CN111445186A - Petri network theory-based vehicle path optimization method with time window - Google Patents

Abstract

A method for optimizing vehicle route with time window based on Petri network theory includes setting up customer data model according to vehicle route problem with time window to be solved, calculating distance between each other in distribution center and customer point, setting up mathematical model of vehicle route problem with time window, setting up Petri network model of vehicle route problem with time window based on customer data model, converting mathematical model into integer linear programming problem based on Petri network model, calling relevant program of integer linear programming problem in MAT L AB, and finally utilizing YA L MIP optimizing tool box to solve program content and carry out result analysis.

Description

Petri network theory-based vehicle path optimization method with time window

Technical Field

The invention belongs to the technical field of logistics, and particularly relates to a Petri net theory-based vehicle path optimization method with a time window.

Background

With the advancement of science and technology and the vigorous development of the internet, electronic commerce enters a high-speed development stage, and the scale of logistics distribution is increased rapidly. The efficiency of logistics distribution is seriously affected due to the increase in the number of logistics orders, the complexity of transportation networks, and the unreasonable distribution routes. Therefore, how to scientifically optimize the distribution route and reduce the transportation cost is an important research content of the logistics industry.

In the logistics industry, Vehicle Routing distribution (VRP) is one of the core problems in logistics management and transportation, and research thereof is receiving a great deal of attention. The vehicle path problem is an optimization problem of enabling a fleet of vehicles to start from a distribution center, distributing goods to customers with different quantity of goods, and meeting certain constraint conditions (such as time limit, vehicle-mounted capacity limit and the like) to achieve the goals of lowest distribution cost, minimum distributed vehicles, shortest distribution distance and the like.

The vehicle path optimization problem belongs to an NP difficult problem, and the solutions thereof comprise a heuristic method and a mathematical analysis method. The heuristic method is a hotspot of current research and comprises a saving method, a simulated annealing method, a tabu search method, a neural network method and the like, but the heuristic method generally requires a designer to have stronger professional knowledge and stronger specificity; in addition, heuristics often do not yield optimal solutions. The common mathematical analysis methods include a branch boundary method, an integer programming method, a dynamic programming method and the like, the operation amount of the method increases exponentially with the increase of the problem scale, and the method can obtain an optimal solution but is not suitable for solving a large-scale problem.

Disclosure of Invention

In order to overcome the defects of the prior art, the invention aims to provide a Petri network theory-based vehicle path optimization method with a time window, which can realize the optimal vehicle distribution route of a distribution center, simultaneously enables the total distribution distance of vehicles to be shortest, reduces the vehicle transportation cost to a greater extent and has good application prospect.

In order to achieve the purpose, the invention adopts the following technical scheme:

a Petri net theory-based vehicle path optimization method with a time window comprises the following steps:

step one, establishing a customer data model: establishing a client data model according to the vehicle path problem with the time window to be solved;

step two, calculating the distance d between each two of the distribution center and the customer points_i，jI ≠ j, and meanwhile, the transportation time between every two is assumed to be 1;

step three, establishing a mathematical model of the vehicle path problem with a time window: according to the distribution demand of the vehicles, the vehicle distribution problem belongs to the vehicle route problem with hard time window constraint, and the vehicles from the distribution center must arrive at the destination according to the time window requirement of the destination, and cannot be earlier than the earliest time requirement and cannot be later than the latest time requirement; designing an objective function, wherein the objective function is used for enabling the total distribution distance of the vehicles to be shortest;

step four, establishing a Petri network model with a time window vehicle path problem based on the client data model;

step five, based on the Petri network model in the step four, converting the mathematical model in the step three into an integer linear programming problem;

step six, calling a relevant program of the integer linear programming problem in the step five in MAT L AB;

and step seven, solving the program content in the step six by using a YA L MIP optimization tool box and analyzing the result.

The Petri network model of the step four-time window vehicle path problem is as follows:

a) distribution center and customer point i 1, 2, n is represented by set P, P₁，p₂，...，p_nDenotes a collection of libraries of distribution centers and customer sites, where each element is denoted by p_iI 1 denotes a distribution center, i 2, 3.

b) In the distribution centre and in all customer points, there may be transport paths between each other, so transition t is used in the Petri network_i，jTo indicate the vehicle slave position p_iTransport to p_jBy t_j，iTo indicate the vehicle slave position p_jTransport to p_i；

c) Based on the reason that each customer point can only be accessed by one vehicle and only once, K is used for representing the number of steps of vehicle movement, and each step of vehicle addition means that the vehicle accesses one point;

d) in order to clarify the transportation state of each step of the vehicle, namely know the position of each step of the vehicle, the identifier M of the Petri net is used for representing; using M_c，k＝[M_c，k(p₁)，M_c，k(p₂)，...，M_c，k(p_n)]^TIndicating that the vehicle c is at each point p at the k-th step_iA position mark, i.e. the position of the vehicle C at the k-th step, where C is 1, 2. K1, 2,. K; 1, 2, n; if the position of the vehicle c at the k step is p_iThen M_c，k(p_i) 1, otherwise M_c，k(p_i)＝0；

e) In Petri nets, using

And

respectively representing a pre-incidence matrix and a post-incidence matrix of the library and the transition; if the output arc of the bank p points to the transition t, Pre (p, t) is 1, otherwise Pre (p, t) is 0; if the output arc of the transition t points to the position p of the library, the Post (p, t) is 1, otherwise, the Post (p, t) is 0;

f) use of v in Petri nets_i，jThis cargo vector represents the vehicle from p_iTo p_jDuring transportation, p_jA required amount r of_j；

g) To represent the transport path of the kth step c of the vehicle, a column vector θ is used in a Petri network_{c，k，i，j}＝[θ_{c，k，0，1}，θ_{c，k，0，2}，...，θ_{c，k，i，j}，...，θ_{c，k，n，n-1}]^TPath vector representing the k-th step of vehicle cWherein: i, j ≠ 1, 2,. n; if the k-th step of vehicle c is from position p_iTo position p_jThen theta_{c，k，i，j}1, otherwise θ_{c，k，i，j}＝0；

h) To satisfy each position p_iUsing tau in a Petri net_c，kTo indicate the time of the vehicle c at the kth step and the initial time τ_c，0＝0。

The integer linear programming problem model of the step five is as follows:

the objective function is:

the constraint conditions are as follows:

constraint 1: m_c，k＝M_c，k-1+(Post-Pre)×θ_{c，k，i，j}

Constraint 2: m_c，k-1-Pre×θ_{c，k，i，j}≥0

Constraint condition 3: 1^T×θ_{c，k，i，j}≤1

Constraint 4:

constraint 5:

constraint 6:

constraint 7: e.g. of the type_i-τ_c，k≤[1-M_c，k(p_i)]×H，i＝2，3，...，n，k＝1，2，...，K-1，c＝1，2，...，C

Constraint condition 8: tau is_c，k-l_i≤[1-M_c，k(p_i)]×H，i＝2，3，...，n，k＝1，2，...，K-1，c＝1，2，...，C

Constraint 9: e.g. of the type₁-τ_c，K≤[1-M_c，K(p₁)]×H，c＝1，2，...，C

Constraint 10: tau is_c，K-l₁≤[1-M_c，K(p₁)]×H，c＝1，2，...，C

Constraint 11: tau is_c，k＝τ_c，k-1+1^T×θ_{c，k，i，j}+D_c，k-1

Constraint 12: tau is_c，0＝0，c＝1，2，...，C

Constraint condition 13: i, j ≠ n and i ≠ j, K ≠ 1, 2,.. K, C ═ 1, 2

Wherein H is a sufficiently large number; other variables are defined as follows:

c is the set of invokable vehicles, C ═ 1, 2,. C };

k is the number of moving steps of the vehicle, K is the number of steps the vehicle moves most, and K is {1, 2.., K };

i, j denotes a customer site or a distribution center, i, j ═ 1, 2.

A Pre-correlation matrix representing the library and the transition, wherein if the output arc of the library p points to the transition t, Pre (p, t) is 1, otherwise Pre (p, t) is 0;

representing a Post incidence matrix of the library and the transition, wherein when an output arc of the transition t points to the library p, Post (p, t) is 1, otherwise, Post (p, t) is 0;

θ_{c，k，i，j}＝[θ_{c，k，0，1}，θ_{c，k，0，2}，...，θ_{c，k，i，j}，...，θ_{c，k，n，n-1}]^Ta path vector representing a Petri net; if the vehicle c departs at the k-th step point i to visit the j point, theta_{c，k，i，j}1, otherwise θ_{c，k，i，j}＝0；

M_c，k＝[M_c，k(p₁)，M_c，k(p₂)，...，M_c，k(p_n)]^TIs the location identification of the Petri net, if the location of the vehicle c at the kth step is p_iThen M_c，k(p_i) 1, otherwise M_c，k(p_i)＝0；

d_i，jRepresents the Euclidean distance between point i and point j;

r is the maximum cargo capacity per vehicle;

v_i，ja freight vector representing a Petri Net, if the vehicle goes from point i to point j, then v_i，j＝r_j；

e_iAnd l_iRespectively representing the earliest service time and the latest service time at each customer point or distribution center i;

D_c，krepresenting the service time of the k step of the vehicle c;

τ_c，kindicating the moment at which the vehicle c steps k.

The invention has the beneficial effects that:

the method combines a Petri network model, converts a mathematical model of a vehicle path problem into an integer linear programming problem, solves the problem in MAT L AB by using a YA L MIP optimization tool box, and finally finds the optimal distribution path meeting distribution requirements.

Drawings

FIG. 1 is a block flow diagram of the method of the present invention.

FIG. 2 is a schematic diagram of a distribution path according to an embodiment of the present invention.

Detailed Description

The invention is further illustrated below with reference to examples and figures.

Referring to fig. 1, a Petri net theory-based vehicle path optimization method with a time window comprises the following steps:

step one, establishing a customer data model: establishing a client data model according to the vehicle path problem with the time window to be solved;

it is known that a distribution center completes distribution tasks for all customer points by using vehicles with the same model and load capacity at most, and all vehicles at the end of distribution should return to the distribution center, each customer point can be visited by one vehicle only and can be visited only once, and the distribution center and the customer points are represented by i (i is 1, 2.., n), wherein i is 1 represents the distribution center, i is 2.., and n represents the customer points; the coordinates for setting the customer site and the distribution center are x_i，y_i(ii) a The maximum cargo capacity of each vehicle is set to R and it is assumed that the demand of the distribution center and the customer site is R_iAnd max r_iR is less than or equal to R; setting the time window of the distribution center and the customer point as [ e ]_i，l_i]，e_iAnd l_iRespectively representing the earliest time and the latest time when the vehicle arrives at the distribution center or the customer site i, the service time of the distribution center is 0, the service time of each customer site is unit 1, and customer information is stored in the form of a data table as shown in the following table.

Step two, calculating the distance d between each two of the distribution center and the customer points_i，j(i ≠ j), meanwhile, assuming that the transportation time between every two is 1;

calculating the distance d between two of the distribution center and the customer site_i，j(i ≠ j; i ≠ 1, 2, …, n, j ═ 1, 2, …, n), it is used

Obtaining the linear distance between any two points; the transit time between any two points is measured in units of 1;

step three, establishing a mathematical model of the vehicle path problem with a time window: according to the distribution demand of the vehicles, the vehicle distribution problem belongs to the vehicle route problem with hard time window constraint, and the vehicles from the distribution center must arrive at the destination according to the time window requirement of the destination, and cannot be earlier than the earliest time requirement and cannot be later than the latest time requirement; designing an objective function, wherein the objective function is used for enabling the total distribution distance of the vehicles to be shortest;

the mathematical model of the vehicle path problem with the time window established in the third step is as follows:

e_i≤T_c，i≤l_i，i＝1，2，3，…，n；c＝1，2，...，C (7)

T_c，1，0＝0，c＝1，2，...，C (8)

x_c，i，j∈{0，1}，i＝1，2，…，n，j＝1，2，…，n；c＝1，2，…，C；i≠j

y_ci∈{0，1}，i＝1，2，…，n；c＝1，2，…，C

wherein:

the meaning of each formula in the mathematical model is as follows:

formula (1) is an objective function, and represents that the total distribution distance of all vehicles in a distribution center is shortest; equation (2) is the loadability constraint of each vehicle, i.e., the total load of each vehicle c on its delivery route cannot exceed its maximum load R by 90; equation (3) indicates that only one vehicle will serve each customer site; the formula (4) ensures that the number of vehicles in the distribution center is C, namely all vehicles start from the distribution center and return to the distribution center after distribution; equations (5) and (6) ensure that the number of vehicles arriving and leaving each customer site is 1; equation (7) indicates that the time when each vehicle arrives at the distribution center or customer point i is always within the time window required by the point; formula (8) shows that each vehicle is at the distribution center at the initial moment;

step four, establishing a Petri network model with a time window vehicle path problem based on the client data model in the step one;

the Petri net model for the vehicle path problem with the time window is as follows:

a) distribution center and customer point i 1, 2, n is represented by set P, P₁，p₂，...，p_nDenotes a collection of libraries of distribution centers and customer sites, where each element is denoted by p_i(i 1, 2.., n), i 1 denotes a distribution center, i 2, 3., n denotes a customer point;

b) in the distribution centre and in all customer points, there may be transport paths between each other, so transition t is used in the Petri network_i，jTo indicate the vehicle slave position p_iTransport to p_jBy t_j，iTo indicate the vehicle slave position p_jTransport to p_i(ii) a Wherein i, j ≠ j, 1, 2., n;

c) based on the reason that each customer point can only be visited by one vehicle and only once, K is used to represent the number of steps that the vehicle moves, and each step that the vehicle increases means that the vehicle visits one point (which may be the same point as the previous step, i.e. it means that the vehicle does not leave the customer point or the distribution center);

d) in order to clarify the transportation state of each step of the vehicle, namely know the position of each step of the vehicle, the identifier M of the Petri net is used for representing; using M_c，k＝[M_c，k(p₁)，M_c，k(p₂)，...，M_c，k(p_n)]^TIndicating that the vehicle c is at each point p at the k-th step_iA position mark, i.e. the position of the vehicle C at the k-th step, where C is 1, 2. K1, 2,. K; 1, 2, n; if the position of the vehicle c at the k step is p_iThen M_c，k(p_i) 1, otherwise M_c，k(p_i)＝0；

e) In Petri nets, using

And

respectively representing a pre-incidence matrix and a post-incidence matrix of the library and the transition; if the output arc of the bank p points to the transition t, Pre (p, t) is 1, otherwise Pre (p, t) is 0; if the output arc of the transition t points to the position p of the library, the Post (p, t) is 1, otherwise, the Post (p, t) is 0;

f) to represent each position p_iDemand of (c), using v in the Petri net_i，jThis cargo vector represents the vehicle from p_iTo p_jDuring transportation, p_jA required amount r of_j(ii) a Wherein i, j ≠ j, 1, 2., n, and i ≠ j;

g) to represent the transport path of the kth step c of the vehicle, a column vector θ is used in a Petri network_{c，k，i，j}＝[θ_{c，k，0，1}，θ_{c，k，0，2}，...，θ_{c，k，i，j}，...，θ_{c，k，n，n-1}]^TA path vector representing the kth step of vehicle c, wherein: i, j ≠ 1, 2,. n; if the k-th step of vehicle c is from position p_iTo position p_jThen theta_{c，k，i，j}1, otherwise θ_{c，k，i，j}＝0；

h) To satisfy each position p_iUsing tau in a Petri net_c，kTo indicate the time of the vehicle c at the kth step and the initial time τ_c，0＝0；

Step five, based on the Petri network model in the step four, converting the mathematical model in the step three into an integer linear programming problem;

the specific method comprises the following steps:

equation (1) translates to:

equation (2) translates to:

equation (3) translates to:

equation (4) translates to:

equation (7) translates to:

e_i-τ_c，k≤[1-M_c，k(p_i)]×H，i＝2，3，...，n，k＝1，2，...，K-1，c＝1，2，...，C (13)

τ_c，k-l_i≤[1-M_c，k(p_i)]×H，i＝2，3，...，n，k＝1，2，...，K-1，c＝1，2，...，C (14)

e₁-τ_c，K≤[1-M_c，K(p₁)]×H，c＝1，2，...，C (15)

τ_c，K-l₁≤[1-M_c，KK(p₁)]×H，c＝1，2，...，C (16)

equation (8) translates to:

τ_c，0＝0，c＝1，2，...，C (17)

therefore, the integer linear programming problem model of step five is as follows:

the objective function is:

the constraint conditions are as follows:

constraint 1: m_c，k＝M_c，k-1+(Post-Pre)×θ_{c，k，i，j}

Constraint 2: m_c，k-1-Pre×θ_{c，k，i，j}≥0

Constraint 3: 1^T×θ_{c，k，i，j}≤1

Constraint 4:

constraint 5:

constraint 6:

constraint 7: e.g. of the type_i-τ_c，k≤[1-M_c，k(p_i)]×H，i＝2，3，...，n，k＝1，2，...，K-1，c＝1，2，...，C

Constraint condition 8: tau is_c，k-l_i≤[1-M_c，k(p_i)]×H，i＝2，3，...，n，k＝1，2，...，K-1，c＝1，2，...，C

Constraint 9: e.g. of the type₁-τ_c，K≤[1-M_c，K(p₁)]×H，c＝1，2，...，C

Constraint 10: tau is_c，K-l₁≤[1-M_c，K(p₁)]×H，c＝1，2，...，C

Constraint 11: tau is_c，k＝τ_c，k-1+1^T×θ_{c，k，i，j}+D_c，k-1

Constraint 12: tau is_c，0＝0，c＝1，2，...，C

Constraint condition 13: i, j ≠ n and i ≠ j, K ≠ 1, 2,.. K, C ═ 1, 2

Wherein H is a sufficiently large number; other variables are defined as follows:

c is the set of invokable vehicles, C ═ 1, 2,. C };

k is the number of moving steps of the vehicle, K is the number of steps the vehicle moves most, and K is {1, 2.., K };

i, j denotes a customer site or a distribution center, i, j ═ 1, 2.

A Pre-correlation matrix representing the library and the transition, wherein if the output arc of the library p points to the transition t, Pre (p, t) is 1, otherwise Pre (p, t) is 0;

representing a Post incidence matrix of the library and the transition, wherein when an output arc of the transition t points to the library p, Post (p, t) is 1, otherwise, Post (p, t) is 0;

θ_{c，k，i，j}＝[θ_{c，k，0，1}，θ_{c，k，0，2}，...，θ_{c，k，i，j}，...，θ_{c，k，n，n-1}]^Ta path vector representing a Petri net; if the vehicle c departs at the k-th step point i to visit the j point, theta_{c，k，i，j}1, otherwise θ_{c，k，i，j}＝0；

M_c，k＝[M_c，k(p₁)，M_c，k(p₂)，...，M_c，k(p_n)]^TIs the location identification of the Petri net, if the location of the vehicle c at the kth step is p_iThen M_c，k(p_i) 1, otherwise M_c，k(p_i)＝0；

d_i，jRepresents the Euclidean distance between point i and point j;

r is the maximum cargo capacity per vehicle;

v_i，ja freight vector representing a Petri net, if a vehicle goes from point i to point j, i, j ≠ 1, 2_i，j＝r_j；

e_iAnd l_iRespectively representing the earliest service time and the latest service time at each customer point or distribution center i;

D_c，krepresenting the service time of the k step of the vehicle c;

τ_c，kindicating the moment at which the vehicle c steps k;

the objective function ensures that the total transportation distance of all vehicles is shortest; constraint 1 represents the transport state of each step; the constraint condition 2 ensures the correctness of the transportation process of each step; the constraint condition 3 ensures that the vehicle has at most one transition trigger in each step, namely the vehicle only goes to one destination at most to complete distribution in the transportation process of each step; the constraint condition 4 ensures that the cargo capacity of each vehicle on the transportation path does not exceed the maximum cargo capacity of each vehicle; constraint 5 ensures that each customer point will only be delivered once; constraint 6 indicates that all vehicle starting points and the last step positions are at the distribution center; constraint conditions 7 and 8 ensure that the time when the vehicle c reaches the client point in the kth step meets the time window requirement of the client point; the constraint conditions 9 and 10 ensure that the time when the vehicle c returns to the distribution center in the last step meets the time window requirement of the distribution center; constraint 11 indicates the time of the kth step of vehicle c; constraint 12 ensures that the initial time of all vehicles is 0; constraint 13 is n-23, K-8, and C-4;

step six, calling a relevant program of the step five integer linear programming problem in MAT L AB, and specifically comprising the following steps:

6.1) constructing a Petri network structure, wherein the Petri network structure comprises a pre-incidence matrix, a post-incidence matrix and an incidence matrix, and meanwhile, adding corresponding demand to each transition;

6.2) inputting the distance between each two of the distribution center and the customer points;

6.3) inputting variables such as the transport path of the vehicle, the initial identification and the final identification of the vehicle;

6.4) converting the objective function and the constraint condition of the integer linear programming problem into a program;

6.5) calling a YA L MIP optimization tool box to solve;

step seven, solving the program content in the step six by using a YA L MIP optimization tool box and analyzing the result;

referring to fig. 2, the vehicle path planning results generated by 1 distribution center in the embodiment, 22 customer point models, using YA L MIP optimization toolbox to solve the procedure of step six are as follows:

the distribution route of the 1 st vehicle is as follows: p is a radical of₁(distribution center) → p₆(client Point 6) → p₄(client Point 4) → p₂₃(client Point 23) → p₁₁(customer Point 11) → p₂₁(client Point 21) → p₁₆(client Point 16) → p₁(distribution center);

delivery route of 2 nd vehicle: p is a radical of₁(distribution center) → p₁₇(client Point 17) → p₁₅(client Point 15) → p₁₂(client Point 12) → p₂₀(client Point 20) → p₁₉(customer site 19) → p₁(distribution center);

delivery route of 3 rd vehicle: p is a radical of₁(distribution center) → p₂₂(client Point 22) → p₉(client Point 9) → p₁₀(client Point 10) → p₁₈(client Point 18) → p₁₄(client Point 14) → p₁₃(client Point 13) → p₇(client Point 7) → p₁(distribution center);

delivery route of 4 th vehicle: p is a radical of₁(distribution center) → p₈(client Point 8) → p₂(client Point 2) → p₃(customer site)3)→p₅(client Point 5) → p₁(distribution center);

the results meet the time window requirements of the examples, as shown in the following table:

TABLE 1 time windows for distribution centers and customer sites

The results were further analyzed as follows: all customer points only have one vehicle to distribute the customer points, 4 vehicles exist in a distribution center and can be called to distribute, the shortest total transportation distance is finally realized, and the transportation cost is greatly saved; the actual cargo capacity of the 1 st vehicle is 90 units, the actual cargo capacity of the 2 nd vehicle is 80 units, the actual cargo capacity of the 3 rd vehicle is 83 units, the actual cargo capacity of the 4 th vehicle is 41 units, the maximum cargo capacity of the vehicles is not exceeded, and the sum of the path tracks of the vehicles achieves the shortest 25.0128 km.

Finally, it should be noted that: the foregoing is only a preferred embodiment of the present invention, and it will be apparent to those skilled in the art that various modifications and improvements can be made without departing from the principle of the invention, and these modifications and improvements should be considered as the protection scope of the present invention.

Claims (3)

1. A Petri net theory-based vehicle path optimization method with a time window is characterized by comprising the following steps:

step one, establishing a customer data model: establishing a client data model according to the vehicle path problem with the time window to be solved;

step two, calculating the distance d between each two of the distribution center and the customer points_i，jI ≠ j, and meanwhile, the transportation time between every two is assumed to be 1;

step three, establishing a mathematical model of the vehicle path problem with a time window: according to the distribution demand of the vehicles, the vehicle distribution problem belongs to the vehicle route problem with hard time window constraint, and the vehicles from the distribution center must arrive at the destination according to the time window requirement of the destination, and cannot be earlier than the earliest time requirement and cannot be later than the latest time requirement; designing an objective function, wherein the objective function is used for enabling the total distribution distance of the vehicles to be shortest;

step four, establishing a Petri network model with a time window vehicle path problem based on the client data model;

step five, based on the Petri network model in the step four, converting the mathematical model in the step three into an integer linear programming problem;

step six, calling a relevant program of the integer linear programming problem in the step five in MAT L AB;

and step seven, solving the program content in the step six by using a YA L MIP optimization tool box and analyzing the result.

2. The Petri net theory-based vehicle path optimization method with the time window is characterized by comprising the following steps of: the Petri network model of the step four-time window vehicle path problem is as follows:

a) distribution center and customer point i 1, 2, n is represented by set P, P₁，p₂，...，p_nDenotes a collection of libraries of distribution centers and customer sites, where each element is denoted by p_iI 1 denotes a distribution center, i 2, 3.

b) In the distribution centre and in all customer points, there may be transport paths between each other, so transition t is used in the Petri network_i，jTo indicate the vehicle slave position p_iTransport to p_jBy t_j，iTo indicate the vehicle slave position p_jTransport to p_i；

c) Based on the reason that each customer point can only be accessed by one vehicle and only once, K is used for representing the number of steps of vehicle movement, and each step of vehicle addition means that the vehicle accesses one point;

d) in order to clarify the transportation state of each step of the vehicle, namely know the position of each step of the vehicle, the identifier M of the Petri net is used for representing; using M_c，k＝[M_c，k(p₁)，M_c，k(p₂)，...，M_c，k(p_n)]^TIndicating that the vehicle c is at each point p at the k-th step_iA position mark, i.e. the position of the vehicle C at the k-th step, where C is 1, 2. K1, 2,. K; 1, 2, n; if the position of the vehicle c at the k step is p_iThen M_c，k(p_i) 1, otherwise M_c，k(p_i)＝0；

e) In Petri nets, using

And

respectively representing a pre-incidence matrix and a post-incidence matrix of the library and the transition; if the output arc of the bank p points to the transition t, Pre (p, t) is 1, otherwise Pre (p, t) is 0; if the output arc of the transition t points to the position p of the library, the Post (p, t) is 1, otherwise, the Post (p, t) is 0;

f) use of v in Petri nets_i，jThis cargo vector represents the vehicle from p_iTo p_jDuring transportation, p_jA required amount r of_j；

g) To represent the transport path of the kth step c of the vehicle, a column vector θ is used in a Petri network_{c，k，i，j}＝[θ_{c，k，0，1}，θ_{c，k，0，2}，...，θ_{c，k，i，j}，...，θ_{c，k，n，n-1}]^TA path vector representing the kth step of vehicle c, wherein: i, j ≠ 1, 2,. n; if the k-th step of vehicle c is from position p_iTo position p_jThen theta_{c，k，i，j}1, otherwise θ_{c，k，i，j}＝0；

h) To satisfy each position p_iUsing tau in a Petri net_c，kTo indicate the time of the vehicle c at the kth step and the initial time τ_c，0＝0。

3. The Petri net theory-based vehicle path optimization method with the time window is characterized by comprising the following steps of: the integer linear programming problem model of the step five is as follows:

the objective function is:

the constraint conditions are as follows:

constraint 1: m_c，k＝M_c，k-1+(Post-Pre)×θ_{c，k，i，j}

Constraint 2: m_c，k-1-Pre×θ_{c，k，i，j}≥0

Constraint 3: 1^T×θ_{c，k，i，j}≤1

Constraint 4:

constraint 5:

constraint 6:

constraint 7: e.g. of the type_i-τ_c，k≤[1-M_c，k(p_i)]×H，i＝2，3，...，n，k＝1，2，...，K-1，c＝1，2，...，C

Constraint condition 8: tau is_c，k-l_i≤[1-M_c，k(p_i)]×H，i＝2，3，...，n，k＝1，2，...，K-1，c＝1，2，...，C

Constraint 9: e.g. of the type₁-τ_c，K≤[1-M_c，K(p₁)]×H，c＝1，2，...，C

Constraint 10: tau is_c，K-l₁≤[1-M_c，K(p₁)]×H，c＝1，2，...，C

Constraint 11: tau is_c，k＝τ_c，k-1+1^T×θ_{c，k，i，j}+D_c，k-1

Constraint 12: tau is_c，0＝O，c＝1，2，...，C

Constraint condition 13: i, j ≠ n and i ≠ j, K ≠ 1, 2,.. K, C ═ 1, 2

Wherein H is a sufficiently large number; other variables are defined as follows:

c is the set of invokable vehicles, C ═ 1, 2,. C };

k is the number of moving steps of the vehicle, K is the number of steps the vehicle moves most, and K is {1, 2.., K };

i, j denotes a customer site or a distribution center, i, j ═ 1, 2.

A Pre-correlation matrix representing the library and the transition, wherein if the output arc of the library p points to the transition t, Pre (p, t) is 1, otherwise Pre (p, t) is 0;

representing a Post incidence matrix of the library and the transition, wherein when an output arc of the transition t points to the library p, Post (p, t) is 1, otherwise, Post (p, t) is 0;

θ_{c，k，i，j}＝[θ_{c，k，0，1}，θ_{c，k，0，2}，...，θ_{c，k，i，j}，...，θ_{c，k，n，n-1}]^Ta path vector representing a Petri net; if the vehicle c departs at the k-th step point i to visit the j point, theta_{c，k，i，j}1, otherwise θ_{c，k，i，j}＝0；

M_c，k＝[M_c，k(p₁)，M_c，k(p₂)，...，M_c，k(p_n)]^TIs the location identification of the Petri net, if the location of the vehicle c at the kth step is p_iThen M_c，k(p_i) 1, otherwise M_c，k(p_i)＝0；

d_i，jRepresents the Euclidean distance between point i and point j;

r is the maximum cargo capacity per vehicle;

v_i，ja freight vector representing a Petri Net, if the vehicle goes from point i to point j, then v_i，j＝r_j；

e_iAnd l_iRespectively representing the earliest service time and the latest service time at each customer point or distribution center i;

D_c，krepresenting the service time of the k step of the vehicle c;

τ_c，kindicating the moment at which the vehicle c steps k.

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