A two-stage hybrid iterative search method for solving soft cluster arc path optimization problem

Abstract

The present invention relates to a two-stage hybrid iterative search method for solving a soft cluster arc path optimization problem, comprising the following steps: step (1), generating an initial solution g of a cluster layer through an initialization program, and recording an optimal solution; step (2), optimizing the cluster layer solution g using a variable neighborhood descent operation based on random sorting; step (3), optimizing the edge layer solution r using a variable neighborhood descent operation based on lower bound guidance, and updating the optimal solution; step (4), using a similarity-driven hybrid perturbation to help the algorithm jump out of a local optimum; step (5), repeating step (2)-step (4) until a set stop condition is reached, and obtaining an optimal solution to the problem; the present invention integrates three efficient modules, and has the advantages of high solution quality, strong robustness, wide application range, and the like.

Description

Technical Field

[0001] The present invention relates to the technical field of graph layout design and planning, and in particular to a two-stage hybrid iterative search method for solving a soft cluster arc path optimization problem. Background Art

[0002] Soft-Clustered Capacitated Arc Routing Problem (SoftCluCARP) is a variant of Capacitated Arc Routing Problem (CARP). It divides the arcs or edges to be served into clusters, and all arcs or edges to be served in the same cluster must be served by the same vehicle. SoftCluCARP well simulates various real-world applications such as garbage collection, postal delivery, snow removal, salt spreading, and transmission line inspection.

[0003] The soft cluster capacity constrained arc path problem is defined on an undirected graph G = (V, E) with a vertex set V and an edge set E, where the vertex v0∈V is a unique warehouse point, and there are m vehicles with the same capacity Q at the warehouse point. The edges in the edge set E are either required service edges E R , or it is a non-essential service side E\E R , the edges that must be served are traversed at least once but only served once. If convenient, the edges that are not required to be served can be traversed. Any traversal without service is called an idle run. Each edge e∈E that must be served R Each edge has a non-negative demand d(e) ≥ 0, and each edge e∈E has a non-negative service cost sc(e) ≥ 0 and a non-negative idle cost dc(e) ≥ 0. SoftCluCARP does not distinguish between service cost and idle cost, i.e., sc(e) = dc(e). All edges with required services are divided into disjoint clusters, i.e., E R =∪ i∈H C i and Where H is the index set of clusters and i,j∈H. Each cluster C i There is a non-negative demand d i Can be considered as belonging to C i The goal of SoftCluCARP is to determine the set of paths with the lowest cost on the graph G, such that all customer edges of the same cluster are served once by the same car, and the total demand of customer edges on each path does not exceed the capacity Q of the car.

[0004] The soft cluster capacity constrained arc path problem is highly complex and takes a long time to solve. A feasible SoftCluCARP solution requires the following four conditions to be met: (1) Each path should start and end at a warehouse; (2) Each required service edge is served only once, but can be traversed multiple times; (3) The total demand for the edges served by each vehicle must not exceed its capacity Q; (4) All customer edges in the same cluster must be served by the same vehicle.

[0005] Figure 1 An example of the soft cluster capacity constrained arc path problem is shown in the figure. The graph G of this example contains 9 vertices and 11 edges, where v0 is a warehouse point. 8 of the 11 edges are required service edges and are divided into 5 clusters, where E1 = {e1, e2}, E2 = {e3, e4}, E3 = {e5}, E4 = {e6, e7}, E5 = {e8}. The corresponding SoftCluCARP solution consists of three paths: r1 = {e1(v0, v1), e2(v2, v0)}, r2 = {e5(v5, v7), e4(v4, v3), e3(v3, v0)} and r3 = {e6(v0, v6), e8(v6, v8), e7(v7, v0)}.

[0006] Therefore, the present invention proposes a two-stage hybrid iterative search method for solving the soft cluster arc path optimization problem to solve the above problem. Summary of the invention

[0007] The technical problem to be solved by the present invention is to provide a two-stage hybrid iterative search method for solving the soft cluster arc path optimization problem, which can achieve high robustness and high quality solution to the soft cluster capacity constrained arc path problem.

[0008] In order to solve the above technical problems, the technical solution of the present invention is: a two-stage hybrid iterative search method for solving the arc path optimization problem of a soft cluster, the innovation of which is: the two-stage hybrid iterative search method represents a solution S of an arc path problem with a capacity constraint of a soft cluster as two groups of solutions of size m, namely, a cluster-level solution g and a customer-level solution r, the cluster-level solution g represents a group of visited clusters, and the customer-level solution r, namely, the edge-level solution, represents the order of serving customer edges, and the steps of the two-stage hybrid iterative search method include:

[0009] Step (1), generate an initial solution g of the cluster layer through the initialization program, and record the optimal solution;

[0010] Step (2), optimizing the clustering layer solution g using a random sorting based variable neighborhood descent operation;

[0011] Step (3), using the variable neighborhood descent operation based on the lower bound guidance to optimize the edge layer solution r, and update the optimal solution;

[0012] Step (4), using similarity-driven hybrid perturbations to help the algorithm escape from the local optimum;

[0013] Step (5): Repeat steps (2) to (4) until the set stop condition is reached and the optimal solution to the problem is obtained.

[0014] Furthermore, the specific steps of initializing the program in step (1) include:

[0015] Step (1.1), sort all clusters in descending order according to demand;

[0016] Step (1.2), select a cluster at a position in the arrangement according to a random factor and add it to the nearest feasible vehicle, requiring that the capacity of the vehicle does not exceed the maximum capacity of each vehicle;

[0017] Step (1.3) and repeat step (1.2) until all clusters are assigned to vehicles, thus obtaining a feasible solution g at the cluster level. If no vehicle has enough capacity to load the next cluster, the reallocation operator is used to reoptimize the current cluster allocation.

[0018] Furthermore, the specific steps of the variable neighborhood descent operation based on random sorting in step (2) include:

[0019] Step (2.1), solve g for the offspring generated in step (1) O Random Order-Based Variable Neighborhood Descent (RO-VND) is used for optimization at the cluster level. RO-VND contains four neighborhoods to optimize the lines within and between paths respectively. The neighborhood for optimizing the lines within the path includes two classic operators, namely the 2-opt operator and the or-opt operator, and the neighborhood for optimizing the lines between paths includes two operators, namely the relocate-sequence operator and the swap-sequence operator.

[0020] Step (2.2), randomly sort the four neighborhoods at the beginning;

[0021] Step (2.3), select the first neighborhood pair solution g after sorting O Optimize. If no improved solution is found in the current neighborhood, move to the next neighborhood. Once a better solution is found, update the current solution and return to the first neighborhood to continue searching.

[0022] Step (2.4) and repeat step (2.3) until all neighborhoods have been explored or the current solution cannot be improved.

[0023] Furthermore, the specific steps of the variable neighborhood descent operation based on the lower bound guidance in step (3) include:

[0024] Step (3.1), through the random transformation operator, the optimized cluster layer solution g O Transformed into edge layer solution O ;

[0025] Step (3.2), the initial solution r of the edge layer generated in step (3.1) O The Lower Bound-guided Variable Neighborhood Descent (LB-VND) is used to optimize the edge layer. Each edge layer path is regarded as a Rural Postman Problem (RPP). The RPP is solved by Iterated Local Search (ILS), i.e. ILS-RPP.

[0026] Step (3.3), use ILS-RPP to optimize the solution of the transformed edge layer within the path;

[0027] Step (3.4), sequentially use the relocate-sequence and swap-sequence neighborhoods to optimize the solution obtained in step (3.3) between paths;

[0028] Step (3.5), when the neighborhood in step (3.4) has improved the solution, the new path is optimized using ILS-RPP;

[0029] Step (3.6), accelerate the search of LB-VND through the lower bound guidance mechanism.

[0030] Furthermore, the specific steps of the similarity-driven mixed disturbance in step (4) include:

[0031] Step (4.1), calculate the solution S′ obtained in step (3) and the optimal solution S * Similarity

[0032] Step (4.2), when When , the Backbone-based Directed Perturbation (BDP) strategy is adopted;

[0033] Step (4.3), when When the random perturbation is performed based on the destruction-repair (Destroy-repair Random Perturbation, DRP) strategy, it is adopted.

[0034] The advantages of the present invention are:

[0035] The hybrid memetic search method for solving the soft cluster capacity constrained arc path problem provided by the present invention, in addition to a generalized initialization procedure, also integrates three efficient modules: a variable neighborhood descent operation based on random sorting to generate high-quality cluster layer solutions, a variable neighborhood descent operation based on lower bound guidance to optimize the cluster layer solution corresponding to the edge layer solution, and a hybrid perturbation driven by similarity to help the algorithm jump out of the local optimum. The method has the advantages of high solution quality, strong robustness, wide application range, etc. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] The present invention will be further described in detail below in conjunction with the accompanying drawings and specific embodiments.

[0037] Figure 1 The figure is a schematic diagram of a calculation example of a soft cluster capacity-constrained arc path problem of the present invention.

[0038] Figure 2 A schematic diagram of encoding and representing a single arc path of the present invention.

[0039] Figure 3 A schematic diagram of the lower bound calculation of the variable neighborhood descent operation based on lower bound guidance of the present invention.

[0040] Figure 4 It is a flow chart of the two-stage hybrid iterative search method of the present invention. DETAILED DESCRIPTION

[0041] The following embodiments can enable those skilled in the art to more fully understand the present invention, but the present invention is not limited to the scope of the embodiments.

[0042] The bilevel hybrid iterative search method (Bilevel Hybrid Iterated Search, BHIS) proposed in this invention is now combined with Figure 2 Explain the representation and evaluation of BHIS solutions.

[0043] BHIS represents the solution S of a soft cluster arc path problem with capacity constraints as two sets of solutions of size m, namely the cluster-level solution g and the customer-level solution r. The cluster-level solution g represents a set of visited clusters, while the customer-level solution r represents the order of serving customer edges, i.e., the edge-level solution. m-tour is used to represent a customer-level solution set r containing m arc paths. Each arc path is represented as a series of served customer edges and their service mode. The service mode represents the service travel direction. In SoftCluCARP, each edge has two service travel directions, i.e., LR and RL. LR means that the service path passes through the left end of the edge first and then the right end of the edge, while RL means the opposite. Therefore, the service sequence of customer edges in the entire arc path and the service mode of each edge are explicitly represented. Figure 1 The encoding and representation of the two arc paths in the example are as follows: Figure 2 Therefore, Figure 1 The feasible solution of the problem can be expressed as S = {g, r}, with g = {g1, g2, g3}, r = {r1, r2, r3}; where g1 = {1}, g2 = {3, 2}, g3 = {4, 5}. i Represents the customer edge sequence and mode selection served by the i-th vehicle, namely, r1 = {1,2}, r2 = {5,4,3} and r3 = {6,8,7}.

[0044] The goal of SoftCluCARP is to determine a set of m routes with minimum cost that serves all clusters and only serves all required service client edges once. Therefore, the objective function f(S) is defined as:

[0045]

[0046] In the formula, |r k | represents the number of edges served in the k-th arc path, c(r k (i)) represents the cost of the edge served by the i-th service in the k-th arc path, represents the right node of the edge of the i-th service in the k-th arc path, represents the left node of the edge of the i+1th service in the kth arc path, represents the shortest distance (minimum cost) between the right node of the i-th service edge and the left node of the i+1-th service edge in the k-th arc path, and v0 represents the warehouse node. Therefore, the first part of the outermost summation symbol represents the total distance (cost) of all edges to be served in the k-th arc path, the second part represents the sum of the shortest distances between the right nodes of all edges to be served (excluding the last edge to be served) in the k-th arc path and the left nodes of the subsequent edges to be served, the third part represents the shortest distance between the warehouse node and the left node of the first edge to be served in the k-th arc path, and the fourth part represents the shortest distance between the right node of the last edge to be served in the k-th arc path and the warehouse node.

[0047] A two-stage hybrid iterative search method for solving the soft cluster arc path optimization problem achieves high robustness and high quality solution to the soft cluster capacity constrained arc path problem, such as Figure 4 As shown, the specific steps include:

[0048] Step (1), generate an initial solution g of the cluster layer through the initialization program, and record the optimal solution;

[0049] Step (2), optimizing the clustering layer solution g using a random sorting based variable neighborhood descent operation;

[0050] Step (3), using the variable neighborhood descent operation based on the lower bound guidance to optimize the edge layer solution r, and update the optimal solution;

[0051] Step (4), using similarity-driven hybrid perturbations to help the solution escape from the local optimum;

[0052] Step (5): Repeat steps (2) to (4) until the set stop condition is reached and the optimal solution to the problem is obtained.

[0053] The specific steps of the initialization procedure in step (1) include:

[0054] Step (1.1), sort all clusters in descending order according to requirements to obtain a list L;

[0055] Calculate the distance between any two customer edges in graph G and the distance between any two clusters, cluster E i and E j The distance is defined as:

[0056]

[0057] Where V i and V j Represents cluster E i and E j The vertex set D(v p ,v q ) is vp and v q The shortest distance between any pair of vertices in .

[0058] Step (1.2), select the position [x θ |L|] is added to the cluster of the nearest feasible vehicle g i , and delete it from L, where θ is a random number greater than 1, x is a random number that follows a uniform distribution between (0,1), and the capacity of the vehicle is required not to exceed the maximum capacity of each vehicle;

[0059] In step (1.3), after all clusters are randomly sorted, clusters are randomly added to a feasible vehicle iteratively, requiring that the capacity of the vehicle does not exceed the maximum capacity of each vehicle.

[0060] Step (1.4) repeats step (1.2) until all clusters are assigned to vehicles, thus obtaining a feasible solution g at the cluster level. If no vehicle has enough capacity to load the next cluster, the reallocation operator is used to reoptimize the current cluster assignment.

[0061] The specific steps of the variable neighborhood descent operation based on random sorting in step (2) include:

[0062] Step (2.1), solve g for the offspring generated in step (1) O Random Order-Based Variable Neighborhood Descent (RO-VND) is used for optimization at the cluster level. RO-VND contains four neighborhoods to optimize the lines within and between paths respectively. The neighborhood for optimizing the lines within the path includes two classic operators, namely the 2-opt operator and the or-opt operator; the neighborhood for optimizing the lines between paths includes two operators, namely the relocate-sequence operator and the swap-sequence operator. The relocate-sequence first deletes x (x∈{1,2,3}) consecutive clusters from one path and then inserts them into another path. The swap-sequence exchanges x (x∈{1,2,3}) consecutive clusters between two paths.

[0063] Step (2.2), randomly sort the four neighborhoods at the beginning;

[0064] Step (2.3), select the first neighborhood pair solution g after sorting O Optimize. If no improved solution is found in the current neighborhood, move to the next neighborhood. Once a better solution is found, update the current solution and return to the first neighborhood to continue searching.

[0065] Step (2.4) and repeat step (2.3) until all neighborhoods have been explored or the current solution cannot be improved.

[0066] The specific steps of the variable neighborhood descent operation based on the lower bound guidance in step (3) include:

[0067] Step (3.1), through the random transformation operator, the cluster layer solution g optimized in step (2) is transformed into O Transformed into edge layer solution O ;

[0068] Step (3.2), the initial solution r of the edge layer generated by the transformation of step (3.1) O The optimization is performed on the edge layer using Lower Bound-guided Variable Neighborhood Descent (LB-VND); each edge layer solution consists of m paths, each of which can be regarded as a single Rural Postman Problem (RPP). In order to optimize the solution, the RPP is solved under the framework of Iterated Local Search (ILS), namely ILS-RPP.

[0069] Step (3.3), use ILS-RPP to optimize the solution of the transformed edge layer within the path;

[0070] Step (3.4), sequentially use the relocate-sequence and swap-sequence neighborhoods to optimize the solution obtained in step (3.3) between paths;

[0071] Step (3.5), when the neighborhood in step (3.4) has improved the solution, the new path is optimized using ILS-RPP;

[0072] Step (3.6), accelerate the search of LB-VND through the lower bound guidance mechanism.

[0073] like Figure 3 The figure shows the acceleration process of the lower bound guidance mechanism of the domain search, including the selection of the entire path edge sequence and the mode selection of each edge, which can be divided into five steps:

[0074] Step 1: Determine the edge to be searched. ILS-RPP uses common relocate-sequence, swap-sequence and 2-opt operators to generate a new sequence of continuous services based on the original sequence. Figure 2In the example, the new sequence (e1, e2, e4), (e5, e3, e6, e7) is generated by the relocate operation based on the original sequence (e1, e2, e3, e4), (e5, e6, e7);

[0075] Step 2: Operate on the target edge. The process of ILS-RPP generating a new sequence will generate breakpoints, and the paths between the breakpoints are grouped as a whole. By ignoring the details within the group, the complexity of the subsequent dynamic programming algorithm is simplified.

[0076] Step 3: Generate a mode selection network, where each edge has two modes, LR and RL. i and e j There are four ways to connect: ILS-RPP generates a connectivity network containing all connectivity modes within and between groups;

[0077] Step 4: Optimal mode selection, including the selection of connection modes between groups and within groups, to form a feasible optimal path starting from the warehouse point and ending at the warehouse point. If the target value of the new path f(r′ i ) is smaller than the original path f(r i ), then ILS-RPP proceeds to the next step, otherwise returns to step 1;

[0078] Step 5: Calculate the lower bound f LB (r′ i ) and f LB (r′ j ), ILS-RPP stores any sequence r from edge e i To the edge j The optimal mode selection of the subsequence can be restored based on this in the new path, and the optimal mode selection between groups has been obtained by the dynamic programming algorithm.

[0079] LB-VND integrates two neighborhoods: relocation neighborhood (relocate) and exchange neighborhood (swap). i and r j , the relocate neighborhood will tentatively move r i All customer edges of a cluster in the path are removed from the current path, and then the customer edges are greedily inserted into another path r in sequence j Two new paths r′ are formed in i and r′ j .

[0080] The swap neighborhood will exchange paths r iThe client edge of a cluster and another path r j The client edge of a cluster in the cluster. Due to the high complexity of optimizing the path using ILS-RPP, LB-VND uses a neighborhood decomposition strategy to speed up the search. Compared with checking the relocate neighborhood, the complexity of checking the swap neighborhood is higher, so LB-VND will prioritize searching the relocate neighborhood.

[0081] The specific steps of the similarity-driven mixed perturbation in step (4) include:

[0082] Step (4.1), calculate the solution S′ obtained in step (3) and the optimal solution S * The similarity φ(S′,S * );

[0083] φ(S′,S * )=|C1∪C2∪…∪C m | / |H|

[0084] Among them, C1, C2…C m To form S′ and S * Corresponding cluster solutions g′ and g * There are m clusters, for each cluster C i , both have and |C1∪C2∪…∪C m | for g′ and g * is the number of complete matching clusters in , and |H| is the total number of all clusters.

[0085] Step (4.2), when φ(S′,S * )<τ,τ∈[0.5,1.0], the Backbone-based Directed Perturbation (BDP) strategy is adopted: according to the cluster matching results of step (4.1), the path is divided into a fully matched path and a partially matched path. All clusters of the fully matched path remain unchanged, and the unmatched clusters in the partially matched path are removed. The removed clusters are randomly inserted into the path that meets the requirement constraints.

[0086] Step (4.3), when φ(S′,S * )>τ, the destruction-repair random perturbation (DRP) strategy is adopted: first, randomly remove clusters with a perturbation coefficient ρ∈(0.1,0.4), and finally the removed clusters are randomly inserted into the path that meets the demand constraints.

[0087] Step (4.4), if no vehicle has enough capacity to load the next cluster, the reallocation operator is used to reoptimize the current cluster allocation.

[0088] The two-stage hybrid iterative search method for solving the soft cluster arc path optimization problem of the present invention integrates three efficient modules: a variable neighborhood descent operation based on random sorting to generate high-quality cluster layer solutions, a variable neighborhood descent operation based on lower bound guidance to optimize the cluster layer solution corresponding to the edge layer solution, and a hybrid perturbation driven by similarity to help the algorithm jump out of the local optimum. Compared with the existing methods, the method of the present invention has high competitive performance and effectiveness.

[0089] Those skilled in the art should understand that the present invention is not limited to the above embodiments, and the above embodiments and descriptions are only for explaining the principles of the present invention. Without departing from the spirit and scope of the present invention, the present invention may have various changes and improvements, and these changes and improvements fall within the scope of the present invention to be protected. The scope of protection of the present invention is defined by the attached claims and their equivalents.

Claims

1. A two-stage hybrid iterative search method for solving the soft cluster arc path optimization problem, characterized by: The two-stage hybrid iterative search method represents a solution S of a soft cluster arc path problem with capacity constraints as two sets of solutions of size m, namely, a cluster-level solution g and a client-level solution r. The cluster-level solution g represents a set of visited clusters, and the client-level solution r, namely, the edge-level solution, represents the order of serving client edges. The steps of the two-stage hybrid iterative search method include: Step (1), generate an initial solution g of the cluster layer through the initialization program, and record the optimal solution; Step (2), optimizing the clustering layer solution g using a random sorting based variable neighborhood descent operation; The specific steps of the variable neighborhood descent operation based on random sorting in step (2) include: Step (2.1), the sub-generation solution g^O generated in step (1) is optimized at the cluster level using the random order based variable neighborhood descent RO-VND, RO-VND contains four neighborhoods to optimize the lines within the path and between the paths respectively, the neighborhood for optimizing the lines within the path includes two classic operators, namely the 2-opt operator and the or-opt operator, and the neighborhood for optimizing the lines between the paths includes two operators, namely the relocate-sequence operator and the swap-sequence operator; Step (2.2), randomly sort the four neighborhoods at the beginning; Step (2.3), select the first neighborhood after sorting to optimize the solution g^O. If no improved solution is found in the current neighborhood, enter the next neighborhood. Once a better solution is found, update the current solution and return to the first neighborhood to continue searching; Step (2.4), repeat step (2.3) until all neighborhoods have been explored or the current solution cannot be improved; Step (3), using the variable neighborhood descent operation based on the lower bound guidance to optimize the edge layer solution r, and update the optimal solution; The specific steps of the variable neighborhood descent operation based on the lower bound guidance in step (3) include: Step (3.1), transform the optimized cluster layer solution g^O into the edge layer solution r^O through random transformation operator; Step (3.2), the initial solution r^O of the edge layer generated in step (3.1) is optimized at the edge layer using variable neighborhood descent LB-VND based on lower bound guidance. The path of each edge layer is regarded as the rural postman problem RPP, and the RPP is solved by iterative local search ILS, that is, ILS-RPP; Step (3.3), use ILS-RPP to optimize the solution of the transformed edge layer within the path; Step (3.4), sequentially use the relocate-sequence and swap-sequence neighborhoods to optimize the solution obtained in step (3.3) between paths; Step (3.5), when the neighborhood in step (3.4) has improved the solution, the new path is optimized using ILS-RPP; Step (3.6), accelerate the search of LB-VND through the lower bound guidance mechanism; Step (4), using similarity-driven hybrid perturbations to help the algorithm escape from the local optimum; The specific steps of the similarity-driven mixed disturbance in step (4) include: Step (4.1), calculate the similarity φ(S^',S^*) between the solution S^' obtained in step (3) and the optimal solution S^*; Step (4.2), when φ(S^',S^*)<τ,τ∈[0.5,1.0], the bone-based directional perturbation BDP strategy is adopted; Step (4.3), when φ(S^',S^*)>τ, adopt the random perturbation DRP strategy based on destruction-repair; Step (5): Repeat steps (2) to (4) until the set stop condition is reached and the optimal solution to the problem is obtained.

2. The two-stage hybrid iterative search method for solving the soft cluster arc path optimization problem according to claim 1, characterized in that: The specific steps of the initialization program in step (1) include: Step (1.1), sort all clusters in descending order according to demand; Step (1.2), select a cluster at a position in the arrangement according to a random factor and add it to the nearest feasible vehicle, requiring that the capacity of the vehicle does not exceed the maximum capacity of each vehicle; Step (1.3) and repeat step (1.2) until all clusters are assigned to vehicles, thus obtaining a feasible solution g at the cluster level. If no vehicle has enough capacity to load the next cluster, the reallocation operator is used to reoptimize the current cluster allocation.

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