Design of intelligent logistics path planning algorithm for operations research

Intelligent logistics has always been an important development direction in the logistics industry, through the introduction of intelligent technology and algorithms, it can improve the efficiency of logistics and transportation, reduce costs, and better meet customer needs. Path planning, as a key link in the intelligent logistics system, aims to determine the optimal logistics path to realize the fast, safe and efficient transportation of goods [1-4].

The design of path planning algorithms is the core part in intelligent logistics systems. When designing the path planning algorithm, the following factors need to be considered. First, the distance measurement index, the primary goal of path planning is to determine the shortest logistics path, so it is necessary to choose the appropriate distance measurement index. Commonly used distance measures are Euclidean distance, Manhattan distance and so on [5-8]. Second, the path search algorithm: the core of the path planning algorithm is to search for the optimal path. Common path search algorithms include depth-first search (DFS), breadth-first search (BFS), Dijkstra’s algorithm, etc. DFS and BFS are suitable for simple scenarios, while Dijkstra’s algorithm can better deal with the path planning problem in complex scenarios [9-12]. Choosing the appropriate algorithm according to the actual needs can improve the search efficiency and the accuracy of path planning. In practical applications, the design of path planning algorithms in intelligent logistics systems needs to be customized according to specific scenarios [13-14]. The complexity of the logistics network, traffic conditions, cargo demand and other factors should be fully considered to find the best path planning scheme as much as possible. At the same time, the design of path planning algorithm needs to be synergized with other modules, such as warehouse management, vehicle scheduling, etc., in order to realize the high efficiency of the overall logistics operation [15-18].

The research proposes a model of logistics transportation path planning problem based on cargo loss, adding cargo loss function to the objective function and considering the problem that cargo will increase with time causing loss during distribution and transportation. The research problem is planned as a distribution center and multiple distribution points, multiple trucks depart from the distribution points, each truck is responsible for a certain amount of distribution points, and then return to the distribution center after completing the transportation process. Factors such as fixed cost, transportation cost, penalty cost and customer satisfaction are also considered in this model. In addition, for the defects of the basic ant colony algorithm that the search time is long and easy to fall into local optimization, combining with the actual problems in logistics operation, we add the waiting factor in the state transfer probability and keep the pheromone local update rule to make positive feedback global update to the path of the optimal ants. At the same time, the pheromone restriction range is set, and the value of pheromone volatility coefficient is changed adaptively to realize the improvement of ant colony algorithm as well as its application in intelligent logistics path planning oriented to operations research. Finally, the effects of planning paths with different operations research methods, planning results without considering customer value and planning paths considering customer value are compared.

2

Intelligent logistics path planning modeling

2.1

Basic assumptions of the problem and description of the parameters

2.1.1

Basic assumptions

According to the classification of the problem affecting vehicle paths and the actual situation in logistics and transportation, this section makes the following assumptions about the model:

1) The impact of non-critical factors such as wind and rain, traffic accidents or force majeure is not considered.

2) The location of distribution centers and customer points are provided, and the trucks need to return to the distribution centers after the service is completed, and only one-way closed-loop research is done.

3) Distribution goods need to include perishable or easily broken items.

4) The vehicles in the distribution center need to be of the same type, the capacity and load capacity of the transported vehicles need to be the same, the weight of the distributed goods should not exceed the total weight of the vehicles, and the work at each point should be completed by only one vehicle.

5) The distribution goods planned for the vehicle path do not exceed the maximum load capacity of the vehicle.

6) Do not consider the case of whole-vehicle distribution, i.e., one vehicle with goods for multiple service points.

7) The length of distribution time is the only factor that affects the quality of the goods.

2.1.2

Description of parameters

From the logistics transportation path planning problem described above and the assumptions made, the parameters used in the model are now defined as follows, as shown in Table 1: Table 1.

Relevant symbols and definitions

Symbol	Define
L_c	Cargo damage assembly
P_c	Cost of punishment due to timeout window
E_c	Energy cost
F_c	Fixed assembly
A_c	Distribution assembly
T_c	Transport assembly
D_c	Unit distance transportation cost
F_k	A fixed cost of distribution
D_ij	Customer point I to j
v₁	Unit price of product
v₂	The price of the product
$Q_{i}^{k}$	The vehicle reaches the customer point I after the remaining product quantity
ω₁	Unit time loss coefficient
ω₂	Unit time loss coefficient
$P_{i}^{k}$	The total amount of the customer point I
$T_{i}^{k}$	The vehicle is arriving at the customer point I
$T_{i j}^{k}$	The loading time of the vehicle at the customer point j
E_c	The cost of fuel consumption generated by vehicle distribution
E_c1	The cost of fuel consumption in the vehicle
E_c2	Fuel consumption cost per unit time
μ₁	The vehicle reaches the customer point penalty coefficient in advance
μ₂	The penalty coefficient of the vehicle late to the customer’s point
C_pi	Customer point I’s penalty cost
[E,L]	Specified delivery time window
[ET,LT]	Acceptable delivery time window

2.2

Logistics planning modeling

2.2.1

Analysis of related distribution costs

Total distribution cost is generally composed of fixed cost, transportation cost and fuel consumption contractor, which will be analyzed in this section. Although the cost of cargo loss is also in the total distribution cost, this paper places the cost of cargo loss in customer satisfaction because it largely reflects customer satisfaction. The three costs are analyzed as follows:

1)

Fixed cost:

The fixed cost of logistics transportation refers to that part of the cost caused with the business operation of the logistics company, such as the vehicle, the depreciation of hardware facilities, maintenance costs, the company’s personnel expenditures and so on. In this paper, we only consider the cost caused by the distribution vehicles, due to other hardware facilities, personnel spending costs involved in the calculation. The fixed cost formula in this paper is shown below: (1) $F_{C} = \sum_{k = 1}^{m} F_{k} Z_{k}$ Where F_k represents the fixed cost of one delivery, Z_k being 1 indicates participation in this delivery, otherwise it indicates non-participation, and F_C represents the fixed cost required for m vehicles to work together.

2)

Transportation cost:

In accordance with the assumptions made in the previous section, this paper only considers the distribution distance as a factor, and the transportation cost is positively correlated with the distribution path distance, and the longer the distribution distance, the higher the transportation cost. Transportation cost calculation formula is as follows: (2) $T_{C} = \sum_{k = 1}^{m} \sum_{i = 0}^{n} \sum_{j = 1}^{n} D_{C} D_{i j k} X_{i j k}$ where D_ij denotes the distance between two customer points on the distribution path, X_ijk denotes whether the vehicle travels from i point to j point, and T_c denotes the transportation cost of the vehicle;

3)

the cost of fuel consumption:

The increase in fuel consumption caused by frequent braking and accelerating during driving, or the increase in fuel consumption due to unnecessary merging and overtaking on the way. The fuel consumption incurred when the vehicle is stopped is mainly caused when loading and unloading is stopped but the engine is not switched off. The energy costs incurred are shown below: (3) $E_{C} = \sum_{k = 1}^{m} \sum_{i = 0}^{n} \sum_{j = 1}^{n} X_{i j k} E_{c 1} T_{i j}^{k} + \sum_{k = 1}^{m} \sum_{i = 0}^{n} \sum_{j = 1}^{n} Y_{i j k} E_{c 2} T_{r j}^{k}$ where X_ijk denotes whether the vehicle traveled from point i to point j, $T_{i j}^{k}$ denotes the time spent while traveling, E_e1 denotes the fuel consumption per unit of time spent while traveling, $T_{r j}^{k}$ denotes whether the vehicle loaded or unloaded at various customer points, E_c2 denotes the fuel consumption per unit of time spent while the vehicle stopped but did not turn off the engine at various points, and $T_{r j}^{k}$ denotes the time spent by the vehicle when it was parked at a customer point.

2.2.2

Customer satisfaction analysis

In the process of vehicle traveling or in the process of loading and unloading goods, this paper only considers the relationship between cargo loss and time. By reviewing the literature, the value of goods and time is an exponential relationship, with the increase of time and decrease, and the trend of reducing gradually, can be expressed by the following formula: (4) $Q_{t} = e^{- θ t} + b$ Where θ represents the cargo damage factor and Q_t changes over time.

The formula for calculating the cost of cargo damage while the vehicle is in motion is as follows: (5) $L_{c 1} = (v_{1} - v_{2}) \sum_{k = 1}^{m} \sum_{i = 0}^{n} \sum_{j = 1}^{n} Y_{j k} Q_{j}^{k} (1 - e^{- ω_{1} T_{i}^{k}})$

The formula for calculating the cost of cargo damage to vehicles during loading and unloading is as follows: (6) $L_{c 2} = (v_{1} - v_{2}) \sum_{k = 1}^{m} \sum_{j = 0}^{n} Y_{j k} Q_{j}^{k} (1 - e^{- ω_{2} T_{r i}^{k}})$ where v₁ indicates the average price of the goods, v₂ indicates the average guaranteed price of the goods, 32 ω₁ and ω₂ indicate the damage factor of the vehicle during transportation and during loading and unloading, respectively, and the damage factor during loading and unloading is greater than the damage factor during transportation. 33 The average price of the goods at the point of loading and unloading is 1, 1 indicates loading and unloading at that point, and 0 indicates not at that point. $Y_{j}^{k}$ indicates whether the vehicle was loaded or unloaded at point j, 1 indicates that the vehicle was loaded or unloaded at that point, and 0 indicates that it was not loaded or unloaded at that point.

Combining the cost of cargo damage during transportation and loading/unloading, the formula for calculating the cost of cargo damage is as follows: (7) $L_{C} = (v_{1} - v_{2}) \sum_{k = 1}^{m} \sum_{i = 0}^{n} \sum_{j = 1}^{n} Y_{j k} Q_{j}^{k} (2 - e^{- ω_{1} T_{j}^{k}} - e^{- ω_{2} T_{n}^{k}})$

Time window penalty cost:

In normal logistics and distribution practical application scenarios, logistics or courier companies will want to deliver the goods to the customer’s point or arrive at the customer’s point to pick up the goods within the agreed time, but due to uncertainties, the vehicle can not be 100% in the time window specified by the customer to arrive, thus generating a certain amount of penalty cost, which is a factor that should be taken into account in the path planning model, to specify the relationship between the penalty cost and the arrival time of the vehicle [19]. The following scenarios exist:

The relationship between penalty cost and time window is shown in the following equation: (8) $P_{C} = {\begin{array}{l} M, E > T_{i}^{k} \\ μ_{1} (T_{i}^{k} - E), E < T_{i}^{k} < E T \\ 0, E T \leq T_{i}^{k} \leq L T \\ μ_{2} (L - T_{i}^{k}), L T < T_{i}^{k} < L \\ M, L < T_{i}^{k} \end{array}$ where $T_{i}^{k}$ refers to the time at which the vehicle k arrives at i the customer’s point, interval [ET,LT] denotes the customer’s desired customer-agreed time of arrival of the vehicle, interval [E,ET] denotes the window of time in which the customer is acceptable to arrive early, interval [LT,L] denotes the window of time in which the customer is acceptable to be late, interval [–∞,ET] denotes the window of time prior to the window of time in which the customer is acceptable to arrive early, and interval [L,–∞] denotes the window of time after the window of time in which the customer is acceptable to be late.

The relationship of customer satisfaction with time window is shown in (9): (9) $S (T_{i}^{k}) = {\begin{cases} 100 %, E T \leq T_{i}^{k} \leq L T \\ (1 - \frac{T_{i}^{k} - E}{E T - E}) \times 100 %, E_{i} < T_{i}^{k} < e_{i} \\ (1 - \frac{L - T_{i}^{k}}{L - L T}) \times 100 %, l_{i} < T_{i}^{k} < L_{i} \\ 0, L < T_{i}^{k} > > o L < E \end{cases}$

The relationship between customer satisfaction and cost of goods loss is represented in equation (10): (10) $S_{h} = (\frac{\sum_{i = 1}^{n} (ν_{1} - ν_{2}) P_{i}^{k} - A_{C}}{\sum_{i = 1}^{n} (ν_{1} - ν_{2}) P_{i}^{k}}) \times 100 %$

The relationship between customer satisfaction and time window and cost of goods loss is shown in equation (11): (11) $S = λ_{1} \frac{\sum_{i = 1}^{n} \sum_{k = 1}^{m} S (T_{i}^{k})}{n} + λ_{2} (\frac{\sum_{i = 1}^{n} (ν_{1} - ν_{2}) P_{i}^{k} - A_{C}}{\sum_{i = 1}^{n} (ν_{1} - ν_{2}) P_{i}^{k}})$

2.2.3

Establishment of the objective function

According to the fixed cost of distribution, transportation cost, fuel cost, cargo damage cost and time window penalty cost analysis function established above, the comprehensive consideration of the various costs and customer satisfaction function, each type of cost will be assigned a weight, the discrimination is recorded as w₁,w₂,w₃,w₄,w₅,w₆, then the expression formula of transportation cost is: (12) $A_{C} = w_{1} F_{C} + w_{2} T_{C} + w_{3} E_{C} + w_{4} L_{C} + w_{5} P_{C}$

Taking the total minimum transportation cost as the ultimate goal of the model i.e: (13) $M I N A_{C} = w_{1} F_{C} + w_{2} T_{C} + w_{3} E_{C} + w_{4} L_{C} + w_{5} P_{C}$

In summary, by bringing the total cost into the above equation, the objective function of the logistics transportation path planning model is expressed as: (14) $M I N A_{C} = {\begin{array}{l} w_{1} \sum_{k = 1}^{m} F_{k} Z_{k} + w_{2} \sum_{k = 1}^{m} \sum_{i = 0}^{n} \sum_{j = 1}^{n} D_{C} D_{i j k} X_{i j k} + \\ w_{3} \sum_{k = 1}^{m} \sum_{i = 0}^{n} \sum_{j = 1}^{n} X_{i j k} E_{c 1} T_{i j}^{k} + \sum_{k = 1}^{m} \sum_{i = 0}^{n} \sum_{j = 1}^{n} Y_{i j k} E_{c 2} T_{r j}^{k} + \\ w_{4} (ν_{1} - ν_{2}) \sum_{k = 1}^{m} \sum_{i = 0}^{n} \sum_{j = 1}^{n} Y_{i k} Q_{j}^{k} (2 - e^{- ω_{1} T_{i j}^{k}} - e^{- ω_{2} T_{i j}^{k}}) + \\ w_{3} λ_{1} \sum_{k = 1}^{m} \sum_{i = 1}^{n} \max [(T_{i}^{k} - E), 0] + λ_{2} \sum_{k = 1}^{m} \sum_{i = 1}^{n} \max [(L - T_{i}^{k}), 0] \end{array}$

2.2.4

Setting constraints

Set the constraints of the model: (15) $t_{i j}^{k} = M i n t_{i j}^{k} (x = 1, 2, ..., x)$ (16) $\sum_{k = 1}^{m} \sum_{i = 0}^{n} \sum_{j = 1}^{n} x_{i j k} = n$ (17) $\sum_{j = 1}^{n} x_{0 j k} = \sum_{i = 1}^{n} x_{i 0 k} \leq 1, k = 1, ..., m$ (18) $\sum_{j = 0}^{n} \sum_{k = 1}^{m} x_{i j k} = 1, i = 0, 1, ..., n$ (19) $\sum_{t = 0}^{n} \sum_{k = 1}^{m} x_{i j k} = 1, j = 0, 1, ..., n$ (20) $\sum_{i = 1}^{n} q_{i} \leq W$

Equation (15) indicates the shortest passage time from customer point i to customer point j , equation (16) indicates that a total of three customer points are passed on the distribution path, equation (17) indicates that the vehicle departs from the distribution center and passes through the customer points before finally returning to the distribution center, equations (18) and (19) indicate that the vehicle delivers for the same customer point at most once, and equation (20) indicates that the going to and from the customer points on the distribution path Deliveries ah total must not exceed the total vehicle capacity.

3

Optimal Logistics Path Planning Algorithm Design for Operations Research

3.1

Classical Ant Colony Algorithm

3.1.1

Modeling the Ant Colony Algorithm

The basic formulation of the ACO algorithm model on the TSP problem is as follows:

1) In the process of transferring from city i to city j, the ant decides its choice of point based on the transfer probability, which is jointly determined by the pheromone concentration and the heuristic information;

2) In order to ensure that the ants will not repeat their visits to the cities they have already visited, a taboo table can be established to add the visited cities and prohibit them from being visited again;

3) When all the ants have visited all the cities, i.e., after completing one traversal, the pheromone needs to be updated, and the new pheromone is used as the basis for further optimization, while the taboo table is emptied;

The selection probability used in the ACO algorithm is determined by both the pheromone and the heuristic information, and its calculation formula is as follows: (21) $p_{i j}^{k} (t) = {\begin{matrix} \frac{{[τ_{i j} (t)]}^{α} \cdot {[η_{i j}]}^{β}}{\sum_{j \in a l l o w e d_{k}} {[τ_{i j} (t)]}^{α} \cdot {[η_{i j}]}^{β}} & i f j \in a l l o w e d k \\ 0 & o t h e r w i s e \end{matrix}$ Where τ_j(t) is the pheromone concentration on the path (i,j) at the moment of t ; allowed_k denotes the set of cities that the ant k has not yet visited, and the city will be deleted from the table each time after choosing a city to avoid repeated choices; and parameters a and β are the pheromone parameter and the inspired information parameter, respectively, which denote the importance of the pheromone and the inspired information in the process of deciding the choice probability of the ants [20]. (In order to avoid the same city being visited multiple times by the same ants during a cycle, a taboo table tabu(k) needs to be established for each ant, which is used to record the city points that the ants k have already visited, and tabu(k) is instantly updated according to the constant movement of the ants.

In order to advance the constant march of the colony ants and to achieve further optimization of the algorithm, the updating rules of the pheromone on the path are set up with the following formulas: (22) $τ_{i j} (t + 1) = (1 - ρ) τ_{i j} (t) + Δ τ_{i j} (t)$ (23) $Δ τ_{i j} (t) = \sum_{i = 1}^{m} Δ τ_{i j}^{k} (t)$ Where ρ is the coefficient of pheromone volatilization over time, the size of which affects the convergence speed of the ACO algorithm and the global optimization ability, and takes a value between (0, 1); Δτ_ij is the pheromone increment on the path (i,j) in the time period from t to t+1.

According to the different pheromone updating strategies, M. Dorigo defines three different ACO algorithm models namely Ant-Density model, Ant-Quantity model and Ant-Cycle model, and the difference of their models is mainly in the calculation method of pheromone increment.

1) In Ant-Density model, the pheromone increment is calculated as: (24) $Δ τ_{i j}^{k} (t) = {\begin{array}{l} Q, & Ant k passes through the path(i, j) \\ 0, & e l s e \end{array}$ where Q is the pheromone increment constant. From this formula, it can be seen that the pheromone increment is only a fixed constant related to the size of Q, independent of other factors.

2) In the Ant-Quantity model, the pheromone increment is calculated by the formula: (25) $Δ τ_{i j}^{k} (t) = {\begin{array}{l} \frac{Q}{d_{i j}}, & Ant k passes through the path(i, j) \\ 0, & e l s e \end{array}$ where d_ij is the length of path (i,j). In this model, the pheromone increment is proportional to the constant Q and inversely proportional to the length of path (i,j).

3) In the Ant-Cycle model, the pheromone increment is calculated as: (26) $Δ τ_{j}^{k} (t) = {\begin{array}{l} \frac{Q}{L_{k}}, & Ant k passes through the path(i, j) \\ 0, & e l s e \end{array}$

In this model, the pheromone increment constant Q is proportional to and inversely proportional to the total length of the path after ant k completes one cycle, i.e., traversal.

In both Ant-Density model and Ant-Quantity model, the pheromone update strategy is a local update strategy, while Ant-Cycle model adopts a global pheromone update strategy. A large number of experimental validations show that Ant-Cycle model is more prominent in terms of algorithm convergence speed and global optimization ability, and the optimization effect is obvious. Therefore, the Ant-Cycle model-based ACO algorithm is often used in practical problem solving.

3.1.2

Ant colony algorithm flow

The steps to implement the ant colony algorithm for TSP problem based on Ant-Density model are as follows:

1) Read data and set initial parameters. Read the relevant information of each node to be fetched by the program; set the initial parameters: the order time t = 0 , the number of cycles Nc = 0, the maximum number of cycles N_max, randomly assign m ants to n cities, set the heuristic information η_ij and the initial pheromone concentration τ_ij(t) = C as well as the heuristic information parameter β and the pheromone parameter α ;

2) Set the number of cycles Nc = Nc+1 to advance the algorithm optimization process;

3) Create a taboo table for each ant to avoid repeated visits;

4) Number of ants k = k+1;

5) Calculate the selection probability of ant k for the city to be visited according to Eq. (22) and determine the next visit point;

6) Writing the new access point into the forbidden table of ant k and updating the list of cities to be visited of k.

7) Determine whether the list of cities to be visited by ant k is empty, if not, continue process 5) until the ant completes one traversal;

8) Determine whether the number of ants has reached the maximum, i.e., whether every ant has completed a traversal, if not, jump to process 4) until all ants have completed a traversal;

9) Determine the shortest path and update the pheromone according to Eqs. (23)(24);

10) Determine whether the end conditions of the algorithm are satisfied, if so, end the loop and output the optimization results, otherwise empty the taboo table and refresh the cities to be visited by ants k, and jump to process 2) to continue the optimization.

The calculation steps are shown in Figure 1:

3.2

Improvement of Ant Colony Algorithm

Too many iterations and the phenomenon of “early maturity” of the solution results are relatively large defects of the basic ACO algorithm, and the improvements made in this paper are mainly aimed at the above two points. On the other hand, the aquatic products logistics and transportation problem model established in this paper is based on the VRPTW problem, and the customers have higher requirements for the delivery timeliness and product quality, which makes the problem structure and customer requirements more complicated than the basic VRP problem.

On the basis of the introduced basic ACO algorithm, in order to improve the transportation efficiency of the overall distribution process, the quality of the optimal solution, etc., the corresponding improvement methods are proposed in the directions of the state transfer rule, global pheromone updating step, and adaptive change of parameters.

1)

Improvement of state transfer rule

The distance between the current point and the next node and the concentration of pheromone on each path are the main basis for selecting the next node in the basic ACO algorithm. This is far from satisfying the requirement of selection probability accuracy in the real problem model. For example, in this paper, due to the perishable characteristics of aquatic goods, customers generally pay more attention to the timeliness of the arrival of the transport vehicle, the transport vehicle arrives earlier than the time frame specified by the customer, it is necessary to wait for a period of time to carry out the distribution service, which will result in a reduction in the overall efficiency of the vehicle distribution, so in order to make the customer with a shorter waiting time to be selected as the next transportation point more likely in the selection of the next customer point. Waiting factor is added to the rules for selecting the next node.

(27)

w_{i j k} = \frac{1}{r t_{j} - t_{i k}}

t_ik - The moment when the knd vehicle reaches point i Also, in order to achieve a good balance between fast convergence of the solution and the diversity of the solutions, a pseudo-randomized proportionality rule is used. Where n₀∈[0.1] is a random number used to control the transfer rule, $n = \frac{2}{N}$ , N are the number of customer points. Combining the above methods to improve the state transfer rule, for the current node i ant k selects node i as the next arrival target with probability: (28) $a = {\begin{array}{l} \max_{j \in a l b w e d} {[τ_{i j} (t)]}^{α} {[η_{i j} (t)]}^{β} {[w_{i j} (t)]}^{2} & i f n \geq n_{0} \\ p_{i j}^{k} (t) & i f n \leq n_{0} \end{array}$ (29) $p_{i j}^{k} (t) = {\begin{array}{l} \frac{{[τ_{i j} (t)]}^{α} {[η_{i j} (t)]}^{β} {[w_{i j} (t)]}^{2}}{\sum_{s \subset N_{i}^{k}} {[τ_{i s} (t)]}^{α} {[η_{i s} (t)]}^{β} {[w_{i s} (t)]}^{2}} & , j \in a l l o w e d \\ 0 & , o t h e r w i s e \end{array}$

2)

Improvement of pheromone global update step

In this paper, the pheromone updating of ACO algorithm is global pheromone updating and is improved on this basis. In order to strengthen the positive feedback effect of the current optimal solution generated by each iteration on the pheromone concentration on each channel, a new updating formula is introduced, which produces feedback effects of varying degrees according to the performance of their target values when facing different feasible solutions. That is, the global pheromone update is performed according to the following formulation, as shown in Fig. 2.

When all m ants have completed a cycle, i.e., when all ants have constructed the feasible path R, it is compared with the globally optimal path R^* , where R^* = min(R_k|k = 1,2,3,…….,m) In order to make each ant’s iterative transportation scheme have a positive or negative feedback on the increase or decrease of pheromone, a global update of pheromone is applied to all the ants R at each iteration according to the rule in the following figure:

(1) Of the pheromone update equation is: (30) $τ_{i j}^{n e w} = ρ^{2} τ_{i j}^{o l d}, \forall (i, j) \in R$

(2) Of the pheromone update equation is: (31) $τ_{i j}^{n e w} = ρ π_{i j}^{o l d}, \forall (i, j) \in R$

(3) Of the pheromone update equation is: (32) ${\begin{array}{l} τ_{i j}^{n e w} = * \max (ρ_{i j}^{o l d} + Δ τ_{i j}^{*}, τ_{i j}^{o l d}) \\ Δ τ_{i j}^{*} = \frac{Q}{L^{*}}, \forall (i, j) \in R^{*} \end{array}$

The amount of pheromone released by the ant traversing all nodes in each iteration is Q; $τ_{i j}^{n e w}$ is used to represent the size of the pheromone between the road section between node i and node j at the beginning of the next iteration; $τ_{i j}^{o l d}$ is used to represent the size of the pheromone between the road section between node i and node j in this iteration; and L^* is the optimal solution in the result of the current iteration.

3)

Introduction of the MMAS idea

In order to guarantee that certain channels with low pheromone concentration still have the value of being searched and selected, when the amount of pheromone decreases below the set minimum value, its pheromone concentration is artificially improved. Its formula is described as shown below.

(33)

τ_{i j} = {\begin{array}{l} τ_{\max}, i f τ_{i j} \geq τ_{\max} \\ \frac{τ_{\max}}{3}, i f τ_{i j} \leq τ_{\min} \end{array}

(34)

τ_{i j} \in [τ_{\min}, τ_{\max}]

4)

Adaptive change of information retention level parameter

The adaptive improvement of ρ parameter not only achieves the above algorithmic improvement effect, but also echoes with the global pheromone updating improvement step, strengthening its positive feedback effect of global pheromone updating after each iteration, while suppressing its negative effect that may cause the algorithm to produce local optimal solutions.

Adjustment of ρ based on the above description.

(35)

ρ_{t} = {\begin{array}{l} 0.95 ρ_{t - 1}, i f 0.95 ρ_{t - 1} \geq ρ_{\min} \\ ρ_{\min}, o t h e r w i s e \end{array}

where ρ_min is a pre-set minimum value of ρ to ensure that the pheromone concentration does not undergo significant evaporation with the number of iterations, causing the algorithm to arrive at a locally optimal solution.

3.3

Improvement of model solving process of ACO algorithm

The implementation steps of the improved mosquito swarm algorithm are as follows:

Step1: Define each parameter, C,Q,α,β,λ,ρ, the pheromone value on the roadway between any two points is initially τ_max, set NC_max as the maximum number of iterations, and the total number of ants is m.

Step2: Set the logistics and transportation center as the initial location of the ants, and at the same time add the center to the taboo table.

Step3: Set the number of iterations NC = NC+1, and adopt the method of constructing the solution sequentially.

Step4: For m ants, calculate the transfer probability, and at the same time require to satisfy the freshness time window as well as the load constraints, choose another demand point other than the current solution set, and then place it in the current solution: if the next node cannot be found to satisfy the load constraints of the transportation vehicle or the time constraints, then return to the aquatic products distribution center.

Step5: Loop step (3) until all m ants have visited all the points and obtained the corresponding circuits starting from the aquatic products distribution center and satisfying the constraints, the several circuits of each ant are equivalent to the distribution paths formed by several transport vehicles issued by the distribution center, and the shortest path R^* is calculated and saved.

Step6: Implement the global update and make adjustments to ρ .

Step7: Determine whether the number of iterations reaches the pre-set value, if yes, terminate the transportation, if not, zero the taboo table and go to step 7.

4

Example analysis

4.1

Data sources

In solving the vehicle path planning problem, Solomon proposed 56 VRPTW test algorithms in 1983, and published the dataset on the website for other scholars to cite in their research on the problem, and in the following, we will refer to this dataset as the Solomon dataset for short. This paper also conducts experiments on the basis of this dataset, which is divided into three categories according to the different locations of customers from the distribution center, namely, R, C and RC, in which the geographic location of R customers is randomly distributed; the distribution of C customers’ geographic location is relatively centralized; and the geographic location of RC customers is generated by combining the geographic locations of R and C customers, with both centralized and decentralized geographic locations of the customers. The geographic location of RC customers is a combination of the geographic locations of R and C customers. And according to the size of the time window is subdivided into six classes, respectively, R1 class, R2 class, C1 class, C2 class, RC1 class and RC2 class, in which the time window of the class 1 data set is short, the capacity of the vehicle is small and the number of customer nodes served by each route is small; and the class 2 data set has a long time window, a large capacity of the vehicle and a large number of customer nodes served by each route.

Since the logistics distribution path problem studied in this paper takes the delivery of agricultural products cold chain transportation and distribution as a case study, and the problem is a special case of the vehicle path planning problem, in order to facilitate the conduct of the experiment and to avoid the chance of the results produced by simple data on the experiment, this paper will select 30 data deformations of the Solomon dataset to carry out the experiment. Therefore, in order to facilitate the experiment, the data obtained after deforming the Solomon dataset are shown in Table 2.

Table 2.

Details of customer points

Serial number	x-coordinate	y-coordinate	Demand	Service time	Time window	Acceptable time window
1	402	61
2	495	14	1.6	16	[7:30,9:00]	[7:00,9:30]
3	498	7	1.8	32	[6:00,8:00]	[5:30,8:30]
4	498	2	1.3	20	[6:30,8:30]	[6:00,9:00]
5	492	8	1.1	13	[8:00,9:00]	[7:30,9:30]
6	496	3	1.2	13	[7:00,9:00]	[6:30,9:30]
7	486	7	0.7	8	[7:40,10:20]	[7:10,10:50]
8	496	23	1.4	18	[8:30,9:00]	[8:00,9:30]
9	435	13	1.6	20	[7:30,9:00]	[7:00,9:30]
10	486	5	1.4	16	[6:30,8:30]	[6:00,9:00]
11	487	0.6	0.6	8	[6:30,9:00]	[6:00,9:30]
12	436	8	1.4	20	[7:00,9:00]	[6:30,9:30]
13	482	25	1.9	19	[8:00,9:00]	[7:30,9:30]
14	435	5	1.3	13	[7:00,9:00]	[6:30,9:30]
15	480	10	0.9	11	[7:30,9:00]	[7:00,9:30]
16	437	20	0.8	8	[7:00,10:00]	[6:30,10:30]
17	478	21	1.3	14	[7:30,9:00]	[7:00,9:30]
18	441	7	1.5	18	[8:30,10:00]	[8:00,10:30]
19	479	5	0.8	10	[7:00,8:30]	[6:30,9:00]
20	482	-3	1.4	20	[8:30,9:00]	[8:00,9:30]
21	489	30	0.9	7	[8:00,10:00]	[7:30,10:30]
22	481	26	1.6	18	[6:00,8:30]	[5:30,9:00]
23	478	24	1.4	16	[6:30,9:00]	[6:00,9:30]
24	474	16	1	13	[7:00,9:00]	[6:30,9:30]
25	474	5	1.7	20	[6:00,10:00]	[5:30,10:30]
26	476	21	0.8	8	[6:30,8:00]	[6:00,8:30]
27	486	32	1	14	[6:30,8:00]	[6:00,8:30]
28	445	8	1.7	20	[8:30,11:00]	[8:00,11:30]
29	474	2	0.7	11	[8:30,10:00]	[8:00,10:30]
30	491	32	1.4	16	[8:00,9:00]	[7:30,9:30]
31	494	37	1.6	25	[6:30,8:00]	[6:00,8:30]

According to the construction of the optimization model of the cold chain logistics distribution path for agricultural products, the parameters of this paper are set as follows, the price of fresh agricultural products is 17 yuan/KG, the penalty for violating the soft time window is P1=10 yuan/hour, P2=6 yuan/hour, P3=6 yuan/hour, P4=10 yuan/hour, the speed of reefer trucks is 40 kilometers/hour, the cost of each vehicle is 120 yuan per trip, and the cost of unloaded and fully loaded normal driving fuel consumption is 0.15 l/km and 0.3 l/km respectively, the cost of refrigerant consumption per hour during transportation and unloading is 5 yuan and 10 yuan respectively, the unit emission cost of carbon is 30 yuan/tonne, the price of diesel fuel is 5.42 yuan/liter, and the carbon emission rate per liter is 2.672KG, the deterioration rate in transportation λ₁ is 0.02, and the deterioration rate in unloading λ₂ is 0.04.

4.2

Results and comparative analysis

4.2.1

Calculations

By analyzing the model of cold chain logistics distribution of agricultural products and the model of ant colony algorithm, the information of each customer point is integrated into the algorithm and Matlab software is used for programming to arrive at the distribution path results of the ant colony algorithm before and after the improvement, as shown in Figure 3 and Figure 4. We can visualize the specific distribution route of each refrigerated truck from the figure.

Using the traditional ACO algorithm, the distribution path of each refrigerated vehicle at each service node is shown in Table 3-Table 8, which contains the service node route of each vehicle, the unloading volume of the vehicle when it arrives and leaves each node, the service time of the vehicle’s start and end, as well as the distance of the node from the previous node, where the service node 1 indicates the distribution center and each vehicle returns to the distribution center at the end.

Table 3.

Optimize the path result of the first car before

First car
Service node	1	16	4	6	5	2	1
Arrival discharge	0	0	0.5	3	3.3	4.5	5.9
Discharge discharge	0	0.9	3	3.4	4.5	5.9	5.90
Service start time	300	303.32	314.78	336.86	354.58	368.27	387.86
Service end time	300	314.52	336.86	352.91	368.01	385.31	387.86
The distance from the previous service node	0	54.31	64.55	1.55	6.75	7.26	108.51

Table 4.

Optimize the path result of the second car

Second car
Service node	1	26	17	21	27	30	14	1
Arrival discharge	0	0	0.8	1.9	2.6	3.6	4.8	5.9
Discharge discharge	0	0.8	1.9	2.6	3.6	4.8	5.9	5.9
Service start time	300	303.12	312.27	328.71	337.82	352.93	372.45	389.21
Service end time	300	312.12	328.27	337.71	352.82	371.93	387.55	389.21
The distance from the previous service node	0	86.45	6.41	18.42	4.71	6.22	65.53	66.81

Table 5.

Optimize the path result of the third car

Third car
Service node	1	24	13	23	22	1
Arrival discharge	0	0	1	2.9	4.1	5.5
Discharge discharge	0	1	2.9	4.1	5.5	5.5
Service start time	300	305.16	319.46	340.76	358.71	381.66
Service end time	300	319.16	340.46	358.76	378.71	381.66
The distance from the previous service node	0	87.00	13.04	8.57	4.67	89.27

Table 6.

Optimize the path result of the fourth car before

Fourth car
Service node	1	18	28	12	9	1
Arrival discharge	0	0	1.5	3.0	4.3	5.6
Discharge discharge	0	1.6	2.9	4.2	5.6	5.6
Service start time	300	303.45	323.51	345.88	368.11	391.55
Service end time	300	323.26	345.51	367.88	390.11	391.55
The distance from the previous service node	0	68.02	6.04	9.12	3.73	61.15

Table 7.

Optimize the path result of the fifth car before

Fifth car
Service node	1	31	8	15	3	29	1
Arrival discharge	0	0	1.6	2.8	4.6	6.2	7
Discharge discharge	0	1.6	2.8	4.6	6.2	7	7
Service start time	300	304.51	331.86	352.43	365.81	396.41	410.70
Service end time	300	331.33	351.81	365.43	395.81	409.41	410.70
The distance from the previous service node	0	97.91	15.15	24.41	18.00	24.62	94.45

Table 8.

The path result of the first sixth car

Sixth car
Service node	1	11	20	10	25	19	7	1
Arrival discharge	0	0	0.9	2.1	4.5	5.2	5.8	6.2
Discharge discharge	0	0.9	2.1	4.5	5.2	5.8	6.2	6.2
Service start time	300	304.71	314.85	337.08	355.41	379.52	388.75	399.24
Service end time	300	314.71	336.85	337.08	379.41	389.52	398.75	399.24
The distance from the previous service node	0	106.89	8.24	9.67	13.59	7.24	9.51	100.54

Using the improved ACO algorithm, the specifics of each refrigerated vehicle’s distribution path at each service node are shown in Table 9-Table 14.

Table 9.

The path result of the first car after optimization

First car
Service node	1	22	23	26	17	24	1
Arrival discharge	0	0	1.6	3.1	4.1	5.1	6.2
Discharge discharge	0	1.6	3.1	4.1	5.1	6.2	6.2
Service start time	300	304.21	324.31	342.39	351.52	367.65	384.75
Service end time	300	324.21	342.31	351.39	367.55	382.65	384.75
The distance from the previous service node	0	89.41	5.56	5.56	6.41	5.00	87.00

Table 10.

The path result of the second car after optimization

Second car
Service node	1	13	21	30	31	27	1
Arrival discharge	0	0	1.9	2.6	4.2	6	7
Discharge discharge	0	1.9	2.6	4.2	6	7	7
Service start time	300	304.42	325.61	334.71	352.81	380.06	397.31
Service end time	300	325.42	334.61	352.71	378.81	395.06	397.31
The distance from the previous service node	0	94.78	10.78	4.56	6.00	10.28	91.31

Table 11.

The path result of the third car is optimized

Third car
Service node	1	8	2	3	5	1
Arrival discharge	0	0	1.6	3.9	5.6	6.1
Discharge discharge	0	1.6	3.9	5.6	6.1	6.1
Service start time	300	304.61	324.93	343.00	373.24	391.80
Service end time	300	324.61	342.93	373.00	388.24	391.80
The distance from the previous service node	0	104.51	15.06	5.27	6.41	108.77

Table 12.

The path result of the fourth car after optimization

Fourth car
Service node	1	6	4	11	10	7	15	1
Arrival discharge	0	0	1.3	2.8	3.3	5.1	6	6.7
Discharge discharge	0	1.3	2.8	3.3	5.1	6	6.7	6.7
Service start time	300	304.82	319.86	342.11	352.16	370.30	379.45	394.82
Service end time	300	319.81	341.86	352.11	370.16	379.30	392.45	394.82
The distance from the previous service node	0	114.00	2.14	11.00	3.44	6.00	7.00	97.21

Table 13.

The path result of the fifth car after optimization

Fifth car
Service node	1	25	29	19	20-	28	1
Arrival discharge	0	0	1.9	3.1	4.1	5.1	6.5
Discharge discharge	0	1.9	3.1	4.1	5.1	6.5	6.5
Service start time	300	304.82	338.41	351.51	372.75	395.75	397.41
Service end time	300	326.32	5.00	6.00	9.66	41.88	68.51
The distance from the previous service node	0	92.75	5.00	6.00	10.54	41.9 2	68.45

Table 14.

The path result of the sixth car after optimization

Sixth car
Service node	1	18	12	9	14	16	1
Arrival discharge	0	0	1.7	3.1	4.2	6	6.8
Discharge discharge	0	1.7	3.1	4.2	6	6.8	6.8
Service start time	300	302.71	323.81	345.86	368.06	383.53	394.84
Service end time	300	323.71	345.81	367.86	383.06	393.53	394.84
The distance from the previous service node	0	68.12	5.14	3.91	9.00	21.12	55.41

The path diagrams and the specific situation of each vehicle in its corresponding service node before and after the improvement of the ACO algorithm show that the number of vehicles used by both algorithms for each customer node is 6, but through the improvement of the distribution process, the cost and the distance traveled have been reduced: the total cost generated by the distribution before the improvement of the total cost of 3389 yuan, and the total cost generated by the improvement of the total cost of 3207 yuan, which is a reduction of 182 yuan; before improvement The total distance traveled by distribution before improvement is 1,456 kilometers, and the total distance traveled by distribution after improvement is 1,311 kilometers, a decrease of 145 kilometers. The cost of distribution and distance traveled after the improvement increased by 6.21% and 8.01%, respectively.

By analyzing the model of cold chain logistics and distribution of agricultural products and the model of ant colony algorithm, the information of each customer point is integrated into the algorithm and Matlab software is used to program to derive the total cost convergence diagrams of the ant colony algorithm before and after the improvement, which are shown in Fig. 5 and Fig. 6. We can visualize the total cost incurred in each iteration from the graphs. From the graphs, we can see that the total cost of the first iteration before improvement is 3630, and the last iteration is 3389, and the total cost of the first iteration after improvement is 3244, and the last iteration is 3207, and the total cost of the path diagram of the improved ACO algorithm is lower.

4.2.2

Comparative analysis

The traditional ant colony algorithm and the improved ant colony algorithm are applied to the cold chain logistics path optimization model of agricultural products constructed in this paper and run in Matlab environment respectively, resulting in the convergence of the total cost of distribution for path optimization of the model in a comparative diagram as shown in Fig. 7. From the figure, it is clear to see the difference between the two algorithms in terms of convergence speed and minimum cost: the traditional ACO algorithm converges to the minimum cost of 3398 yuan only in the 135th iteration, while the improved ACO algorithm converges to the minimum cost of 3210 yuan in the 19th iteration. Obviously, the convergence speed of the improved ACO algorithm is much higher than that of the pre-improved one, and it also improves by 6.23% in terms of the total cost. This also proves the reliability and reasonableness of the improved ACO algorithm for path optimization in this paper.

The path optimization model of cold chain logistics for agricultural products constructed in this paper was run in Matlab environment by using traditional ant colony algorithm and improved ant colony algorithm respectively, which resulted in the distribution shortest path convergence comparison diagram of this path optimization, as shown in Figure 8. From the figure, we can clearly see the difference between the two algorithms in terms of convergence speed and minimum cost: the traditional ACO algorithm converges to the shortest path of 1341 kilometers in the 137th iteration, while the improved ACO algorithm converges to the shortest path of 1235 kilometers in the 19th iteration, which is an improvement of 8.03% compared with the pre-improvement. This also reflects that the in-transit loss rate of agricultural products is reduced in the process of carrying out cold chain distribution of agricultural products.

Through the comparative analysis of the two algorithms before and after the improvement, it can be seen that the convergence speed of the improved ACO algorithm and the optimization degree of the results are greatly improved compared with the pre-improvement period, and the improved ACO algorithm can better improve the search range of the solution space, thus avoiding the traditional ACO algorithm from falling into the local optimum, and then optimizing the distribution paths effectively. Better for the enterprise to save distribution costs and improve the efficiency of the enterprise.

4.3

Comparison of path planning considering customer value

4.3.1

Path planning without considering customer value

Without considering customer classification and customer value, with the objective of minimizing the total cost, the results of the program run are shown in Figure 9. Vehicle scheduling without considering customer value requires 18 refrigerated trucks, at which time the total cost required is 2,681.23, and the total value of the customer is 21,251.45. The percentage of customer satisfaction at all levels is shown in Tables 15 and 16, with 58% of customers satisfied with less than 50%, and 11% of customers satisfied with 100%.

Table 15.

Don’t consider customer value satisfaction

Satisfaction <50%		50%<= Satisfaction<80%	80%<= Satisfaction<100%	Satisfaction 100%
Satisfaction <50%		<80%	<100%	Satisfaction 100%
r	18%	35%	18%	29%
SS	58%	10%	21%	11%

Table 16.

The average satisfaction of different customer categories

	Focus on customers	Consolidated customer	Maintenance customer	Chicken ribs
r	68.90%	69.85%	77.71%	76.68%
SS	60.91%	22.51%	43.71%	46.67%

4.3.2

Path planning considering customer value

After running in Matlab with the processed objective benefit function as the final objective, the results are displayed as shown in Figure 10 and Table 17. From the table it can be seen that this distribution task requires a total of 18 vehicles to complete, most of the vehicles are loaded more than 91%, the average load factor is 96.67%, at this time, the total cost is 2894.54%, the customer value is 26765.42.

Table 17.

Distribution of vehicle distribution

Delivery number	Delivery path	Load quantity	Loading rate□□
1	0-91-70-54-61-87-55-0	98	98%
2	0-81-90-93-96-58-65-68-0	97	97%
3	0-66-83-98-95-57-65-67-0	100	100%
4	0-69-62-82-56-97-95-72-0	98	98%
5	0-95-68-52-36-76-0	100	100%
6	0-84-24-21-50-27-63-85-0	97	97%
7	0-71-100-5-47-0	75	75%
8	0-13-79-75-48-78-30-0	99	99%
9	0-21-24-22-47-26-0	100	100%
10	0-8-9-7-47-4-1-15-0	100	100%
11	0-83-55-77-91-0	94	94%
12	0-73-43-39-45-45-40-0	97	97%
13	0-3-6-37-19-34-0	100	100%
14	0-43-41-37-38-31-0	100	100%
15	0-13-16-17-11-14-0	100	100%
16	0-32-28-27-29-33-0	98	98%
17	0-32-28-27-30-33-0	91	91%
18	0-60-98-18-0	76	76%

Next, the percentage of customer satisfaction at all levels is calculated as shown in Table 18 and Table 19. From the table, it can be seen that customer satisfaction (SS) for delivery time is basically polarized, with customers whose satisfaction is greater than 80% or even equal to 100% being equal to those whose satisfaction is less than 50%, and the number of customers between 50% and 80% being less, and 76% of the customers receiving cold chain product shipments with an acceptable level of quality greater than 50%, which indicates that the vast majority of the customers receiving the good quality products.

Table 18.

Satisfaction ratio

Satisfaction<50%		50%<= Satisfaction<80%	80%<= Satisfaction<100%	Satisfaction= 100%
Satisfaction<50%		<80%	<100%	Satisfaction= 100%
r	24%	31%	18%	27%
SS	45%	12%	13%	30%

Table 19.

The average satisfaction of different customer categories

	Focus on customers	Consolidated customer	Maintenance customer	Chicken ribs
r	73.12%	83.61%	76.41%	67.18%
SS	92.51%	61.45%	53.56%	45.66%

From Table 19, we can see that the average freshness of the products received by the focusing customers, maintaining customers and consolidating customers is greater than 70%, and the average freshness of the products received by the chicken-ribs customers is slightly lower, but also 67.18%, which also has the factor of the delivery time window limitation, so it can be said that the vehicle planning paths established by this paper’s model basically ensures that the customers can receive the good quality and fresh cold chain products.

5

Conclusion

In the study of intelligent logistics distribution path optimization model, factors such as power consumption cost, loss cost and penalty cost unique to logistics are considered. Through reasonable assumptions, a logistics distribution cost model is established. An improved ACO algorithm is proposed based on the basic ACO algorithm. Through example analysis, experiments were conducted on the ant colony algorithm before and after the improvement, and the results obtained from the algorithm before and after the improvement were compared, which showed that the convergence speed of the improved ant colony algorithm and the optimization degree of the results were greatly improved compared with that before the improvement. In the comparison with the results without considering the customer value, it is found that the model considering the customer value increases the total value of the customer substantially by increasing a small amount of cost, and 76% of the customers receive product goods with a satisfaction level greater than 50%, which fully proves the effectiveness of the model.

Funding:

Teaching and Research Project of Qingdao University of Technology: Construction of university-level first-class courses in 2024 (No.: Offline course 13-Operations Research).

Idioma:: Inglés

Calendario de la edición:: 1 veces al año
Temas de la revista:: Ciencias de la vida, Ciencias de la vida, otros, Matemáticas, Matemáticas aplicadas, Matemáticas generales, Física, Física, otros

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Design of intelligent logistics path planning algorithm for operations research

Kexia Liu

Publicado en línea: 17 mar 2025

Recibido: 31 oct 2024

Aceptado: 03 feb 2025

DOI: https://doi.org/10.2478/amns-2025-0232

Palabras claveOperations research, Objective function, Intelligent logistics path planning, Low cost high customer satisfaction, Ant colony algorithm

© 2025 Kexia Liu, published by Sciendo

This work is licensed under the Creative Commons Attribution 4.0 International License.

Palabras clave
Operations research, Objective function, Intelligent logistics path planning, Low cost high customer satisfaction, Ant colony algorithm