2.1 Problem description and basic assumptions

The distribution center, refrigerated transportation vehicles, and customer nodes constitute the fresh food distribution network. The problem is as follows. The distribution center distributes fresh food to customers. The locations, demands, and time windows of customer nodes are known. Distribution vehicles can arrive at customer nodes in advance but must wait until the start of service time to serve. There is one type of fresh food to distribute, and its safety constantly deteriorates. The city traffic congestion periods and the congestion levels can be obtained from the transportation department. The problem is to comprehensively consider the logistics distribution costs and determine the optimal vehicle dispatching and routing plan that minimizes the total distribution costs and maximizes the food safety. To facilitate analysis and study, the following assumptions are made: (1) There is only one distribution center, which has sufficient supplies. There are many refrigerated transportation vehicles, which are of the same type. (2) Distribution vehicles depart from the cold chain logistics distribution center and return to the center immediately after completing the delivery task. (3) The demand of each customer node is less than the vehicle capacity. There is also a service time-window requirement. A vehicle can reach a client point to provide service at an earlier or later time but then will pay a certain penalty. (4) Considering the time-varying nature of network traffic, vehicle speeds in all the sections of the road network are updated every 10 min. (5) The temperature inside refrigerated vehicles are assumed to be stable and desirable during distribution. (6) Each vehicle has its own fixed cost, and time cost related to travel time in the distribution process. (7) The maximum load capacity of each vehicle is fixed. Each customer node has only one vehicle to serve it. While waiting to serve the customer, the engine is turned off to avoid fuel consumption and carbon emissions, but the temperature control equipment operates normally.

2.2 Symbols and variables

N = {0,1,…n}: Set of all nodes in the cold chain logistics distribution network; 0 denotes the distribution center, N` = N\{0} denotes the set of customers nodes

K = {1,2,…k}: Set of cold chain distribution vehicles

Q: Maximum load capacity of vehicle (kg)

W = {1,2…w}: Set of controlled temperatures w (Celsius)

T = {T₁,T₂,…,T_m}: Set of the time period throughout the day, m is the total number of time periods.

d_ij: Driving distance (km) of the vehicle from node i to node j, i,j∈N

t_ijk: Travel time (min) of the k^th vehicle on the road (i, j), k∈K, i,j∈N

: Time at which the k^th vehicle arrives at point i

: Time at which the k^th vehicle leaves point i

c₁: Fixed cost of per vehicle (yuan/vehicle)

c₂: Time cost of vehicle per unit time(yuan/h)

q_i: Demand of customer node i, (kg), q_i<Q

p: Unit cost of transporting fresh food(yuan/kg)

[ET_i,LT_i]: Time window when the customer i expects to be served

: Unloading time needed when serving at customer node i(min)

ϑ: Length of a sufficiently short route section in partitioning the route(km)

: Travel distance(km) of the k^th vehicle on the route section R of road (i, j)

: Travel speed(km/h) of the k^th vehicle on the route section R of road (i, j)

: Start time of the k^th vehicle on the route section R of road (i, j)

: Travel time(min) of the k^th vehicle on the route section R of road (i, j)

ds: Safety influence coefficient

W_min: Temperature at which no microorganisms grow (Celsius)

w: Temperature of controlled temperatures (Celsius)

COP: Conversion rate between energy and heat

W_o: Ambient thermodynamic temperature outside, thermodynamic temperature (K)

W_L: Thermodynamic temperature of controlled temperatures w (K), W_L = 273+w

f_kw: Fuel consumption of unit delivery time and unit fresh product where the k^th vehicle travels in the control temperature of w (L/kg.h)

g_kw: Fuel consumption cost of unit delivery time and unit fresh product where the

k^th vehicle travels in the control temperature of w (yuan/kg.h)

θ_w: Cooling cost coefficient in the control temperature of w

WC_w: Total refrigeration fuel consumption cost of temperature control w for all vehicles during the distribution process(yuan)

FS_ijkw: Food safety loss of the k^th vehicle travelling from node i to node j at a controlled Celsius temperature w

ω₀,ω₁,ω₂,ω₃,ω₄,ω₅,ω₆: Correction factors, whose values are dependent on the type of the loaded vehicle

χ₀, χ₁, χ₂, χ₃,χ₄, χ₅, χ₆, χ₇: Loading correction factors, whose values depend on the type of the loaded vehicle

: Fuel consumption rate of the k^th vehicle on the route section R of the road (i, j), (L/km)

FC_ij: Fuel consumption cost of the vehicle on the road (i, j), (yuan)

: Carbon emission of the k^th vehicle on the route section R of the road (i, j), (kg/km)

CY_ij: Carbon emissions cost of the vehicle on the road (i, j), (yuan)

ce: Carbon emissions generated by 1 L gas(kg/L)

cf: Unit cost of carbon emissions(yuan/kg)

gf: Unit cost of fuel consumption(yuan/L)

P_e: Cost of waiting for the unit time if the vehicle arrives at customer node in advance (yuan/min)

P_l: Cost of punishing for the unit time if the vehicle is late to the customer node (yuan/min)

x_kw: Dummy variable (0 and 1): x_kw = 1 if the vehicle k is used at a controlled temperature of w, otherwise x_kw = 0

y_ikw: Dummy variable (0 and 1): y_ikw = 1 if the vehicle k with a controlled temperature of w serves customer node i, otherwise y_ikw = 0

z_ijkw: Dummy variable (0 and 1): z_ijkw = 1 if the vehicle k with a controlled temperature of w travels on the route (i, j), otherwise z_ijkw = 0

2.3 Solving the driving time using route partitioning strategy

In a time-varying network, vehicle speeds vary with the time period. Therefore, travel time is difficult to calculate and needs to be rationally dealt with. Based on the previously proposed cross-time period travel time calculation methods [23, 27, 28], the real-time speed of a vehicle within a sufficiently short distance is used as its fixed speed within the time period. The following shows a step function for the speed (v_ij(t)) at various times on the road (i, j): (1)

This paper analyzes the relationships between driving time, vehicle velocity, route section, and driving distance based on the existing literature and designs a route partitioning method to solve the problem.

Step 1. Partition the route. When the k^th vehicle runs on the route (i, j), first the length of a sufficiently short route section is set to be ϑ (ϑ = 0.2km), and the vehicle speed on this route section is assumed to be constant. The route (i, j) is partitioned to a number of H = ⌈d_ij/ϑ⌉ route sections (⌈⌉ means rounding up to the nearest integer) according to ϑ. The first H-1 route sections have a length of ϑ, while the last section has a length of d_ij-ϑ* (H-1), so (2)

Step 2. Calculate the driving time on the first H-1 route sections. The speed of the route section is set according to the starting time of the route section within the time period. Calculate the travel time of the k^th vehicle during the first route section on the road (i, j): , and determine the time period of . Calculate it according to formula (1), and . The time that the vehicle arrives at this route section is used as the departure time from the second route section, . The driving times on the middle sections can be expressed in the same manner.

Step 3. Calculate the driving time on the last route section. , .

Step 4. Calculate the total driving time of the k^th vehicle on route (i, j): . The computation terminates.

2.4 Food safety loss measurement function

The primary issue in the process of cold chain distribution is the hygiene and safety of food. Once a food safety problem occurs during transportation, it could induce health events and cause panic. The foodborne diseases caused by pathogenic microorganisms have become the number-one factor affecting food safety. The assumption proposed in Ref. [26], the temperature inside distribution vehicles keeps constant during the distribution process, is employed. The food safety loss F_ijw is: (3) where ds denotes the safety influence coefficient, whose value is determined by the type of food, the cold chain logistics environment, and the type of pathogenic microorganism. If w≥W_min, it will lead to the growth of microorganisms in food and increase the cost of food safety loss; otherwise the loss of food safety remains the same with the absence of microbial growth, but the cost of temperature control will increase significantly with the decrease of temperature. Due to the differences in customer demand, the total safety loss not only relates to the controlled temperature and the distribution time, but also depends on the customer service order.

The total safety loss cost of all vehicles during distribution FSC_w can be expressed as: (4)

2.5 Calculation of temperature control cost

Since different types of refrigerated food have different requirements of temperature control, the appropriate temperature needs to be set based on the characteristics of food during transportation to ensure food quality. According to the method of Ref. [22, 29], COP is used to describe the conversion rate between energy and heat: (5)

W_L and W_o are thermodynamic temperatures, corresponding to absolute temperatures, denoted as W(K), and the unit is Kelvin (K). The relationship between thermodynamic temperature W(K) and Celsius temperature temp(°C) is: W(K) = 273+temp(°C). If the outside thermodynamic temperature W_o is 303 K(outside temperature is 30°C), and the target controlled thermodynamic temperature W_L is 283K(w = 10°C), the values of COP is . According to formula (5), COP at different temperature-controlled temperatures w can be calculated. According to the method of Ref. [22, 29], and to make the calculations more realistic, we set the benchmark: Assuming that the outside temperature is 25°C, and the cost of cooling to 10°C is one unit, i.e., θ_w = 1 (w = 10°C). In this case, the refrigeration equipment of refrigerated vehicle adopts independent units, and the unit time fuel consumption of refrigerated equipment under full load is 2.4(L/h) which is obtained through experimental simulation, and f_kw = 1.2×10⁻³ (L/kg.h). then the cooling cost coefficient of 9°C, θ_w(w = 9°C) is 18.87/17.63 = 1.07, and the cooling cost coefficients for other controlled temperatures can be calculated. The f_kw of different temperature control w can be calculated: (6)

Table 1 lists the values of COP, f_kw, and θ_w at different temperatures w (°C). The cost of temperature control per unit time at different temperatures can be obtained by solving the equation: (7)

Then, the total refrigeration fuel consumption cost WC_w is: (8)

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Table 1. The values of COP, f_kw, and θ_w at different temperatures w.

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2.6 Vehicle fuel consumption and carbon emission calculation

The cost of carbon emissions during cold chain distribution of food mainly includes the cost caused by fuel consumption for vehicle running and for temperature control. In this model, the carbon emissions generated by fuel consumption during transportation are mainly calculated with the methodologies to estimate emissions from transport (MEET) model [2]. The specific procedures are as follows: the carbon emission rate (g/km) of the k^th vehicle on the route section R of road (i,j) is (9) where is the carbon emission rate of an empty vehicle driving on a road without slope at a speed of v (). The loading correction factors of the MEET model are: (10) where γ_ij is the ratio between the real loading and the capacity on the route(i, j). Thus, the carbon emission rate of the k^th vehicle on the route section R is . According to ce = 2.32 [23], and the fuel consumption rate of the k^th vehicle on the route section R is ; therefore, the carbon emissions cost of the k^th vehicle on the route(i, j) is: (11)

The fuel consumption cost of the k^th vehicle on the road (i, j) is: (12)

Meanwhile, the cost of carbon emissions due to temperature control during the distribution process is .

2.7 Vehicle penalty cost

This penalty cost is incurred by the vehicle as a result of violating the client’s time window during the distribution process. Client i accepts service outside the time window, but there is a penalty cost for failing to provide service within the appointed time, and the total penalty cost PC equals to (13)

2.8 Mathematical model

The optimization goals of this paper are the minimum economic cost of logistics distribution(consisting of the fixed vehicle cost, time cost, time penalty cost, the cost of fuel consumption and carbon emission in transportation, the cost of fuel consumption and carbon emission in refrigeration, and cost of food safety loss). The vehicle routing problem model with time windows in time-varying road network for cold chain food distribution is as follows: (14) (15) (16) (17) (18) (19) (20) (21) (22) (23) (24) (25)

Eq (14) is the objective function. Eq (15) means that each customer must be served exactly once. Eq (16) indicates that the vehicle arrives at and leaves from the same node. Eq (17) shows the relationship between the travel time on the whole route and the travel times on route sections. Eq (18) gives the vehicle capacity constraints. Eq (19) means that each vehicle can only be used once. Eqs (20 and 21) express the relationships between the arrival time, travel time, service time, and leaving time. Eqs (22 and 23) express the relationships between decision variables z_ijkw and y_ikw. Eq (24) shows that all vehicles start and end at the distribution center. Eq (25) gives the variable value constraints.

4.1 Experiment setup

The data of the test example were from the international standard Solomon’s VRPTW databases [2]. Considering various factors of the fresh food distribution, we used the data of the type C (concentrated distribution), type R (random distribution) and type RC (mixed distribution) examples of the international standard VRPTW database. In each example, there are 100 customer points and 1 distribution center. To be more realistic, the above cases were modified as follows: A fresh milk production enterprise distribution center needed to distribute 100 customer points within the city. The starting and ending times of the model are set to 7:00 (corresponding to time 0, i.e., 0 min) and 24:00 (corresponding to 1,020 min), respectively. The length of each time period is 10 min. According to urban traffic patterns, the period 7:30–9:00 (i.e., 4th–12th periods) and the period 17:30–19:00 (i.e., 64th–72nd periods) are set as traffic congestion periods, and the other periods are set as normal periods. According to Refs. [27, 31], let ϖ = 0.05. Three vehicle speeds [VR(1−ϖ),VR(1+2ϖ),VR(1−3ϖ)] are allowed. Based on the T value of the time period, the remainder function π = mod(T,3) is used. The π values of 0, 1, and 2 correspond to three vehicle speeds during the normal time periods, VR = 60(km/h), and that during traffic congestion hours was 30 km/h.

The algorithm was programmed with Matlab R2016a and run on a computer with 8 GB of RAM and a 1.90-GHz CPU. According to the methods in Refs. [2, 17, 23, 26, 29], the variable parameters were set as follows: The basic parameters of the vehicle are Q = 200 units; let one unit load be 10 kg; then the vehicle capacity is 2 tons; the correction factors in MEET model ω₀ = 110, ω₁ = 0, ω₂ = 0, ω₃ = 0.000375, ω₄ = 8702, ω₅ = 0, ω₆ = 0; the loading correction factors in MEET model χ₀ = 1.27, χ₁ = 0.0614, χ₂ = 0, χ₃ = -0.0011, χ₄ = -0.00235, χ₅ = 0, χ₆ = 0, χ₇ = -1.33; and the model parameters are shown in Table 2, and the values of the algorithm parameters were taken from Ref. [31], as shown in Table 3.

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Table 2. Model parameters.

https://doi.org/10.1371/journal.pone.0235950.t002

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Table 3. Ant colony algorithm parameters.

https://doi.org/10.1371/journal.pone.0235950.t003

4.2 Experimental results

4.2.1 Large-sized data example programming.

The example was calculated with the dataset R208. The running time was 314.7 seconds, and the overall distribution cost obtained was 4258.0 yuan. A total of 8 vehicles were dispatched for delivery. The fixed vehicle cost was 1200 yuan. The time cost was 1029.9 yuan. The time penalty cost was 714.3 yuan. The cost of fuel consumption and carbon emission in transportation was 938.60 yuan. The cost of fuel consumption and carbon emission in refrigeration was 367.9 yuan. The cost of food safety loss was 7.31yuan. The specific routing optimization plan is presented in Table 4. In Table 4, SN denotes the number of the vehicle, VR represents the driving route of the vehicle (0 denotes the distribution center and 0–100 denote customer nodes), VAT represents the time that the vehicle arrives at each customer node, and ST represents the departure time period.

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Table 4. Vehicle route optimization results of the example with the R208 dataset.

https://doi.org/10.1371/journal.pone.0235950.t004

It can be seen from the calculation that (1) the algorithm in this paper can yield optimal route planning in a relatively short time. (2) Combining VR and VAT, it can be seen that due to the constraints of city traffic congestion, customer service time windows, and vehicle capacity, obvious differences exist between the cold chain logistics vehicle distribution routes. Vehicle 3 and 6 served the most customer nodes, 15; Vehicle 8 served the least number of customer nodes, only 9. The reason is that the time window of each node is different from the others. Vehicle route planning must ensure a minimal cost while satisfying the customer nodes’ time windows. (3) From VAT and ST, it can be seen that vehicles 1, 3, 4, and 7 depart at time 0 from the distribution center to deliver, and other vehicles depart at different times. If departing at time 0, a vehicle needs to bear a relatively large cost of waiting due to early arrival. This suggests that it is necessary for logistics enterprises to consider time dependence when planning the routes. They should scientifically plan the routes based on real situations, such as the road network conditions and the customer node time windows, and comprehensively consider the costs, such as those for temperature control and fuel consumption during transportation. (4) According to VAT, Vehicle 2 and 8 completely avoided the congestion periods. Vehicles 1,3, 4, 6, and 7 only entered the congestion period of the morning rush hour but did not enter the evening rush hour. Vehicle 5 experienced the evening rush hour congestion, but not the morning one. This indicates that the method proposed in this paper can help to appropriately avoid the traffic congestion periods and increase the distribution efficiency of vehicles.

4.2.2 Example simulation at different controlled temperatures.

Take the section 4.2.1 as an example, assuming that the temperature in refrigerated vehicles varies within a certain range, and other characteristics, such as the properties of goods and customer points, customer point service order and distribution route planning, keep unchanged. The temperature control cost and the food safety loss change during transportation, affecting the logistics distribution cost and lowering the safety and value of food. The temperature in fresh food distribution vehicles was controlled within the range from −5°C to 10°C. The sensitivity relationships between refrigerated temperature, temperature control cost (The cost of fuel consumption and carbon emission in refrigeration), safety loss cost, and logistics cost are illustrated in Figs 1 and 2.

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Fig 1. The relation between safety loss and temperature control cost.

https://doi.org/10.1371/journal.pone.0235950.g001

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Fig 2. Sensitivity analysis of distribution cost and refrigerating temperature.

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As can be seen from Fig 1, when the refrigerated temperature gradually decreased from 10°C, the safety loss cost gradually declined, but the temperature control cost increased accordingly. At a refrigerated temperature of 10°C, the temperature control cost during milk distribution was minimum, but the food safety loss was maximum; on the contrary, when the refrigerated temperature went below 0°C, the safety loss of milk was zero since no microorganisms grew, but the temperature control cost of consumption and carbon emissions increased as temperatures went down. According to Fig 2, as the refrigerated temperature decreased, the total logistics distribution cost was reduced first. At a refrigerated temperature of 1°C, the logistics cost was the lowest. Then, as the refrigerated temperature decreased, the logistics cost increased, suggesting that 1°C is preferable to other refrigerated temperatures, 1°C is the optimal refrigerated temperature for transporting milk. Neither increasing nor decreasing the temperature can optimize the total logistics distribution cost.

4.2.3 Optimization simulation on examples with different customer distributions.

Different types of examples were used to solve the milk cold chain transportation problem. The controlled refrigerated temperature was 1°C. Each example was solved 10 times, and the average value was taken as the final result. Table 5 shows the experimental results, in which EX represents the type of example, TC means the total distribution cost, TMC denotes the fixed costs of the vehicles, the time cost, and the time penalty cost, WC is the fuel consumption cost generated by temperature control, FC is the fuel consumption cost during driving, CF is the carbon emission cost generated by driving and temperature control, SC is the safety loss cost, and VN is the number of vehicles. The route plan of datasets C103, R204, and RC207 are illustrated in Fig 3.

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Fig 3. Vehicle route planning for examples with different distribution types.

https://doi.org/10.1371/journal.pone.0235950.g003

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Table 5. Simulation results of examples with different customer distributions.

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The following can be seen from Fig 3 and Table 5: (1) The example of type C had the highest total logistics distribution cost, the fixed costs of the vehicles, the time cost, and the time penalty cost, safety loss cost, temperature control cost, and number of vehicles used among all types, but its fuel consumption cost generated by driving was lower than that of other types. This is mainly because the customer nodes of the type C distribution were mainly concentrated at several regions, and the driving distances between the customer nodes were relatively short, which lowered the fuel consumption cost generated by driving. Nevertheless, the customers’ time windows were relatively narrow. In the Solomon dataset, the service time at a customer node of the type C model was 90 minutes, as opposed to the 10-minute service time at the customer node of types R and RC. Therefore, a higher requirement was placed on vehicles to arrive at customer nodes on time and satisfy the time windows, and the time penalty cost is relatively high. At the same time, the total delivery time is the longest because the customer point service time is too long, as a result, the refrigeration time is the longest and the temperature control cost is the highest. (2) Compared to type C, types R and RC had a lower total distribution cost, the time cost, the time penalty cost, safety loss cost, temperature control cost, and number of vehicles. This can mainly be attributed to the 10-minute service time at the customer nodes of these two types. The customers’ time window requirements were relatively loose, so the vehicles could visit many customers, and the total driving times were relatively short. At the same time, the refrigeration time is shorter and the refrigeration cost is lower. (3) For all types of examples, the TMC made up a relatively high proportion in the total distribution cost, making up approximately 70% of the total. For type C103, the proportion reached 79.2%. This indicates that the cost of vehicle usage is one of the major sources of logistics distribution cost. Since driving time is the main affecting factor, in order lower the logistics costs, the primary task should be reducing the total transportation time. (4) The fuel consumption costs generated by temperature control and driving also accounted for a relatively high proportion of the logistics costs, making up 19.7% and 20.9% of the total logistics costs for type C103 and C104. Although the fuel consumption cost during driving of type C is less, mainly because the driving distance is the least, the driving time is the longest due to the customer point service time of 90 minutes, so it needs a higher cost of temperature control fuel consumption. The total fuel consumption costs are more than 20% for types R and RC. This suggests that the fuel consumption due to temperature control and driving is also a major source of the logistics costs. Since the main affecting factors are driving time and the controlled refrigerated temperature, in order to reduce the logistics costs, the main task is to control the transportation time and properly adjust the refrigerated temperature. (5) The proportion of carbon emissions was very low, accounting for only 0.4–0.6% of the total costs, implying that from the economic perspective, the carbon emission cost caused by the current carbon tax can hardly effectively promote the logistics enterprises to save energy and reduce emissions.

4.2.4 Comparative analysis of different algorithms.

The fresh food cold distribution optimization problem is solved by the improved ant colony algorithm (IACA), conventional ant colony algorithm (ACA) [32] and tabu search algorithm (TS) [33]. The starting time of vehicles designed in ACA and TS is 0, and there is no time factor and saving matrix in the ACA compared to IACA. We used the data of C104、R203 and RC208 to contrast experiment. Note that all of the algorithms in comparison are well tuned to achieve the best performance. The results obtained are shown in Table 6, CPUT means the computation time.

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Table 6. Calculation results solved by three algorithms.

https://doi.org/10.1371/journal.pone.0235950.t006

As can be seen from Table 6: (1) the distribution cost by the IACA is lower than those by the two conventional algorithms. (2) In terms of computing time, compared with the two conventional algorithms, the IACA in this paper mainly considers the starting time calculation and the improvement of state transition probability, but the time consuming of the three algorithms is similar, and all of them can get the optimal results in relatively less time. (3) Although the calculation time is relatively fast in the TS. However, it is highly dependent on the initial value, and the algorithm suffers from convergence problems.