A Monte Carlo Optimization Method for a Stochastic Demand Multi-Objective Green Vehicle Routing Problem

Abstract

This invention proposes a Monte Carlo optimization method for a stochastic multi-objective green vehicle path problem. It utilizes a path-based crossover method to achieve chromosome gene exchange, employs Monte Carlo simulation to evaluate the objective function, and utilizes a multi-directional neighborhood search method to obtain a better Pareto set. This solves the technical problem that traditional neighborhood search methods are difficult to apply to the improvement of high-dimensional multi-objective evolutionary algorithms on multiple objectives, thereby improving the accuracy of objective calculation and enhancing the efficiency of objective function evaluation.

Description

Technical Field

[0001] This invention relates to the field of smart logistics technology, and in particular to a Monte Carlo optimization method for a stochastic demand multi-objective green vehicle routing problem. Background Technology

[0002] With the surge in urban population, the urban delivery market is booming. Faced with increasingly complex logistics scenarios, traditional optimization methods can no longer meet the needs of multiple stakeholders (drivers, customers, merchants, and governments, etc.).

[0003] Traditional multi-objective optimization methods are difficult to apply directly to solving real-world multi-objective vehicle routing problems. On the one hand, customer needs are uncertain in reality, and customer satisfaction is difficult to describe accurately. On the other hand, there are many stakeholders in real-world logistics and distribution scenarios. Under the background of low-carbon operation, the number of stakeholders will be greater than or equal to four, which brings considerable challenges to traditional multi-objective optimization methods, especially algorithms with Pareto set selection and maintenance.

[0004] Previous research on the vehicle routing problem (VRP) has largely been limited to deterministic models, while research on stochastic demand vehicle routing problems has often focused on optimizing a single objective. The stochastic demand vehicle routing problem (VRPSD) was proposed by Bertsimas in 1992. Unlike the classic VRP, customer demand in the stochastic demand vehicle routing problem follows a certain probability distribution, and the exact demand information is only known when the vehicle arrives at the customer's location. Since multiple stakeholders exist in real-world logistics scenarios, their objectives are often conflicting, making it difficult for single-objective optimization techniques to balance the interests of all parties. Customer satisfaction is a difficult indicator to characterize, and industries and enterprises often use time windows as a reference to describe satisfaction. In multi-objective optimization scenarios, another difficult indicator to characterize is fuel consumption. Fuel consumption is closely related to carbon emissions, and for ease of modeling, practitioners and scholars generally use simpler fuel consumption models, such as the linear fuel consumption model of Xiao (2012). However, the fuel consumption calculated using this method differs significantly from the actual fuel consumption, lacking a certain degree of accuracy. Traditional multi-objective optimization algorithms have certain limitations when dealing with high-dimensional optimization problems: as the number of optimization objectives increases, the number of solutions in the non-dominated solution set generally increases sharply, causing the algorithm to lose its evolutionary pressure. On the other hand, research on using high-dimensional multi-objective evolutionary algorithms to solve high-dimensional combinatorial optimization problems is still rare, and even less so in the field of vehicle path optimization. Summary of the Invention

[0005] The purpose of this invention is to propose a Monte Carlo optimization method for the stochastic demand multi-objective green vehicle path problem that effectively improves the accuracy of fuel consumption algorithms and enhances the evaluation efficiency of objective functions.

[0006] To achieve the above objectives, this invention proposes a Monte Carlo optimization method for the stochastic demand multi-objective green vehicle routing problem, which optimizes four objectives: total delivery cost, route equilibrium, customer satisfaction, and driver waiting time.

[0007] The total delivery cost includes fuel consumption cost (FC).

[0008] Let e ​​be a complete path, in which the order of nodes is Ω(e)=<0,n1(e),...,n l (e), 0 > ; where n l (e) represents the last customer point on path e;

[0009] The FC includes both determined fuel consumption cost and random fuel consumption cost, and the calculation method is as follows:

[0010] Wherein, F(e) k ) is path e k The total fuel cost function includes deterministic fuel cost and stochastic fuel cost; w1 is the fuel price; the total fuel cost for path e is:

[0011] in, Let be the fuel consumption on edge (i,j); EF(e) is the random fuel consumption cost of path e;

[0012] The calculation of the random fuel consumption cost EF(e) requires the specification of a compensation strategy for delivery failures. It is calculated using a Monte Carlo sampling method, which randomly generates multiple demand scenarios according to the probability distribution of customer demand, and completes the delivery according to the specified compensation strategy. In multiple scenarios, the average of the non-fixed fuel consumption cost of path e is used as the random fuel consumption cost EF(e).

[0013] Furthermore, the total delivery cost TC also includes vehicle dispatch cost WC and carbon emission cost CC; TC = WC + FC + CC.

[0014] Furthermore, the vehicle dispatch cost (WC) is related to the number of routes, and the calculation method is as follows:

[0015] In the formula, fixed_cost is the fixed cost per vehicle per day; y k Let be the decision variable for vehicle k; m is the size of the candidate vehicle set.

[0016] Furthermore, the carbon emission cost CC is:

[0017] In the formula, w2 is the carbon tax price, ρ is the oil-to-carbon conversion rate, and y k Let f(k) be the decision variable for vehicle k; m be the size of the candidate vehicle set; and F(e) be the decision variable for vehicle k. k ) is path e k Total fuel cost.

[0018] Furthermore, the path balancing objective is represented by the difference between the maximum and minimum travel times:

[0019] In the formula: Dur k The travel time of the kth vehicle is composed of the fixed travel time and the expected travel time; the expected travel time is calculated using the Monte Carlo sampling method.

[0020] Furthermore, the customer satisfaction target is based on the varying importance of customers and their different sensitivities to delayed delivery; the customer satisfaction target employs a calculation method that balances customer priority and on-time delivery.

[0021] In the formula, pr i Priority values ​​are assigned to customers i∈C, using an expert scoring method and standardized to the [1,100] range, with higher values ​​indicating greater customer importance. i The calculation method;

[0022]

[0023] Among them, l i LST is the latest pickup time that customer i has scheduled in the system. i The latest acceptable pickup time for the customer; arr i The expected value of the service obtained for customer i is obtained through the Monte Carlo sampling method; early arrival has a smaller impact on satisfaction, while satisfaction decreases non-linearly after the upper limit of the time window submitted by the customer, and delivery within the time window results in complete satisfaction.

[0024] The delivery plan needs to meet two constraints: the amount of goods loaded in the vehicle is less than the maximum load capacity, and the expected transit time is less than or equal to 10 hours.

[0025] Furthermore, the driver waiting time target is:

[0026] In the formula, C is the set of customer points; e i Set the lower bound of the client's time window.

[0027] Furthermore, the delivery route planning uses a variable-length matrix chromosome to represent the set of delivery routes, and then a path-based crossover method is used to generate offspring. The principle of the path-based crossover method is: exchanging superior genes in chromosomes, with genes equivalent to paths, to achieve population evolution, that is, to optimize delivery routes. The steps are as follows:

[0028] S1: Randomly select two sets of delivery routes as the parent chromosome;

[0029] S2: Identify the optimal routes from the two sets of delivery routes;

[0030] S3: Insert the superior routes into another set of delivery routes respectively;

[0031] S4: For customer points that exist in both the original delivery route set and the newly inserted delivery route set, remove them from the original delivery route set.

[0032] Furthermore, the target is optimized by embedding a multi-objective evolutionary algorithm with multi-directional neighborhood search.

[0033] First, a population-based multi-directional neighborhood search algorithm is established; the multi-directional neighborhood search algorithm sorts all individuals in the current population, and multiple Pareto fronts participate in the multi-directional variable neighborhood search; the local search operators used by the multi-directional variable neighborhood search algorithm are respectively aimed at multiple objectives of the high-dimensional multi-objective vehicle path problem model;

[0034] The multi-directional neighborhood search algorithm is as follows:

[0035]

[0036] The multi-directional variable neighborhood search algorithm is embedded into a high-dimensional multi-objective evolution algorithm, which is as follows:

[0037]

[0038]

[0039] Furthermore, for the total delivery cost, intra-path and inter-path insertion operations, as well as path merging operations, are used to discover a more cost-efficient solution.

[0040] For the aforementioned path balancing, under the constraint of ensuring the path balance, the customer point on the path with the longest travel time will be inserted into the path with the shortest travel time.

[0041] Regarding customer satisfaction, customers are ranked from highest to lowest according to their delivery weight, and are inserted into other positions through inter-path operation operators to improve service quality;

[0042] For the driver waiting time, customers are sorted from longest to shortest waiting time, and inserted into other positions using inter-path operation operators to improve the quality of driver waiting.

[0043] Compared with the prior art, the advantages of the present invention are:

[0044] 1. The algorithm for fuel consumption in this invention effectively improves the accuracy of fuel consumption measurement.

[0045] 2. Compared with the original multi-directional neighborhood search algorithm, the population-based multi-directional neighborhood search algorithm proposed in this invention can be perfectly integrated with some classic multi-objective optimization frameworks, and can also make full use of population information to discover more Pareto solutions.

[0046] 3. The Monte Carlo simulation method for vehicle routing problems in stochastic demand scenarios proposed in this invention can effectively improve the evaluation efficiency of the objective function. Attached Figure Description

[0047] Figure 1 This is a schematic diagram illustrating the representation of a delivery path set using a variable-length matrix chromosome in an embodiment of the present invention;

[0048] Figure 2 This is a schematic diagram of the path intersection method in an embodiment of the present invention. Detailed Implementation

[0049] To make the objectives, technical solutions, and advantages of the present invention clearer, the technical solutions of the present invention will be further described below.

[0050] For ease of description, the problem solved by this invention is called the multi-objective green vehicle routing problem with stochastic demands (MOGVRPSD), and the four objectives to be optimized are: total delivery cost, route balancing, customer satisfaction, and driver waiting time.

[0051] To facilitate the description of the above four objectives, the following symbolic description is given: Assume that e is a complete path, and the order of the nodes on the path is Ω(e) = <0,n1(e),...,n l (e), 0 > . Where, n l (e) represents the last customer point on path e.

[0052] The costs optimized by MOGVRPSD include vehicle delivery costs, fuel consumption costs, and carbon emission costs.

[0053] TC=WC+FC+CC (1)

[0054] WC is a fixed dispatch cost, which is related to the number of routes. The calculation method is as follows:

[0055]

[0056] Where fixed_cost is the fixed cost per vehicle per day; y k Let be the decision variable for vehicle k; m is the size of the candidate vehicle set.

[0057] FC represents fuel consumption cost, which includes deterministic fuel consumption cost and random fuel consumption cost. The calculation method is as follows:

[0058]

[0059] Wherein, F(e) k ) is path e k The total fuel cost function includes deterministic fuel cost and stochastic fuel cost; w1 is the fuel price. The total fuel consumption for path e is:

[0060]

[0061] in, Let be the fuel consumption on edge (i,j); EF(e) is the random fuel consumption cost of path e.

[0062] To calculate the stochastic fuel consumption cost EF(e), a compensation strategy for delivery failures must first be defined. This invention uses a return trip heavy-load strategy as an example to illustrate the proposed technique. It is worth noting that the technique involved in this invention can also be extended to the optimization of stochastic vehicle routing problems under other compensation strategies.

[0063] The random fuel consumption cost EF(e) is calculated using a Monte Carlo sampling method, which involves randomly generating multiple demand scenarios according to the probability distribution of customer demand and completing delivery according to a prescribed compensation strategy. The average of the non-fixed fuel consumption costs for path e across multiple scenarios is used as the random fuel consumption cost EF(e).

[0064] CC represents carbon emission costs, which are related to the carbon tax price w2, the oil-to-carbon conversion rate ρ, and fuel consumption. The calculation method is as follows:

[0065]

[0066] The path balancing objective in MOGVRPSD is represented by the difference between the maximum and minimum travel times.

[0067]

[0068] Among them, Dur k The travel time of vehicle k is the sum of the fixed travel time and the expected travel time; the expected travel time is calculated using the Monte Carlo sampling method.

[0069] Considering the heterogeneity of customers in real-world logistics and delivery processes—on the one hand, customers have varying degrees of importance; on the other hand, customers have varying sensitivities to delayed delivery—this invention employs a customer satisfaction calculation method that balances customer priority and on-time delivery.

[0070]

[0071] Among them, pr i Priority values ​​are assigned to customers i∈C, using an expert scoring method and standardized to the [1,100] range, with higher values ​​indicating greater customer importance. i The calculation method is as follows: arriving early has a smaller impact on satisfaction; arriving later than the upper limit of the customer's submission time window results in a non-linear decrease in satisfaction; delivery within the time window results in complete satisfaction. The calculation method is shown in formula (8).

[0072]

[0073] Among them, l i LST is the latest pickup time that customer i has scheduled in the system. i The latest acceptable pickup time for the customer; arr i The expected value of the service obtained for customer i (obtained by the Monte Carlo sampling method).

[0074] To characterize delivery convenience from a time perspective, service wait time needs to be considered, with a target wait time of [missing information].

[0075]

[0076] Where C is the set of customer points; e i Set the lower bound of the client's time window.

[0077] The delivery plan must meet the following constraints: (1) the amount of goods loaded on the vehicle is less than the maximum load capacity; (2) the expected transit time is not allowed to exceed 10 hours.

[0078] This invention employs an optimization framework combining evolutionary algorithms and neighborhood search, and utilizes a variable-length matrix chromosome to represent the delivery path set, illustrated in the diagram. Figure 1 As shown.

[0079] For variable-length matrix chromosomes, a path-based crossover method is used to generate offspring, as illustrated in the diagram below. Figure 2 The principle behind this method is to achieve population evolution by exchanging preferred pathways (genes) in chromosomes.

[0080] like Figure 2 As shown, the specific steps of the path-based intersection method are as follows:

[0081] • Randomly select two chromosomes as the parent chromosomes to form the delivery path set (Parent 1 and Parent 2 in the figure);

[0082] • Identify the optimal pathways for each chromosome. The general criterion for selecting optimal pathways is the unit customer service cost (total service cost per pathway / number of customer points). For simplicity, assume that the optimal pathway for Parent 1 is "6-10" and the optimal pathway for Parent 2 is "1-3-7".

[0083] • Insert the superior path into another chromosome, that is, insert it into another set of delivery paths;

[0084] • To address the issue of duplicate points, chromosome repair is performed: For customer points that exist in both the original chromosome path and the newly inserted path, they are removed from the old path.

[0085] Based on the above encoding and crossover methods, this invention proposes an evolutionary search framework with multi-directional neighborhood search for solving the aforementioned high-dimensional multi-objective vehicle path problem.

[0086] To facilitate the explanation of the multi-directional neighborhood search strategy embedded in the high-dimensional multi-objective evolutionary algorithm proposed in this invention, the NSGA-III algorithm is adopted as the basic algorithm. Generally speaking, NSGA-III and NSGA-II have similar frameworks, with the main difference lying in their selection mechanisms. NSGA-II primarily relies on crowding for ranking, while NSGA-III maintains population diversity by introducing widely distributed reference points. Furthermore, NSGA-III employs a constraint dominance criterion: 1) feasible solutions dominate infeasible solutions; 2) solutions with small violations dominate solutions with large violations.

[0087] A high-dimensional multi-objective evolutionary algorithm with multi-directional neighborhood search embedding:

[0088]

[0089]

[0090] The multi-directional neighborhood search conditions in the algorithm are: 25%, 50%, 75%, and 100% of the evolutionary process.

[0091] To fully utilize population information and avoid the population from prematurely falling into local optima, this invention proposes a population-based multi-directional neighborhood search algorithm, denoted as PMDLS (Population-based MDLS). The main technical points are: 1) ranking all individuals in the current population, with multiple Pareto fronts participating in the multi-directional variable neighborhood search; 2) the local search operators used in the multi-directional variable neighborhood search are specifically designed for the MOGVRPSD objective. The PMDLS algorithm framework is shown below.

[0092]

[0093] For the total delivery cost, we use intra-path and inter-path insertion operations, as well as path merging operations, to find a more cost-effective solution.

[0094] For path balancing, under the constraint of ensuring the path is balanced, the customer point on the path with the longest travel time will be inserted into the path with the shortest travel time.

[0095] To improve customer satisfaction, customers are ranked from highest to lowest according to their delivery weight, and then inserted into other locations using inter-path operation operators to enhance service quality.

[0096] The system calculates driver waiting time, ranks customers by their waiting time from longest to shortest, and uses inter-path operation operators to insert them into other positions to improve driver waiting quality.

[0097] The above are merely preferred embodiments of the present invention and do not constitute any limitation on the present invention. Any equivalent substitutions or modifications made by those skilled in the art to the technical solutions and content disclosed in the present invention without departing from the scope of the present invention shall be deemed to have remained within the protection scope of the present invention.

Claims

1. A Monte Carlo optimization method for a stochastic multi-objective green vehicle routing problem, characterized in that, Four objectives are optimized using a multi-objective evolutionary algorithm embedded with multi-directional neighborhood search. First, a population-based multi-directional neighborhood search algorithm is established. This algorithm sorts all individuals in the current population, and multiple Pareto fronts participate in the multi-directional variable neighborhood search. The local search operators used in the multi-directional variable neighborhood search algorithm are respectively targeted at multiple objectives of the high-dimensional multi-objective vehicle path problem model. The four objectives include: total delivery cost, route balance, customer satisfaction, and driver wait time; The total delivery cost includes fuel consumption cost (FC). Let e ​​be a complete path, in which the order of nodes is Ω(e)=<0,n1(e),...,n l (e), 0 > where n l (e) represents the last customer point on path e; The FC includes both determined fuel consumption cost and random fuel consumption cost, and the calculation method is as follows: Wherein, F(e) k ) is path e k The total fuel cost function includes deterministic fuel cost and stochastic fuel cost; w1 is the fuel price; the total fuel cost for path e is: in, for and The collective term represents the fuel consumption on edge (i,j); EF(e) is the random fuel consumption cost of path e. The calculation of the random fuel consumption cost EF(e) requires the specification of a compensation strategy for delivery failures. It is calculated using the Monte Carlo sampling method, which randomly generates multiple demand scenarios according to the probability distribution of customer demand, and completes the delivery according to the specified compensation strategy. In multiple scenarios, the average of the non-fixed fuel consumption cost of path e is used as the random fuel consumption cost EF(e). Delivery routes are planned using a variable-length matrix chromosome to represent the set of delivery routes. Then, a path-based crossover method is used to generate offspring. The principle of this method is to exchange superior genes within chromosomes; genes are analogous to paths, thus achieving population evolution, i.e., optimizing delivery routes. The steps are as follows: S1: Randomly select two sets of delivery routes as the parent chromosome; S2: Identify the optimal routes from the two sets of delivery routes; S3: Insert the superior routes into another set of delivery routes respectively; S4: For customer points that exist in both the original delivery route set and the newly inserted delivery route set, remove them from the original delivery route set. For the total delivery cost, intra-path and inter-path insertion operations, as well as path merging operations, are used to find a more cost-effective solution. For the aforementioned path balancing, under the constraint of ensuring the path balance, the customer point on the path with the longest travel time will be inserted into the path with the shortest travel time. Regarding customer satisfaction, customers are ranked from highest to lowest according to their delivery weight, and are inserted into other positions through inter-path operation operators to improve service quality; For the driver waiting time, customers are sorted from longest to shortest waiting time, and inserted into other positions using inter-path operation operators to improve the quality of driver waiting.

2. The Monte Carlo optimization method for the stochastic demand multi-objective green vehicle path problem according to claim 1, characterized in that, The total delivery cost TC also includes vehicle dispatch cost WC and carbon emission cost CC; TC = WC + FC + CC.

3. The Monte Carlo optimization method for the stochastic demand multi-objective green vehicle path problem according to claim 2, characterized in that, The vehicle dispatch cost (WC) is related to the number of routes, and the calculation method is as follows: In the formula, fixed_cost is the fixed cost per vehicle per day; y k Let be the decision variable for vehicle k; m is the size of the candidate vehicle set.

4. The Monte Carlo optimization method for the stochastic demand multi-objective green vehicle path problem according to claim 2, characterized in that, The carbon emission cost CC is: In the formula, w2 is the carbon tax price, ρ is the oil-to-carbon conversion rate, and y k Let f(k) be the decision variable for vehicle k; m be the size of the candidate vehicle set; and F(e) be the decision variable for vehicle k. k ) is path e k Total fuel cost.

5. The Monte Carlo optimization method for the stochastic demand multi-objective green vehicle path problem according to claim 1, characterized in that, The path balancing objective is represented by the difference between the maximum and minimum travel times: In the formula: Dur k The travel time of vehicle k is the sum of the fixed travel time and the expected travel time; the expected travel time is calculated using the Monte Carlo sampling method.

6. The Monte Carlo optimization method for the stochastic demand multi-objective green vehicle routing problem according to claim 1, characterized in that, The customer satisfaction target is based on the varying importance of customers and their different sensitivities to delayed delivery; the customer satisfaction target employs a calculation method that balances customer priority and on-time delivery. In the formula, pr i Priority values ​​are assigned to customers i∈C, using an expert scoring method and standardized to the [1,100] range, with higher values ​​indicating greater customer importance. i The calculation method; Among them, l i LST is the latest pickup time that customer i has scheduled in the system. i The latest acceptable pickup time for the customer; arr i For customer i, the expected value of the service, r i The sensitivity of customers to delayed delivery is determined by the Monte Carlo sampling method. Early arrival has a smaller impact on satisfaction, while delivery later than the upper limit of the customer's submission time window results in a non-linear decrease in satisfaction. Delivery within the time window results in complete satisfaction. The delivery plan needs to meet two constraints: the amount of goods loaded in the vehicle is less than the maximum load capacity, and the expected transit time is less than or equal to 10 hours.

7. The Monte Carlo optimization method for the stochastic demand multi-objective green vehicle routing problem according to claim 6, characterized in that, The target for driver waiting time: In the formula, C is the set of customer points; e i Set the lower bound of the client's time window.

8. The Monte Carlo optimization method for the stochastic demand multi-objective green vehicle routing problem according to claim 1, characterized in that, The multi-directional neighborhood search algorithm is as follows: The multi-directional variable neighborhood search algorithm is embedded into a high-dimensional multi-objective evolution algorithm, which is as follows:

Images