'} \sum_{k = 1}^{K} \sum_{j = 1}^{N} z_{j}^{k} t_{j}^{s}$ (10)

where, p₃ is unit refrigeration cost generated in transportation process, p₃ is unit refrigeration cost during loading and unloading process.

The time cost includes the waiting cost when the vehicle arrives at the farmer’s location earlier and the punishment cost when the vehicle exceeds the latest pick-up time. Therefore, the time cost of the vehicle during the entire transportation is by Equation (10): $C_{4} = p_{4} \sum_{k = 1}^{K} \sum_{i = 1}^{N} \max {0, l_{i} - a_{i}^{k}} + p_{4}^{'} \sum_{k = 1}^{K} \sum_{i = 1}^{N} \max {0, a_{i}^{k} - u_{i}}$ (11)

where, p₄ is the waiting cost of early arriving, and p₄ is the punishment cost of vehicles arriving late. $min C = C_{1} + C_{2} + C_{3} + C_{4}$ (12)

Subject to: $\sum_{k \in T_{i}} z_{i}^{k} = 1, \forall i \in N$ (13) $\sum_{j \in N} x_{ji}^{k} = \sum_{j \in N} x_{ij}^{k}, \forall i \in N, \forall k \in K$ (14) $2 x_{ij}^{k} ⩽ z_{i}^{k} + z_{j}^{k}, \forall i, j \in N, \forall k \in K$ (15) $\sum_{k \in T_{i} \cap T_{j}} \sum_{j \in N} x_{ij}^{k} ⩽ 1, \forall i \in N$ (16) $\sum_{k \in T_{i} \cap T_{j}} \sum_{j \in N} x_{ji}^{k} ⩽ 1, \forall i \in N$ (17) $y_{ij}^{k} ⩽ {Qx}_{ij}^{k}, \forall i, j \in N, \forall k \in K$ (18) $\sum_{j \in N} y_{i, i + 1}^{k} - \sum_{i \in N} y_{i - 1, i}^{k} = q_{i}^{k} e_{i}^{k}, \forall k \in K$ (19) $\sum_{j \in N} y_{ji}^{k} - \sum_{j \in N} y_{ij}^{k} ⩾ q_{i} z_{i}^{k}, \forall i \in N, \forall k \in K$ (20) $a_{j}^{k} = b_{i}^{k} + t_{ij}, \forall i, j \in V, \forall k \in K$ (21) $r^{k} ⩾ z_{i}^{k}, \forall i \in N, \forall k \in K$ (22)

The objective function is to minimize the total cost, in which C1–C4 are defined by Equations (7)–. Constraints indicate that each customer can only be serviced by one vehicle. Constraints represent the flow balance in each node. Constraints represent that the vehicle k passes through the arc (i, j) until the customers i and j are served by the vehicle simultaneously. Constraints (15)-(16) indicate that each customer can only be serviced by one vehicle. Constraints (17) indicates that the load of the vehicle k on the arc (i, j) cannot exceed the maximum load of the vehicle. Constraints (18)-(19) indicate that customer needs must be met. Constraints (20) indicate that the arrival time of the vehicle k must be equal to the departure time of i plus the travel time $t_{ij}^{k}$ . Constraints (21) indicate that only when the vehicle is in use, the customer can get the service of the vehicle.

Given all parameters, the above models can be solved by CPLEX (academic edition: 12.10). However, the optimal solution could not be obtained by the solver in a reasonable time with the increase of the problem scale. Therefore, a hybrid simulated annealing algorithm integrated into the genetic algorithm is designed in the next section.

5 Algorithm design

The advantages of using simulated annealing in hybrid heuristic algorithms to solve optimization problems are mainly: first, simulated annealing can accelerate the convergence speed of the algorithm, allowing the algorithm to find a region close to the optimal solution more quickly. Second, simulated annealing has strong global search capabilities, which can to some extent avoid the algorithm from falling into local optimal solutions. Third, simulated annealing achieves flexible acceptance through acceptance probability formulas, which helps explore more regions in the solution space. Fourth, combining simulated annealing with search methods in genetic algorithms allows for collaborative search in multiple search directions, improving search efficiency. In conclusion, hybrid heuristic algorithms based on simulated annealing can improve the convergence speed and global search capabilities of the algorithm, effectively solving optimization problems.

The solution approach included two independent parts is designed, According to features of the problem proposed in this paper. In the first part, the gray prediction algorithm is used to predict the amount of goods collected by farmers. In the second part, a Hybrid Simulated Annealing (HSA) Algorithm with genetic operators is designed to solve the pick-up routing problem. The SA algorithm has strong robustness and is suitable for parallel processing [58]. The combination of genetic operators can improve the global search ability and convergence rate of the algorithm [59]. The framework of the solution approach is shown in Fig. 3.

Fig. 3

Grey Prediction Flowchart.

5.1 Demand forecasting

Fig. 4

An overview of the solution approach.

5.2 Coding

In this paper, natural numbers are used to encode distribution center, customers (i.e. farmers). For example, a case contains 10 customers, if we use 1–10 representing customers and 0 representing distribution center, then [0-1-2-3-4-5-0-6-7-0] means a feasible solution, including two routes.

5.3 Initial solution

Step 1. According to the historical data of farmers’ pick-up demand, the GM (1, 1) is used to predict farmers’ demand within the current horizon.

Step 2. Constrain the loading capacity of the truck to determine whether the demand of farmers on the route meets the capacity constraint of the truck. If the capacity limit is exceeded, it is split into more paths by the split operator.

Step 3. If the time window requirements of the farmers are met, the initial feasible solution is generated.

5.4 SA framework

SA uses the idea of physics [47]. The change rule of entropy is similar to cost functions in combinatorial optimization problems. The basic idea of SA is as follows: The particles inside the object will become disordered with the increase of temperature based on the physics theory. The particles move slowly to release the internal energy and become stable and orderly if the temperature is slowly lowered. According to physics, the steady state will be achieved at environment temperature, and the energy of particles is minimum, i.e., the cost-friendly solution can be obtained. The simulated annealing algorithm designed in this paper adopts random perturbation operator, and its steps are as follows:

Step 1. Generating a new state s_j randomly.

Step 2. If min{ 1, exp [- (C (s_j) - C (s))] } ⩾ random [0, 1), then s = s_j.

Step 3. Acceptance criteria for the state function: min{ 1, exp(- ΔC/t) }.

5.5 Genetic operators

Genetic operators include selection operator, crossover operator and mutation operator. A hybrid selection operator without duplicate substring is proposed in this paper, which is generated by the combination of two selection methods in a certain proportion, and then reproduces steady-state without duplicate substring. The selection methods used are the tournament selection and roulette selection.

Crossover operator employs the strategy of retaining the best substring. The principle is to select two chromosomes as parent 1 and parent 2. Then, putting the substring with the largest load in parent 1 in the front of the new gen, and selecting the other two random gene segments in parent 1 as the fixed gene. Meanwhile, removing the genes other than the selection gene and the fixed gene. The same part of the undeleted gene in parent 2 was deleted, and the remaining gene in parent 2 except the depot 0 is inserted into parent 1.

The mutation operator is to randomly a mutation position in the chromosome. If the position is not the depot 0, a new gene will be randomly generated to replace the original gene. Otherwise, the mutation location is selected again. The mutation method used in this paper is similar to the [59].

6 Computational experiment

6.1 Benchmark test

The gray forecasting algorithm based on data transformation is used for quantitative analysis. The discrete data scattered on the time axis is regarded as a series of continuous change. Then the unknown factors in the gray system are weakened and the influence degree of known factors is strengthened by the way of accumulation and subtraction. Finally, a continuous differential equation with time as the variable is constructed, and the parameters in the equation are determined by mathematical method, so as to achieve accurate prediction of small sample data. In order to verify the solving quality of the HSA algorithm, the well-known Solomon benchmark instance is used to test it in this section. The Solomon instances are classified into three categories. Instance names start with “C”, “R”, and “RC” respectively. “C” represents the clustering distribution of all customer nodes, “R” represents the random distribution of all customer nodes, and “RC” represents the combination of the two distributions. The Number of Vehicles (NV), and vehicle capacity vary with different instance. The number of nodes in the tested instances is 50 and 100. The number of available vehicles and vehicle capacity vary with different instances. The data files for all the instances can be downloaded from http://web.cba.neu.edu/msolomon/problems.htm, and the Best Known Solution (BKS) can be obtained from http://web.cba.neu.edu/msolomon/heuristi.htm. The result is used to compare with HAS, which is represented as C&S [60]. The performance of HAS algorithm is measured in terms of gap percentage which is defined by the following Equations (22)–(24).

G a p^{a} = \frac{(C S_{T D} - B K S_{T D})}{B K S_{T D}} \times 100

(23)

G a p^{b} = \frac{(H S A_{T D} - B K S_{T D})}{B K S_{T D}} \times 100

(24)

G a p^{c} = \frac{(H S A_{N V} - B K S_{N V})}{B K S_{N V}} \times 100

(25)

Since the benchmark instances are only simulations of realistic distribution scenarios, when applying HAS algorithm, Euclidian distance is used to calculate the distance between nodes, and the travel time is assumed to be equal to the corresponding Total Distance (TD), that is, the vehicle speed is 1. In addition, since the objective function designed in this paper represents the total cost and is designed for real cases, it needs to be changed to minimize the total distance. The algorithm is run 3 times on each benchmark instance, and the best results are summarized in Table 2.

Table 2

The results based on the Solomon instances (N = 50)

Instance	TD				NV
	BKS	C&S	HAS	Gap ^a	Gap ^b	BKS	HAS	Gap ^c
C101-50	362.4	418.42	362.4	15.46%	0.00%	5	5	0.00%
C102-50	361.4	417.34	361.4	15.48%	0.00%	5	5	0.00%
C103-50	361.4	416.06	378.9	15.12%	4.84%	5	5	0.00%
C104-50	358	360.83	358	0.79%	0.00%	5	5	0.00%
C201-50	360.2	373.55	361.4	3.71%	0.33%	3	3	0.00%
C202-50	360.2	366.78	362.2	1.83%	0.56%	3	3	0.00%
C203-50	359.8	371.81	364.5	3.34%	1.31%	3	3	0.00%
C204-50	350.1	365.86	353.6	4.50%	1.00%	2	2	0.00%
R101-50	1044	1046.7	1147	0.26%	9.87%	12	14	16.67%
R102-50	909	911.44	977	0.27%	7.48%	11	13	18.18%
R103-50	361.4	/	388.9	/	7.61%	9	10	11.11%
R104-50	625.4	642.13	639.1	2.68%	2.19%	6	6	0.00%
R201-50	791.9	808.53	798.8	2.10%	0.87%	6	6	0.00%
R202-50	698.5	714.19	706.4	2.25%	1.13%	5	5	0.00%
R203-50	605.3	620.59	614.5	2.53%	1.52%	5	5	0.00%
R204-50	506.4	515.12	519.7	1.72%	2.63%	2	2	0.00%
RC101-50	944	958.59	951	1.55%	0.74%	8	9	12.50%
RC102-50	822.5	886.47	822.5	7.78%	0.00%	7	7	0.00%
RC103-50	710.9	823.98	710.9	15.91%	0.00%	6	6	0.00%
RC104-50	545.8	639.28	545.8	17.13%	0.00%	5	5	0.00%
RC201-50	684.8	686.31	694.6	0.22%	1.43%	5	5	0.00%
RC202-50	613.6	615.04	614.3	0.23%	0.11%	5	5	0.00%
RC203-50	555.3	559.68	604.1	0.79%	8.79%	4	4	0.00%
RC204-50	444.2	471.82	456.7	6.22%	2.81%	3	3	0.00%
Avg.				5.30%	2.30%			2.44%

As can be seen from Table 2, the average Gap ^a between the results obtained by the HAS algorithm and the C&S for the three types of instances (N = 50) is 5.30%. Meanwhile, the average Gap ^b between the results obtained by the HSA algorithm and the BKS for the three types of instances (N = 50) “C”, “R” and “RC” is 2.30%, and the average Gap ^c of vehicles enabled is 2.44%.

As can be seen from Table 3, in terms of total distance (N = 100), the average gap between the solution in this paper and BKS is 4.48%, and the average gap between C&S’s solution and BKS is 7.15%. In terms of the number of vehicles (N = 100), the average gap between the solution in this study and BKS is 2.98%.

Table 3

The results based on the Solomon instances (N = 100)

Instance	TD				NV
	BKS	C&S	HAS	Gap ^a	Gap ^b	BKS	HAS	Gap ^c
C101-100	827.3	944.32	896.4	14.14%	8.35%	10	11	10.00%
C102-100	827.3	952.65	838.7	15.15%	1.38%	10	10	0.00%
C103-100	826.3	963.26	828.1	16.58%	0.22%	10	10	0.00%
C104-100	822.9	934.46	826.2	13.56%	0.40%	10	10	0.00%
C201-100	589.1	609.22	589.1	3.42%	0.00%	3	3	0.00%
C202-100	589.1	591.56	589.1	0.42%	0.00%	3	3	0.00%
C203-100	588.7	605.21	588.7	2.80%	0.00%	3	3	0.00%
C204-100	588.1	666.75	597.3	13.37%	1.56%	3	3	0.00%
R101-100	1637.7	1692.29	1658.9	3.33%	1.29%	20	20	0.00%
R102-100	1466.6	1541.15	1623.6	5.08%	10.71%	18	20	11.11%
R103-100	1208.7	1255.6	1382.3	3.88%	14.36%	14	16	14.29%
R104-100	971.5	1036.32	1046.2	6.67%	7.69%	11	12	9.09%
R201-100	1143.2	1191.36	1206.7	4.21%	5.55%	8	8	0.00%
R202-100	/	1084.14	1159.4	/	/	/	8	/
R203-100	/	902.62	933.8	/	/	/	8	/
R204-100	/	797.09	754.8	/	/	/	5	/
RC101-100	1619.8	1702.69	1697	5.12%	4.77%	15	15	0.00%
RC102-100	1457.4	1581.29	1498.6	8.50%	2.83%	14	14	0.00%
RC103-100	1258	1376.89	1365.5	9.45%	8.55%	11	12	9.09%
RC103-100	/	1229.81	1056.4	/	16.42%	/	8	/
RC201-100	1261.8	1282.35	1261.8	1.63%	0.00%	9	9	0.00%
RC202-100	1092.3	1108.01	1104.7	1.44%	1.14%	8	8	0.00%
RC203-100	/	1014.43	986.9	/	/	/	8	/
RC203-100	/	834.73	809.1	/	/	/	5	/
Avg.				7.15%	4.48%			2.98%

The results of the above two experiments show that the HSA algorithm can obtain higher solution quality on the instance sets with different scales. The solution performance of HSA algorithm is also significantly improved by comparing with the results of reference [60] under the same conditions.

The experimental results of the above Solomon example show that the optimal value obtained by the hybrid simulated annealing algorithm is better than that of the C&S algorithm.

6.2 Real case and Forecasting on the pick-up demand

Baby mustard (Brassica juncea var. gemmifera) is a special fresh agricultural product in China, rich in Sichuan province and Chongqing city with very strict quality assurance [61]. In this paper, 100 locations of farmers (numbered 1 –100) who grow baby mustard are selected as the research object of the optimization of the pick-up route.

According to the historical data of the harvest quantity of baby mustard in the past five years, the grey forecasting method is used to predict the current collection quantity (unit, kg). Take farmer 1 as an example, the aggregate quantity of baby mustard in recent five years is 268, 325, 314, 330 and 327, respectively. Considering the sequence as the initial series, the new series 268, 318, 320, 322 and 326 are obtained by the accumulation operation one time. The grey parameters are solved by constructing the matrix B. Then the grey parameters are put into the differential equation, and the predicted value is 328 according to Eq.. Finally, the rounding operation is performed to obtain the forecast aggregate demand of the farmer in the current horizon, i.e., 328. Similarly, the prediction results of the aggregate demand of 100 farmers are shown in Table 4.

Table 4
The predicted result

NO. of Predicted NO. of Predicted NO. of Predicted NO. of Predicted NO. of Predicted

farmer yield farmer yield farmer yield farmer yield farmer yield

1 328 21 300 41 395 61 216 81 193

2 425 22 336 42 313 62 284 82 406

3 262 23 281 43 250 63 440 83 180

4 338 24 182 44 271 64 456 84 448

5 257 25 308 45 193 65 353 85 389

6 335 26 319 46 234 66 401 86 186

7 333 27 321 47 280 67 303 87 233

8 397 28 263 48 280 68 243 88 229

9 242 29 413 49 309 69 361 89 293

10 418 30 414 50 280 70 342 90 257

11 338 31 280 51 336 71 277 91 238

12 294 32 370 52 331 72 349 92 279

13 190 33 275 53 305 73 291 93 239

14 292 34 317 54 257 74 271 94 350

15 330 35 288 55 299 75 304 95 330

16 247 36 476 56 272 76 367 96 318

17 454 37 304 57 249 77 378 97 290

18 250 38 340 58 251 78 255 98 236

19 245 39 226 59 370 79 248 99 238

20 243 40 324 60 424 80 326 100 253

NO. of	Predicted	NO. of	Predicted	NO. of	Predicted	NO. of	Predicted	NO. of	Predicted
1	328	21	300	41	395	61	216	81	193
2	425	22	336	42	313	62	284	82	406
3	262	23	281	43	250	63	440	83	180
4	338	24	182	44	271	64	456	84	448
5	257	25	308	45	193	65	353	85	389
6	335	26	319	46	234	66	401	86	186
7	333	27	321	47	280	67	303	87	233
8	397	28	263	48	280	68	243	88	229
9	242	29	413	49	309	69	361	89	293
10	418	30	414	50	280	70	342	90	257
11	338	31	280	51	336	71	277	91	238
12	294	32	370	52	331	72	349	92	279
13	190	33	275	53	305	73	291	93	239
14	292	34	317	54	257	74	271	94	350
15	330	35	288	55	299	75	304	95	330
16	247	36	476	56	272	76	367	96	318
17	454	37	304	57	249	77	378	97	290
18	250	38	340	58	251	78	255	98	236
19	245	39	226	59	370	79	248	99	238
20	243	40	324	60	424	80	326	100	253

6.3 Results in the real case

Given the initial cost of the pick-up vehicle is 100 CNY per vehicle, the variable cost is 2 CNY per km, the time window penalty cost is 2 CNY per minute, and the speed is 30 km per hour. The location and time windows of farmers (see the Supplementary Materials Fig. S1 and Table S1), the predicted pick-up demand, and other parameters are input into the HAS algorithm designed in this paper. Then, the algorithm is run 20 times to get the best result. The routes are listed in Table 5.

Table 5
The best tour plan obtained by the proposed algorithm

Route Visit sequence

1 0->87->99->86->74->75->94->96->0

2 0->27->80->93->91->49->46->18->0

3 0->36->98->85->95->55->100->0

4 0->13->45->20->4->1->10->23->0

5 0->56->71->77->34->22->50->0

6 0->47->89->90->26->25->92->88->0

7 0->58->70->78->79->73->72->57->0

8 0->31->32->33->35->63->28->0

9 0->14->16->8->6->69->65->0

10 0->38->97->84->15->53->24->0

11 0->60->66->67->68->17->61->0

12 0->40->29->48->43->59->21->0

13 0->12->7->11->9->5->2->0

14 0->64->76->81->30->82->0

15 0->37->42->19->44->3->41->0

16 0->39->52->54->83->51->62->0

Route	Visit sequence
1	0->87->99->86->74->75->94->96->0
2	0->27->80->93->91->49->46->18->0
3	0->36->98->85->95->55->100->0
4	0->13->45->20->4->1->10->23->0
5	0->56->71->77->34->22->50->0
6	0->47->89->90->26->25->92->88->0
7	0->58->70->78->79->73->72->57->0
8	0->31->32->33->35->63->28->0
9	0->14->16->8->6->69->65->0
10	0->38->97->84->15->53->24->0
11	0->60->66->67->68->17->61->0
12	0->40->29->48->43->59->21->0
13	0->12->7->11->9->5->2->0
14	0->64->76->81->30->82->0
15	0->37->42->19->44->3->41->0
16	0->39->52->54->83->51->62->0

To illustrate the advantages of the proposed hybrid algorithm, a comparative experiment is done. The costs obtained by SA, GA and HSA algorithms are summarized in Table 6 based on the real case.

Table 6

The cost comparation based on the real case

Algorithm	C₁	C₂	C₃ + C₄	C
GA	768.51	1670	948.54	3387.05
SA	749.44	1659	941.34	3349.78
HSA	727.84	1600	918.94	3246.78

As can be seen in Table 6, the HSA algorithm has the lowest cost, which is 4% and 3% lower than GA and SA respectively. However, due to the complex structure of the hybrid algorithm, it requires more time consumption under the same conditions. The experimental results indicate that, the optimal solution obtained by the hybrid simulated annealing algorithm is better than those of other heuristic algorithms such as genetic algorithm and simulated annealing algorithm in the fresh food collection path problem.

7 Management insights

The design of an improved gray prediction model based on data transformation is incorporated into the optimization of fresh logistics pick-up routing problems. The reduction of the total cost and the improvement of transportation efficiency demonstrate the benefits of addressing the fresh logistics pick-up routing problems. In this section, the management insights of this study are described as follows:

The economic optimization model and the proposed hybrid simulated annealing algorithm with genetic operators have significant managerial implications for cost reduction in the fresh logistics industry. By minimizing transportation costs through efficient pickup routing, companies can enhance their profitability and competitiveness.

The accurate prediction of pickup demand using the improved grey forecasting method enables better planning and coordination of logistics operations. This, in turn, leads to improved customer service by ensuring timely and reliable delivery of fresh agricultural products, thereby enhancing customer satisfaction and loyalty.

The optimization model and routing methods proposed in this study can help optimize resource utilization, such as vehicle capacity and driver scheduling. By efficiently allocating resources based on accurate demand forecasts, logistics companies can minimize idle time, reduce fuel consumption, and enhance overall operational efficiency.

This study has a significant practical implication for the optimization of logistics networks. The optimization of the fresh logistics pick-up routes greatly alleviates the phenomenon of violations of customers’ time windows. The adoption of an improved gray prediction strategy in fresh logistics pick-up distribution network can reduce the number of required pick-up vehicles, thus facilitating the construction of resource-efficient and environmentally friendly logistics networks.

The proposed solution method in this study to solve the fresh logistics pick-up routing problems can provide a methodological reference for the research of the Pick-up routing problems. The nonlinear programming model of the minimum total cost balances the operation of the logistics network from two conflicting aspects of economy and efficiency. The hybrid algorithm, including a gray prediction algorithm and a heuristic algorithm, is developed to solve the fresh logistics pick-up routing problems with time windows. Therefore, the proposed solution methods contribute to building an economic and sustainable logistics network and promoting the enterprise’s competitiveness.

The optimization of the fresh logistics pick-up distribution networks plays a great role in elevating the agricultural digitalization system. With the integration and utilization of information technology and digital economy in fresh production, distribution and operation, Smart food markets and Path planning technologies s are widely adopted by fresh enterprises and logistics companies to construct a sustainable logistics distribution network for coping with the increasing competition. Meanwhile, many government departments have enacted a series of policies to promote the popularity and development of agriculture digital economy to push China from a large agricultural country to a powerful agricultural country.

In order to further build the supply chain of fresh agricultural products and improve the quality of fresh agricultural products, the government should strengthen the location planning and operation management of agricultural production bases. As many regulations and standards as possible should be issued to allow farmers to plant agricultural products at the same time. More technical training and guidance should be provided to individual farmers so that more high-quality fresh products can be brought to market from the source.

8 Conclusion

With the improvement of material living standard, the demand of urban residents for fresh agricultural products is increasing sharply. In order to reduce the transportation cost of collecting agricultural products from the source area, this paper firstly adopts an improved grey prediction method with nonlinearity and less volatility to estimate the pickup demand of fresh agricultural products. Then a mathematical model of the cargo damage cost caused by the spoilage rate of fresh fruits and vegetables transportation is established with a nonlinear objective function. To solve this model, an integrated genetic operator and hybrid simulated annealing algorithm are designed to solve this problem. Numerical experiments on a set of benchmark instances show the usability of the proposed algorithm. Finally, a case study is used to verify the model and algorithm designed and developed in this paper. This paper introduces the optimization research on the “first mile” of fresh agricultural products pick-up routing, which is of practical significance to shorten the transportation time and ensure the supply quality and safety.

Although the models and solution approach have many features, there are still some limitations in this study. First of all, this paper only considers the single cycle of fresh goods collection problem, the multi-horizon pick-up problem could be considered. In addition, the impact of environmental temperature and vehicle speed on variable cost are ignored in the design of simulation experiment. More actual environmental factors could also be applied to evaluate the variable cost of vehicles more accurately. In addition, the use of hybrid fleet to complete the collection process of agricultural products, while considering the constraints of carbon emissions, are worthy of further study.

Supplementary Materials

Fig. S1. The location of farmers based on latitude and longitude coordinates; Table S1. The pick-up time windows of farmers.

The supplementary material is available in the electronic version of this article: https://dx-doi-org-s.web.bisu.edu.cn/10.3233/JIFS-235260

Author Contributions

Methodology, writing–original draft, Yonghong Liang; Conceptualization, supervision, Xianlong Ge; Formal analysis, editing, Yuanzhi Jin; Software, Yating Zhang; Funding acquisition, Zhong Zheng; Writing literature review, Yunyun Jiang. All authors have read and agreed to the published version of the manuscript.

Funding

This research was funded by the humanities and social science research project of Chongqing Education Commission [grant number 20JD059], and the Chongqing Jiaotong University graduate scientific research innovation project [grant number 2022S0069]. This research was also funded by the Young and middle-aged teachers basic ability improvement project of Guangxi University “Research on Key Technologies of Urban and Rural Logistics Integration Operation Based on Big Data and Cloud Computing” (2023KY1862).

Footnotes

Acknowledgments

This research was funded by the Key scientific and technological innovation project of “Construction of Chengdu-Chongqing Economic Circle” of Chongqing Municipal Education Commission [grant number KJCXZD2020031], the humanities and social science research project of Chongqing Education Commission [grant number 20JD059], and the Chongqing Jiaotong University graduate scientific research innovation project [grant number 2022S0069]. This research was also funded by the Young and middle-aged teachers basic ability improvement project of Guangxi University “Research on Key Technologies of Urban and Rural Logistics Integration Operation Based on Big Data and Cloud Computing” (2023KY1862).

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