A multimodal transport path planning method based on adaptive large-scale neighborhood search, a storage medium and an equipment

Abstract

The application discloses a multimodal transport path planning method based on adaptive large-scale neighborhood search, a storage medium and equipment, and comprises the following steps: according to the start and end nodes of a cargo transport task, using road network data, determining the transport nodes of multimodal transport, and establishing a multimodal transport network; according to the transport nodes and transport modes in the multimodal transport network, obtaining the transport cost and the degree of preference of the cargo owner between the transport nodes, and determining the transfer cost of each transport node; according to whether the transport path between the transport nodes and the transfer mode on the transport node exist, establishing a constraint condition of multimodal transport, and combining the transport cost, the transfer cost and the degree of preference of the cargo owner to establish a first-level objective function and a second-level objective function; using an adaptive large-scale neighborhood search algorithm to iteratively optimize and solve the first-level objective function and the second-level objective function, and finding the optimal multimodal transport path. The method realizes more intelligent and comprehensive path planning by introducing adaptive large-scale neighborhood search and the preference of the cargo owner.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of multimodal transport path planning, in particular to a multimodal transport path planning method based on adaptive large-scale neighborhood search, a storage medium and equipment. BACKGROUND

[0002] In modern society, with the rapid development of globalization and the logistics industry, efficient freight transportation and distribution have become increasingly important. In order to achieve optimal resource allocation, reduce costs, and improve transportation efficiency, multimodal transport has become a widely used logistics strategy. Multimodal transport refers to the combination of different transportation modes such as road, rail, waterway, etc. to complete the transportation of goods, thereby fully utilizing the advantages of various transportation modes, reducing the use of high-cost modes such as air transport and road transport, and achieving the goal of reducing logistics costs and environmental impact.

[0003] However, in practical applications, multimodal transport faces a series of challenges and problems. Traditional path planning methods such as Dijkstra's algorithm and A* algorithm usually only consider transportation cost or time to find the shortest path or optimal path, making it difficult to take into account the preferences of shippers and the conversion costs between different transportation modes. In addition, the complex relationship between nodes and transportation modes in a large-scale network makes path planning more difficult. Therefore, how to intelligently plan the optimal path of goods in a multimodal transport network, taking into account various factors, has become a research hotspot in the logistics field. SUMMARY

[0004] In view of the problems existing in the prior art, the present application provides a multimodal transport path planning method based on adaptive large-scale neighborhood search, a storage medium and equipment, which introduces adaptive large-scale neighborhood search and shipper preference factors to achieve more intelligent and comprehensive path planning, thereby optimizing the use of logistics resources, reducing costs, improving efficiency, and meeting the individual needs of shippers.

[0005] To achieve the above technical purpose, the present application adopts the following technical scheme: a multimodal transport path planning method based on adaptive large-scale neighborhood search, specifically comprising the following steps:

[0006] Step 1: According to the start and end nodes of the freight transportation task, using road network data, determining the transportation nodes of multimodal transport, and establishing a multimodal transport network;

[0007] Step 2: According to the transportation nodes and transportation modes in the multimodal transport network, obtaining the transportation cost and shipper preference degree between the transportation nodes, and determining the transfer cost of each transportation node;

[0008] Step 3, according to the transport path between the transport nodes and whether there is a constraint condition for establishing multimodal transport for the transfer mode on the transport node, and combining the transport cost, transfer cost and the preference degree of the cargo owner to establish a primary objective function and a secondary objective function;

[0009] Step 4, using the adaptive large-scale neighborhood search algorithm to iteratively optimize and solve the primary objective function and the secondary objective function, and finding the optimal multimodal transport path.

[0010] Further, the multimodal transport network established in step 1 is represented by a graph G=(N,R), wherein N represents a set of transport nodes in the multimodal transport network, and R represents a set of transport modes supported by the multimodal transport network.

[0011] Further, the primary objective function C1 in step 3 is:

[0012]

[0013] wherein c i,j,r represents the transport cost of transporting from the transport node i to the transport node j by the transport mode r, i,j∈N; x i,j,r represents whether there is a transport path between the transport node i and the transport node j, t i,r,r′ represents the transfer cost of changing the transport mode r on the transport node i to another transport mode r', r,r'∈R; y i,r,r′ represents whether there is a transfer mode on the transport node i,

[0014] Further, the secondary objective function C2 in step 3 is:

[0015]

[0016] wherein c min represents the minimum value of the primary objective function C1, N represents the number of intermediate transfers between the start and end nodes of the cargo transport task when the primary objective function C1 takes the minimum value at the current iteration, p i,j,r represents the preference degree of the cargo owner for transporting from the transport node i to the transport node j by the transport mode r.

[0017] Further, the constraint condition for multimodal transport in step 3 is:

[0018]

[0019] wherein o represents the starting node of the transport task, d represents the target node of the transport task, x j,h,r represents the transport from the transport node j to the transport intermediate node h by the transport mode r, I represents a set of transportation nodes, I / (o, d) represents a set of transportation nodes excluding the starting node o and the target node d, y j,r,r′ represents whether there is a transfer mode on the transportation node j, x j,h,r′ represents whether there is a transfer path from the transportation node j to the transportation intermediate node h.

[0020] Further, step 4 includes the following sub-steps:

[0021] Step 4.1, randomly generate a number of solutions satisfying the constraint conditions [x i,j,r , y i,r,r′ ] into the feasible solution pool according to the multimodal transport network established by the cargo transport task, set the destruction operator and the repair operator, initialize the scores of the destruction operator and the repair operator, initialize the annealing temperature, set the maximum number of iterations, and set the acceptance probability threshold of each iteration;

[0022] Step 4.2, randomly select a destruction operator and a repair operator according to the score weight of the destruction operator and the score weight of the repair operator respectively using the roulette method to destroy and rebuild the solution in the feasible solution pool, and add the updated solution to the feasible solution pool;

[0023] Step 4.3, calculate the first-level objective function C1 using the solutions in the feasible solution pool, find the minimum value c min of the first-level objective function C1 at the current iteration number and its corresponding number of transfers N;

[0024] Step 4.4, use the solutions in the feasible solution pool and c min and N at the current iteration number again to calculate the second-level objective function C2, sort the calculated values of the second-level objective function C2 in ascending order, and take the solution corresponding to the first place as the optimal solution at the current iteration number, and take the solution corresponding to the second place as the suboptimal solution at the current iteration number;

[0025] Step 4.5, generate the probability of accepting the optimal solution at the current iteration number according to the annealing temperature, if the probability is less than the acceptance probability threshold of the current iteration, accept the optimal solution at the current iteration number, and add 1 to the scores of the destruction operator and the repair operator used by the optimal solution at the current iteration number, update the score weight of the destruction operator and the score weight of the repair operator; otherwise, accept the suboptimal solution, and the scores of the destruction operator and the repair operator used by the optimal solution at the current iteration number remain unchanged;

[0026] Step 4.6, in the next iteration, repeat steps 4.2-4.5 using the updated score weight of the destruction operator and the score weight of the repair operator for the accepted solution, until the maximum number of iterations is reached, take the solution when the second-level objective function C2 is the minimum value as the final solution, and obtain the optimal multimodal transport path.

[0027] Further, the destruction operator includes a random destruction operator, a cluster destruction operator, a path cutting operator and a worst destruction operator, and the repair operator includes a greedy repair operator, a near neighbor repair operator and a path reconstruction operator.

[0028] Further, the probability p that the optimal solution in the current iteration is accepted according to the annealing temperature in step 4.5 a The specific process is as follows:

[0029]

[0030] Wherein, represents the secondary objective function value corresponding to the optimal solution in the a th iteration, T a represents the annealing temperature in the a th iteration, T a = αT a-1 , and α represents the annealing temperature drop coefficient.

[0031] Further, the present application also provides a computer readable storage medium storing a computer program, wherein the computer program enables a computer to execute the multi-modal transport path planning method based on adaptive large-scale neighborhood search.

[0032] Further, the present application also provides an electronic device comprising a memory, a processor and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to realize the multi-modal transport path planning method based on adaptive large-scale neighborhood search.

[0033] Compared with the prior art, the present application has the following beneficial effects: the multi-modal transport path planning method based on adaptive large-scale neighborhood search of the present application introduces the algorithm of adaptive large-scale neighborhood search, integrates the cargo owner preference with the transportation cost and the transfer cost for comprehensive optimization, not only reduces the logistics cost, but also improves the practicability and intelligence of path planning; by effectively coordinating different transportation modes, minimizing the transfer cost and meeting the cargo owner demand, the method has significant innovation and application potential in the field of modern logistics, and provides a beneficial solution for improving resource utilization efficiency and reducing environmental impact. BRIEF DESCRIPTION OF DRAWINGS

[0034] Figure 1 is a flow chart of the multi-modal transport path planning method based on adaptive large-scale neighborhood search of the present application;

[0035] Figure 2 is a flow chart of the adaptive large-scale neighborhood search algorithm used for iterative optimization and solution of the primary objective function and the secondary objective function in the present application. DETAILED DESCRIPTION

[0036] The technical solutions of the present application will be further explained in combination with the drawings.

[0037] As Figure 1 A flow chart of the multi-modal transport path planning method based on adaptive large-scale neighborhood search of the present application, which specifically comprises the following steps:

[0038] Step 1, according to the start and end nodes of the cargo transportation task, using the road network data, determining the transportation nodes of the multi-modal transport, and establishing a multi-modal transport network; the multi-modal transport network established in the present application is represented by graph G=(N,R), wherein N represents the set of transportation nodes in the multi-modal transport network, and R represents the set of transportation modes supported by the multi-modal transport network.

[0039] Step 2, according to the transportation nodes and transportation modes in the multi-modal transport network, obtaining the transportation cost and the degree of preference of the cargo owner between the transportation nodes, and determining the transfer cost of each transportation node; the transportation cost refers to the actual cargo transportation cost required in the multi-modal transport network from one transportation node to another transportation node; the transfer cost refers to the fee generated by transferring the cargo from one transportation mode to another transportation mode in the multi-modal transport network, including the loading and unloading, warehousing and other fees at the transfer node, and the cost related to the transfer of cargo between different transportation modes; the preference of the cargo owner refers to the preference and requirement of the shipper or cargo owner on different transportation modes, routes, etc.

[0040] Step 3, according to the transportation path between the transportation nodes and whether the transfer mode on the transportation node exists to establish the constraint condition of the multi-modal transport, since the preference of the cargo owner lacks a clear quantitative dimension, it is difficult to directly measure, if the cost of the preference of the cargo owner is set by artificial, the result is easily affected by the setting method, therefore, the present application establishes a first-level objective function by the transportation cost and the transfer cost, uses the obtained result to naturally quantify the cost of the preference of the cargo owner, and constructs a second-level objective function combining the transportation cost, the transfer cost and the cost of the preference of the cargo owner.

[0041] The first-level objective function C1 in the present application is:

[0042]

[0043] Wherein, c i,j,r represents the transportation cost of transporting from transportation node i to transportation node j by transportation mode r, i,j∈N; x i,j,r represents whether there is a transportation path between transportation node i and transportation node j, t i,r,r′ represents the transfer cost of changing the transportation mode r on the transportation node i to another transportation mode r', r,r'∈R; y i,r,r′ represents whether there is a transfer mode on the transportation node i,

[0044] The secondary objective function C2 in the application is:

[0045]

[0046] Wherein, c min represents the minimum value of the primary objective function C1, N represents the number of transfer operations between the start and end nodes of the cargo transportation task when the primary objective function C1 takes the minimum value at the current iteration, p i,j,r represents the degree of preference of the cargo owner for transportation from transportation node i to transportation node j using transportation mode r, p i,j,r ∈ [0, 1], 1 represents strong recommendation, and 0 represents strong non-recommendation. It is a cargo owner preference penalty cost, and the path with high preference degree will have lower penalty cost, while the path with low preference degree will have higher penalty cost, and the secondary objective function will be optimized considering the cargo owner preference to find a balanced solution between the total transportation cost and the cargo owner preference.

[0047] The constraint condition of intermodal transportation in the application is:

[0048]

[0049] Wherein, o represents the starting node of the transportation task, d represents the target node of the transportation task, x j,h,r represents the transportation from transportation node j to transportation intermediate node h using mode r, I represents the set of transportation nodes, I / (o, d) represents the set of transportation nodes excluding the starting node o and the target node d, y j,r,r′ represents whether there is a transfer mode on the transportation node j, x j,h,r′ represents whether there is a transfer path from the transportation node j to the transportation intermediate node h. Formulas (1) and (2) represent that the transportation starting node and the target node must use a transportation mode to leave or enter the node in a unique path, and formula (3) represents the flow balance constraint of the transportation node, represents the number of path modes entering the transportation node j, and The number of ways of representing the paths leaving the transport node j, and both are equal, meaning the transport node in and out conservation, specifically, if the number of entering the transport node j is zero, the number of leaving the transport node j is also zero; if the number of entering the transport node j is 1, the number of leaving the transport node j is also 1; formula (4) and (5) represent the internal relationship between transfer and transport, formula (4) limits the consistency of transport activities and corresponding transport mode selection, prevents unreasonable transport arrangement, for example, not selecting a certain mode but arranging transport, and limits the number of mode conversion at the node level, ensures the rationality and simplicity of the scheme; formula (5) binds mode conversion and transport flow, ensures that mode conversion can only occur when the transport path and flow arrangement are reasonable.

[0050] Step 4, using adaptive large-scale neighborhood search algorithm to iteratively optimize the first-level objective function and the second-level objective function, finding the optimal multimodal transport path; such as Figure 2 , specifically including the following sub-steps:

[0051] Step 4.1, randomly generating a number of solutions [x i,j,r ,y i,r,r′ ] that meet the constraint conditions in the feasible solution pool according to the multimodal transport network established by the cargo transport task, setting a destruction operator and a repair operator, initializing the score of the destruction operator and the repair operator, initializing the annealing temperature, setting the maximum number of iterations, and setting the acceptance probability threshold of each iteration;

[0052] Step 4.2, using roulette method according to the score weight of the destruction operator and the score weight of the repair operator, dynamically adapting the change of the operator score weight, randomly selecting a destruction operator and a repair operator to destroy and reconstruct the solution in the feasible solution pool, and adding the updated solution to the feasible solution pool, wherein the score weight of the destruction operator is the proportion of the score of the destruction operator to the scores of all destruction operators, and the score weight of the repair operator is the proportion of the score of the repair operator to the scores of all repair operators; the destruction operator includes: random destruction operator, cluster destruction operator, path cutting operator and worst destruction operator, the random destruction operator refers to randomly removing some existing transport paths to break the structure of the current solution; the cluster destruction operator refers to removing transport paths that form a tight cluster to destroy the local structure of the solution; the path cutting operator refers to dividing the path into smaller segments to destroy the continuity of the current solution; the worst destruction operator refers to removing the highest cost section in the current path, i.e. the part that has the greatest impact on the objective function value; the repair operator includes: greedy repair operator, near neighbor repair operator and path reconstruction operator, the greedy repair operator refers to using the greedy method to reinsert the destroyed path into the solution to try to repair its structure; the near neighbor repair operator refers to inserting the destroyed element into the nearest position to maintain the connectivity of the solution; the path reconstruction operator refers to reconstructing the destroyed path or route segment to meet the constraint conditions.

[0053] Step 4.3, calculate the first-level objective function C1 using the solutions in the feasible solution pool, find the minimum value c of the first-level objective function C1 at the current iteration number min and its corresponding transit number N;

[0054] Step 4.4, again use the solutions in the feasible solution pool and c min and N at the current iteration number to calculate the second-level objective function C2, sort the calculated values of the second-level objective function C2 in ascending order, take the corresponding solution ranked first as the optimal solution at the current iteration number, and take the corresponding solution ranked second as the suboptimal solution at the current iteration number;

[0055] Step 4.5, generate the probability of accepting the optimal solution at the current iteration number according to the annealing temperature so as to reduce the randomness of the search and improve the convergence speed of the algorithm, wherein, represents the second-level objective function value corresponding to the optimal solution at the a-th iteration number, T a represents the annealing temperature at the a-th iteration number, T a = αT a-1 , and a represents the annealing temperature drop coefficient;

[0056] If the acceptance probability is less than the acceptance probability threshold of the current iteration, accept the optimal solution at the current iteration number, and add 1 to the scores of the destruction operator and the repair operator used by the optimal solution at the current iteration number, and update the score weights of the destruction operator and the repair operator; otherwise, accept the suboptimal solution, and the scores of the destruction operator and the repair operator used by the optimal solution at the current iteration number remain unchanged.

[0057] Step 4.6, in the next iteration, repeat steps 4.2-4.5 using the accepted solution and the updated score weights of the destruction operator and the repair operator, until the maximum iteration number is reached, take the solution when the second-level objective function C2 is the minimum value as the final solution, and obtain the optimal multimodal transport path. The multimodal transport path planning method based on adaptive large-scale neighborhood search of the present application combines simulated annealing, roulette technology, and adaptive parameter adjustment, can efficiently search for the optimal solution in a large-scale neighborhood, not only avoids local optimal trap, but also balances the exploration and development efficiency, and significantly improves the robustness, convergence speed and applicability to complex problems of the algorithm.

[0058] The method introduces an adaptive large-scale neighborhood search algorithm, integrates the freight owner preference with the transportation cost and the transfer cost, and performs comprehensive optimization, so as to not only reduce the logistics cost, but also improve the practicability and intelligence of the path planning; by effectively coordinating different transportation modes, minimizing the transfer cost and meeting the freight owner demand, the method has significant innovation and application potential in the modern logistics field, and provides a beneficial solution for improving resource utilization efficiency and reducing environmental impact.

[0059] In one of the technical solutions of the present application, a computer readable storage medium is provided, which stores a computer program, and the computer program enables a computer to execute the multi-modal transport path planning method based on adaptive large-scale neighborhood search.

[0060] In one of the technical solutions of the present application, an electronic device is provided, which includes a memory, a processor, and a computer program stored in the memory and executable on the processor, and when the processor executes the computer program, the multi-modal transport path planning method based on adaptive large-scale neighborhood search is implemented.

[0061] In the embodiments disclosed in the present application, the computer storage medium can be a tangible medium, which can contain or store programs for use by or in connection with an instruction execution system, apparatus or device. The computer storage medium can include, but is not limited to, electronic, magnetic, optical, electromagnetic, infrared, or semiconductor system, apparatus or device, or any suitable combination of the above. More specific examples of the computer storage medium can include one or more wires, portable computer disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fiber, compact disk read-only memory (CD-ROM), optical storage device, magnetic storage device, or any suitable combination of the above.

[0062] Those skilled in the art can realize that the units and algorithm steps of the examples described in combination with the embodiments disclosed in the present application can be realized by electronic hardware or a combination of computer software and electronic hardware. Whether the functions are realized in hardware or software depends on the specific application and design constraints of the technical solution. Those skilled in the art can use different methods to implement the described functions for each specific application, but such implementation should not be considered beyond the scope of the present application.

[0063] The above are only preferred embodiments of the present application, and the protection scope of the present application is not limited to the above-mentioned embodiments. Any technical scheme falling within the concept of the present application shall fall within the protection scope of the present application. It should be noted that, for ordinary skilled in the art, some improvements and refinements without departing from the principles of the present application shall be considered as the protection scope of the present application.

Claims

1. A method for multi-modal route planning based on adaptive large neighborhood search, characterized in that, Specifically comprising the following steps: Step 1, according to the beginning and end nodes of the cargo transportation task, using the road network data, determine the transportation nodes of multimodal transport, and establish a multimodal transport network; Step 2, according to the transportation nodes and transportation modes in the multimodal transport network, obtain the transportation cost and the degree of preference of the consignor between the transportation nodes, and determine the transfer cost of each transportation node; Step 3, according to the transportation path between the transportation nodes and whether the transfer mode on the transportation node exists to establish the constraint condition of multimodal transport, and combine the transportation cost, the transfer cost and the degree of preference of the consignor to establish a primary objective function and a secondary objective function; Step 4, using the adaptive large-scale neighborhood search algorithm to iteratively optimize and solve the primary objective function and the secondary objective function, and find the optimal multimodal transport path. 2.The method of claim 1, wherein, The multimodal transport network established in step 1 is represented by graph G=(N,R), wherein N represents the set of transportation nodes in the multimodal transport network, and R represents the set of transportation modes supported by the multimodal transport network. 3.The method of claim 2, wherein, The primary objective function C1 in step 3 is: wherein c i,j,r denotes the transportation cost from transportation node i to transportation node j using transportation mode r, i, j ∈ N; x i,j,r denotes whether there is a transportation path between transportation node i and transportation node j, t i,r,r′ denotes the transshipment cost from changing transportation mode r at transportation node i to another transportation mode r', r, r' ∈ R; y i,r,r′ denotes whether there is a transshipment mode at transportation node i, 4. The method of claim 3, wherein, The secondary objective function C2 in step 3 is: wherein c min represents the minimum value of the first objective function C1, N represents the number of intermediate transportations between the start and end nodes of the cargo transportation task when the first objective function C1 takes the minimum value at the current iteration, p i,j,r represents the degree of preference of the cargo owner for transportation from the transportation node i to the transportation node j by the transportation mode r.

5. The method of claim 3 or 4, wherein, The constraint condition of multimodal transport in step 3 is: wherein o represents a starting node of the transportation task, d represents a target node of the transportation task, x j,h,r represents a transportation from a transportation node j to a transportation intermediate node h in a mode r, I represents a set of transportation nodes, I / (o, d) represents a set of transportation nodes excluding the starting node o and the target node d, y j,r,r′ represents whether there is a transfer mode on the transportation node j, x j,h,r′ represents whether there is a transfer path from the transportation node j to the transportation intermediate node h.

6. The method of claim 5, wherein, Step 4 includes the following sub-steps: Step 4.1, randomly generate a number of solutions [x i,j,r ,y i,r,r′ ] satisfying the constraint conditions according to the multimodal transport network established according to the cargo transport task, put the solutions into a feasible solution pool, set a destruction operator and a repair operator, initialize the scores of the destruction operator and the repair operator, initialize an annealing temperature, set a maximum number of iterations, and set an acceptance probability threshold value for each iteration; Step 4.2, according to the score weights of the destruction operator and the score weights of the repair operator, respectively using the roulette method to randomly select a destruction operator and a repair operator to destroy and rebuild the solution in the feasible solution pool, and add the updated solution to the feasible solution pool; Step 4.

3. Calculate the first-level objective function C1 with the solutions in the pool of feasible solutions, find the minimum value c of the first-level objective function C1 at the current iteration number min and its corresponding transit number N; Step 4.4, use the solution in the pool of feasible solutions and c at the current iteration number again min and N, calculate the secondary objective function C2, sort the calculated values of the secondary objective function C2 in ascending order, take the solution corresponding to the first one as the optimal solution at the current iteration number, and take the solution corresponding to the second one as the suboptimal solution at the current iteration number; Step 4.5, according to the annealing temperature to generate the probability of accepting the optimal solution at the current iteration number, if the probability is less than the acceptance probability threshold of the current iteration, accept the optimal solution at the current iteration number, and add 1 to the score of the destruction operator and the repair operator used by the optimal solution at the current iteration number, update the score weights of the destruction operator and the repair operator; Otherwise, accept the suboptimal solution, and the score of the destruction operator and the repair operator used by the optimal solution at the current iteration number remains unchanged; Step 4.6, in the next iteration, repeat steps 4.2-4.5 using the updated score weights of the destruction operator and the repair operator to accept the solution, until the maximum iteration number is reached, the solution when the secondary objective function C2 is the minimum value is taken as the final solution, and the optimal multimodal transport path is obtained.

7. The method of claim 6, wherein, The destruction operator includes a random destruction operator, a cluster destruction operator, a path cutting operator and a worst destruction operator, and the repair operator includes a greedy repair operator, a near neighbor repair operator and a path reconstruction operator. 8.The method of claim 6, wherein, The probability p that the optimal solution generated in step 4.5 according to the annealing temperature is accepted for the current iteration number a The specific process is as follows: wherein, T a represents the value of the secondary objective function corresponding to the optimal solution at the a-th iteration, T a T a represents the annealing temperature at the a-th iteration, T a = αT a-1 , α represents the annealing temperature reduction coefficient.

9. A computer readable storage medium storing a computer program, characterized in that, The computer program enables the computer to execute the multimodal transport path planning method based on adaptive large-scale neighborhood search as claimed in any one of claims 1-8.

10. An electronic device, comprising: It comprises: A memory, a processor and a computer program stored on the memory and executable on the processor, when the processor executes the computer program, the multimodal transport path planning method based on adaptive large-scale neighborhood search as claimed in any one of claims 1-8 is realized.

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