An integrated optimization method for process planning and resource scheduling of batch tasks

Abstract

The present invention belongs to the field of resource allocation technology, and specifically relates to a process planning and resource scheduling integrated optimization method for batch tasks, comprising the following steps: Step 1: describing the process flow, optional operations, and operation time, as follows: a resource allocation system, such as a manufacturing module and a logistics module, can simultaneously execute multiple types of tasks J = {J1, J2, ..., J q}, resource set R = {R1, R2, …, R k}, where each task J i It can be divided into different steps#imgabs0# which can be expressed as follows:#imgabs1# Specifically, each step S i,j It can be composed of different process routes, recorded as: #imgabs2# That is, step S i,j There is i,j Process Routing (u i,j =|S i,j |), process routing L i,j,u It can be expressed as different operation sequences. The present invention uses the transition delay S 3 PR models and analyzes the resource allocation system, plans the process and resource scheduling scheme with the goal of minimizing the task completion time.

Description

Technical Field

[0001] The present invention belongs to the technical field of resource configuration, and in particular relates to a process planning and resource scheduling integrated optimization method for batch tasks. Background Art

[0002] Systems involving hardware or software resources (e.g., robots, machines, computing resources, etc.) such as manufacturing modules, logistics and transportation systems, and multi-threaded software systems can all be abstracted as resource allocation systems. In the research related to resource allocation systems, process planning and resource scheduling are core technologies that aim to achieve higher system throughput by optimizing parameters such as the number of resources and the order of processes.

[0003] Process planning focuses on selecting and sequencing production operations without considering the quantity or status of resources, while resource scheduling problems require allocating resources to selected operations. The integrated optimization problem of process planning and scheduling, called Integrated Process Planning and Scheduling, IPPS, has better system performance than individual optimization and has been widely studied in recent years.

[0004] IPPS solution methods can be divided into two types: exact methods and inexact methods, as follows:

[0005] For the former, branch-constrained methods, branch-and-cut procedures, and mixed-integer linear programming problems (MILPPs) are common techniques, among which MILPPs is the most widely used. However, these exact methods can only handle single-batch tasks, and it becomes complicated when considering batch tasks;

[0006] Therefore, imprecise algorithms such as heuristic algorithms and intelligent algorithms are gradually applied. In essence, heuristic algorithms are optimized based on the search direction guided by rules summarized from actual process planning and scheduling. The efficiency and diversity of intelligent optimization algorithms in global optimization make them more popular than heuristic techniques. Some new methods (such as machine learning, deep learning, and reinforcement learning) are also being tried. These algorithms can provide approximations in a very short time, and the results are very dependent on the encoding, decoding, and operations used.

[0007] Existing research focuses more on new approximate solution algorithms without the need to establish mathematical models. For example, the Redis method proposed from the data cache level divides different job channels to allocate resources, but the resource coupling relationship between batch tasks is more complex and the processing time is longer, which will be a great challenge. How to model and analyze the resource allocation system and realize the process planning of batch tasks and the optimal planning of resource scheduling integration has become an urgent problem to be solved. To this end, we propose a method that uses the transition delay S 3 PR models and analyzes the resource allocation system, with the goal of minimizing task completion time, and integrates optimization methods for process planning and resource scheduling of batch tasks. Summary of the invention

[0008] The purpose of the present invention is to provide a process planning and resource scheduling integrated optimization method for batch tasks, which can utilize the transition delay S 3 PR models and analyzes the resource allocation system, plans the process and resource scheduling scheme with the goal of minimizing the task completion time.

[0009] The technical solution adopted by the present invention is as follows:

[0010] A method for integrated optimization of process planning and resource scheduling of batch tasks, comprising the following steps:

[0011] Step 1: Describe the process flow, available operations, and operation time as follows:

[0012] Resource allocation systems, such as manufacturing modules and logistics modules, can simultaneously execute multiple types of tasks J = {J1, J2, ..., J q}, resource set R = {R1, R2, …, R k}, where each task J i Can be divided into different steps It can be expressed as follows:

[0013]

[0014] Specifically, each step S i,j It can be composed of different process routes, recorded as:

[0015]

[0016] That is, step S i,j There is i,j Process Routing (u i,j =|S i,j |), process routing L i,j,u It can be expressed as different sequences of operations, recorded as:

[0017]

[0018] Each operation requires specifying a resource in Represents a mapping from operations O to resources R; operations that require the same resources compete for resources, such as machine tools, automated guided vehicles, robots, etc.

[0019] Step 2: Establish the transition delay S based on the description of tasks, steps, routes, operations, and resources 3 PR model;

[0020] Among them, the transition delay S 3 PR Model Defined as a set of subnets A combination of and They are the starting place, the ending place, the activity place and the resource place; the description of tasks, steps, routes, operations and resources. The transition delay S is established according to Algorithm 1 in Table A as follows 3 PR model;

[0021] Table A Algorithm 1

[0022]

[0023]

[0024] Step 3: Establish a mixed integer linear programming model that integrates process planning and resource scheduling, as follows:

[0025] 3.1) Objective function:

[0026] This study aims to reasonably determine the process path and manage resources to minimize the task completion time. The optimization objectives are as follows:

[0027] min z (1)

[0028] 3.2) State transition constraints:

[0029] The state of the resource allocation system can be expressed using the transition delay S 3 The finite tag sequence M0,M1,…,M of the PR model h To express:

[0030]

[0031] Formula (2) ensures the correctness of state transition, where σ i =[σ i (t1),...,σ i (t m )] T is the emission count vector of the i-th stage; in other words, σi (t) ≥ 1 means executing the operation corresponding to transition t and occupying the corresponding resources;

[0032] 3.3) Task completion time constraints:

[0033] Let the emission completion vector τ i =[τ i (t1),τ i (t2),…,τ i (t m )] T represents the emission completion time of the i-th stage transition, where τ i (t) = 0 means that transition t is not enabled in this stage, and the global clock starts from 0; the maximum completion time is defined as the maximum completion time of all tasks, that is, the maximum value of the maximum emission completion time of all transitions, which can be expressed as:

[0034]

[0035] Considering the objective function (2), the above nonlinear constraints can be transformed into:

[0036]

[0037] 3.4) Transition emission completion time constraint:

[0038] Given a transition delay S 3 PR model N =<P,T,F,δ> , can be divided into process subnet N o = <P O ,T,F O ,δ> and resource subnet N r = <P R ,T,F R ,δ>, where P O =P B ∪P E ∪P A , P R =P R ;

[0039] Given a transition delay S 3 PR model N =<P,T,F,δ> , if and only if transition t is in subnet N o and N r When both are enabled, the transition t has a transition delay S 3 PR is enabled; in other words, the transition sets for the two subnets are the same, and the enabling of the transitions is subject to the common constraints of the two subnets; then the process subnet N o and resource subnet N r The transition constraints in are defined as process constraints and resource constraints respectively;

[0040] Since no resources have been used before stage 1, the first emission completion vector τ1 is expressed as:

[0041] τ1=σ1⊙δ (4)

[0042] Among them, ⊙ represents the Hadamard product, that is,

[0043] Starting from phase 2, the emission completion time of transitions will be subject to two constraints, as follows:

[0044] First, consider the process constraints:

[0045] Defined as the set of initial transitions, transition t∈T B The enablement is limited only by resource constraints; the transfer between processes is determined by the matrix Indicates that if the transition t of stage i j Changes in stage i′ After execution,

[0046] Therefore, the following constraints need to be satisfied:

[0047]

[0048] The above formula can be expressed as:

[0049]

[0050] Where D(x) is defined as the sum of a matrix x or a vector x;

[0051] In order to satisfy the process constraints, the transition t j′ Needs to be satisfied Right now,

[0052]

[0053] Among them, G o ∈{0,1} m×m It is defined as a transition-predecessor-transition matrix, where:

[0054]

[0055] In addition, the following constraints need to be met:

[0056]

[0057] Based on the above constraints, subnet N o Medium Change j The completion time constraint can be expressed as:

[0058]

[0059] Furthermore, the above constraints can be transformed into the following linear constraints:

[0060]

[0061] in, τ=[τ1,τ2,…,τ h ], H is a very large number;

[0062] Second, consider resource constraints:

[0063] Since the number of resources may be greater than 1, transition t is only valid if every resource is used. j You need to use the transition t j′ The resources released in the i-th stage transition t j With other changes k The relationship between i,j ∈{0,1} m×1 Indicates; if the transition t k Launched in stage i, and the launch time is no later than t j , then y i,j (k)=1, otherwise y i,j (k) = 0, that is,

[0064]

[0065] The above formula can be further transformed into the following linear constraint:

[0066]

[0067] Among them, ε is a very small number, and H is a very large number;

[0068] Therefore, when the transition t j When the trigger starts, all triggered transitions are represented as The corresponding resources used The cumulative number is described as follows:

[0069]

[0070] Furthermore, the above constraints can be transformed into the following linear constraints:

[0071]

[0072] Among them, G r ∈{0,1} k×m The transition-resource relationship matrix representing the relationship between resources and transitions is:

[0073]

[0074] Assume that the transition t in stage i j Use the transition t of stage i′ j′ Released resources, using matrix Indicates; if in the transition t j All resources when triggered is used, a transition t must be selected j′ ,Right now:

[0075]

[0076] in, For the quantity of each resource, it can be expressed as:

[0077]

[0078] Furthermore, the above constraints can be transformed into the following linear constraints:

[0079]

[0080] Among them, ε is a very small number, and H is a very large number;

[0081] In addition, the following constraints need to be met:

[0082]

[0083] in, is a row vector with all elements equal to 1;

[0084] Based on the above constraints, subnet N r Medium Change j The completion time constraint can be expressed as:

[0085]

[0086] The above formula can be further transformed into the following linear constraint:

[0087]

[0088] 3.5) Integrate the mixed integer linear programming model:

[0089] Through equations (1)-(13), the mixed integer linear programming model can be described as follows:

[0090]

[0091] Step 4: Use the Gurobi optimization toolbox in MATLAB to solve the optimal solution of the linear model and conduct experimental analysis.

[0092] The technical effects achieved by the present invention are:

[0093] The present invention provides an integrated optimization method for process planning and resource scheduling of batch tasks, which can select process operations and reasonably schedule resources for different batch tasks according to the workflow to minimize the task completion time, making up for the defects of existing precise mathematical models and providing theoretical support for non-precise algorithms, planning processes and resource scheduling solutions.

[0094] The present invention provides an integrated optimization method for process planning and resource scheduling of batch tasks, which realizes a resource allocation system such as a manufacturing module and a logistics module through the Internet of Things technology. It can simultaneously execute multiple types of tasks and complete resource collection. It is suitable for process planning and resource scheduling of batch tasks to cope with the challenges brought about by more complex resource coupling relationships between batch tasks.

[0095] The present invention provides a batch task process planning and resource scheduling integrated optimization method, given the description of the task, step, route, operation, resource, according to Algorithm 1 to establish the transition delay S 3 The PR model,considers these factors, meets the needs of handling various emergencies,on the batch task site, effectively reduces concurrency, and prolongs the working time of this,method.

[0096] The process planning and resource scheduling integrated optimization method for batch tasks of the present invention takes process constraints and resource constraints into consideration, and can refine the precise mathematical model to make the planning of process and resource scheduling schemes more accurate. BRIEF DESCRIPTION OF THE DRAWINGS

[0097] Figure 1 It is a flow chart of a method for integrated optimization of process planning and resource scheduling of batch tasks of the present invention;

[0098] Figure 2 is a flow chart of two tasks of an embodiment of the present invention;

[0099] Figure 3 is the transition delay S of the present invention 3 Schematic diagram of the PR model;

[0100] Figure 4 It is a statistical diagram of the process planning and resource scheduling results of the present invention. DETAILED DESCRIPTION

[0101] In order to make the purpose and advantages of the present invention more clearly understood, the present invention is specifically described below in conjunction with embodiments. It should be understood that the following text is only used to describe one or several specific embodiments of the present invention, and does not strictly limit the scope of protection of the specific claims of the present invention.

[0102] like Figure 1-4 As shown, a method for integrated optimization of process planning and resource scheduling of batch tasks includes the following steps:

[0103] Step 1: Describe the process flow, available operations, and operation time as follows:

[0104] Resource allocation systems, such as manufacturing modules and logistics modules, can simultaneously execute multiple types of tasks J = {J1, J2, ..., J q}, resource set R = {R1, R2, ..., R k}, realized through the Internet of Things technology, suitable for process planning and resource scheduling of batch tasks, where each task J i Can be divided into different steps It can be expressed as follows:

[0105]

[0106] Specifically, each step S i,j It can be composed of different process routes, recorded as:

[0107]

[0108] That is, step S i,j There is i,j Process Routing (u i,j =|S i,j |), process routing L i,j,u It can be expressed as different sequences of operations, recorded as:

[0109]

[0110] Each operation requires specifying a resource in Represents a mapping from operations O to resources R; operations that require the same resources compete for resources, such as machine tools, automated guided vehicles, robots, etc.

[0111] In this embodiment, the system consists of two tasks J = {J1, J2} and six resources R = {R1, R2, C1 ..., C4}. The details of the process are as follows:

[0112]

[0113]

[0114] Step 2: Establish the transition delay S based on the description of tasks, steps, routes, operations, and resources 3 PR model;

[0115] Among them, the transition delay S 3 PR Model Defined as a set of subnets A combination of and They are the starting place, the ending place, the activity place and the resource place; the description of tasks, steps, routes, operations and resources. The transition delay S is established according to Algorithm 1 in Table A as follows 3 The PR model takes into account multiple factors to meet the needs of dealing with various emergencies on the batch task site, effectively reducing concurrency and extending the working time of this method;

[0116] Table A Algorithm 1

[0117]

[0118]

[0119] In this embodiment, if Figure 3 As shown, the transition delay S established according to Algorithm 1 3 PR model, where P B ={p1,p7},P E ={p6,p 12}, P A ={p2,p3,p4,p5,p8,p9,p 10 ,p 11}, P R ={p 13 ,p 14}, T={t1,t2,...,t 11}, the relationship between operation, transition and delay time is shown in Table B below:

[0120] Table B Relationship between operation, transition and delay time

[0121]

[0122] Step 3: Establish a mixed integer linear programming model that integrates process planning and resource scheduling, as follows:

[0123] 3.1) Objective function:

[0124] This study aims to reasonably determine the process path and manage resources to minimize the task completion time. The optimization objectives are as follows:

[0125] min z (1)

[0126] 3.2) State transition constraints:

[0127] The state of the resource allocation system can be expressed using the transition delay S 3 The finite tag sequence M0,M1,…,M of the PR model h To express:

[0128]

[0129] Formula (2) ensures the correctness of state transition, where σ i =[σ i (t1),...,σ i (t m )] T is the emission count vector of the i-th stage; in other words, σ i (t) ≥ 1 means executing the operation corresponding to transition t and occupying the corresponding resources;

[0130] 3.3) Task completion time constraints:

[0131] Let the emission completion vector τ i =[τ i (t1),τ i (t2),…,τ i (t m )] T represents the emission completion time of the i-th stage transition, where τ i (t) = 0 means that transition t is not enabled in this stage, and the global clock starts from 0; the maximum completion time is defined as the maximum completion time of all tasks, that is, the maximum value of the maximum emission completion time of all transitions, which can be expressed as:

[0132]

[0133] Considering the objective function (2), the above nonlinear constraints can be transformed into:

[0134]

[0135] 3.4) Transition emission completion time constraint:

[0136] Given a transition delay S 3 PR model N =<P,T,F,δ> , can be divided into process subnet N o = <P O ,T,F O ,δ> and resource subnet N r = <P R ,T,F R ,δ>, where P O =PB ∪P E ∪P A , P R =P R ;

[0137] Given a transition delay S 3 PR model N =<P,T,F,δ> , if and only if transition t is in subnet N o and N r When both are enabled, the transition t has a transition delay S 3 PR is enabled; in other words, the transition sets for the two subnets are the same, and the enabling of the transitions is subject to the common constraints of the two subnets; then the process subnet N o and resource subnet N r The transition constraints in are defined as process constraints and resource constraints respectively;

[0138] Since no resources have been used before stage 1, the first emission completion vector τ1 is expressed as:

[0139] τ1=σ1⊙δ (4)

[0140] Among them, ⊙ represents the Hadamard product, that is,

[0141] Starting from phase 2, the emission completion time of transitions will be subject to two constraints, as follows:

[0142] First, consider the process constraints:

[0143] Defined as the set of initial transitions, transition t∈T B The enablement is limited only by resource constraints; the transfer between processes is determined by the matrix Indicates that if the transition t of stage i j Changes in stage i′ After execution,

[0144] Therefore, the following constraints need to be satisfied:

[0145]

[0146] The above formula can be expressed as:

[0147]

[0148] Where D(x) is defined as the sum of a matrix x or a vector x;

[0149] In order to satisfy the process constraints, the transition t j′ Needs to be satisfied Right now,

[0150]

[0151] Among them, G o ∈{0,1} m×m It is defined as a transition-predecessor-transition matrix, where:

[0152]

[0153] In addition, the following constraints need to be met:

[0154]

[0155] Based on the above constraints, subnet N o Medium Change j The completion time constraint can be expressed as:

[0156]

[0157] Furthermore, the above constraints can be transformed into the following linear constraints:

[0158]

[0159] in, τ=[τ1,τ2,...,τ h ], H is a very large number;

[0160] Second, consider resource constraints:

[0161] Since the number of resources may be greater than 1, transition t is only valid if every resource is used. j You need to use the transition t j′ The resources released in the i-th stage transition t j With other changes k The relationship between i,j ∈{0,1} m×1 Indicates; if the transition t k Launched in stage i, and the launch time is no later than t j , then y i,j (k)=1, otherwise y i,j (k) = 0, that is,

[0162]

[0163] The above formula can be further transformed into the following linear constraint:

[0164]

[0165] Among them, ε is a very small number, and H is a very large number;

[0166] Therefore, when the transition t j When the trigger starts, all triggered transitions are represented as The corresponding resources used The cumulative number is described as follows:

[0167]

[0168] Furthermore, the above constraints can be transformed into the following linear constraints:

[0169]

[0170] Among them, G r ∈{0,1} k×m The transition-resource relationship matrix representing the relationship between resources and transitions is:

[0171]

[0172] Assume that the transition t in stage i j Use the transition t of stage i′ j′ Released resources, using matrix Indicates; if in the transition t j All resources when triggered is used, a transition t must be selected j′ ,Right now:

[0173]

[0174] in, For the quantity of each resource, it can be expressed as:

[0175]

[0176] Furthermore, the above constraints can be transformed into the following linear constraints:

[0177]

[0178] Among them, ε is a very small number, and H is a very large number;

[0179] In addition, the following constraints need to be met:

[0180]

[0181] in, is a row vector with all elements equal to 1;

[0182] Based on the above constraints, subnet N r Medium Change j The completion time constraint can be expressed as:

[0183]

[0184] The above formula can be further transformed into the following linear constraint:

[0185]

[0186] 3.5) Integrate the mixed integer linear programming model:

[0187] Through equations (1)-(13), the mixed integer linear programming model can be described as follows:

[0188]

[0189] Step 4: Use the Gurobi optimization toolbox in MATLAB to solve the optimal solution of the linear model and conduct experimental analysis.

[0190] In this embodiment, consider that the batch size of both tasks is 3, that is, both task Job1 and task Job2 need to be completed three times. The process planning and resource scheduling results are as follows: Figure 4 The results show that the first and third operations of Job1 are completed by resource C1, while the second operation is completed by resource C2. The time occupied by each resource is as follows: Figure 4 shown.

[0191] In summary, the present invention can select process operations for different batch tasks according to the workflow and reasonably schedule resources to minimize the task completion time, which makes up for the defects of the existing precise mathematical model and provides theoretical support for the imprecise algorithm.

[0192] The above is only a preferred embodiment of the present invention. It should be noted that, for those skilled in the art, several improvements and modifications can be made without departing from the principles of the present invention, and these improvements and modifications should also be considered as the protection scope of the present invention. The structures, devices and operating methods not specifically described and explained in the present invention shall be implemented according to the conventional means in the art unless otherwise specified and limited.

Claims

1. A method for integrated optimization of process planning and resource scheduling for batch tasks, characterized by: Includes steps: Step 1: Describe the process flow, available operations, and operation time; Step 2: Establish the transition delay S based on the description of tasks, steps, routes, operations, and resources 3 PR model; Step 3: Establish a mixed integer linear programming model integrating process planning and resource scheduling; Step 4: Use the Gurobi optimization toolbox in MATLAB to solve the optimal solution of the linear model and conduct experimental analysis; The step one comprises: A resource allocation system, wherein the resource allocation system simultaneously executes several types of tasks J={J1, J2, ..., J q }, resource set R = {R1, R2, ..., R k }, where each task J i Divide into different steps It is expressed as follows: Specifically, each step S i,j It consists of different process routes, recorded as: That is, step S i,j There is i,j Process Routing, Process Routing L i,j,u It is represented as a sequence of operations, recorded as: Furthermore, each operation specifies a resource in Represents the mapping from operation O to resource R; The resource allocation system includes a manufacturing module for performing several types of tasks and a logistics module for resource aggregation; The transition delay S 3 The PR model is: and is defined as a set of subnets A combination of and Subnet Starting place, ending place, activity place and resource place; The step three comprises the following steps: Step 1: Establish the objective function; Step 2: state transition constraints; Step 3: Task completion time constraints; Step 4: Change the time constraint of the launch completion; Step 5: Integrate the mixed integer linear programming model; The objective function is: minz(1) is used as the optimization target; The state of the resource allocation system in step 2 is represented by the transition delay S 3 The finite tag sequence M0,M1,...,M of the PR model h To express it, as follows: Formula (2) ensures the correctness of state transition, where σ i =[σ i (t1),...,σ i (t m )] T is the emission count vector of the i-th stage; and, σ i (t) ≥ 1 means executing the operation corresponding to transition t and occupying the corresponding resources; In step 3, the emission completion vector τ is set i =[τ i (t1),τ i (t2),...,τ i (t m )] T represents the launch completion time of the i-th stage transition; where τ i (t) = 0 means that transition t is not enabled in this phase, and the global clock starts from 0; the maximum completion time is defined as the maximum completion time of all tasks, or the maximum value of the maximum emission completion time of all transitions, as follows: Considering the objective function (1), the above nonlinear constraints Translates to: In step 4, a transition delay S is set 3 PR model N =<P,T,F,δ> , divided into process subnet N o = <P O ,T,F O ,δ> and resource subnet N r = <P R ,T,F R ,δ>, where P O =P B ∪P E ∪P A ; Set a transition delay S 3 PR model N =<P,T,F,δ> , only when transition t is in subnet N o and N r When both are enabled, the transition t has a transition delay S 3 PR, so that the transition sets for the two subnets are the same, and the enabling of transitions is subject to the common constraints of the two subnets; then the process subnet N o and resource subnet N r The transition constraints in are defined as process constraints and resource constraints respectively; Since no resources have been used before stage 1, the first emission completion vector τ1 is expressed as: τ1=σ1⊙δ (4) Among them, ⊙ represents the Hadamard product; Starting from phase 2, the emission completion time of transitions will be subject to two constraints, as follows: First, consider the process constraints: Defined as the set of initial transitions, transition t∈T B The enablement is limited only by resource constraints; the transfer between processes is determined by the matrix Indicates that if the transition t of stage i j The transition t in the i′th stage j′ ∈ ·· t j After execution, Therefore, the following constraints need to be met: It is expressed as: Where D(x) is defined as the sum of a matrix x or a vector x; In order to satisfy the process constraints, the transition t j′ To meet t j′ ∈ ·· t j ,Right now: Among them, G o ∈{0,1} m×m It is defined as a transition-predecessor-transition matrix, where: In addition, the following constraints must be met: Subnet N o Medium Change j The completion time constraint is expressed as: Transformed into the following linear constraints: in, τ=[τ1,τ2,...,τ h ], H is a very large number; Second, consider resource constraints: When every resource is utilized, the transition t j Need to use the transition t j′ The resources released in the i-th stage transition t j With other changes k The relationship between i,j ∈{0,1} m×1 express; If the change t k Launched in stage i, and the launch time is no later than t j , then y i,j (k) = σ i (k), otherwise y i,j (k) = 0, that is, Transformed into the following linear constraints: Among them, ε is a very small number, and H is a very large number; Therefore, when the transition t j When the trigger starts, all triggered transitions are represented as The corresponding resources used The cumulative number is described as follows: Transformed into the following linear constraints: Among them, G r ∈{0,1} k×m The transition-resource relationship matrix representing the relationship between resources and transitions is: Assume that the transition t in stage i j Use the transition t of stage i′ j′ Released resources, using matrix Indicates; if in the transition t j All resources when triggered is used, a transition t must be selected j′ ,Right now: in, For the quantity of each resource, it is expressed as: Transformed into the following linear constraints: Among them, ε is a very small number, and H is a very large number; In addition, the following constraints are met: in, is a row vector with all elements equal to 1; Subnet N r Medium Change j The completion time constraint is expressed as: Transformed into the following linear constraints: In step 5, the mixed integer linear programming model is described as follows through equations (1)-(13):

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