CN110991917A

CN110991917A - Multi-mode resource-limited project scheduling optimization method adopting two-stage genetic algorithm

Info

Publication number: CN110991917A
Application number: CN201911259880.3A
Authority: CN
Inventors: 单晓杭; 王嘉梁; 李研彪; 张利; 叶必卿; 谢毅
Original assignee: Zhejiang University of Technology ZJUT
Current assignee: Zhejiang University of Technology ZJUT
Priority date: 2019-12-10
Filing date: 2019-12-10
Publication date: 2020-04-10
Anticipated expiration: 2039-12-10
Also published as: CN110991917B

Abstract

The invention discloses a multi-mode resource-limited project scheduling optimization method adopting a two-stage genetic algorithm, which comprises the following steps of: acquiring information required by scheduling; judging whether a feasible scheme exists; carrying out pretreatment; calculating a level value of the task; initializing a contemporary population; evolution is carried out in two stages: stage 1 adopts a hierarchical-based scheduling sequence cross variation operation and adopts an individual execution mode list generated based on the earliest task completion time and calculates the fitness value of the individual execution mode list, so that the algorithm is quickly converged near the optimal solution, stage 2 adopts the execution mode and scheduling sequence list cross variation operation and uses an FBI & D method to improve the population and calculate the fitness value, and the neighborhood expansion search is carried out to find the optimal solution; outputting a scheduling optimization result; compared with a single-stage search strategy, the method has higher search efficiency and optimizing capability.

Description

Multi-mode resource-limited project scheduling optimization method adopting two-stage genetic algorithm

Technical Field

The invention relates to the fields of computer technology, information technology and system engineering, in particular to a multi-mode resource-limited project scheduling optimization method, and more particularly relates to a multi-mode resource-limited project scheduling optimization method adopting a two-stage genetic algorithm.

Background

The Resource-Constrained Project scheduling problem RCPSP (Resource-Constrained Project scheduling Project) refers to how to scientifically and reasonably allocate resources and arrange the execution sequence of tasks to determine the start and completion time thereof under the constraint of satisfying the Resource and task timing relationship, so as to achieve the given objectives, such as: optimization of construction period, cost, etc. With the adoption of project-oriented organizational structures and management modes by more and more modern enterprises, the RCPSP has a strong engineering background and is widely applied to enterprises in single-piece or small-batch production modes such as construction engineering, software development, airplane and ship manufacturing and the like. The RCPSP can be further divided into a single-mode resource-limited project scheduling problem SRCPSP and a multi-mode resource-limited project scheduling problem MRCPSP, wherein each task in the SRCPSP only has one execution mode, the resource is renewable resource, and the construction period and the resource demand of the task are determined; the tasks in the MRCPSP have a plurality of execution modes, resources comprise renewable resources and non-renewable resources, and the construction period and the resource demand amount corresponding to each execution mode of the tasks are different. SRCPSP can therefore be considered a special case of MRCPSP, which is a more complex problem than SRCPSP, but is more of a practical significance.

The method for solving the RCPSP at the present stage mainly comprises the following steps: 1) the precise algorithm, such as: 0-1 integer programming and branch-and-bound; 2) heuristic algorithms, such as: heuristic algorithm, separation arc method and local search calculation based on priority rule; 3) intelligent algorithms, such as: genetic algorithm, particle swarm optimization algorithm, ant colony algorithm and the like. Although the accurate algorithm can obtain the optimal solution in theory, the calculation time is generally unacceptable and is generally only suitable for small-scale problems, the heuristic algorithm can quickly obtain the solution of the problem based on a certain heuristic rule but generally cannot ensure the quality of the solution, the intelligent algorithm is based on the calculation intelligence, the near-optimal solution or the satisfactory solution is found in reasonable time by adopting an improved heuristic rule and a search mode, and the efficiency generally depends on the design of the algorithm and the type of the problem.

Disclosure of Invention

In order to overcome the defects that the existing precise algorithm is long in calculation time and cannot be suitable for large-scale problems, and the heuristic algorithm, the search space imperfection combined with the heuristic semi-intelligent algorithm, the intelligent algorithm efficiency depend on the design of the algorithm and the type of the problems and the like, the invention provides a multi-mode resource-limited project scheduling optimization method adopting a two-stage genetic algorithm, so that the solving time of the resource-limited project scheduling problem is effectively reduced, and the solving quality is improved.

The technical scheme adopted by the invention for solving the technical problems is as follows: a multi-mode resource-limited project scheduling optimization method adopting a two-stage genetic algorithm comprises the following steps:

step 1: formalizing a multi-mode resource limited project scheduling problem and acquiring information required by scheduling optimization;

acquiring a project task set T ═ T₀,t₁,...,t_J,t_J+1}; wherein t is_jIndicating task j, i.e. task numbered j, t₀And t_J+1The method is a virtual task which is artificially increased, namely, the project construction period and the resource are not occupied, and J is the actual task quantity to be scheduled;

obtaining the time sequence relation between tasks, namely a parent task set PR of the task j_jAnd a set of subtasks SC_jTask j at its parent task j^-∈PR_jCannot be started until all the completion, J is 0,1, …, J +1, t₀Without a parent task, i.e.

t_J+1Without subtasks, i.e.

Obtaining the usable quantity R of renewable resources k at any time_kK is 1, …, K is the renewable resource type number; acquiring non-updatable resource l in whole project construction periodUsable amount of N in_lL ═ 1, …, L is the number of non-updateable resource types;

acquiring the time needed by the task j to execute in the mode m, namely the construction period d_j,mAnd the number r of occupied renewable resources k_j,m,kThe number n of non-updatable resources l_j,m,l；j＝1,…,J，m＝1,…,M_jK1, …, K, L1, …, L, wherein M is_jIs the number of modes available for task j to choose to employ;

step 2: if there is an un-updateable resource/satisfy

Or that there is one task j and renewable resources k satisfying

Then the problem has no feasible scheduling scheme, the solution is finished, otherwise, the step 3 is switched to;

and step 3: carrying out pretreatment: carrying out redundancy and invalidation check on the mode and the resource, and removing the redundancy mode, the invalidation mode and the redundancy resource;

step 3.1: let the pattern flag value mflg ═ 0, resource flag value rflg ═ 0; if r is present_j,m,k＞R_kOr n_j,m,l＞N_lThen mode m is invalid for task j, removing mode m in task j: order to

Wherein:

k1, …, K, L1, …, L; let M_j＝M_j-1；

Step 3.2: if mflg is equal to 0, then let mflg be equal to 1, go to step 3.3; otherwise go to step 3.4;

step 3.3: if there is task j, patterns m and m' satisfy: d_j,m′≤d_j,m，r_j,m′,_k≤r_j,m,k，n_j,m′,_l≤n_j,m,lM' ≠ m, K ≠ 1, …, K, L ═ 1, …, L; then pattern m is redundant for task j, removing pattern m in task j: order to

Wherein:

k1, …, K, L1, …, L; let M_j＝M_j-1，rflg＝0；

Step 3.4: if rflg is equal to 0, let rflg be equal to 1, go to step 3.5; otherwise go to step 3.7;

step 3.5: if the renewable resource k satisfies:

then the renewable resource is redundant, removing renewable resource k: order to

Wherein: j-1, …, J,

let K-1, mflg-0;

step 3.6: if the non-updateable resource/satisfies:

then the non-updateable resource is redundant, removing non-updateable resource/: order to

Wherein: j-1, …, J,

let L-1, mflg-0;

step 3.7: if mflg ═ 1 ^ rflg ═ 1, go to step 3.8, otherwise go to step 3.2;

step 3.8: finishing the pretreatment;

and 4, step 4: calculating the hierarchy value of the task:

for a starting task 0 without a parent task, its hierarchy value is:

lvl₀＝0 (1)

the hierarchy values of other tasks are calculated using the following recursive formula:

and 5: initializing a contemporary population;

generating N different individuals by an individual generation method based on the hierarchy and the earliest task completion time to form an initial contemporary population, wherein N is the population scale and is a positive even number;

the individual adopts 2J bit integer coding, and the method comprises the following steps: ch ═ g₁,…,g_J；g_J+1,…,g_2J}, gene g_jIs a positive integer, where g₁,…,g_JIs the execution mode list, g_jIndicates the mode, g, adopted by task j_j＝1,...,M_jJ — 1, …, J, for example: g₁2 indicates that task 1 employs mode 2; { g_J+1,…,g_2JIs a scheduling order list, g_J+1,…,g_2JIs an arrangement of 1, …, J and satisfies the timing constraint of a task, i.e. no task can be placed in front of its parent, g_J+jIndicating the jth scheduled task, i.e. task g_J+jIs the jth scheduled, e.g. g_J+13, the 1 st scheduled task is the task number 3;

the individual generation method based on the hierarchy and the earliest task completion time comprises the following steps:

step A1: randomly arranging and dividing t according to task hierarchy value from small to large₀And t_J+1Other tasks, i.e., those with smaller hierarchy value in the front of larger hierarchy value and those with the same hierarchy value are arranged randomly to form a gene { g } which is an individual scheduling order list_J+1,…,g_2J}；

Step A2: individual serial decoding method based on earliest completion time of task generates individual execution mode list namely gene { g₁,…,g_JObtaining a fitness value fit thereof;

step A3: outputting an individual ch ═ g₁,…,g_J,g_J+1,…,g_2JFourthly, the operation is finished;

the individual serial decoding method based on the earliest completion time of the task comprises the following steps:

step B1: initializing a system state: order completion time f of task 0₀0, F₀Lf, lf representing the latest completion time of the item:

the elements in F are arranged from small to large; making a cycle control variable delta equal to 1; let the remaining available amount of the renewable resource corresponding to each completion time in F:

step B2: fetch task j equals g_J+δLet mode m equal to 1;

step B3: calculating the completion time f of task j_jNamely:

step B3.1: let start time of task j

Let flag value flag be 1;

step B3.2: finding rt or more in F_jAnd form a set TF, i.e., TF ═ { F ∈ F | F ≧ rt_j}；

Step B3.3: take the element with the smallest value from TF, do not set to f_j′；

Step B3.4: if flag is 1 and f_j′-s_j,m≥d_j,mThen go to step B3.7; otherwise go to stepStep B3.5;

step B3.5: if flag is equal to 0, let s_j,m＝f_j′，flag＝1；

Step B3.6: judgment of f_j′At the moment, whether the remaining available amount of all renewable resources meets the requirement of the task j is determined, that is, whether all K is 1, … and K meets the requirement

If not, the flag is 0; go to step B3.3;

step B3.7: let f_j,m＝s_j,m+d_j,m；

Step B3.8: making m equal to m + 1; if M is less than or equal to M_jGo to step B3.1, otherwise from

Taking out with minimum time of completion

Order to

Go to step B4;

step B4: if f is_jIf F is absent, F is F ∪ { F ═ F_jIs increased by f_jRemaining available amount of renewable resources at the moment:

wherein: f. of_j″Is f_jThe previous time of day;

step B5: update [ s ]_j,f_j) Remaining available amount of renewable resources within the time period:

step B6: let δ be δ +1, go to step B2 if δ ≦ J, otherwise go to step B7;

step B7: obtain execution mode list g₁,…,g_J-calculating a fitness value fit of the individual,finishing the operation;

wherein:

representing the remaining available amount of resource k at time instant tau,

indicating the ready time of task j; step 6: carrying out N/2 times of scheduling sequence cross operation based on the hierarchy on the contemporary population to generate a new population, and carrying out scheduling sequence variation operation based on the hierarchy on each individual in the new population;

the scheduling order list interleaving operation based on the hierarchy comprises the following steps:

step C1: randomly selecting two different individuals from the contemporary population as a father 1 and a father 2 based on a ranking round-robin method;

step C2: randomly selecting a hierarchy without setting to be l;

step C3: exchanging the scheduling sequence of the task of the level l in the scheduling sequence list of the parent 1 and the scheduling sequence list of the parent 2, emptying the execution mode list, generating two children 1 and 2, and finishing the operation;

the hierarchical based scheduling order list mutation operation comprises the following steps:

step D1: generating a random number λ ∈ [0,1), if λ < p_mGo to step D2, otherwise go to step D4;

step D2: if the number of tasks in the layer is larger than 1, randomly selecting one of the layers, not setting the layer as l, and going to step D3; otherwise go to step D4;

step D3: randomly selecting two tasks from the layer l, and exchanging the two tasks in a scheduling sequence list;

step D4: finishing the operation of the variation of the scheduling sequence list based on the hierarchy;

wherein: p is a radical of_m∈(0,1]Is the rate of variation;

and 7: generating an individual execution mode list and calculating the fitness value of each individual in the new population by adopting an individual serial decoding method based on the earliest task completion time;

and 8: selecting N different individuals from the current generation population and the new population from good to bad to form a new current generation population;

and step 9: judging whether a first-stage iteration termination condition is met, if so, ending the first-stage evolution, and turning to the step 10, otherwise, turning to the step 6;

the first stage termination condition is that the iteration is carried out to a specified algebraic TG₁Or successive iterations GG₁No improvement was made for the best individuals;

step 10: performing N/2 times of cross operation of the execution mode list and the scheduling sequence list on the contemporary population to generate a new population, and performing variation operation on the execution mode list and the scheduling sequence list on each individual in the new population;

the operation of crossing the execution mode list and the scheduling order list comprises the following steps:

step E1: two different individuals were randomly selected from the contemporary population as father 1 and father 2 based on a ranking round-robin method, which is not set as:

the daughter 1 and daughter 2 generated were not:

step E2: if it is not

And

if not, go to step E3, otherwise

Go to step E8;

step E3: finding out from front to back

And

position delta of the 1 st different gene of (1)₁Finding out from back to front

And

position delta of the 1 st different gene of (1)₂(ii) a If delta₁≠δ₂Go to step E4, otherwise

Go to step E7;

step E4: randomly generating a delta₁To delta₂-a positive integer α of 1;

step E5: ch (channel)^c1The first α genes in the execution Pattern List part of (1) were from ch^p1The first α genes of the execution mode list part of (1), i.e.

The last J- α genes of the execution Pattern List part were from ch^p2The last J- α genes of the execution mode list part of (1), i.e.

Step E6: ch (channel)^c2The first α genes in the execution Pattern List part of (1) were from ch^p2The first α genes of the execution mode list part of (1), i.e.

The last J- α genes of the execution Pattern List part were from ch^p1The last J- α genes of the execution mode list part of (1), i.e.

Step E7: if it is not

And

if not, go to step E8; otherwise

Go to step E12;

step E8: finding out from front to back

And

position delta of the 1 st different gene of (1)₃From the back to the front

And

position delta of the 1 st different gene of (1)₄；

Step E9: randomly generating a delta₃To delta₄-a positive integer β of 1;

step E10: ch (channel)^c1The first β genes in the dispatch order list from ch^p1The first β genes of the scheduling order list, i.e.

The last J- β genes are from ch^p2Scheduling order list of

Mid-delete ch^p1Scheduling order ofThe first β genes of the Table

The latter gene list;

step E11: ch (channel)^c2The first β genes in the dispatch order list from ch^p2The first β genes of the scheduling order list, i.e.

The last J- β genes are from ch^p1Scheduling order list of

Mid-delete ch^p2The first β genes of the scheduling order list of

The latter gene list;

step E12: to obtain

Finishing the cross operation;

the execution mode list and scheduling order list mutation operation comprises the following steps:

step F1: generating a random number lambda₁E [0,1), if λ₁＜p_mGo to step F2; otherwise go to step F3;

step F2: from the execution mode list g₁,…,g_JRandomly selecting one gene, not setting as g_jJ is 1, …, J, and if m is not set, g is the execution mode selected for task J again randomly_j＝m；

Step F3: generating a random number lambda₂E [0,1), if λ₂＜p_mGo to step F4; otherwise go to step F5;

step F4: from the scheduling order list g_J+1,…,g_2JRandomly selecting one gene, not setting as g_jJ +1, …,2J if task g_jThere is a parent task other than task 0Then from g_jBegin finding the first parent g ahead_j′Let the position value pos₁J' +1, otherwise let pos₁J + 1; if task g_jThere are subtasks other than task J +1, then from g_jStarting to find the first subtask g backwards_j″Let the position value pos₂J "-1, otherwise let pos₂2J; in [ pos ]₁,pos₂]Randomly selects a position to insert g_j；

Step F5: finishing the mutation operation;

step 11: improving all individuals in the new population by adopting an FBI & D method and calculating the fitness value of the individuals;

the FBI & D method comprises the steps of:

step G1: making a counting variable theta equal to 1; performing serial individual decoding based on insertion mode on individual ch to obtain all task completion time f_jAnd its fitness value fit;

step G2: changing task 0 into task J +1 and task J +1 into task 0, reversing the time sequence relationship between tasks, i.e. changing the father task set of tasks into the son task set, changing the son task set into the father task set, and according to the task completion time f, the scheduling sequence list in individual ch_jRearranging from large to small, i.e. g genes in individuals_J+jSet to the jth last completed task, J ═ 1, …, J, forming an individual ch;

step G3: let θ be θ +1, perform serial individual decoding based on the insertion mode on the individual ch to obtain the completion time f of all tasks_jJ ═ 1, …, J, and its fitness value fit;

step G4: if fit < fit, then ch, fit, f_j＝f_jJ is 1, …, J, go to step G2, otherwise go to step G5;

step G5: if theta is an even number, outputting the individual ch, otherwise outputting the individual ch, the fitness value fit of the individual ch, and ending the operation;

the serial individual decoding based on the insertion mode includes the steps of:

step H1: initializing a system state:order completion time f of task 0₀0, F₀Lf, the elements in F are arranged from small to large; making a cycle control variable delta equal to 1; let the remaining available amount of the renewable resource corresponding to each completion time in F:

step H2: fetch task j equals g_J+δAnd the mode m ═ g adopted by it_j；

Step H3: calculating the completion time of task j

Namely:

step H3.1: let start time s of task j_j＝rt_jLet flag value flag be 1;

step H3.2: finding rt or more in F_jAnd form a set TF, i.e., TF ═ { F ∈ F | F ≧ rt_j}；

Step H3.3: take the element with the smallest value from TF, do not set to f_j′；

Step H3.4: if flag is 1 and f_j′-s_j≥d_j,mThen go to step H3.7; otherwise go to step H3.5;

step H3.5: if flag is equal to 0, let s_j＝f_j′，flag＝1；

Step H3.6: judgment of f_j′At the moment, whether the remaining available amount of all renewable resources meets the requirement of the task j is determined, that is, whether all K is 1, … and K meets the requirement

If not, if the flag is 0, the step goes to a step H3.3;

step H3.7: let f_j＝s_j+d_j,m；

Step H4: if f is_jIf F is absent, F is F ∪ { F ═ F_jIs increased by f_jRemaining available amount of renewable resources at the moment:

wherein: f. of_j″Is f_jThe previous time of day;

step H5: update [ s ]_j,f_j) Remaining available amount of renewable resources within the time period:

step H6: let δ be δ +1, go to step H2 if δ ≦ J, otherwise go to step H7;

step H7: obtaining all task completion times f_j(ii) a Calculating the fitness value fit of the target, and finishing the operation;

step 12: selecting N different individuals from the current generation population and the new population from good to bad to form a new current generation population;

step 13: judging whether a second-stage evolution termination condition is met, if so, turning to a step 14, otherwise, turning to a step 10;

the second stage termination condition is that the iteration is carried out to a specified algebraic TG₂Or successive iterations GG₂No improvement was made for the best individuals;

step 14: if the optimal individual in the contemporary population is a feasible individual, outputting a scheduling scheme corresponding to the optimal individual and the project execution time thereof

Otherwise the problem has no feasible scheduling scheme.

Further, a specific calculation method of the fitness value fit is as follows: if the individual is an infeasible individual, i.e., there is an infeasible resource,/, satisfied

Then let the individual fitness value fit be lf + ms, otherwise let the individual fitness value fit be ms, where:

executing time for the project;

the smaller the fitness value fit, the better the individual.

Further, the specific steps of randomly selecting two different individuals from the contemporary population as father 1 and father 2 based on the ranking round-robin in the steps C1 and E1 are as follows:

step I1: sequencing the individuals in the contemporary population according to the priority to obtain the sequencing value rk of the individual n_nN is 1, …, N, where the first row takes 1, the second row takes 2, and so on, and the last row takes N;

step I2: calculating the probability that the individual n is selected

R is greater than 1 and is a discrimination coefficient;

step I3: calculating the cumulative probability: a. the₀＝0，

Step I4: generating a random number lambda₁E [0,1), if A_n-1≤λ₁＜A_nThen individual n is selected as father 1;

step I5: generating a random number lambda₂E [0,1), if A_n′-1≤λ₂＜A_n′And n '≠ n, then selects individual n' as parent 2, goes to step I6, otherwise goes to step I5;

step I6: the individual selection operation ends.

The invention has the beneficial effects that:

(1) compared with the traditional multi-mode resource limited project scheduling optimization method, the method provided by the invention has the advantages that the preprocessing step is added, so that the problem scale can be reduced and the solving time can be shortened when a redundant mode, an invalid mode and redundant resources exist.

(2) Compared with a heuristic algorithm, a semi-intelligent algorithm combined with the heuristic algorithm and an existing intelligent algorithm based on layered coding, the coding method adopted by the invention is designed, and any scheduling scheme has a coding scheme corresponding to the coding scheme, so that the search space is complete, and the global search can be realized;

(3) compared with an accurate algorithm, the algorithm designed by the invention can be suitable for solving large-scale problems.

(4) Compared with a common coding mode based on priority, the task scheduling sequence adopted by the invention adopts an integer coding method based on topological sorting, and the time sequence relation among tasks is considered, so that the decoding is simpler, the decoding efficiency can be effectively improved, and the overall efficiency of the algorithm is further improved.

(5) The serial decoding method, which is designed to be used in the present invention and which arranges the execution of tasks as early as possible based on the insertion mode, generally finds a better scheduling scheme than the decoding method based on the non-insertion mode and the parallel mode.

(6) Compared with the common one-way decoding method, the forward and backward individual decoding improvement strategy FBI & D adopted by the design of the invention enhances the neighborhood optimizing capability of the individual, thereby improving the optimizing capability and the searching efficiency of the whole algorithm.

(7) Compared with the existing single-stage intelligent search method, the method adopts a two-stage adaptive evolution strategy, the stage 1 adopts a mixed heuristic semi-intelligent algorithm based on a layering technology and the earliest task completion time, so that the algorithm can be converged near the optimal solution as soon as possible, and the stage 2 adopts a full intelligent algorithm to perform neighborhood expansion search on the basis of the previous stage, so that better individuals can be found, and the method has higher search efficiency and optimization capability.

(8) The invention adopts the strategy of generating the next generation population by combining the current generation population and the new population, thereby ensuring that the optimal individual is inherited to the next generation and leading the algorithm to be monotonously converged.

(9) Compared with the traditional crossing algorithm, the method increases the validity judgment of the crossing point in the crossing operation, and improves the crossing efficiency.

(10) Compared with the traditional random initialization method, the invention adopts the individual generation method based on the hierarchical topological sorting and the earliest task completion time when initializing the seed group, so that the algorithm can start searching from the vicinity of a better individual meeting the hierarchical coding, the searching efficiency can be improved, and the searching time can be shortened.

Drawings

FIG. 1 is a flow chart of a multi-mode resource-constrained project scheduling optimization method using a two-stage genetic algorithm according to the present invention.

FIG. 2 is a timing diagram of tasks according to an embodiment of the present invention.

Detailed Description

The present invention will be described in further detail with reference to the drawings and examples, but the present invention is not limited to the examples.

A project is composed of 20 tasks numbered 0 to 19, and the task structure and the time sequence relation are shown in FIG. 2, wherein t₀And t₁₉Is a virtual task artificially increased, namely, the project construction period and the resource are not occupied, t₁To t₁₈The executable pattern of (a), and the time required for execution, the number of occupied updatable resources and the number of non-updatable resources in that pattern are shown in table 1. In this project, the available amount of the updatable resource 1 at any time is 12, the available amount of the updatable resource 2 at any time is 13, the available amount of the non-updatable resource 1 in the entire project period is 70, and the available amount of the non-updatable resource 2 in the entire project period is 85.

TABLE 1

For the above case, as shown in fig. 1, a method for scheduling and optimizing a multi-mode resource-constrained project using a two-stage genetic algorithm includes the following steps:

executing the step 1: formalizing a multi-mode resource limited project scheduling problem and acquiring information required by scheduling optimization;

acquiring task set T ═ T of project₀,t₁,t₂,t₃,t₄,t₅,t₆,t₇,t₈,t₉,t₁₀,t₁₁,t₁₂,t₁₃,t₁₄,t₁₅,t₁₆,t₁₇,t₁₈,t₁₉Where task 0 and task 19, i.e. t₀And t₁₉Are virtual tasks, task 1 through task 18, i.e., t₁,…,t₁₈Is a task that actually needs to be scheduled;

obtaining the time sequence relation between tasks, namely a parent task set PR of the task j_jAnd a set of subtasks SC_j：

SC₀＝{t₁,t₂,t₃}；PR₁＝{t₀}，SC₁＝{t₄,t₆,t₁₄}；PR₂＝{t₀}，SC₂＝{t₈}；PR₃＝{t₀}，SC₃＝{t₄,t₅,t₆}；PR₄＝{t₁,t₃}，SC₄＝{t₁₀,t₁₁,t₁₃}；PR₅＝{t₃}，SC₅＝{t₇,t₁₂,t₁₃}；PR₆＝{t₁,t₃}，SC₆＝{t₁₀,t₁₈}；PR₇＝{t₅}，SC₇＝{t₈}；PR₈＝{t₂,t₇}，SC₈＝{t₉,t₁₄,t₁₅}；PR₉＝{t₈}，SC₉＝{t₁₁,t₁₆}；PR₁₀＝{t₄,t₆}，SC₁₀＝{t₁₂,t₁₆}；PR₁₁＝{t₄,t₉}，SC₁₁＝{t₁₈}；PR₁₂＝{t₅,t₁₀}，SC₁₂＝{t₁₇}；PR₁₃＝{t₄,t₅}，SC₁₃＝{t₁₅,t₁₇}；PR₁₄＝{t₁,t₈}，SC₁₄＝{t₁₇}；PR₁₅＝{t₈,t₁₃}，SC₁₅＝{t₁₆,t₁₈}；PR₁₆＝{t₉,t₁₀,t₁₅}，SC₁₆＝{t₁₉}；PR₁₇＝{t₁₂,t₁₃,t₁₄}，SC₁₇＝{t₁₉}；PR₁₈＝{t₆,t₁₁,t₁₅}，SC₁₈＝{t₁₉}；PR₁₉＝{t₁₆,t₁₇,t₁₈}，

Acquiring the available quantity R of the renewable resource k in the project at any time_k：R₁＝12，R ₂13, acquiring the available quantity N of the non-updatable resource l in the whole project period_l：N₁＝70，N₂＝85；

Acquiring the time needed by the task j to execute in the mode m, namely the construction period d_j,mAnd the number r of occupied renewable resources_j,m,kNumber of non-updatable resources n_j,m,l：d_1,1＝2，r_1,1,1＝6，r_1,1,2＝9，n_1,1,1＝0，n_1,1,2＝8；d_1,2＝3，r_1,2,1＝5，r_1,2,2＝7，n_1,2,1＝4，n_1,2,2＝0；d_1,3＝5，r_1,3,1＝5，r_1,3,2＝6，n_1,3,1＝0，n_1,3,2＝6；……；d_18,1＝2，r_18,1,1＝6，r_18,1,2＝6，n_18,1,1＝0，n_18,1,2＝10；d_18,2＝4，r_18,2,1＝3，r_18,2,2＝4，n_18,2,1＝8，n_18,2,2＝0；d_18,3＝5，r_18,3,1＝3，r_18,3,2＝1，n_18,3,1＝2，n_18,3,2＝0。

And (3) executing the step 2: if there is an un-updateable resource/satisfy

Or that there is one task j and renewable resources k satisfying

for a non-updateable resource 1,

for the non-updateable resource 2,

for renewable resource 1, all tasks j are satisfied

For renewable resource 2, all tasks j are satisfied

Therefore, there may be a feasible scheduling scheme for this problem, going to step 3. And (3) executing the step: carrying out pretreatment: carrying out redundancy and invalidation check on the mode and the resource, and removing the redundancy mode, the invalidation mode and the redundancy resource;

step 3.1 is executed: mflg ═ 0, rflg ═ 0; due to the absence of r_j,m,k＞R_kAnd n_j,m,l＞N_l，j＝1,…,18，m＝1,…,M_jK is 1,2, l is 1,2, so there is no invalid pattern in this item;

step 3.2 is executed: making mflg equal to 1 and going to step 3.3 because mflg is equal to 0;

step 3.3 is executed; since there is a task 12 and its mode 2 and mode 3 are satisfied: d_12,2＝8＜d_12,3＝9，r_12,2,1＝6≤r_12,3,1＝6，r_12,2,2＝1≤r_12,3,2＝1，n_12,2,1＝3＜n_12,3,1＝6，n_12,2,2＝2＜n_12,3,24, so pattern 3 is redundant to task 12, removing pattern 3 from task 12: let M₁₂＝3-1＝2，rflg＝0；

Step 3.4 is executed: making rflg ═ 1 because rflg is 0, go to step 3.5;

step 3.5 is executed: for renewable resource 1:

for renewable resource 2:

there are therefore no renewable resources k that satisfy:

the project does not have redundant renewable resources;

step 3.6 is executed: for non-updateable resource 1:

for non-updateable resource 2:

so that there is no non-updateable resource/that satisfies:

redundant non-updatable resources do not exist in the project;

step 3.7 is executed: since mflg ═ 1 ^ rflg ═ 1, go to step 3.8;

step 3.8 is executed: and finishing the pretreatment.

And (4) executing: calculating a level value of the task;

t₀without parent task, with a hierarchy value of lvl₀＝0；t₁Only one parent task 0, then

Similarly, the hierarchy values of other tasks may be obtained: lvl₂＝lvl₃＝1；lvl₄＝lvl₅＝lvl₆＝2；lvl₇＝lvl₁₀＝lvl₁₃＝3；lvl₈＝lvl₁₂＝4；lvl₉＝lvl₁₄＝lvl₁₅＝5；lvl₁₁＝lvl₁₆＝lvl₁₇＝6；lvl₁₈＝7；lvl₁₉＝8。

And 5, executing the step: initializing a contemporary population;

taking the population size N as 10;

the process of generating 1 individual based on the individual generation method of the hierarchy and the earliest completion time of the task is as follows:

step a1 is executed: randomly arranging tasks in level 1, which are 1,2,3, randomly arranging tasks in level 2, which are 4,5,6, randomly arranging tasks in level 3, which are 10,13,7, randomly arranging tasks in level 4, which are 8,12, randomly arranging tasks in level 5, which are 9,14,15, randomly arranging tasks in level 6, which are 11,16,17, randomly arranging tasks in level 7, which are 18, sequentially connecting the arrangements of tasks in level 1 to level 7 to form an individual scheduling order list { g }₁₉,…,g₃₆}＝{1,2,3,4,5,6,10,13,7,8,12,9,14,15,11,16,17,18}；

Step a2 is executed: individual serial decoding method based on earliest completion time of task generates individual execution mode list { g₁,…,g₁₈}＝{1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,2,1,2}；

Step a3 is executed: output individual ch₁1,1,1,1,1,1,1,3,1,1,1,1,1,1,2,1, 2; 1,2,3,4,5,6,10,13,7,8,12,9,14,15,11,16,17,18}, the fitness value fit thereof₁The operation ends 66.

The specific implementation process of generating the individual execution mode list based on the individual serial decoding method of the earliest task completion time is as follows:

step B1 is executed: initialization of the system state f₀＝0，

Task completion time set F ═ F₀-lf } - {0,122 }; the remaining available amount of the renewable resource corresponding to each completion time in F is:

δ＝1；

step B2 is executed: fetch task j equals g_J+δ＝g₁₉1, making the mode m 1;

step B3 is executed: calculating the completion time of task 1; i.e. step B3.1 is performed: let start time of task 1

Let flag be 1; and step B3.2 is executed: where F is ═ F₀And if is greater than or equal to rt is found in {0,122}₁0, the composition set TF ═ { F ∈ F | F ≧ rt₁＝0}＝{f₀-lf } - {0,122 }; step B3.3 is performed: from TF ═ f₀The element with the smallest value is taken from {0,122}, which is f₀0; step B3.4 is performed: due to f₀-s_1,1＝0-0＝0＜d_1,1Go to step B3.5, 2; step B3.5 is performed: since flag is 1 at this time, go directly to step B3.6; step B3.6 is performed: due to the fact that

f₀At time 0, the remaining available amount of all renewable resources meets the requirements of task 1, so the process goes directly to step B3.3; step B3.3 is performed: taking an element with the smallest value from TF { lf } {122}, wherein the element is lf-122; step B3.4 is performed: since flag is 1 and lf-s_1,1＝122-0＝122＞d_1,1Go to step B3.7, 2; step B3.7 is performed: f. of_1,1＝s_1,1+d_1,10+ 2-2; step B3.8 is performed: m + 1-2, M2 ≦ M₁If 3, go to step B3.1; step B3.1 is performed: let start time s of task 1_1,2＝rt₁＝max{f₀}＝f₀0,1 is given as flag; and step B3.2 is executed: where F is ═ F₀Finding rt or more from lf {0,122}₁0, the composition set TF ═ { F ∈ F | F ≧ rt₁＝0}＝{f₀-lf } - {0,122 }; step B3.3 is performed: from TF ═ f₀If is, take out the element with the smallest value (0,122)Element which is f₀0, then TF 122; step B3.4 is performed: due to f₀-s_1,2＝0-0＝0＜d_1,2Go to step B3.5, No. 3; step B3.5 is performed: since flag is 1 at this time, go directly to step B3.6; step B3.6 is performed: due to the fact that

f₀At time 0, the remaining available amount of all renewable resources k meets the requirements of task 1, so the process goes directly to step B3.3; step B3.3 is performed: if the element with the smallest value is selected from TF { lf } {122}, which is lf 122, then

Step B3.4 is performed: since flag is 1 and lf-s_1,2＝122-0＝122＞d_1,2Go to step B3.7, No. 3; step B3.7 is performed: f. of_1,2＝s_1,2+d_1,20+ 3-3; step B3.8 is performed: m is 2+1 is 3, because M is 3 ≦ M₁If 3, go to step B3.1; … …, and repeating steps B3.1 to B3.8 until M is 4 > M₁Is 3, gives s_1,1＝0，s_1,2＝0，s_1,3＝0，f_1,1＝2，f_1,2＝3，f_1,3From f to 5_1,1＝2，f_1,2＝3，f_1,3Minimum removal completion time f of 5_1,12, let f₁＝f_1,1＝2，s₁＝f₁-d_1,1＝2-2＝0，g₁Go to step B4, if 1;

step B4 is executed: due to the fact that

F is F ∪ { F ═ F₁}＝{f₀,lf}∪f₁＝{f₀,f₁And if is {0,2,122}, increasing f₁The remaining available amount of resources can be updated at 2:

step B5 is executed: updating[s₁,f₁) Remaining available amount of renewable resource k in time period [0, 2):

step B6 is executed: if δ +1 is 2, then go to step B2, since δ 2 ≦ J ≦ 18;

step B2 is executed: fetch task j equals g₁₈₊₂＝g₂₀2, making the mode m equal to 1;

step B3 is executed: calculating the completion time of task 2; i.e. step B3.1 is performed: let start time s of task 2_2,1＝rt₂＝max{f₀}＝f₀0,1 is given as flag; and step B3.2 is executed: find out rt or more in F ═ {0,2,122}₂0, the composition set TF ═ {0,2,122 }; step B3.3 is performed: taking the element with the smallest value from TF {0,2,122}, and taking the element with the smallest value as 0, then TF {2,122 }; step B3.4 is performed: due to 0-s_2,1＝0＜d_2,1Turning to step B3.5, if 6; step B3.5 is performed: since flag is 1 at this time, go directly to step B3.6; step B3.6 is performed: due to the fact that

That is, at time 0, the remaining available amount of the renewable resource 1 cannot meet the requirement of the task 2, if flag is equal to 0, go to step B3.3; step B3.3 is performed: taking the element with the smallest value from TF {2,122}, wherein the element is 2, and TF {122 }; step B3.4 is performed: if flag is 0, go to step B3.5; step B3.5 is performed: since flag is equal to 0, s is_2,12,1 is flag; step B3.6 is performed: due to the fact that

2, the residual available quantity of all renewable resources meets the requirement of the task 2, so the step B3.3 is directly carried out; step B3.3 is performed: the element with the smallest value is taken from TF 122, which is 122,

step B3.4 is performed: since flag is 1 and 122-s_2,1＝120＞d_2,1Turning to step B3.7, if 6; step B3.7 is performed: f. of_2,1＝s_2,1+d_2,12+ 6-8; step B3.8 is performed: m + 1-2, M2 ≦ M₂If 3, go to step B3.1; … …, and repeating steps B3.1 to B3.8 until M is 4 > M₂Is 3, gives s_2,1＝2，s_2,2＝2，s_2,3＝2，f_2,1＝8，f_2,2＝10，f_2,311 from f_2,1＝8，f_2,2＝10，f_2,3Minimum removal completion time f of 11_2,1Let f be 8₂＝f_2,1＝8，s₂＝f₂-d_2,1＝8-6＝2，g₂Go to step B4, if 1;

step B4 is executed: due to the fact that

Then F ═ F₀,f₁,f₂And if is {0,2,8,122}, increasing f₂The remaining available amount of resources can be updated at time 8:

step B5 is executed: update [ s ]₂,f₂) 2,8) remaining available amount of renewable resources in the time period:

step B6 is executed: if δ 2+1 is 3, go to step B2, since δ 3 ≦ J ≦ 18;

……

thus, the steps B2 to B6 are repeatedly executed until δ 19 > J18, and then the process goes to step B7;

step B7 is executed: obtain execution mode list g₁,…,g₁₈-1, 1,1,1,1,1,1,1,3,1,1,1,1,1,1,2,1,2}, calculating an individual fitness value: due to the fact that

The individual is thus a viable individual, the fitness value of which

The operation is ended.

Similarly, the remaining 9 different individuals that can generate the contemporary population are:

ch₂＝{2,2,1,1,1,2,1,1,1,1,1,2,1,1,1,1,3,1；3,1,2,4,6,5,13,7,10,12,8,15,9,14,16,11,17,18}；

ch₃＝{2,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,3,1；2,3,1,5,4,6,10,13,7,12,8,14,15,9,16,11,17,18}；

ch₄＝{2,2,1,1,1,1,1,1,2,1,1,2,1,1,1,2,1,1；3,1,2,5,4,6,10,13,7,12,8,14,9,15,17,16,11,18}；

ch₅＝{1,1,1,1,1,1,1,1,2,1,1,1,3,1,1,2,1,1；2,1,3,5,4,6,7,13,10,12,8,14,9,15,17,16,11,18}；

ch₆＝{2,2,1,1,1,1,1,1,1,1,1,1,1,1,3,1,3,1；3,1,2,4,5,6,13,7,10,8,12,15,9,14,11,17,16,18}；

ch₇＝{2,1,1,1,1,1,1,1,2,1,1,1,3,1,1,1,1,1；2,3,1,5,4,6,7,10,13,12,8,14,9,15,16,17,11,18}；

ch₈＝{1,2,1,1,1,1,1,1,1,1,1,1,1,1,3,1,1,1；3,2,1,5,4,6,13,7,10,8,12,15,9,14,16,17,11,18}；

ch₉＝{2,1,1,1,1,1,1,1,3,1,1,1,1,1,3,1,3,1；2,3,1,4,6,5,13,10,7,8,12,15,14,9,16,11,17,18}；

ch₁₀＝{1,2,1,1,1,1,1,1,2,1,1,1,3,1,1,2,1,2；1,3,2,6,5,4,7,10,13,12,8,14,15,9,11,16,17,18}；

the fitness values are respectively as follows: fit₂＝53，fit₃＝179，fit₄＝178，fit₅＝184，fit₆＝58，fit₇＝186，fit₈＝185，fit₉＝61，fit₁₀＝55；

The initial current generation population finally generated in this way is CP＝{ch₁,ch₂,ch₃,ch₄,ch₅,ch₆,ch₇,ch₈,ch₉,ch₁₀}。

And 6, executing the step: carrying out N/2 times of scheduling sequence cross operation based on the hierarchy on the contemporary population to generate a new population, and carrying out scheduling sequence variation operation based on the hierarchy on each individual in the new population;

the specific implementation process of performing one-time cross operation on the contemporary population is as follows:

step C1 is executed: rank-based betting rounds randomly select two different individuals from the contemporary population as father 1 and father 2, namely: step I1 is executed: sequencing the individuals in the current generation population according to the priority, and obtaining the sequencing value of each individual: rk₁＝5，rk₂＝1，rk₃＝7，rk₄＝6，rk₅＝8，rk₆＝3，rk₇＝10，rk₈＝9，rk₉＝4，rk₁₀2; step I2 is executed: let R1 + 1/N1.1, calculate the 1 st individual ch₁Probability of being selected

Same principle A₂＝0.148，A₃＝0.083，A₄＝0.092，A₅＝0.076，A₆＝0.122，A₇＝0.063，A₈＝0.069，A₉＝0.111，A₁₀0.135; step I3 is executed: calculating cumulative probability A₀＝0，

A₃＝0.332，A₄＝0.424，A₅＝0.500，A₆＝0.622，A₇＝0.685，A₈＝0.754，A₉＝0.865，A₁₀1.000; step I4 is executed: generate a random number of [0,1), which is 0.433, since A₄＝0.424≤0.433<0.500＝A₅Thus selecting ch₅As a father 1; step I5 is executed: randomly generating a random number of [0,1), which is 0.840 due to A₈＝0.754≤0.840＜0.865＝A₉And 9 ≠ 4, so ch is selected₉As father 2, go to step I6; step I6 is executed: the individual selection operation ends.

Step C2 is executed: randomly selecting a level, which is level 3;

step C3 is executed: exchanging the scheduling order of the task of level 3 in the scheduling order list of parent 1 and the scheduling order list of parent 2, as shown in table 2, and emptying the execution mode list to generate child 1:

ch′₁{,,,,,,,,,,,,,,,; 2,1,3,5,4,6,13,10,7,12,8,14,9,15,17,16,11,18} and the subvolume 2:

ch′₂{,,,,,,,,,,,,,,,; 2,3,1,4,6,5,7,13,10,8,12,15,14,9,16,11,17,18 }; finishing the operation;

gene	g₁₉	g₂₀	g₂₁	g₂₂	g₂₃	g₂₄	g₂₅	g₂₆	g₂₇	g₂₈	g₂₉	g₃₀	g₃₁	g₃₂	g₃₃	g₃₄	g₃₅	g₃₆
																			Father body 1	2	1	3	5	4	6	7	13	10	12	8	14	9	15	17	16	11	18
Father body 2	2	3	1	4	6	5	13	10	7	8	12	15	14	9	16	11	17	18
																			Sub-body 1	2	1	3	5	4	6	13	10	7	12	8	14	9	15	17	16	11	18
Sub-body 2	2	3	1	4	6	5	7	13	10	8	12	15	14	9	16	11	17	18

TABLE 2

Similarly, the remaining individuals in the new population generated after crossover are:

ch′₃＝{,,,,,,,,,,,,,,,,,；1,2,3,4,5,6,10,13,7,8,12,9,14,15,11,16,17,18}；

ch′₄＝{,,,,,,,,,,,,,,,,,；3,2,1,5,4,6,13,7,10,8,12,15,9,14,16,17,11,18}；

ch′₅＝{,,,,,,,,,,,,,,,,,；3,1,2,4,5,6,7,13,10,8,12,15,9,14,11,17,16,18}；

ch′₆＝{,,,,,,,,,,,,,,,,,；2,1,3,5,4,6,13,7,10,12,8,14,9,15,17,16,11,18}；

ch′₇＝{,,,,,,,,,,,,,,,,,；3,1,2,4,6,5,10,13,7,12,8,14,9,15,17,16,11,18}；

ch′₈＝{,,,,,,,,,,,,,,,,,；2,3,1,5,4,6,13,10,7,8,12,15,14,9,16,11,17,18}；

ch′₉＝{,,,,,,,,,,,,,,,,,；3,1,2,4,5,6,10,13,7,12,8,14,9,15,17,16,11,18}；

ch′₁₀＝{,,,,,,,,,,,,,,,,,；3,1,2,5,4,6,13,7,10,8,12,15,9,14,11,17,16,18}；

the new population thus finally generated is NP ═ ch'₁,ch′₂,ch′₃,ch′₄,ch′₅,ch′₆,ch′₇,ch′₈,ch′₉,ch′₁₀}。

The specific implementation process of performing mutation operation on the new population is as follows:

taking the rate of variation p_m＝0.2；

To individual ch'₁{,,,,,,,,,,,,,,,; 2,1,3,5,4,6,13,10,7,12,8,14,9,15,17,16,11,18} performing mutation operations as follows:

step D1 is executed: randomly generating a [0,1) random number λ, which is 0.35, since λ 0.35 > p_mIf 0.2, go to step D4;

step D4 is executed: finishing the operation of the variation of the scheduling sequence list based on the hierarchy;

……

to individual ch'₄{,,,,,,,,,,,,,,,; 3,2,1,5,4,6,13,7,10,8,12,15,9,14,16,17,11,18} performing mutation operations as follows:

step D1 is executed: randomly generating a [0,1) random number λ, which is 0.01, since λ 0.01 < p_mIf 0.2, go to step D2;

step D2 is executed: if the number of tasks in the levels 1,2,3,4,5 and 6 is more than 1, randomly selecting a level from the levels 1,2,3,4,5 and 6, wherein the level is level 5, and turning to the step D3;

step D3 is executed: follow from level 5The machine selects two tasks, which are t₉And t₁₅The two tasks are swapped in the scheduling order list, and the mutated individuals become ch'₄＝{,,,,,,,,,,,,,,,,,；3,2,1,5,4,6,13,7,10,8,12,9,15,14,16,17,11,18}；

gene	g₁₉	g₂₀	g₂₁	g₂₂	g₂₃	g₂₄	g₂₅	g₂₆	g₂₇	g₂₈	g₂₉	g₃₀	g₃₁	g₃₂	g₃₃	g₃₄	g₃₅	g₃₆
																			Before mutation	3	2	1	5	4	6	13	7	10	8	12	15	9	14	16	17	11	18
After mutation	3	2	1	5	4	6	13	7	10	8	12	9	15	14	16	17	11	18

TABLE 3

……

Thus, the individuals in the new population after mutation become:

ch′₁＝{,,,,,,,,,,,,,,,,,；2,1,3,5,4,6,13,10,7,12,8,14,9,15,17,16,11,18}；

ch′₂＝{,,,,,,,,,,,,,,,,,；2,3,1,4,6,5,7,13,10,8,12,15,14,9,16,11,17,18}；

ch′₇＝{,,,,,,,,,,,,,,,,,；3,1,2,4,6,5,13,10,7,12,8,14,9,15,17,16,11,18}；

ch′₁₀＝{,,,,,,,,,,,,,,,,,；3,1,2,5,4,6,13,7,10,8,12,15,9,14,11,17,16,18}。

and 7, executing the step: generating an individual execution mode list based on the earliest task completion time for each individual in the new population and calculating the fitness value of the individual execution mode list;

after all individuals in the new population generate an individual execution mode list based on the earliest task completion time, the method is changed into the following steps:

ch′₁＝{1,1,1,1,1,1,1,1,2,1,1,2,1,1,1,2,1,1；2,1,3,5,4,6,13,10,7,12,8,14,9,15,17,16,11,18}；

ch′₂＝{2,1,1,1,1,1,1,1,3,1,1,1,3,1,3,1,3,1；2,3,1,4,6,5,7,13,10,8,12,15,14,9,16,11,17,18}；

ch′₃＝{1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,2,1,2；1,2,3,4,5,6,10,13,7,8,12,9,14,15,11,16,17,18}；

ch′₄＝{1,2,1,1,1,1,1,1,3,1,1,1,1,3,1,1,1,1；3,2,1,5,4,6,13,7,10,8,12,9,15,14,16,17,11,18}；

ch′₅＝{2,2,1,1,1,1,1,1,1,1,1,1,3,1,3,1,3,1；3,1,2,4,5,6,7,13,10,8,12,15,9,14,11,17,16,18}；

ch′₆＝{1,1,1,1,1,1,1,1,2,1,1,2,1,1,1,2,1,1；2,1,3,5,4,6,13,7,10,12,8,14,9,15,17,16,11,18}；

ch′₇＝{2,2,1,1,1,2,1,1,2,1,1,2,1,1,1,2,1,1；3,1,2,4,6,5,13,10,7,12,8,14,9,15,17,16,11,18}；

ch′₈＝{2,1,1,1,1,1,1,1,3,1,1,1,1,1,3,1,3,1；2,3,1,5,4,6,13,10,7,8,12,15,14,9,16,11,17,18}；

ch′₉＝{2,2,1,1,1,1,1,1,2,1,1,2,1,1,1,2,1,1；3,1,2,4,5,6,10,13,7,12,8,14,9,15,17,16,11,18}；

ch′₁₀＝{2,2,1,1,1,1,1,1,1,1,1,1,1,1,3,1,3,1；3,1,2,5,4,6,13,7,10,8,12,15,9,14,11,17,16,18}；

the fitness values are respectively as follows: fist'₁＝183，fit′₂＝57，fit′₃＝66，fit′₄＝182，fit′₅＝54，fit′₆＝183，fit′₇＝55，fit′₈＝60，fit′₉＝178，fit′₁₀＝58。

And step 8 is executed: selecting N different individuals from the current generation population and the new population from good to bad to form a new current generation population;

selecting the ch of the current generation population from good to bad according to the fitness value₁，ch₂，ch₆，ch₉，ch₁₀And ch 'of the New population'₂，ch′₅，ch′₇，ch′₈，ch′₁₀Forming the next generation of population, i.e. GP ═ ch₁,ch₂,ch₆,ch₉,ch₁₀,ch′₂,ch′₅,ch′₇,ch′₈,ch′₁₀}。

Let CP be GP, the contemporary population becomes:

ch₁＝{1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,2,1,2；1,2,3,4,5,6,10,13,7,8,12,9,14,15,11,16,17,18}；

ch₃＝{2,2,1,1,1,1,1,1,1,1,1,1,1,1,3,1,3,1；3,1,2,4,5,6,13,7,10,8,12,15,9,14,11,17,16,18}；

ch₄＝{2,1,1,1,1,1,1,1,3,1,1,1,1,1,3,1,3,1；2,3,1,4,6,5,13,10,7,8,12,15,14,9,16,11,17,18}；

ch₅＝{1,2,1,1,1,1,1,1,2,1,1,1,3,1,1,2,1,2；1,3,2,6,5,4,7,10,13,12,8,14,15,9,11,16,17,18}；

ch₆＝{2,1,1,1,1,1,1,1,3,1,1,1,3,1,3,1,3,1；2,3,1,4,6,5,7,13,10,8,12,15,14,9,16,11,17,18}；

ch₇＝{2,2,1,1,1,1,1,1,1,1,1,1,3,1,3,1,3,1；3,1,2,4,5,6,7,13,10,8,12,15,9,14,11,17,16,18}；

ch₈＝{2,2,1,1,1,2,1,1,2,1,1,2,1,1,1,2,1,1；3,1,2,4,6,5,13,10,7,12,8,14,9,15,17,16,11,18}；

ch₉＝{2,1,1,1,1,1,1,1,3,1,1,1,1,1,3,1,3,1；2,3,1,5,4,6,13,10,7,8,12,15,14,9,16,11,17,18}；

ch₁₀＝{2,2,1,1,1,1,1,1,1,1,1,1,1,1,3,1,3,1；3,1,2,5,4,6,13,7,10,8,12,15,9,14,11,17,16,18}；

the fitness values are respectively as follows: fit₁＝66，fit₂＝53，fit₃＝58，fit₄＝61，fit₅＝55，fit₆＝57，fit₇＝54，fit₈＝55，fit₉＝60，fit₁₀＝58。

And step 9 is executed: judging whether a first-stage iteration termination condition is met, if so, ending the first-stage evolution, and turning to the step 10, otherwise, turning to the step 6;

the first stage iteration termination condition is set as population evolution TG₁15 generation;

and (4) since the current population is iteratively evolved for two generations and does not meet the iteration termination condition of the first stage, the step 6 is carried out.

……

Thus, the steps 6 to 9 are continuously and repeatedly executed, and after 15 generations of iteration, the contemporary population becomes:

ch₁＝{2,2,1,1,1,2,1,1,1,1,1,2,1,1,1,1,3,1；3,1,2,4,6,5,13,7,10,12,8,15,9,14,16,11,17,18}；

ch₂＝{2,1,1,1,1,1,1,1,3,1,1,1,3,1,3,1,3,1；2,3,1,4,6,5,7,13,10,8,12,15,14,9,16,11,17,18}；

ch₃＝{2,1,1,1,1,1,1,1,3,1,1,1,1,1,3,1,3,1；2,3,1,5,4,6,13,10,7,8,12,15,14,9,16,11,17,18}；

ch₄＝{2,2,1,1,1,1,1,1,1,1,1,1,1,1,3,1,3,1；3,1,2,4,5,6,10,13,7,8,12,15,9,14,11,17,16,18}；

ch₅＝{2,2,1,1,1,2,1,1,3,1,1,1,3,1,3,1,3,1；3,1,2,4,6,5,7,13,10,8,12,15,14,9,16,11,17,18}；

ch₆＝{2,1,1,1,1,1,1,1,3,1,1,1,1,1,3,1,3,1；2,3,1,4,6,5,13,10,7,8,12,15,14,9,16,11,17,18}；

ch₇＝{2,2,1,1,1,1,1,1,1,1,1,1,1,1,3,1,3,1；3,1,2,4,5,6,13,7,10,8,12,15,9,14,11,17,16,18}；

ch₈＝{2,2,1,1,1,1,1,1,1,1,1,1,3,1,3,1,3,1；3,1,2,4,5,6,7,13,10,8,12,15,9,14,11,17,16,18}；

ch₉＝{1,2,1,1,1,1,1,1,2,1,1,1,3,1,1,2,1,2；1,3,2,6,5,4,7,10,13,12,8,14,15,9,11,16,17,18}；

ch₁₀＝{1,1,1,1,1,1,1,1,3,1,1,1,1,1,1,2,1,2；1,2,3,4,5,6,10,13,7,8,12,9,14,15,11,16,17,18}；

the fitness values are respectively as follows: fit₁＝53，fit₂＝57，fit₃＝60，fit₄＝59，fit₅＝52，fit₆＝61，fit₇＝58，fit₈＝54，fit₉＝55，fit₁₀＝66；

And (4) because the population is iteratively evolved for 15 generations and meets the iteration termination condition of the first stage, ending the evolution of the first stage and turning to the step 10.

Executing the step 10: performing N/2 times of cross operation of the execution mode list and the scheduling sequence list on the contemporary population to generate a new population, and performing variation operation on the execution mode list and the scheduling sequence list on each individual in the new population;

the specific implementation process of performing 1-time cross operation on the contemporary population is as follows:

step E1 is executed: randomly selecting two different individuals from the contemporary population as a father 1 and a father 2 by adopting a ranking-based round-robin method, wherein the two different individuals are respectively as follows:

ch₅2,2,1,1,1,2,1,1,3,1,1,1,3,1,3,1,3, 1; 3,1,2,4,6,5,7,13,10,8,12,15,14,9,16,11,17,18} and

step E2 is executed: since {2,2,1,1,1,2,1,1,3,1,1,1,3,1,3,1, 1,1, 1} is different from {2,2,1,1,1,1,1,1,1,1,1,1,3,1,3, 1}, go to step E3;

step E3 is executed: finding the position of the first different gene of the execution mode list {2,2,1,1,1,1, 2,1,1,3,1,1,1, 1,3,1,3,1} of the father 1 and the execution mode list {2,2,1,1,1,1,1,1, 3,1,3,1,3,1} of the father 2 from the front to the back, which is 6, finding the position of the first different gene of the execution mode list {2,2,1,1,1,1, 3,1,1,1,1,1,1, 1,3,1,3,1} of the father 1 and the execution mode list {2,2,1,1,1,1,1,1,1,1,1, 3,1} of the father 2 from the back to the front; as 6 ≠ 9, it goes to step E4;

step E4 is executed: randomly generating a positive integer between 6 and 8, which is 8;

step E5 is executed: as shown in table 4, the first 8 genes of the execution pattern list part of the daughter 1 are from the first 8 genes of the execution pattern list part of the father 1, and the last 10 genes of the execution pattern list part are from the last 10 genes of the execution pattern list part of the father 2;

step E6 is executed: as shown in table 4, the first 8 genes of the execution pattern list part of the daughter 2 are from the first 8 genes of the execution pattern list part of the father 2, and the last 10 genes of the execution pattern list part are from the last 10 genes of the execution pattern list part of the father 1;

TABLE 4

Step E7 is executed: since {3,1,2,4,6,5,7,13,10,8,12,15,14,9,16,11,17,18} is different from {3,1,2,4,5,6,7,13,10,8,12,15,9,14,11,17,16,18}, go to step E8;

step E8 is executed: finding the position of the first different gene of the scheduling order list {3,1,2,4,6,5,7,13,10,8,12,15,14,9,16,11,17,18} of parent 1 and the scheduling order list {3,1,2,4,5,6,7,13,10,8,12,15,9,14,11,17,16,18} of parent 2 from the front to the back, which is 5, finding the position of the first different gene of the scheduling order list {3,1,2,4,6,5,7,13,10,8,12,15,14,9,16,11,17,18} of parent 1 and the scheduling order list {3,1,2,4,5,6,7,13,10,8,12,15,9,14,11,17,16,18} of parent 2 from the back to the front, which is 17;

step E9 is executed: randomly generating a positive integer between 5 and 16, which is 10;

step E10 is executed: as shown in table 5, the first 10 genes of the schedule order list of child 1 are from the first 10 genes of the schedule order list of father 1, and the last 8 genes are from the schedule order list of father 2 {3,1,2,4,5,6,7,13,10,8,12,15,9,14,11,17,16,18} in which the first 10 genes 3,1,2,4,6,5,7,13,10,8 of the schedule order list of father 1 are deleted;

step E11 is executed: as shown in table 5, the first 10 genes of the schedule order list of the daughter 2 are from the first 10 genes of the schedule order list of the father 2, and the last 8 genes are from the schedule order list of the father 1 {3,1,2,4,6,5,7,13,10,8,12,15,14,9,16,11,17,18} in which the first 10 genes 3,1,2,4,5,6,7,13,10,8 of the schedule order list of the father 2 are deleted;

gene	g₁₉	g₂₀	g₂₁	g₂₂	g₂₃	g₂₄	g₂₅	g₂₆	g₂₇	g₂₈	g₂₉	g₃₀	g₃₁	g₃₂	g₃₃	g₃₄	g₃₅	g₃₆
																			Father body 1	3	1	2	4	6	5	7	13	10	8	12	15	14	9	16	11	17	18
Father body 2	3	1	2	4	5	6	7	13	10	8	12	15	9	14	11	17	16	18
																			Sub-body 1	3	1	2	4	6	5	7	13	10	8	12	15	9	14	11	17	16	18
Sub-body 2	3	1	2	4	5	6	7	13	10	8	12	15	14	9	16	11	17	18

TABLE 5

Step E12 is executed: end of the crossover operation, output

ch′₁2,2,1,1,1,2,1,1,1,1,1, 3,1,3,1,3, 1; 3,1,2,4,6,5,7,13,10,8,12,15,9,14,11,17,16,18} and

ch′₂＝{2,2,1,1,1,1,1,1,3,1,1,1,3,1,3,1,3,1；3,1,2,4,5,6,7,13,10,8,12,15,14,9,16,11,17,18}。

ch′₃＝{2,1,1,1,1,1,1,1,3,1,1,1,3,1,3,1,3,1；2,3,1,4,6,5,7,13,10,8,12,15,14,9,16,11,17,18}；

ch′₄＝{2,2,1,1,1,2,1,1,3,1,1,1,1,1,3,1,3,1；3,1,2,4,6,5,13,10,7,8,12,15,14,9,16,11,17,18}；

ch′₅＝{2,2,1,1,1,2,1,1,1,1,1,1,3,1,3,1,3,1；3,1,2,4,6,5,13,7,10,8,12,15,14,9,16,11,17,18}；

ch′₆＝{2,2,1,1,1,2,1,1,3,1,1,2,1,1,1,1,3,1；3,1,2,4,6,5,7,13,10,12,8,15,9,14,16,11,17,18}；

ch′₇＝{2,2,1,1,1,1,1,1,2,1,1,1,3,1,1,2,1,2；2,3,1,4,6,5,7,10,13,12,8,14,15,9,11,16,17,18}；

ch′₈＝{1,1,1,1,1,1,1,1,3,1,1,1,3,1,3,1,3,1；1,3,2,6,4,5,7,13,10,8,12,15,14,9,16,11,17,18}；

ch′₉＝{2,2,1,1,1,1,1,1,2,1,1,1,3,1,1,2,1,2；3,1,2,6,5,4,7,10,13,12,8,14,15,9,11,16,17,18}；

ch′₁₀＝{1,2,1,1,1,1,1,1,1,1,1,1,1,1,3,1,3,1；1,3,2,4,5,6,13,7,10,8,12,15,9,14,11,17,16,18}；

taking the rate of variation p_m＝0.2；

For ch'₁＝{2,2,1,1,1,2,1,1,1,1,1,1,3,1,3,1,3,1；3,1,2,4,6,5,7,13,10,8,12,15,9,14,11,17,16,18}；

Step F1 is performed: generating a random number of [0,1), which is 0.18, since 0.18 < p_mIf 0.2, go to step F2;

step F2 is performed: randomly selecting a gene from the execution pattern list {2,2,1,1,1,2,1,1,1,1,1, 3,1,3,1}, which is g₈1, a pattern is randomly re-selected from the executable patterns of task 8, which is pattern 2, then g₈＝2；

Step F3 is performed: generating a random number of [0,1), which is 0.03, since 0.03 < p_mIf 0.2, go to step F4;

step F4 is performed: randomly selecting a gene from the scheduling order list {3,1,2,4,6,5,7,13,10,8,12,15,9,14,11,17,16,18}, which is g₂₇Since task 10 has a parent task other than task 0 and a child task other than task 19, the first parent task g is found forward₂₃When the average molecular weight is 6, pos is₁23+ 1-24, find the first subtask g backward₂₉＝12，pos₂At [24,28 ] 29-1-28]Randomly selects a position for insertion, which is g₂₄；

Step F5 is performed: finishing the mutation operation;

post mutation ch'_１＝{2,2,1,1,1,2,1,2,1,1,1,1,3,1,3,1,3,1；3,1,2,4,6,10,5,7,13,8,12,15,9,14,11,17,16,18}。

Similarly, other individuals in the new population become:

ch′₂＝{2,2,1,1,1,1,1,1,3,1,1,1,3,1,3,1,3,1；3,1,2,4,5,6,7,13,10,8,15,14,9,16,11,12,17,18}；

ch′₃＝{2,3,1,1,1,1,1,1,3,1,1,1,3,1,3,1,3,1；2,3,1,6,4,5,7,13,10,8,12,15,14,9,16,11,17,18}；

ch′₆＝{2,2,1,3,1,2,1,1,3,1,1,2,1,1,1,1,3,1；3,1,2,4,6,5,7,13,10,12,8,15,9,14,16,11,17,18}；

ch′₇＝{2,2,1,1,1,1,1,1,2,1,1,2,3,1,1,2,1,2；2,3,1,4,6,5,7,10,13,12,8,14,15,9,11,16,17,18}；

ch′₈＝{1,1,1,1,1,1,1,1,3,1,1,1,3,1,3,2,3,1；1,3,2,6,4,5,13,7,10,8,12,15,14,9,16,11,17,18}；

ch′₉＝{2,2,1,1,1,1,2,1,2,1,1,1,3,1,1,2,1,2；3,1,2,6,5,4,7,10,13,12,8,14,15,9,11,16,17,18}；

ch′₁₀＝{2,2,1,1,1,1,1,1,1,1,1,1,1,1,3,1,3,1；1,3,2,4,5,6,7,13,10,8,12,15,9,14,11,17,16,18}。

executing step 11: improving all individuals in the new population by adopting an FBI & D method and calculating the fitness value of the individuals;

to individual ch'₆2,2,1,3,1,2,1,1,3,1,1,2,1,1,1, 3, 1; 3,1,2,4,6,5,7,13,10,12,8,15,9,14,16,11,17,18} employs FBI&The method D comprises the following specific implementation processes:

step G1 is executed: let θ be 1, for individual ch'₆Performing serial individual decoding based on the insertion mode to obtain all task completion times: f'₁＝3，f′₂＝11，f′₃＝8，f′_４＝11，f′₅＝17，f′₆＝14，f′₇＝24，f′₈＝44，f′₉＝49，f′₁₀＝18，f′₁₁＝58，f′₁₂＝37，f′₁₃＝29，f′₁₄＝49，f′₁₅＝45，f′₁₆＝52，f′₁₇＝61，f′₁₈60 and a fitness value of fit'₆＝61；

Step G2 is executed: changing task 0 to task 19 and task 19 to task 0, reversing the timing relationship between the tasks, i.e.

SC₀＝{t₁₆,t₁₇,t₁₈}；PR₁₈＝{t₀}，SC₁₈＝{t₆,t₁₁,t₁₅}；PR₁₇＝{t₀}，SC₁₇＝{t₁₂,t₁₃,t₁₄}；PR₁₆＝{t₀}，SC₁₆＝{t₉,t₁₀,t₁₅}；PR₁₅＝{t₁₆,t₁₈}，SC₁₅＝{t₈,t₁₃}；PR₁₄＝{t₁₇}，SC₁₄＝{t₁,t₈}；PR₁₃＝{t₁₅,t₁₇}，SC₁₃＝{t₄,t₅}；PR₁₂＝{t₁₇}，SC₁₂＝{t₅,t₁₀}；PR₁₁＝{t₁₈}，SC₁₁＝{t₄,t₉}；PR₁₀＝{t₁₂,t₁₆}，SC₁₀＝{t₄,t₆}；PR₉＝{t₁₁,t₁₆}，SC₉＝{t₈}；PR₈＝{t₉,t₁₄,t₁₅}，SC₈＝{t₂,t₇}；PR₇＝{t₈}，SC₇＝{t₅}；PR₆＝{t₁₀,t₁₈}，SC₆＝{t₁,t₃}；PR₅＝{t₇,t₁₂,t₁₃}，SC₅＝{t₃}；PR₄＝{t₁₀,t₁₁,t₁₃}，SC₄＝{t₁,t₃}；PR₃＝{t₄,t₅,t₆}，SC₃＝{t₁₉}；PR₂＝{t₈}，SC₂＝{t₁₉}；PR₁＝{t₄,t₆,t₁₄}，SC₁＝{t₁₉}；PR₁₉＝{t₁,t₂,t₃}，

Prepared from ch'₆According to the task completion time f 'in the scheduling sequence list'_jRearranging from large to small, i.e. g genes in individuals_J+jSet to the jth last completed task, form individual ch'₆＝{2,2,1,3,1,2,1,1,3,1,1,2,1,1,1,1,3,1；17,18,11,16,14,9,15,8,12,13,7,10,5,6,4,2,3,1}；

Step G3 is executed: theta ═ 2, ch 'for individuals'₆Performing serial individual decoding based on the insertion mode obtains the completion time of all tasks: f'₁＝52，f′₂＝49，f′₃＝52，f′₄＝41，f′₅＝41，f′₆＝44，f′₇＝31，f′₈＝24，f′₉＝17，f′₁₀＝38，f′₁₁＝8，f′₁₂＝32，f′₁₃＝37，f′₁₄＝13，f′₁₅＝14，f′₁₆＝12，f′₁₇＝9，f′₁₈2 and its fitness value of fit'₆＝52；

Step G4 is executed: due to fact'₆＝52＜fit′₆61, then ch'₆＝ch′₆，fit′₆＝fit′₆，f′_j＝f′_jGo to step G2;

step G2 is executed: changing task 0 to task 19, changing task 19 to task 0, reversing the timing relationship between the tasks, i.e.

Prepared from ch'₆According to the task completion time f 'in the scheduling sequence list'_jRearranging from large to small, i.e. g genes in individuals_J+jSet to the jth last completed task, form individual ch'₆＝{2,2,1,3,1,2,1,1,3,1,1,2,1,1,1,1,3,1；1,3,2,6,4,5,10,13,12,7,8,9,15,14,16,17,11,18}；

Step G3 is executed: theta-3, ch 'for individual'₆Performing serial individual decoding based on the insertion mode obtains the completion time of all tasks: f'₁＝3，f′₂＝11，f′₃＝8，f′₄＝14，f′₅＝14，f′₆＝11，f′₇＝27，f′₈＝35，f′₉＝40，f′₁₀＝15，f′₁₁＝49，f′₁₂＝28，f′₁₃＝20，f′₁₄＝40，f′₁₅＝36，f′₁₆＝43，f′₁₇＝52，f′₁₈51 and its fitness value fit'₆＝52；

Step G4 is executed: due to fact'₆＝fit′₆Go to step G5, 52;

step G5 is executed: since θ is an odd number, 3 is output as the individual ch'₆＝ch′₆＝{2,2,1,3,12,1,1,3,1,1,2,1,1,1, 3, 1; 1,3,2,6,4,5,10,13,12,7,8,9,15,14,16,17,11,18}, and fitness value of fit'₆When 52, the operation is finished;

ch 'to the above individuals'₆2,2,1,3,1,2,1,1,3,1,1,2,1,1,1, 3, 1; 3,1,2,4,6,5,7,13,10,12,8,15,9,14,16,11,17,18}, where θ is 1, the implementation procedure of the serial individual decoding based on the insertion mode is as follows:

step H1 is performed: initialization of the system state f₀0, lf 122, and a task completion time set F₀-lf } - {0,122 }; the remaining available amount of renewable resources corresponding to each completion time in F is:

δ＝1；

step H2 is performed: fetch task j equals g_J+δ＝g₁₉3, and the mode m employed is g₃＝1；

Step H3 is performed: calculating the completion time of task 3; i.e. step H3.1: let start time s of task 3₃＝rt₃＝f₀0,1 is given as flag; step H3.2 is performed: where F is ═ F₀And if is greater than or equal to rt is found in {0,122}₃0, the composition set TF {0,122 }; step H3.3 is performed: taking the element with the smallest value from TF {0,122}, wherein the element is 0, and then TF {122 }; step H3.4 is performed: due to 0-s₃＝0＜d_3,1Go to step H3.5, 8; step H3.5 is performed: since flag is 1 at this time, go directly to step H3.6; step H3.6 is performed: due to the fact that

At time 0, the remaining available amount of all renewable resources meets the requirements of task 3, so the process goes directly to step H3.3; step H3.3 is performed: take one value from TF {122}The smallest element, which is 122, then

Step H3.4 is performed: since flag is 1 and 122-s₃＝122＞d_3,1Go to step H3.7, 8; step H3.7 is performed: f. of₃＝s₃+d_3,1＝0+8＝8；

Step H4 is performed: due to the fact that

Then F ═ F₀,f₃And if is 0,8,122, f is increased₃The remaining available amount of resources can be updated at time 8:

step H5 is performed: update [ s ]₃,f₃) 1, [0,8) remaining available amount of renewable resource during the time period:

step H6 is performed: if δ is 1+1 — 2, go to step H2, since δ is 2 ≦ J ≦ 18;

step H2 is performed: fetch task j equals g₁₈₊₂＝g₂₀1, and the mode m employed is g₁＝2；

Step H3 is performed: calculating the completion time of task 1; i.e. step H3.1: let start time s of task 1₁＝rt₁＝f₀0,1 is given as flag; step H3.2 is performed: where F is ═ F₀,f₃In {0,8,122}, rt or more is found₁0, the composition set TF ═ {0,8,122 }; step H3.3 is performed: taking the element with the smallest value from TF {0,8,122}, wherein the element is 0, and then TF {8,122 }; step H3.4 is performed: due to 0-s₁＝0＜d_1,2Go to step H3.5, No. 3; step H3.5 is performed: since flag is 1 at this time, go directly to step H3.6; step H3.6 is performed: due to the fact that

At time 0, the remaining available amount of all renewable resources meets the requirement of task 1, so the process directly goes to step H3.3; step H3.3 is performed: taking the element with the smallest value from TF {8,122}, wherein the element is 8, and then TF {122 }; step H3.4 is performed: since flag is 1 and 8-s₁＝8＞d_1,2Go to step H3.7, No. 3; step H3.7 is performed: f. of₁＝s₁+d_1,2＝0+3＝3；

Step H4 is performed: due to the fact that

Then F ═ F₀,f₁,f₃And if is {0,3,8,122}, increasing f₁The remaining available amount of resources can be updated at time 3:

step H5 is performed: update [ s ]₁,f₁) 1, [0,3) remaining available amount of renewable resource during the time period:

step H6 is performed: if δ 2+1 is 3, go to step H2, since δ 3 ≦ J is 18;

step H2 is performed: fetch task j equals g₁₈₊₃＝g₂₁2, and the mode m employed is g₂＝2；

Step H3 is performed: calculating the completion time of task 2; i.e. step H3.1: let start time s of task 2₂＝rt₂＝f₀0,1 is given as flag; step H3.2 is performed: where F is ═ F₀,f₁,f₃Where rt is equal to or greater than f, found from {0,3,8,122}₃0, the composition set TF ═ {0,3,8,122 }; step H3.3 is performed: taking the element with the smallest value from TF {0,3,8,122}, and taking the element with the smallest value as 0, then TF ═ {3,8,122 }; step H3.4 is performed: due to 0-s₂＝0＜d_2,2Go to step H3.5, 8; step H3.5 is performed: since flag is 1 at this time, go directly to step H3.6; step H3.6 is performed: due to the fact that

That is, at time 0, the remaining available amount of renewable resource 1 cannot meet the requirement of task 2, if flag is equal to 0, go to step H3.3; step H3.3 is performed: taking the element with the smallest value from TF {3,8,122}, wherein the element is 3, and then TF {8,122 }; step H3.4 is performed: since flag is 0, go to step H3.5; step H3.5 is performed: since flag is equal to 0, s is₂3,1 is flag; step H3.6 is performed: due to the fact that

3, the residual available quantity of all renewable resources meets the requirement of the task 2, so the step H3.3 is directly carried out; step H3.3 is performed: taking the element with the smallest value from TF {8,122}, wherein the element is 8, and then TF {122 }; step H3.4 is performed: due to 8-s₂＝5＜d_2,2Go to step H3.5, 8; step H3.5 is performed: since flag is 1 at this time, go directly to step H3.6; step H3.6 is performed: due to the fact that

At time 8, the residual available amount of all renewable resources meets the requirement of the task 2, so the step H3.3 is directly carried out; step H3.3 is performed: if the element with the smallest value is 122 out of TF 122, then

Step H3.4 is performed: since flag is 1 and 122-s₁＝119＞d_2,2Go to step H3.7, 8; step H3.7 is performed: f. of₂＝s₂+d_2,2＝3+8＝11；

Step H4 is performed: due to the fact that

Then F ═ F₀,f₁,f₃,f₂And if is {0,3,8,11,122}, is increasedf₂The remaining available amount of resources can be updated at time 11:

step H5 is performed: update [ s ]₂,f₂) Remaining available amount of renewable resources in [3,11) time period:

step H6 is performed: if δ is 3+1 — 4, go to step H2, since δ is 4 ≦ J ≦ 18;

……

step H2 to step H6 are repeated until δ 19 > J18, and go to step H7;

step H7 is performed: obtaining completion times for all tasks: f'₁＝3，f′₂＝11，f′₃＝8，f′₄＝11，f′₅＝17，f′₆＝14，f′₇＝24，f′₈＝44，f′₉＝49，f′₁₀＝18，f′₁₁＝58，f′₁₂＝37，f′₁₃＝29，f′₁₄＝49，f′₁₅＝45，f′₁₆＝52，f′₁₇＝61，f′₁₈60; calculating the fitness value: due to the fact that

That is, the subject is a viable subject, whose fitness value is fit'₆＝ms′₆＝max{f′₁₆,f′₁₇,f′₁₈The operation ends 61.

Similarly, other individuals in the new population become:

ch′₁＝{2,2,1,1,1,2,1,2,1,1,1,1,3,1,3,1,3,1；3,1,2,4,6,10,5,7,13,8,12,15,9,14,11,17,16,18}；

ch′₂＝{2,2,1,1,1,1,1,1,3,1,1,1,3,1,3,1,3,1；1,2,3,5,4,6,13,7,8,15,10,9,14,16,12,17,11,18}；

ch′₃＝{2,3,1,1,1,1,1,1,3,1,1,1,3,1,3,1,3,1；2,3,1,6,5,4,13,7,8,15,10,12,9,14,16,17,11,18}；

ch′₄＝{2,2,1,1,1,2,1,1,3,1,1,1,1,1,3,1,3,1；1,2,3,5,4,13,7,6,8,15,10,12,9,14,16,17,11,18}；

ch′₅＝{2,2,1,1,1,2,1,1,1,1,1,1,3,1,3,1,3,1；1,3,2,6,5,4,13,7,10,8,12,15,14,9,16,17,11,18}；

ch′₆＝{2,2,1,3,1,2,1,1,3,1,1,2,1,1,1,1,3,1；1,3,2,6,4,5,10,13,12,7,8,9,15,14,16,17,11,18}；

ch′₇＝{2,2,1,1,1,1,1,1,2,1,1,2,3,1,1,2,1,2；2,3,1,6,5,4,10,13,12,7,8,14,9,15,11,16,17,18}；

ch′₈＝{1,1,1,1,1,1,1,1,3,1,1,1,3,1,3,2,3,1；1,3,2,6,5,4,13,7,8,15,10,12,9,14,16,17,11,18}；

ch′₁₀＝{2,2,1,1,1,1,1,1,1,1,1,1,1,1,3,1,3,1；1,3,2,4,5,6,7,13,10,8,12,15,9,14,11,17,16,18}；

the fitness values are respectively as follows: fist'₁＝54，fit′₂＝55，fit′₃＝53，fit′₄＝55，fit′₅＝55，fit′₆＝52，fit′₇＝57，fit′₈＝57，fit′₉＝57，fit′₁₀＝58。

And step 12 is executed: selecting N different individuals from the current generation population and the new population from good to bad to form a new current generation population;

selecting the ch of the current generation population from good to bad according to the fitness value₁，ch₅，ch₈，ch₉And ch 'of the New population'₁，ch′₂，ch′₃，ch′₄，ch′₅，ch′₆Forming the next generation of population, i.e. GP ═ ch₁,ch₅,ch₈,ch₉,ch′₁,ch′₂,ch′₃,ch′₄,ch′₅,ch′₆}；

Let CP be GP; the contemporary population becomes:

ch₂＝{2,2,1,1,1,2,1,1,3,1,1,1,3,1,3,1,3,1；3,1,2,4,6,5,7,13,10,8,12,15,14,9,16,11,17,18}；

ch₃＝{2,2,1,1,1,1,1,1,1,1,1,1,3,1,3,1,3,1；3,1,2,4,5,6,7,13,10,8,12,15,9,14,11,17,16,18}；

ch₄＝{1,2,1,1,1,1,1,1,2,1,1,1,3,1,1,2,1,2；1,3,2,6,5,4,7,10,13,12,8,14,15,9,11,16,17,18}；

ch₅＝{2,2,1,1,1,2,1,2,1,1,1,1,3,1,3,1,3,1；3,1,2,4,6,10,5,7,13,8,12,15,9,14,11,17,16,18}；

ch₆＝{2,2,1,1,1,1,1,1,3,1,1,1,3,1,3,1,3,1；1,2,3,5,4,6,13,7,8,15,10,9,14,16,12,17,11,18}；

ch₇＝{2,3,1,1,1,1,1,1,3,1,1,1,3,1,3,1,3,1；2,3,1,6,5,4,13,7,8,15,10,12,9,14,16,17,11,18}；

ch₈＝{2,2,1,1,1,2,1,1,3,1,1,1,1,1,3,1,3,1；1,2,3,5,4,13,7,6,8,15,10,12,9,14,16,17,11,18}；

ch₉＝{2,2,1,1,1,2,1,1,1,1,1,1,3,1,3,1,3,1；1,3,2,6,5,4,13,7,10,8,12,15,14,9,16,17,11,18}；

ch₁₀＝{2,2,1,3,1,2,1,1,3,1,1,2,1,1,1,1,3,1；1,3,2,6,4,5,10,13,12,7,8,9,15,14,16,17,11,18}；

the fitness values are respectively fit₁＝53，fit₂＝52，fit₃＝54，fit₄＝55，fit₅＝54，fit₆＝55，fit₇＝53，fit₈＝55，fit₉＝55，fit₁₀＝52；

And step 13 is executed: judging whether a second-stage evolution termination condition is met, if so, turning to a step 14, otherwise, turning to a step 10;

the second stage evolution termination condition is set as population evolution TG₂15 generation;

since the current population has evolved only one generation in the second stage and does not satisfy the second stage iteration termination condition, go to step 10.

……

Thus, steps 10 to 13 are repeated, and after 15 generations of second stage iteration, the contemporary population becomes:

ch₁＝{1,2,1,1,1,1,1,1,1,1,1,1,3,3,1,2,3,1；1,3,2,4,6,5,7,10,13,8,12,14,15,9,11,17,16,18}；

ch₂＝{2,2,1,1,1,2,1,1,3,1,1,1,3,3,1,1,3,1；3,1,2,6,4,5,10,7,13,8,12,9,15,14,16,11,17,18}；

ch₃＝{2,2,1,1,1,2,1,1,1,3,1,1,3,3,2,2,3,1；1,3,2,6,5,4,13,7,8,10,12,14,9,15,17,11,16,18}；

ch₄＝{2,2,1,1,1,2,1,1,1,1,1,2,3,3,1,2,3,1；1,3,2,6,5,4,13,7,8,14,10,12,9,15,17,11,16,18}；

ch₅＝{1,2,1,1,1,1,1,1,1,1,1,1,3,3,1,2,1,2；1,3,2,6,5,4,7,10,13,8,12,14,15,9,11,16,17,18}；

ch₆＝{2,2,1,1,1,2,1,1,1,1,1,1,3,3,2,2,1,2；3,1,2,6,5,4,7,10,13,8,12,14,9,15,11,16,17,18}；

ch₇＝{2,2,1,1,1,2,1,1,1,1,1,1,3,3,2,2,3,1；3,1,2,4,6,5,7,10,13,8,12,14,9,15,11,17,16,18}；

ch₈＝{2,2,1,1,1,2,1,1,1,1,1,1,3,3,2,2,3,1；3,1,2,6,5,4,7,10,13,8,12,14,9,15,11,17,16,18}；

ch₉＝{2,2,1,1,1,1,1,1,3,1,1,1,3,3,1,1,3,1；3,1,2,5,4,6,10,7,13,8,12,9,15,14,16,11,17,18}；

ch₁₀＝{1,2,1,1,1,1,1,1,1,1,1,1,3,3,1,2,1,2；1,3,2,4,6,5,7,10,13,8,12,14,15,9,11,16,17,18}；

the fitness values are respectively as follows: fit₁＝53，fit₂＝49，fit₃＝49，fit₄＝48，fit₅＝54，fit₆＝52，fit₇＝51，fit₈＝50，fit₉＝51，fit₁₀＝55；

Since the population in the second stage is evolved for 15 generations, and the termination condition of the second stage evolution is satisfied, the second stage evolution is ended, and the step 14 is carried out.

And step 14 is executed: if the optimal individual in the contemporary population is a feasible individual, outputting a scheduling scheme corresponding to the optimal individual and the project execution time thereof

Otherwise the problem has no feasible scheduling scheme;

the best individual in the contemporary population is

ch₄2,2,1,1,1,2,1,1,1,1, 2,3,3,1,2,3, 1; 1,3,2,6,5,4,13,7,8,14,10,12,9,15,17,11,16,18}, which are feasible individuals, with an item execution time ms equal to 48, and the corresponding scheduling scheme is shown in table 6.

TABLE 6

The above embodiments are only preferred embodiments of the present invention, and are not intended to limit the technical solutions of the present invention, so long as the technical solutions can be realized on the basis of the above embodiments without creative efforts, which should be considered to fall within the protection scope of the patent of the present invention.

Claims

1. A multi-mode resource-limited project scheduling optimization method adopting a two-stage genetic algorithm is characterized by comprising the following steps of: the method comprises the following steps:

acquiring a project task set T ═ T₀,t₁,...,t_J,t_J+1}; wherein t is_jIndicating task j, i.e. task numbered j, t₀And t_J+1Is a virtual task artificially increased, namely, the project period and the resources are not occupied, J is the requirementThe actual number of tasks to be scheduled;

t_J+1Without subtasks, i.e.

Obtaining the usable quantity R of renewable resources k at any time_kK is 1, …, K is the renewable resource type number; acquiring the available quantity N of the non-updatable resource l in the whole project period_lL ═ 1, …, L is the number of non-updateable resource types;

step 2: if there is an un-updateable resource/satisfy

Or that there is one task j and renewable resources k satisfying

step 3.1: let the pattern flag value mflg ═ 0, resource flag value rflg ═ 0; if r is present_j,m,k>R_kOr n_j,m,l>N_lThen mode m is invalid for task j, removing mode m in task j: order to

Wherein:

k1, …, K, L1, …, L; let M_j＝M_j-1；

step 3.3: if there is task j, patterns m and m' satisfy: d_j,m′≤d_j,m，r_j,m′,k≤r_j,m,k，n_j,m′,l≤n_j,m,lM' ≠ m, K ≠ 1, …, K, L ═ 1, …, L; then pattern m is redundant for task j, removing pattern m in task j: order to

Wherein:

k1, …, K, L1, …, L; let M_j＝M_j-1，rflg＝0；

step 3.5: if the renewable resource k satisfies:

Wherein: j-1, …, J,

let K-1, mflg-0;

step 3.6: if the non-updateable resource/satisfies:

Wherein: j-1, …, J,

let L-1, mflg-0;

step 3.7: if mflg ═ 1 ^ rflg ═ 1, go to step 3.8, otherwise go to step 3.2;

step 3.8: finishing the pretreatment;

and 4, step 4: calculating the hierarchy value of the task:

for a starting task 0 without a parent task, its hierarchy value is:

lvl₀＝0 (1)

and 5: initializing a contemporary population;

the individual adopts 2J bit integer coding, and the method comprises the following steps: ch ═ g₁,…,g_J；g_J+1,…,g_2J}, gene g_jIs a positive integer, where g₁,…,g_JIs the execution mode list, g_jIndicating task j adoptionMode (1), g_j＝1,...,M_j，j＝1,…,J；{g_J+1,…,g_2JIs a scheduling order list, g_J+1,…,g_2JIs an arrangement of 1, …, J and satisfies the timing constraint of a task, i.e. no task can be placed in front of its parent, g_J+jIndicating the jth scheduled task, i.e. task g_J+jIs jth scheduled;

k＝1,…,K；

step B2: fetch task j equals g_J+δLet mode m equal to 1;

step B3: calculating the completion time f of task j_jNamely:

step B3.1: let start time of task j

Let flag value flag be 1;

Step B3.4: if flag is 1 and f_j′-s_j,m≥d_j,mThen go to step B3.7; otherwise go to step B3.5;

step B3.5: if flag is equal to 0, let s_j,m＝f_j′，flag＝1；

If not, the flag is 0; go to step B3.3;

step B3.7: let f_j,m＝s_j,m+d_j,m；

Taking out with minimum time of completion

Order to

Go to step B4;

wherein: f. of_j″Is f_jThe previous time of day;

step B6: let δ be δ +1, go to step B2 if δ ≦ J, otherwise go to step B7;

step B7: obtain execution mode list g₁,…,g_JCalculating the fitness value fit of the individual, and finishing the operation;

wherein:

representing the remaining available amount of resource k at time instant tau,

indicating the ready time of task j;

step 6: carrying out N/2 times of scheduling sequence cross operation based on the hierarchy on the contemporary population to generate a new population, and carrying out scheduling sequence variation operation based on the hierarchy on each individual in the new population;

step C2: randomly selecting a hierarchy without setting to be l;

step D1: generating a random number λ ∈ [0,1) if λ<p_mGo to step D2, otherwise go to step D4;

wherein: p is a radical of_m∈(0,1]Is the rate of variation;

the first stage termination condition is that the iteration is carried out to a specified algebraic TG₁Or successive iterations GG₁No improvement was made for the best individuals; step 10: performing N/2 times of cross operation of the execution mode list and the scheduling sequence list on the contemporary population to generate a new population, and performing variation operation on the execution mode list and the scheduling sequence list on each individual in the new population;

the daughter 1 and daughter 2 generated were not:

step E2: if it is not

And

if not, go to step E3, otherwise

Go to step E8;

step E3: finding out from front to back

And

And

Go to step E7;

step E4: randomly generating a delta₁To delta₂-1 isThe integers α;

Step E7: if it is not

And

if not, go to step E8; otherwise

Go to step E12;

step E8: finding out from front to back

And

position delta of the 1 st different gene of (1)₃From the back to the front

And

position delta of the 1 st different gene of (1)₄；

Step E9: randomly generating a delta₃To delta₄-a positive integer β of 1;

The last J- β genes are from ch^p2Scheduling order list of

Mid-delete ch^p1The first β genes of the scheduling order list of

The latter gene list;

The last J- β genes are from ch^p1Scheduling order list of

Mid-delete ch^p2The first β genes of the scheduling order list of

The latter gene list;

step E12: to obtain

Finishing the cross operation;

step F1: generating a random number lambda₁E [0,1), if λ₁<p_mGo to step F2; otherwise go to step F3;

Step F3: generating a random number lambda₂E [0,1), if λ₂<p_mGo to step F4; otherwise go to step F5;

step F4: from the scheduling order list g_J+1,…,g_2JRandomly selecting one gene, not setting as g_jJ +1, …,2J if task g_jThere is a parent task other than task 0, then from g_jBegin finding the first parent g ahead_j′Let the position value pos₁J' +1, otherwise let pos₁J + 1; if task g_jThere are subtasks other than task J +1, then from g_jStarting to find the first subtask g backwards_j″Let the position value pos₂J "-1, otherwise let pos₂2J; in [ pos ]₁,pos₂]Randomly selects a position to insert g_j；

Step F5: finishing the mutation operation;

the FBI & D method comprises the steps of:

step G2: changing task 0 into task J +1 and task J +1 into task 0, reversing the time sequence relationship between tasks, i.e. changing the father task set of tasks into the son task set, changing the son task set into the father task set, and according to the task completion time f, the scheduling sequence list in individual ch_jRearranging from large to small, i.e. g genes in individuals_J+jSet to the jth last completed task, J1, …, J, forming an individual

Step G3: let theta be theta +1, for individual

Performing serial individual decoding based on insertion mode to obtain completion time of all tasks

And its fitness value

Step G4: if it is

Then

Go to step G2, otherwise go to step G5;

step G5: if theta is even number, then output individual ch, otherwise output individual ch

The fitness value fit is obtained, and the operation is finished;

step H1: initializing a system state: order completion time f of task 0₀0, F₀Lf, the elements in F are arranged from small to large; making a cycle control variable delta equal to 1; let the remaining available amount of the renewable resource corresponding to each completion time in F:

step H2: fetch task j equals g_J+δAnd the mode m ═ g adopted by it_j；

Step H3: calculating the completion time of task j

Namely:

step H3.1: let start time s of task j_j＝rt_jLet flag value flag be 1;

step H3.5: if flag is equal to 0, let s_j＝f_j′，flag＝1；

If not, if the flag is 0, the step goes to a step H3.3;

step H3.7: let f_j＝s_j+d_j,m；

wherein: f. of_j″Is f_jThe previous time of day;

step H6: let δ be δ +1, go to step H2 if δ ≦ J, otherwise go to step H7;

Otherwise the problem has no feasible scheduling scheme.

2. The method for scheduling and optimizing the multi-mode resource-constrained project by using the two-stage genetic algorithm as claimed in claim 1, wherein: the specific calculation method of the fitness value fit is as follows: if the individual is an infeasible individual, i.e., presentOne non-updatable resource, < i > satisfy >

executing time for the project;

the smaller the fitness value fit, the better the individual.

3. The method for scheduling and optimizing the multi-mode resource-constrained project by using the two-stage genetic algorithm as claimed in claim 1, wherein: the specific steps of randomly selecting two different individuals from the contemporary population as the father 1 and the father 2 based on the sorted round-robin in the steps C1 and E1 are as follows:

step I2: calculating the probability that the individual n is selected

R>1 is a discrimination coefficient;

step I3: calculating the cumulative probability:

step I4: generating a random number lambda₁E [0,1) if

Then individual n is selected as father 1;

step I5: generating a random number lambda₂E [0,1) if

And n '≠ n, then individual n' is selected as parent 2, proceeding to step I6, otherwise to step I5;

step I6: the individual selection operation ends.