CN111325389B - Vehicle path optimization method based on Petri network and integer linear programming - Google Patents

Abstract

A vehicle path optimization method based on Petri network and integer linear programming is characterized in that a mathematical model of the vehicle path optimization method is established according to description of a vehicle path problem; then, establishing a Petri network model of the vehicle path problem based on the mathematical model; converting the mathematical model into a program of an integer linear programming problem by combining with a Petri network model; then, importing a program of an integer linear programming problem into the MATLAB, and inputting the distance between the client points and the cargo demand; finally, solving the integer linear programming problem in the fourth step by using a YALMIP optimization tool box, and carrying out experiment and result analysis; the invention can obtain the optimal route of the vehicle distribution route, and the obtained total distribution route has the shortest distance, thereby effectively reducing the vehicle distribution cost and having good application prospect.

Description

Vehicle path optimization method based on Petri network and integer linear programming

Technical Field

The invention belongs to the technical field of logistics distribution, and particularly relates to a vehicle path optimization method based on a Petri network and integer linear programming.

Background

The Vehicle Routing Problem (VRP) is one of the key problems in the logistics distribution process, and as the logistics distribution industry is increasingly competitive and customers have higher requirements on the logistics distribution timeliness, the Vehicle Routing Problem is researched more and more deeply.

The vehicle path optimization problem belongs to an NP (network processor) difficult problem, a solution method comprises an accurate method and a heuristic method, the heuristic method is a research hotspot at present, but the heuristic method usually needs a designer to have more complete professional knowledge and has stronger specificity; in addition, heuristics often do not yield optimal solutions.

The calculation amount of the conventional accurate method, such as a linear programming method, a dynamic programming method, a branch and bound method and the like, is exponentially increased along with the expansion of the problem scale, the accurate method can obtain an optimal solution, but the method is not suitable for solving a large-scale problem and has a complex solving process.

Disclosure of Invention

In order to overcome the defects of the prior art, the invention aims to provide a vehicle path optimization method based on a Petri network and integer linear programming, which can obtain the optimal path of a vehicle distribution path, and meanwhile, the obtained total distribution path has the shortest distance, thereby effectively reducing the vehicle distribution cost and having good application prospect.

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

a vehicle path optimization method based on a Petri network and integer linear programming comprises the following steps:

step one, establishing a mathematical model of a vehicle path problem according to the description of the vehicle path problem;

step two, establishing a Petri network model of the vehicle path problem based on the mathematical model in the step one;

step three, combining the Petri network model in the step two, converting the mathematical model in the step one into an integer linear programming problem;

step four, importing a program of the integer linear programming problem in the step three into the MATLAB, and inputting the distance between the client points and the cargo demand;

and step five, solving the integer linear programming problem in the step four by using a YALMIP optimization tool box, and carrying out experiment and result analysis.

The Petri network model established in the second step is as follows:

a)P＝{p₀,p₁,…,p_np denotes a set of libraries for customer sites and distribution centers, where the distribution center uses a library P₀Indicating that the customer site i uses the library p_iRepresents, i ═ 1.., n;

b) h represents the maximum moving step number of the vehicle, and the vehicle is regarded as moving one step when visiting one point;

c)M_kh＝[M_kh(p₀)，M_kh(p₁)，…，M_kh(p_n)]representing the location identity of the Petri net. If the position of the vehicle k at the h-th step is a point i (i ═ 0, 1.. times.n), then M is determined_kh(p_i) 1, otherwise M_kh(p_i)＝0；

d) There is a path between the distribution center and any customer i, respectively using transition t_0iAnd transition t_i0Indicating that the vehicle departs from the distribution center point to the customer point i, and the vehicle departs from the customer point i to the distribution center;

e) for any two client points i and j, i ≠ j, using transition t_jiAnd transition t_ijIndicating that the vehicle departs from customer point j to customer point i, and that the vehicle departs from customer point i to customer point j;

f)

a Pre-correlation matrix representing the library and the transition, if the transition t is the output transition of the library p, namely the transition t is a route sent from the library p, then Pre (p, t) is 1, otherwise Pre (p, t) is 0;

g)

a Post-correlation matrix representing the library and the transition, wherein if the transition t is the input transition of the library p, namely the destination pointed by the transition t is the library p, the Post (p, t) is 1, otherwise, the Post (p, t) is 0;

h)ν_ijrepresenting the cargo vector of the Petri net, the cargo capacity of the vehicle increased from point i to point j, i, j being 0,1_ij＝q_j；

i)Θ_khij＝[θ_kh01，θ_kh02，...，θ_khij，...，θ_kHnn-1]，Θ_khijRepresenting a path vector of the Petri network, if the vehicle k sends a sending access j point at the h-th step through the point i, i is not equal to j, then theta_khij1, otherwise θ_khij＝0。

The integer linear programming problem model established in the third step is as follows:

the objective function is: min f ═ Σ_k∑_hw_i，jΘ_khij

The constraint conditions are as follows:

constraint 1: m_kh＝M_kh-1+(Post-Pre)×Θ_khij

Constraint 2: m_kh-1-Pre×Θ_khij≥0

Constraint 3: sigma_kM_k0(p₀)＝∑_kM_kH(p₀)＝K

Constraint 4: v is_ij∑_hΘ_khij≤Q

Constraint 5: sigma_k∑_hM_kh(p_c)＝1

Constraint 6: k is 1,2, …, K; h is 1,2, …, H; i, j ≠ 0,1, …, n, i ≠ j;

c＝1，…，n；

wherein the variables are defined as follows:

k is the set of vehicles for recall, K ═ {1,2,3, …, K };

h is the number of steps the vehicle moves, H is the number of steps the vehicle moves at most, and H is {1,2,3, …, H }; i, j denotes a customer point or a distribution center, i ═ j ═ 0,1, …, n };

a Pre-correlation matrix representing the library and the transition, if the transition t is the output transition of the library p, then Pre (p, t) is 1, otherwise Pre (p, t) is 0;

a Post-incidence matrix representing the library and the transition, wherein if the transition t is the input transition of the library p, the Post (p, t) is 1, otherwise, the Post (p, t) is 0;

Θ_khij＝[θ_kh01，θ_kh02，...，θ_khij，...，θ_kHnn-1]a path vector representing a Petri net; if the vehicle k sends an access j point (i ≠ j) through the i point in the h step, then theta is_khij1, otherwise θ_khij＝0； M_kh＝[M_kh(p₀)，M_kh(p₁)，...，M_kh(p_n)]A location identifier representing a Petri net; if the position of the vehicle k at the h-th step is a point i (i ═ 0, 1.. times.n), then M is determined_kh(p_i) 1, otherwise M_kh(p_i)＝ 0；

ν_ijRepresenting the cargo vector of the Petri net, the cargo capacity of the vehicle increased from point i to point j, i, j being 0,1_ij＝q_j。

The invention has the beneficial effects that:

the vehicle path optimization method based on the Petri network and the integer linear programming is characterized in that a corresponding mathematical model and a Petri network model are established according to a vehicle distribution path problem, then the mathematical model and the Petri network model are combined to obtain an integer linear programming problem, and finally an MATLAB is used for solving the problem to find an optimal distribution path of a vehicle. The method can quickly find the optimal route of the vehicle distribution route, and meanwhile, the obtained total distribution route is shortest in distance, so that the vehicle distribution cost is effectively reduced, and the method has a good application prospect.

Drawings

FIG. 1 is a schematic flow diagram of the present invention.

Detailed Description

The present invention will be described in further detail below with reference to the accompanying drawings.

As shown in fig. 1, a vehicle path optimization method based on a Petri net and integer linear programming includes the following steps:

step one, establishing a mathematical model of a vehicle path problem according to the description of the vehicle path problem;

the method comprises the following steps that a distribution center and n customer points are shared in a distribution network, wherein 0 represents the distribution center, and each customer point is i, i is 1, …, n; knowing the location of each customer site i and the demand q_iThe distribution center reaches all customer points from the distribution center by K vehicle pairs at most, each vehicle starts from the distribution center and finally returns to the distribution center, the maximum cargo capacity of each vehicle is Q, and the distance between customers i and j is w_ijI ≠ j, the optimization aims at the shortest total distance of the distribution paths of all vehicles, constraint conditions for establishing a mathematical model are added, and relevant signs and variables of the mathematical model are defined as follows:

q_idemand at customer point i;

q maximum load capacity per vehicle;

k is the set of vehicles available for recall, K ═ {1,2,3, …, K };

k, the number of available vehicles is the same, and all vehicles have the same load capacity;

the mathematical model is as follows:

min f＝∑_i∑_j∑_kw_i，jx_ijk (1)

s.t.∑_iq_iy_ki≤Q，k＝1，2，...，K (2)

∑_ky_ki＝1，i＝1，2，...，n (3)

∑_ky_k0＝K (4)

∑_ix_ijk＝y_ki，j＝1，2，...，n (5)

∑_jx_ijk＝y_ki，i＝1，2，…，n (6)

x_ijk∈{0，1}，i，j＝0，1，...，n；k＝1，2，...，K

y_ki∈{0，1}，i＝0，1，...，n；k＝1，2，...，K

in the mathematical model, the formula (1) means that the total distance of the vehicle distribution route is shortest; equation (2) is a vehicle loadability constraint, i.e., the delivery volume of k on each delivery route cannot exceed its maximum payload; equation (3) ensures that each customer will be serviced by only one vehicle; formula (4) ensures that vehicles start from the distribution center and return to the distribution center; equations (5), (6) ensure that if customer point i, j is on the travel path of vehicle k, then customer point i, j will be serviced by vehicle k;

the mathematical model in the first step comprises the following constraints:

a) the starting point and the end point of each path are distribution centers;

b) each customer can only be serviced by a vehicle once;

c) the sum of all customer demands in each path cannot exceed the vehicle's payload Q;

step two, establishing a Petri network model of the vehicle path problem based on the mathematical model in the step one;

determining a library set, a transition set, a position identifier, a pre-incidence matrix, a post-incidence matrix and a path vector of the Petri network according to the mathematical model established in the step one;

the Petri net model is as follows:

a)P＝{p₀,p₁,…,p_np denotes a set of libraries for customer sites and distribution centers, where the distribution center uses a library P₀Indicating that the customer site i uses the library p_iRepresents, i ═ 1.., n;

b) h represents the maximum moving step number of the vehicle, and the vehicle is regarded as moving one step when visiting one point;

c)M_kh＝[M_kh(p₀)，M_kh(p₁)，…，M_kh(p_n)]representing the location identity of the Petri net. If the position of the vehicle k at the h-th step is a point i (i ═ 0, 1.. times.n), then M is determined_kh(p_i) 1, otherwise M_kh(p_i)＝0；

d) There is a path between the distribution center and any customer i, respectively using transition t_0iAnd transition t_i0Indicating vehicle toStarting from the distribution center point to a customer point i, and starting from the customer point i to the distribution center by the vehicle;

e) for any two client points i and j, i ≠ j, using transition t_jiAnd transition t_ijIndicating that the vehicle departs from customer point j to customer point i, and that the vehicle departs from customer point i to customer point j;

f)

a Pre-correlation matrix representing the library and the transition, if the transition t is the output transition of the library p, namely the transition t is a route sent from the library p, then Pre (p, t) is 1, otherwise Pre (p, t) is 0;

g)

a Post-correlation matrix representing the library and the transition, wherein if the transition t is the input transition of the library p, namely the destination pointed by the transition t is the library p, the Post (p, t) is 1, otherwise, the Post (p, t) is 0;

h)ν_ijrepresenting the cargo vector of the Petri net, the cargo capacity of the vehicle increased from point i to point j, i, j being 0,1_ij＝q_j；

i)Θ_khij＝[θ_kh01，θ_kh02，...，θ_khij，...，θ_kHnn-1]，Θ_khijRepresenting a path vector of the Petri network, if the vehicle k sends a sending access j point at the h-th step through the point i, i is not equal to j, then theta_khij1, otherwise θ_khij＝0；

Step three, combining the Petri network model in the step two, converting the mathematical model in the step one into an integer linear programming problem;

converting the constraints in the mathematical model established in the step one into the following linear constraints by combining the Petri network model in the step two;

equation (2) is expressed as:

ν_ij∑_hΘ_khij≤Q，k＝1，2，…，K (7)

v in the formula (7)_ijIndicates that the vehicle is moving from point i to point j (i, j ≠ 0.., n, i ≠ g)j) The increased cargo capacity, Q represents the maximum cargo capacity of each vehicle, and the formula (7) represents that each vehicle k cannot exceed the maximum cargo capacity from the 1 st step to the H th step in the process of executing the distribution task;

equation (3) is expressed as:

∑_k∑_hM_kh(p_c)＝1，c＝1，2，…，n (8)

m in formula (8)_kh(p_c) The position information of the vehicle k at the H-th step is represented, H is 1, and if the position of the vehicle k at the H-th step is a customer point c, c is 1, 1_kh(p_c) 1, otherwise M_kh(p_c) When the value is 0, equation (8) indicates that for the customer point c ∈ {1,2, …, n }, there is one and only one vehicle to deliver service to the customer point c;

equation (4) is expressed as:

∑_kM_k0(p₀)＝∑_kM_kH(p₀)＝K (9)

the start and end of all vehicles in equation (9) are at the distribution center p₀Point;

the integer linear programming problem model is as follows:

the objective function is: min f ═ Σ_kΣ_hw_i，jΘ_khij

The constraint conditions are as follows:

constraint 1: m_kh＝M_kh-1+(Post-Pre)×Θ_khij

Constraint 2: m_kh-1-Pre×Θ_khij≥0

Constraint 3: sigma_kM_k0(p₀)＝Σ_kM_kH(p₀)＝K

Constraint 4: v is_ij∑_hΘ_khij≤Q

Constraint 5: sigma_k∑_hM_kh(p_c)＝1

Constraint 6: k is 1,2, …, K; h is 1,2, …, H; i, j ≠ 0,1, …, n, i ≠ j;

c＝1，…，n；

wherein the variables are defined as follows:

k is the set of vehicles for recall, K ═ {1,2,3, …, K };

h is the number of steps the vehicle moves, H is the number of steps the vehicle moves at most, and H is {1,2,3, …, H };

i, j denotes a customer point or a distribution center, i ═ j ═ 0,1, …, n };

a Pre-correlation matrix representing the library and the transition, if the transition t is the output transition of the library p, then Pre (p, t) is 1, otherwise Pre (p, t) is 0;

a Post-incidence matrix representing the library and the transition, wherein if the transition t is the input transition of the library p, the Post (p, t) is 1, otherwise, the Post (p, t) is 0;

Θ_khij＝[θ_kh01，θ_kh02，...，θ_khij，...，θ_kHnn-1]a path vector representing a Petri net; if the vehicle k sends an access j point (i ≠ j) through the i point in the h step, then theta is_khij1, otherwise θ_khij＝0； M_kh＝[M_kh(p₀)，M_kh(p₁)，...，M_kh(p_n)]A location identifier representing a Petri net; if the position of the vehicle k at the h-th step is a point i (i ═ 0, 1.. times.n), then M is determined_kh(p_i) 1, otherwise M_kh(p_i)＝ 0；

ν_ijRepresenting the cargo vector of the Petri net, the cargo capacity of the vehicle increased from point i to point j, i, j being 0,1_ij＝q_j；

The target function represents that the total distance of vehicle distribution is shortest, and the constraint condition 1 and the constraint condition 2 ensure the correctness of each step in the task execution process of the vehicle; constraint 3 indicates that the starting and ending positions of the vehicle are both at the distribution center; constraint 4 indicates that each vehicle cannot exceed its maximum cargo capacity during the delivery task; constraint 5 indicates that there is one and only one vehicle to deliver service c to customer point c, 1.. multidot.n;

step four, importing a program of the integer linear programming problem in the step three into the MATLAB, and inputting the distance between the client points, the cargo demand and the like;

in this embodiment, 22 data randomly selected from a vehicle route optimization problem (VRP) international standard data sample att are used as experimental solution examples, as shown in table 1:

TABLE 1 all customer site location and demand

Numbering	X	Y	Demand volume	Numbering	X	Y	Demand volume
								0	4.5	2.0	0	12	6.3	2.9	19
1	2.7	1.5	10	13	6.8	2.7	23
								2	5.5	2.4	7	14	5.6	2.0	20
3	3.0	1.4	13	15	3.3	2.7	8
								4	5.0	1.8	19	16	5.1	1.7	19
5	3.1	1.6	26	17	7.6	3.2	2
								6	5.6	2.5	3	18	4.5	3.6	12
7	3.6	1.7	5	19	5.7	2.7	17
								8	6.3	1.3	9	20	1.9	2.6	9
9	6.9	1.9	16	21	5.5	1.4	11
								10	1.1	2.0	16	22	3.1	1.1	18
11	5.5	2.6	12

Vehicle path problem analysis of the present embodiment:

the number of clients to be distributed is set to n as 22, and the plane coordinate of the client i is (x)_i,y_i),w_i，j(i ≠ j) represents the euclidean distance between clients i and j, the maximum load of the vehicle is Q ═ 200 units, the available vehicle K is 4, the coordinates of the o point of the distribution center are (4.5, 2.0), the distance is km, and one decimal is reserved;

according to the Petri network model established in the step two, establishing a Petri network model with two mutually communicated clients (distribution centers), wherein the Petri network model has 23 places and 506 transitions; and according to the integer linear programming problem constraint condition 6 established in the step three, K is 4, H is 8, and n is 22:

step five, solving the integer linear programming problem in the step four by using a YALMIP optimization tool box, and carrying out experiment and result analysis:

vehicle 1 motion trajectory: p is a radical of₀(distribution center) → p₂(client Point 2) → p₁(customer Point 1) → p₄(client Point 4) → p₀(distribution center);

vehicle 2 motion trajectory: p is a radical of₀(distribution center) → p₁₈(client Point 18) → p₁₂(client Point 12) → p₁₃(client Point 13) → p₁₇(client Point 17) → p₉(client Point 9) → p₈(client Point 8) → p₂₁(client Point 21) → p₀(distribution center);

vehicle 3 motion trajectory: p is a radical of₀(distribution center) → p₁₅(client Point 15) → p₂₀(client Point 20) → p₁₀(client Point 10) → p₂₂(client Point 22) → p₃(client Point 3) → p₅(client Point 5) → p₇(client Point 7) → p₀(distribution center);

movement locus of vehicle 4: p is a radical of₀(distribution center) → p₁₁(customer Point 11) → p₁₉(customer site 19) → p₆(client Point 6) → p₁₄(client Point 14) → p₁₆(client Point 16) → p₀(distribution center);

the results were analyzed as follows: all customer sites have and only one vehicle to perform the corresponding distribution task, the actual cargo capacity of the vehicle 1 is 36 units, the actual cargo capacity of the vehicle 2 is 92 units, the actual cargo capacity of the vehicle 3 is 95 units, the actual cargo capacity of the vehicle 4 is 71 units, the maximum cargo capacity of the vehicle is not exceeded, and the sum of the path tracks of the vehicles is 23.89km in the shortest.

Claims (2)

1. A vehicle path optimization method based on a Petri network and integer linear programming is characterized by comprising the following steps:

step one, establishing a mathematical model of a vehicle path problem according to the description of the vehicle path problem;

step two, establishing a Petri network model of the vehicle path problem based on the mathematical model in the step one;

step three, combining the Petri network model in the step two, converting the mathematical model in the step one into an integer linear programming problem;

step four, importing a program of the integer linear programming problem in the step three into the MATLAB, and inputting the distance between the client points and the cargo demand;

step five, solving the integer linear programming problem in the step four by using a YALMIP optimization tool box, and carrying out experiment and result analysis;

the Petri network model established in the second step is as follows:

a)P＝{p₀,p₁,…,p_np denotes a set of libraries for customer sites and distribution centers, where the distribution center uses a library P₀Indicating that the customer site i uses the library p_iRepresents, i ═ 1.., n;

b) h represents the maximum moving step number of the vehicle, and the vehicle is regarded as moving one step when visiting one point;

c)M_kh＝[M_kh(p₀),M_kh(p₁),…,M_kh(p_n)]a position identifier representing a Petri net, wherein if the position of the vehicle k at the h step is a point i, i is 0,1_kh(p_i) 1, otherwise M_kh(p_i)＝0；

d) There is a path between the distribution center and any customer i, respectively using transition t_0iAnd transition t_i0Indicating that the vehicle departs from the distribution center point to the customer point i, and the vehicle departs from the customer point i to the distribution center point;

e) for any two client points i and j, i ≠ j, using transition t_jiAnd transition t_ijIndicating that the vehicle departs from customer point j to customer point i, and that the vehicle departs from customer point i to customer point j;

f)

a Pre-correlation matrix representing the library and the transition, if the transition t is the output transition of the library p, namely the transition t is a route sent from the library p, then Pre (p, t) is 1, otherwise Pre (p, t) is 0;

g)

a Post-correlation matrix representing the library and the transition, wherein if the transition t is the input transition of the library p, namely the destination pointed by the transition t is the library p, the Post (p, t) is 1, otherwise, the Post (p, t) is 0;

h)ν_ijrepresenting the cargo vector of the Petri net, the cargo capacity of the vehicle increased from point i to point j, i, j being 0,1_ij＝q_j，q_jIs the demand at customer point j;

i)Θ_khij＝[θ_kh01，θ_kh02，...，θ_khij，...，θ_kHnn-1]，Θ_khijrepresenting a path vector of the Petri network, if the vehicle k sends a sending access j point at the h-th step through the point i, i is not equal to j, then theta_khij1, otherwise θ_khij＝0。

2. The Petri net and integer linear programming based vehicle path optimization method according to claim 1, wherein the Petri net and integer linear programming based vehicle path optimization method comprises the following steps: the integer linear programming problem model established in the third step is as follows:

the objective function is: min f ═ Σ_k∑_hw_i，jΘ_khij,w_i,jIs the distance between clients i, j;

the constraint conditions are as follows:

constraint 1: m_kh＝M_kh-1+(Post-Pre)×Θ_khij

Constraint 2: m_kh-1-Pre×Θ_khij≥0

Constraint 3: sigma_kM_k0(p₀)＝∑_kM_kH(p₀)＝K

Constraint 4: v is_ij∑_hΘ_khijQ is less than or equal to Q, and Q represents the maximum cargo capacity of each vehicle;

constraint 5: sigma_k∑_hM_kh(p_c)＝1，M_kh(p_c) The method includes the steps that position information of a vehicle k in the H-th step is represented, H is 1, and if the position of the vehicle k in the H-th step is a customer point c, c is 1, n, M is the position of the vehicle k in the H-th step_kh(p_c) 1, otherwise M_kh(p_c)＝0；

Constraint 6: k is 1,2, …, K; h is 1,2, …, H; i, j ≠ 0,1, …, n, i ≠ j; c is 1, …, n;

wherein the variables are defined as follows:

k is the set of vehicles for recall, K ═ {1,2,3, …, K };

h is the number of steps the vehicle moves, H is the number of steps the vehicle moves at most, and H is {1,2,3, …, H };

i, j denotes a customer point or a distribution center point, i ═ 0,1, …, n, and j ═ 0,1, …, n.

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