CN111078381A - Cloud workflow scheduling optimization method based on two-dimensional coding genetic algorithm - Google Patents

Abstract

The invention discloses a cloud workflow scheduling optimization method based on a two-dimensional coding genetic algorithm, which comprises the following steps of: acquiring scheduling information; calculating a task level value; initializing a current generation population based on the hierarchy and the number of subtasks; evolution is carried out: improving the current generation population and calculating a fitness value by adopting FBI & D and LDI methods, carrying out parameterization uniform cross operation based on preference and two-dimensional topological sorting to form a new population, carrying out mutation operation on the new population, improving the new population and calculating the fitness value by adopting the FBI & D and LDI methods, and selecting N different individuals from the current generation population and the new population to form a new current generation population until an evolution termination condition is met; and outputting the scheduling optimization scheme. The two-dimensional integer coding method adopted by the invention can reduce the coding space of the algorithm and improve the efficiency and quality of solving.

Description

Cloud workflow scheduling optimization method based on two-dimensional coding genetic algorithm

Technical Field

The invention relates to the field of computer technology, information technology and system engineering, in particular to a cloud workflow scheduling optimization method, and more particularly relates to a cloud workflow scheduling optimization method based on a two-dimensional coding genetic algorithm.

Background

The workflow under the cloud computing environment, called the cloud workflow for short, is the integration of cloud computing and related technologies of the workflow, and has wide application prospects in the fields of cross-organization business collaboration, scientific computing and the like which need high-efficiency computing performance and large-scale storage support. In the cloud workflow, timing constraints exist among tasks, and the tasks are generally received and processed by taking a virtual machine as a minimum allocation unit of computing resources during execution. The cloud workflow scheduling refers to how to allocate tasks in the cloud workflow to appropriate virtual machines and how to arrange execution sequences of the tasks allocated to the virtual machines under the condition that the task timing and user requirement constraints are met, namely, the problem in two aspects is solved: task allocation and task execution order. The cloud workflow scheduling directly determines the performance of the whole cloud workflow system, and becomes an important research content of the cloud workflow system.

The current cloud workflow scheduling optimization method can be divided into three categories:

1) the heuristic method means that the distribution and execution sequence of the workflow tasks are generated by the heuristic method, such as: hetereogenous early stage Finish Time (HEFT), Critical Path On a Processors (CPOP), Leveled Min Time (LMT), Dynamic Level Scheduling (DLS), Dynamic Critical Path (DCP), and Localized Dynamic Critical Path (LDCP);

2) the intelligent computing method is characterized in that the distribution and execution sequence of the workflow tasks are searched and generated by the intelligent computing method; such as: genetic algorithm GA, particle swarm optimization PSO, simulated annealing SA and other methods;

3) the semi-intelligent calculation method combined with the heuristic method is that the workflow task allocation is generated by searching through an intelligent calculation method, the task execution sequence is generated by searching through the intelligent calculation method according to a task allocation scheme generated by searching through the intelligent calculation method by adopting a heuristic method based on priority, or the workflow task execution sequence is generated by searching through the intelligent calculation method, and the task allocation is generated by searching through the task execution sequence generated by searching through the intelligent calculation method through a heuristic method based on the earliest completion time.

However, the existing cloud workflow scheduling optimization methods have the following disadvantages:

1) heuristic methods can obtain a scheduling optimization in a short time, but their quality is usually not very high and depends on the type of workflow;

2) the algorithm efficiency of the intelligent calculation method depends on the design of coding and decoding, evolution iteration strategies, selection of control parameters and the like, wherein a solution space, namely a scheduling scheme, searched by combining a heuristic semi-intelligent calculation method is incomplete, so that the possibility that the optimal scheduling scheme cannot be searched theoretically exists, and meanwhile, the time efficiency is not very high because the heuristic method is required to be continuously called in the algorithm; the intelligent calculation method can theoretically realize global search, but the adoption of global search can reduce the search efficiency;

therefore, with the increase of the complexity of the cloud workflow and the application requirements thereof, it is urgently needed to design a more efficient method to solve the problem of cloud workflow scheduling optimization.

Disclosure of Invention

In order to overcome the defects that the quality of a heuristic method solution is usually not very high and depends on the type of a workflow, the imperfection of a heuristic semi-intelligent computing method and an intelligent computing method coding search space based on layered coding are combined, a large number of redundant coding search spaces exist due to the fact that a large number of many-to-one relations exist between individuals and a scheduling scheme, the search efficiency is reduced due to the adoption of global search, and the like in the intelligent computing method based on one-dimensional coding, the cloud workflow scheduling optimization method based on the two-dimensional coding genetic algorithm is provided, and the efficiency and the quality of solving are effectively improved.

The technical scheme adopted by the invention for solving the technical problems is as follows: a cloud workflow scheduling optimization method based on a two-dimensional coding genetic algorithm comprises the following steps:

step 1: formalizing a scheduling problem, and acquiring information required by scheduling optimization;

get task set T ═ T₁,t₂,...,t_IWhere I is the number of tasks, t_iRepresenting a task i, namely a task with the number i;

acquiring a time sequence relation between tasks: parent task set PR of task i_iTask ofi subtask set SC_iWherein I is 1,2 …, I;

acquiring task related parameters: length t of task i_iLength, i.e. the number of instructions that need to be consumed by a virtual machine when task i is processed, list t of input files that need to be processed when task i is processed_iIFL, output File List t generated after task i is processed_i-OFL, and the size of the file in the file list, I ═ 1,2 …, I; task i is task i⁺The requirements of the parent task of (2) are: there is a file that is the output file of task i and is also task i⁺The input file of (a), namely:

obtaining a virtual machine set VM ═ VM in a cloud computing environment₁,vm₂,…,vm_JWhere J is the number of virtual machines, vm_jRepresents virtual machine j, i.e., the virtual machine numbered j;

acquiring related parameters of the virtual machine: computing power vm of virtual machine j_jPs, bandwidth vm of virtual machine j_jBw, wherein J ═ 1,2 …, J;

acquiring a support relationship between the task and the virtual machine: task set T that virtual machine j can handle_jWherein J is 1,2 …, J; virtual machine set VM capable of processing task i_iWherein I is 1,2 …, I;

step 2: calculating the level value level of the task;

for a starting task i without a parent task, the hierarchy value is:

level_i＝1 (1)

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

and step 3: initializing a contemporary population;

initializing a current generation population, wherein the current generation population comprises 1 individual generated by an individual generation method based on HEFT-baseline and the remaining N-1 different individuals generated by an individual random generation method based on a hierarchy, and N is the population scale;

the individuals adopt two-dimensional integer coding, and the method comprises the following steps:

wherein g is_j,iIndicating the number assigned to virtual machine j and in virtual machine j the ith scheduled task, e.g. g_3,25 indicates that task 5 is assigned to virtual machine 3 and is scheduled 2 nd among all tasks assigned to virtual machine 3;

is a sequence of task numbers assigned to virtual machine j and satisfies a task timing constraint, I_jRepresenting the number of tasks assigned to virtual machine j;

the individual generating method based on HEFT-baseline comprises the following steps:

step A1: make virtual machine available a time period list vatl_j＝{[0,M]}, variable k_j1, J-1, 2, …, J, where M is a near infinite number; enabling a task set UT to be processed to be T; let ready times rt of all tasks_i＝0，i＝1,…,I；

Step A2: finding out the task with the minimum level value in the UT, taking out the task with the maximum number of subtasks from the task, and if the task with the maximum number of subtasks is more than one, randomly taking one of the tasks without setting t as t_i；

Step A3: make available virtual machine set AVM_i＝VM_iCalculating t_iAre respectively allocated to AVM_iT after each virtual machine in (1)_iCompletion time of (d):

step A3.1: slave AVM_iGet one virtual machine out of it, set to vm_j；

Step A3.2: calculating t_iTo vm_jExecution time after processing

Step A3.3: in vatl_jFinding out an idle time period [ v ] from morning to evening_j,υ_j]Satisfy upsilon_j-ν_j≥et_i,jAnd upsilon_j-et_i,j≥rt_i；

Step A3.4: calculating t_iTo vm_jStart time after treatment s_i,j＝max{ν_j,rt_iH, completion time f_i,j＝s_i,j+et_i,j；

Step A3.5: if AVM_iIf not, go to step A3.1, otherwise go to step A4;

step A4: finding earliest achievable t in virtual machine order_iVirtual machine of (2) is not set to vm_j(ii) a Order to

Handle t_iTo vm_j：

Step A4.1: let t_iStart time s of_i＝s_i,j，t_iCompletion time f_i＝f_i,j；

Step A4.2: updating t_iReady time of subtask of (2)

Step A4.3: list of time slots available in virtual machine, vatl_jDeletion of [ v ]_j,υ_j]V, with insertion interval length greater than 0_j,s_i]And [ f_i,υ_j]；

Step A5: if UT is not empty, go to step A2, otherwise go to step A6;

step A6: outputting an individual based on HEFT-baseline

Finishing the operation;

wherein:

ω_i,j: is vm_jTreatment t_iThe time of (a) is,

is a handle t_iTo vm_jThe processing needs to obtain the file transfer time of the input file from other virtual machines,

is to treat

The virtual machine of (1);

τ_i,j: is a handle t_iTo vm_jThe process requires obtaining the file transfer time of the input file from the shared database,

the individual random generation method based on the hierarchy comprises the following steps:

step B1: let UT be T, let k be variable_j＝1，j＝1,2,…,J；

Step B2: randomly fetch a task with the minimum hierarchy value from UT, and set t as t_i；

Step B3: slave VM_iRandomly selecting one virtual machine in the virtual network, and setting the virtual machine not to be vm_j；

Step B4: order to

k_j＝k_j+1；

Step B5: if UT is not empty, go to step B2; otherwise go to step B6;

step B6: outputting an individual

Finishing the operation;

and 4, step 4: improving all individuals in the contemporary population by adopting FBI & D and LDI methods and calculating the fitness value of the individuals;

the fitness value is workflow response time rs, and the calculation method is as follows:

wherein: rf (radio frequency)_iIs the response time of the task i,

SFL_iis the set of output files that task i outputs to the shared database, i.e.

j is the virtual machine number of processing task i, i.e. satisfies

J of (1);

the smaller the individual fitness value is, the better the individual is;

the FBI & D method comprises the steps of:

step C1: make reverse workflow response time

Wherein M is a number approaching infinity;

step C2: decoding the individual ch by adopting a serial individual decoding method based on an insertion mode to obtain the completion time f of all tasks₁,…,f_IAnd its workflow response time rs; if rs is less than

Go to step C3, otherwise, go to step C6;

step C3: corresponding gene sequence of each virtual machine j in individual ch

Rearranging according to the task completion time from large to small, J is 1, …, J, forming reverse individuals

Step C4: to pair

Decoding by adopting a serial reverse individual decoding method based on an insertion mode to obtain reverse completion time of all tasks

And its reverse workflow response time

If it is

If the value is less than rs, go to step C5, otherwise, go to step C6;

step C5: make the opposite direction single body

The gene sequence corresponding to each virtual machine j in the database

Rearranging according to the reverse completion time of the task from large to small, wherein J is 1, …, J, forming an individual ch, and turning to the step C2;

step C6: outputting the individual ch and the fitness value thereof, namely the workflow response time rs, and finishing the operation;

the LDI method comprises the following steps:

step D1: ordering task lists assigned to virtual machines

Calculating each virtual machine load

Step D2: finding out a virtual machine j' with the minimum load; if ld is_j′>0, go to step D3, otherwise go to step D4;

step D3: order task set

Go to step D5;

step D4: order task set ST_j′＝T_j′Go to step D5;

step D5: if ST_j′If not, from ST_j′Sequentially fetching a task i 'with the highest load of the virtual machine j' to go to step D6; otherwise go to step D7;

step D6: in that

Searching the last ancestor task number and the first descendant task number of the task i 'from front to back, and if both the ancestor task number and the first descendant task number are found, randomly searching a position between the last ancestor task number and the first descendant task number to insert the task i'; if the descendant task number does not exist and the ancestor task number exists, randomly finding a position behind the last ancestor task number to insert i'; if the descendant task number exists and the ancestor task number does not exist, randomly finding a position before the first descendant task number and inserting i'; if the descendant task number and the ancestor task number do not exist, randomly finding a position to insert i'; in that

Deleting i' to form new individual, using FBI&D, the method carries out decoding improvement on the new individual, if the new individual is improved, the improved individual is used for replacing the original individual, and the step D7 is carried out; otherwise go to step D5;

step D7: the LDI operation is finished;

the definition of the descendant task and the ancestor task is described as follows: if there is a task sequence

Satisfy the requirement of

Is that

Where 1 is not more than k<n is then

Is that

The task of the ancestor of (c),

is that

The descendant task of (2);

and 5: judging whether a termination condition is met, if so, turning to a step 10 after the evolution is finished, otherwise, turning to a step 6;

the termination condition is that the optimal individual is not improved after iteration to a designated generation TG or continuous iteration GG generation;

step 6: carrying out N times of parameterized uniform cross operations based on preference and two-dimensional topological sorting on the contemporary population to form a new population;

the parameterized uniform crossover operation based on preference and two-dimensional topological ordering comprises the following steps:

step E1: randomly selecting two different individuals from the contemporary population as parents by adopting a ranking-based round-robin method, and setting the individuals as parents

And ch^p1Has a fitness value of ch or less^p2Wherein: task number list

Task number list

J is 1, …, J, order the task number list

Making a cycle control variable delta equal to 1;

step E2: if δ ≦ I, go to step E3, otherwise go to step E6;

step E3: generating a random number λ ∈ [0,1) if λ<p_bGo to step E4, otherwise go to step E5;

step E4: from ch^p1Randomly selecting one task which is not empty and is numbered as the first element of the task, wherein the parent task does not exist in the task or the parent task is scheduled

From

Take out the first element i' and add it to

And from ch^p2If i', δ is δ +1, go to step E2;

step E5: from ch^p2Randomly selecting one task which is not empty and is numbered as the first element of the task, wherein the parent task does not exist in the task or the parent task is scheduled

From

Take out the first element i' and add it to

And from ch^p1Deleting i ", δ ═ δ +1, go to step E2;

step E6: output sub-body

Finishing the operation;

wherein: p is a radical of_bE (0.5,1) is the preference rate;

and 7: performing mutation operation on each individual in the new population;

the mutation operation comprises the following steps:

step F1: randomly generating 1 random number lambda epsilon [0,1) if lambda is less than or equal to p_mGo to step F2, otherwise go to step F5;

step F2: randomly selecting a task number i, and deleting a gene with the value of i in an individual;

step F3: slave VM_iRandomly selecting a virtual machine again, and setting the virtual machine as a virtual machine j;

step F4: in that

Searching the last ancestor task number and the first descendant task number of the task i from front to back, and if both the last ancestor task number and the first descendant task number are found, randomly finding a position between the last ancestor task number and the first descendant task number to insert the i; if the descendant task number does not exist and the ancestor task number exists, randomly finding a position behind the last ancestor task number to insert i; if the descendant task number exists and the ancestor task number does not exist, randomly finding a position before the first descendant task number to insert i; if the descendant task number and the ancestor task number do not exist, randomly finding a position to insert i;

step F5: finishing individual variation operation;

wherein: p is a radical of_mE [0,1) is the mutation rate;

and 8: improving all individuals in the new population by adopting FBI & D and LDI methods and calculating the fitness value of the individuals;

and step 9: selecting N different individuals from the current generation population and the new population from the superior to the inferior to form a new current generation population, and turning to the step 5;

step 10: and outputting the optimal individuals in the contemporary population, wherein the corresponding scheduling scheme is an optimal scheme.

Further, the serial individual decoding method based on the insertion mode in step C2 includes the following specific steps:

step G1: let ready times rt of all tasks_i0, I-1, …, I; make all virtual machines available a time period list vatl_j＝{[0,M]And the task number list allocated to the virtual machine

Wherein M is a number approaching infinity;

step G2: selecting a VTI which is not empty and has no parent task or has the scheduled parent task in the first element according to the numbering sequence of the virtual machine_jFrom VTI_jTaking out the first element i;

step G3: assigning task i to virtual machine j based on the insertion pattern;

step G3.1: computing the execution time of task i

Step G3.2: in vatl_jFinding out an idle time period [ v ] from morning to evening_j,υ_j]Satisfies upsilon_j-ν_j≥et_iAnd upsilon_j-et_i≥rt_i；

Step G3.3: calculating the start time and the completion time of task i: s_i＝max{ν_j,rt_i}，f_i＝s_i+et_i(ii) a Updating the Ready time of a subtask of task i

Step G3.4: in vatl_jDeletion of [ v ]_j,υ_j]V, with insertion interval length greater than 0_j,s_i]And [ f_i,υ_j]；

Step G4: if VTI₁,…,VTI_JIf not, go to step G2, otherwise, go to step G5;

step G5: obtain start and completion times for all tasks: s_i、f_iI1, …, I, calculating a fitness value; the operation is ended.

Further, the specific steps of the serial reverse individual decoding method based on the insertion mode in step C4 are as follows:

step H1: make reverse ready time of all tasks

Wherein: i is 1, …, and I, j is the virtual machine number that processes task I, i.e., satisfies

J of (1); make virtual machine available a time period list vatl_j＝{[0,M]And the task number list allocated to the virtual machine

Wherein M is a number approaching infinity;

step H2: selecting a VTI which is not empty and has no subtask numbered first element or has all subtasks scheduled according to the numbering sequence of the virtual machine_jFrom VTI_jTaking out the first element i;

step H3: assigning task i to virtual machine j based on insertion patterns:

step H3.1: computing the execution time of task i

Step H3.2: in vatl_jFinding out an idle time period [ v ] from morning to evening_j,υ_j]Satisfy upsilon_j-ν_j≥et_iAnd

step H3.3: calculating a reverse start time for task i

Reverse completion time

Updating the Ready time of the parent task of task i

Step H3.4: list of time slots available in virtual machine, vatl_jDeletion of [ v ]_j,υ_j]With an insertion interval length greater than 0

And

step H4: if VTI₁,…,VTI_JIf not, go to step H2, otherwise, go to step H5;

step H5: obtaining reverse start times and reverse completion times for all tasks:

calculating reverse workflow response times

The operation is ended.

Further, the specific steps of randomly selecting two different individuals from the contemporary population as parents based on the ranking round-robin in the step E1 are as follows:

step E1.1: sequencing the individuals in the contemporary population from good to bad, namely, the fitness value is from small to large, and obtaining the sequencing value rk of the individual n_nN is 1, …, N, where the first row takes 1 and the second row takes 2, toBy analogy, the last name is selected as N;

step E1.2: calculating the probability that the individual n is selected

Wherein R is>1 is the discrimination coefficient;

step E1.3: calculating cumulative probability

Step E1.4: generating a random number lambda₁E [0,1) if

Then the individual n is selected;

step E1.5: generating a random number lambda₂E [0,1) if

And n '≠ n, then individual n' is selected, go to step E1.6, otherwise go to step E1.5;

step E1.6: two different individuals n and n' are obtained as parents and the individual selection operation ends.

The invention has the beneficial effects that:

(1) compared with a heuristic method, a semi-intelligent calculation method combined with the heuristic method and an existing intelligent calculation method based on layered coding, the two-dimensional coding method adopted by the design of the invention has the advantage that any one scheduling scheme can have an individual corresponding to the two-dimensional coding method, so that the search space is complete and the global search can be realized.

(2) Compared with a general coding mode based on priority, the two-dimensional integer coding method for performing topological sorting on task scheduling sequence tasks according to the virtual machine, which is adopted by the invention, considers the time sequence relation among tasks with resource competition relation, so that the decoding method is simpler, the decoding efficiency can be effectively improved, and the overall efficiency of the algorithm is further improved.

(3) Compared with the existing two-stage one-dimensional coding method adopting a virtual machine distribution list and a task scheduling sequence list, the two-dimensional integer coding method which adopts the virtual machine to carry out topological ordering on the task scheduling sequence only limits the scheduling sequence of the tasks which are allocated to the same virtual machine and have resource competition conflict, thus the coding space can be effectively reduced compared with the two-stage one-dimensional coding method adopting the virtual machine distribution list and the task scheduling sequence list, for example, 3 tasks which can be in parallel exist in a cloud workflow, if the two-stage one-dimensional coding method adopting the virtual machine distribution list and the task scheduling sequence list is adopted, the topological ordering among the 3 tasks in the task scheduling sequence list part has 6 types, but if the 3 tasks are allocated to 3 different virtual machines, the scheduling schemes corresponding to the 6 types of topological ordering are the same, compared with a two-section one-dimensional coding method, the two-dimensional coding method can reduce the coding space of the algorithm and improve the searching efficiency of the algorithm.

(4) Compared with the traditional non-insertion mode and parallel decoding methods, the serial individual decoding method which is used for arranging the task execution as early as possible based on the insertion mode can generally find a better corresponding scheduling scheme.

(5) Compared with the common one-way decoding method, the invention designs the forward and backward individual decoding and improving strategy FBI & D and the load balancing strategy LDI considering the transmission time, thereby enhancing the neighborhood optimizing capability of the individual, and further improving the optimizing capability and the searching efficiency of the whole algorithm.

(6) Compared with the traditional random initialization method, the method disclosed by the invention has the advantages that an individual based on HEFT-baseline is introduced into the initial population, and a hierarchy-based individual random generation method is adopted for the rest individuals, so that the algorithm can start searching near the optimal scheme, and the convergence time of the algorithm is shortened.

(7) Aiming at the proposed two-dimensional integer coding method, the invention designs a new simple and effective intersection and mutation method, and if the father is valid and legal, the method can ensure that the generated daughter is also valid and legal.

Drawings

Fig. 1 is a schematic flow chart of a cloud workflow scheduling optimization method based on a two-dimensional coding genetic algorithm.

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

Detailed Description

The present invention will be described in further detail with reference to fig. 1 and 2 and examples, but the present invention is not limited to the examples.

Suppose a cloud computing center has 6 virtual machines numbered 1 to 6, a virtual machine vm₁，vm₂，…，vm₆The processing power and bandwidth of (a) are shown in table 1; the time sequence relation between tasks of a single workflow is shown in figure 2 and consists of 15 tasks with the serial numbers of 1 to 15, and the task t₁，t₂，…，t₁₅The execution length of (1), the names of the input file and the output file after processing required for the processing, the length, and the virtual machine that can be processed are shown in table 2.

TABLE 1

TABLE 2

For the above case, as shown in fig. 1, a cloud workflow scheduling optimization method based on a two-dimensional coding genetic algorithm includes the following implementation steps:

executing the step 1: formalizing a scheduling problem, and acquiring information required by scheduling optimization;

get task set T ═ T₁,t₂,t₃,t₄,t₅,t₆,t₇,t₈,t₉,t₁₀,t₁₁,t₁₂,t₁₃,t₁₄,t₁₅}；

Obtaining the time sequence relation between tasks, namely a parent task set PR of the task i_iAnd a set of subtasks SC_i：

PR₄＝{t₁}，PR₅＝{t₁,t₂}，PR₆＝{t₁,t₃}，PR₇＝{t₄,t₅,t₆}，PR₈＝{t₇}，PR₉＝{t₁,t₈}，PR₁₀＝{t₂,t₈}，PR₁₁＝{t₃,t₈}，PR₁₂＝{t₉,t₁₀,t₁₁}，PR₁₃＝{t₁₂}，PR₁₄＝{t₁₃}，PR₁₅＝{t₁₄}；

SC₁＝{t₄,t₅,t₆,t₉}，SC₂＝{t₅,t₁₀}，SC₃＝{t₆,t₁₁}，SC₄＝{t₇}，SC₅＝{t₇}，SC₆＝{t₇}，SC₇＝{t₈}，SC₈＝{t₉,t₁₀,t₁₁}，SC₉＝{t₁₂}，SC₁₀＝{t₁₂}，SC₁₁＝{t₁₂}，SC₁₂＝{t₁₃}，SC₁₃＝{t₁₄}，SC₁₄＝{t₁₅}，

Acquiring relevant parameters of the task: t is t₁.length＝126000MI，t₁.IFL＝{f_d1,f_d2}，t₁.OFL＝{f_1-1,f_1-2}；

t₂.length＝138000MI，t₂.IFL＝{f_d1,f_d3}，t₂.OFL＝{f_2-1,f_2-2}；t₃.length＝132000MI，t₃.IFL＝{f_d1,f_d4}，t₃.OFL＝{f_3-1,f_3-2}；t₄.length＝102000MI，t₄.IFL＝{f_d1,f_1-1,f_1-2}，t₄.OFL＝{f_4-1,f_4-2}；……；t₁₅.length＝7800MI，t₁₅.IFL＝{f_14-1}，t₁₅.OFL＝{f_15-1}；

f_d1.size＝36MB，f_d2.size＝4320MB，f_1-1.size＝3960MB，f_1-2.size＝3960MB，……，f_14-1.size＝1560MB，f_15-1.size＝420MB；

Acquiring a virtual machine set in a cloud computing environment: VM ═ VM₁,vm₂,vm₃,vm₄,vm₅,vm₆}；

Acquiring related parameters of the virtual machine: vm₁.ps＝1000MI/s，vm₁.bw＝200Mbit/s；vm₂.ps＝1000MI/s，vm₂.bw＝200Mbit/s；vm₃.ps＝2000MI/s，vm₃.bw＝300Mbit/s；vm₄.ps＝2000MI/s，vm₄.bw＝300Mbit/s；vm₅.ps＝3000MI/s，vm₅.bw＝400Mbit/s；vm₆.ps＝3000MI/s，vm₆.bw＝400Mbit/s；

Acquiring a support relationship between the task and the virtual machine: t is₁＝{t₁,t₂,t₃,t₄,t₅,t₆,t₉,t₁₃,t₁₅}，T₂＝{t₃,t₅,t₇,t₉,t₁₀,t₁₁,t₁₄}，T₃＝{t₂,t₃,t₄,t₆,t₉,t₁₁,t₁₂}，T₄＝{t₁,t₂,t₄,t₆,t₇,t₈,t₉,t₁₀,t₁₁,t₁₂,t₁₄}，T₅＝{t₁,t₂,t₃,t₄,t₆,t₇,t₈,t₉,t₁₂,t₁₄}，T₆＝{t₁,t₄,t₅,t₈,t₁₁,t₁₃,t₁₄,t₁₅}；VM₁＝{vm₁,vm₄,vm₅,vm₆}，VM₂＝{vm₁,vm₃,vm₄,vm₅}，VM₃＝{vm₁,vm₂,vm₃,vm₅}，VM₄＝{vm₁,vm₃,vm₄,vm₅,vm₆}，VM₅＝{vm₁,vm₂,vm₆}，VM₆＝{vm₁,vm₃,vm₄,vm₅}，VM₇＝{vm₂,vm₄,vm₅}，VM₈＝{vm₄,vm₅,vm₆}，VM₉＝{vm₁,vm₂,vm₃,vm₄,vm₅}，VM₁₀＝{vm₂,vm₄}，VM₁₁＝{vm₂,vm₃,vm₄,vm₆}，VM₁₂＝{vm₃,vm₄,vm₅}，VM₁₃＝{vm₁,vm₆}，VM₁₄＝{vm₂,vm₄,vm₅,vm₆}，VM₁₅＝{vm₁,vm₆}。

And (3) executing the step 2: calculating the level value level of the task;

if the task 1, the task 2 and the task 3 have no father task, the level is₁＝level₂＝level₃＝1；

Task 4 has only one parent task 1, then

Similarly, the hierarchy values of other tasks may be obtained: level₅＝level₆＝2；level₇＝3；level₈＝4；level₉＝level₁₀＝level₁₁＝5；level₁₂＝6；level₁₃＝7；level₁₄＝8；level₁₅＝9。

And (3) executing the step: initializing a contemporary population;

taking the population size N as 10; initializing a contemporary population comprising 1 individual generated by an individual generation method based on HEFT-baseline and 9 different individuals generated by an individual random generation method based on a hierarchy;

the specific implementation process for generating an individual based on the HEFT-baseline individual generation method is as follows:

step a1 is executed: initializing all virtual machine available time period list vatl_j＝{[0,M]}，k_j1, j is 1,2,3,4,5,6, where M is a near infinite number; UT ═ T₁,t₂,t₃,t₄,t₅,t₆,t₇,t₈,t₉,t₁₀,t₁₁,t₁₂,t₁₃,t₁₄,t₁₅}; initializing the ready time rt of all tasks_i＝0，i＝1,…,15；

Step a2 is executed: the smallest level task in UTs is t₁、t₂、t₃Due to SC₁＝{t₄,t₅,t₆,t₉}，SC₂＝{t₅,t₁₀}，SC₃＝{t₆,t₁₁I.e. | SC₁|＝4，|SC₂|＝2，|SC₃2, so from UT to t₁,t₂,t₃,t₄,t₅,t₆,t₇,t₈,t₉,t₁₀,t₁₁,t₁₂,t₁₃,t₁₄,t₁₅Get t out₁；

Step a3 is executed: let AVM₁＝VM₁＝{vm₁,vm₄,vm₅,vm₆H, calculating t₁Allocated to AVM₁＝{vm₁,vm₄,vm₅,vm₆T after each virtual machine in the₁The completion time of (c), namely: step a3.1 is performed: slave AVM₁＝{vm₁,vm₄,vm₅,vm₆Get vm out of₁(ii) a Step a3.2 is performed: calculating t₁To vm₁Execution time after processing: omega₁,₁＝126000/1000＝126，

τ_1,18 × (36+4320)/200 ═ 174.24, then

Step a3.3 is performed: in vatl₁Find out an idle time period [0, M ] from morning to evening]Satisfy M-0 ≥ et_1,1300.24 and M-300.24 ≧ rt₁0; step a3.4 is performed: calculating t₁To vm₁Start time after treatment s_1,1＝max{v₁,rt₁Max {0,0}, 0, completion time f_1,1＝s_1,1+et_1,10+ 300.24-300.24; step a3.5 is performed: since the AVM is now₁＝{vm₄,vm₅,vm₆If not, go to step A3.1; step a3.1 is performed: slave AVM₁＝{vm₄,vm₅,vm₆Get vm out of₄(ii) a Step a3.2 is performed: calculating t₁To vm₄Execution time after processing, omega_1,4＝63，

τ_1,4116.16, then

Step a3.3 is performed: in vatl₄Find out an idle time period [0, M ] from morning to evening]M-0 is equal to or more than 179.16 and M-179.16 is equal to or more than 0; step a3.4 is performed: calculating t₁To vm₄Start time after treatment s₁,₄＝max{v₄,rt₁Max {0,0}, 0, completion time f_1,4＝s_1,4+et_1,40+ 179.16-179.16; step a3.5 is performed: due to thisTime-of-flight AVM₁＝{vm₅,vm₆If not, go to step A3.1; … …, the steps A3.1 to A3.5 are repeated until the step A is finished

To obtain s_1,1＝0，s_1,4＝0，s_1,5＝0，s_1,6＝0，f_1,1＝300.24，f₁,₄＝179.16，f_1,5＝129.12，f_1,6When the value is 129.12, the step A4 is switched to;

step a4 is executed: finding earliest achievable t in virtual machine order₁Is vm, which is₅，

k₅＝k₅+ 1-2, and t₁To vm₅Namely: step a4.1 is performed: let t₁Start time s of₁＝s_1,50, completion time f₁＝f_1,5129.12; step a4.2 is performed: updating t₁Ready time of subtask(s):

rt₄＝max{rt₄,f₁}＝max{0,129.12}＝129.12，rt₅＝129.12，rt₆＝129.12，rt₉129.12; step a4.3 is performed: list of time slots available in virtual machine, vatl₅Deletion of [0, M]Insertion interval length greater than 0 [129.12, M]Then, vall₅＝{[129.12,M]}；

Step a5 is executed: since UT is t₂,t₃,t₄,t₅,t₆,t₇,t₈,t₉,t₁₀,t₁₁,t₁₂,t₁₃,t₁₄,t₁₅If not, go to step A2;

step a2 is executed: the smallest level task in UTs is t₂、t₃Due to SC₂＝{t₅,t₁₀}，SC₃＝{t₆,t₁₁I.e. | SC₂|＝2，|SC₃2, so at t₂、t₃Is randomly selected one of t₃From UT ═ t₂,t₃,t₄,t₅,t₆,t₇,t₈,t₉,t₁₀,t₁₁,t₁₂,t₁₃,t₁₄,t₁₅Get t out of₃；

Step a3 is executed: let AVM₃＝VM₃＝{vm₁,vm₂,vm₃,vm₅H, calculating t₃Allocated to AVM₃＝{vm₁,vm₂,vm₃,vm₅T after each virtual machine in the₃The completion time of (c), namely: step a3.1 is performed: slave AVM₃＝{vm₁,vm₂,vm₃,vm₅Get vm out of₁(ii) a Step a3.2 is performed: calculating t₃To vm₁Execution time after processing: omega_3,1＝132，

τ_3,1174.24, then et_3,1306.24; step a3.3 is performed: in vatl₁Find out an idle time period [0, M ] from morning to evening]M-0 is equal to or more than 306.24 and M-306.24 is equal to or more than 0; step a3.4 is performed: calculating t₃To vm₁Start time after treatment s_3,1Max {0,0}, 0, completion time f_3,1＝s_3,1+et_3,1306.24; step a3.5 is performed: since the AVM is now₃＝{vm₂,vm₃,vm₅If not, go to step A3.1; … …, the steps A3.1 to A3.5 are repeated until the step A is finished

To obtain s_3,1＝0，s_3,2＝0，s_3,3＝0，s_3,5＝129.12，f_3,1＝306.24，f_3,2＝306.24，f_3,3＝182.16，f_3,5260.24, go to step a 4;

step a4 is executed: finding earliest achievable t in virtual machine order₃Is vm, which is₃，

k₃＝k₃+ 1-2, and t₃To vm₃Namely: step a4.1 is performed: let t₃Start time s of₃＝s_3,30, completion time f₃＝f_3,3182.16; step a4.2 is performed: updating t₃Ready time of subtask(s):

rt₆＝max{rt₆,f₃}＝max{129.12,182.16}＝182.16，rt₁₁182.16; step a4.3 is performed: list of time slots available in virtual machine, vatl₃Deletion of [0, M]Insertion interval length greater than 0 [182.16, M]Then, vall₃＝{[182.16,M]}；

Step a5 is executed: since UT is t₂,t₄,t₅,t₆,t₇,t₈,t₉,t₁₀,t₁₁,t₁₂,t₁₃,t₁₄,t₁₅If not, go to step A2;

step a2 is executed: the smallest level task in UTs is t₂From UT ═ t₂,t₄,t₅,t₆,t₇,t₈,t₉,t₁₀,t₁₁,t₁₂,t₁₃,t₁₄,t₁₅Get t out of₂；

Step a3 is executed: let AVM₂＝VM₂＝{vm₁,vm₃,vm₄,vm₅H, calculating t₂Allocated to AVM₂＝{vm₁,vm₃,vm₄,vm₅T after each virtual machine in the₂Completion time of (d); namely: step a3.1 is performed: slave AVM₂＝{vm₁,vm₃,vm₄,vm₅Get vm out of₁(ii) a Step a3.2 is performed: calculating t₂To vm₁Execution time after processing: omega_2,1＝138，

τ_2,1174.24, then et_2,1312.24; step a3.3 is performed: in vatl₁Find out an idle time period [0, M ] from morning to evening]M-0 is equal to or more than 312.24 and M-312.24 is equal to or more than 0; step a3.4 is performed: calculating t₂To vm₁Start time after treatment s_2,10, completion time f_2,1312.24; step a3.5 is performed: since the AVM is now₂＝{vm₃,vm₄,vm₅If not, go to step A3.1; … …, the steps A3.1 to A3.5 are repeated until the step A is finished

To obtain s_2,1＝0，s_2,3＝182.16，s_2,4＝0，s_2,5＝129.12，f_2,1＝312.24，f_2,3＝367.32，f_2,4＝185.16，f_2,5262.24, go to step a 4;

step a4 is executed: finding earliest achievable t in virtual machine order₂Is vm, which is₄，

k₄＝k₄+ 1-2, and t₂To vm₄(ii) a Namely: step a4.1 is performed: let t₂Start time s of₂＝s_2,40, completion time f₂＝f_2,4185.16; step a4.2 is performed: updating t₂Ready time of subtask(s):

rt₅＝max{rt₅,f₂}＝max{129.12,185.16}＝185.16，rt₁₀185.16; step a4.3 is performed: list of time slots available in virtual machine, vatl₄Deletion of [0, M]Insertion interval length greater than 0 [185.16, M]Then, vall₄＝{[185.16,M]}；

Step a5 is executed: since UT is t₄,t₅,t₆,t₇,t₈,t₉,t₁₀,t₁₁,t₁₂,t₁₃,t₁₄,t₁₅If not, go to step A2;

……

the steps A2 to A5 are repeatedly executed until the step A2 and the step A5 are executed

Go to step A6;

step a6 is executed: outputting an individual ch based on HEFT-baseline₁{ (), (), (3,11), (2,10,12), (1,6,4,7,8,9), (5,13,14,15) }, the operation ends;

the specific implementation process of generating an individual by the individual random generation method based on the hierarchy is as follows:

step B1 is executed: UT ═ T₁,t₂,t₃,t₄,t₅,t₆,t₇,t₈,t₉,t₁₀,t₁₁,t₁₂,t₁₃,t₁₄,t₁₅}；k₁＝1，k₂＝1，k₃＝1，k₄＝1，k₅＝1，k₆＝1；

Step B2 is executed: from UT ═ t₁,t₂,t₃,t₄,t₅,t₆,t₇,t₈,t₉,t₁₀,t₁₁,t₁₂,t₁₃,t₁₄,t₁₅Randomly taking a task with the minimum level value, which is t₂；

Step B3 is executed: slave VM₂＝{vm₁,vm₃,vm₄,vm₅Randomly selecting a virtual machine which is vm₄；

Step B4 is executed: order to

k₄＝k₄+1＝1+1＝2；

Step B5 is executed: since UT is t₁,t₃,t₄,t₅,t₆,t₇,t₈,t₉,t₁₀,t₁₁,t₁₂,t₁₃,t₁₄,t₁₅If not, go to step B2;

step B2 is executed: from UT ═ t₁,t₃,t₄,t₅,t₆,t₇,t₈,t₉,t₁₀,t₁₁,t₁₂,t₁₃,t₁₄,t₁₅Randomly taking a task with the minimum level value, which is t₃；

Step B3 is executed: slave VM₃＝{vm₁,vm₂,vm₃,vm₅Randomly selecting a virtual machine which is vm₃；

Step B4 is executed: order to

k₃＝k₃+1＝1+1＝2；

Step B5 is executed: since UT is t₁,t₄,t₅,t₆,t₇,t₈,t₉,t₁₀,t₁₁,t₁₂,t₁₃,t₁₄,t₁₅If not, go to step B2;

step B2 is executed: from UT ═ t₁,t₄,t₅,t₆,t₇,t₈,t₉,t₁₀,t₁₁,t₁₂,t₁₃,t₁₄,t₁₅Randomly taking a task with the minimum level value, which is t₁；

Step B3 is executed: slave VM₁＝{vm₁,vm₄,vm₅,vm₆Randomly selecting a virtual machine which is vm₄；

Step B4 is executed: order to

k₄＝k₄+1＝2+1＝3；

Step B5 is executed: since UT is t₄,t₅,t₆,t₇,t₈,t₉,t₁₀,t₁₁,t₁₂,t₁₃,t₁₄,t₁₅If not, go to step B2;

step B2 is executed: from UT ═ t₄,t₅,t₆,t₇,t₈,t₉,t₁₀,t₁₁,t₁₂,t₁₃,t₁₄,t₁₅Randomly taking a task with the minimum level value, which is t₆；

Step B3 is executed: slave VM₆＝{vm₁,vm₃,vm₄,vm₅Randomly selecting a virtual machine which is vm₅；

Step B4 is executed: order to

k₅＝k₅+1＝1+1＝2；

Step B5 is executed: since UT is t₄,t₅,t₇,t₈,t₉,t₁₀,t₁₁,t₁₂,t₁₃,t₁₄,t₁₅If not, go to step B2;

……

the steps B2 to B5 are repeatedly executed until the step B2 and the step B5 are executed

Go to step B6;

step B6 is executed: output individual ch₂{ (4,15), (5,7,10,9,14), (3), (2,1,11,12), (6), (8,13) }, the operation ends;

similarly, the hierarchy-based individual random generation method generates other different individuals in the contemporary population as follows:

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

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

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

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

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

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

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

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

the initial current generation population thus finally generated is CP ═ ch₁,ch₂,ch₃,ch₄,ch₅,ch₆,ch₇,ch₈,ch₉,ch₁₀}。

And (4) executing: improving all individuals in the contemporary population by adopting FBI & D and LDI methods and calculating the fitness value of the individuals;

first, FBI was applied to all individuals in the contemporary population&D method improvements, e.g. to ch in contemporary populations₁₀As { (2,1,5), (3), (), (6,7,11,10,12), (8,9), (4,13,14,15) } FBI is used&The method D comprises the following specific implementation processes:

step C1 is executed: make reverse workflow response time

Wherein M is a number approaching infinity;

step C2 is executed: individual ch adopting serial individual decoding method based on insertion mode₁₀Decoding is carried out, and the completion time of all tasks is obtained: f. of₁＝612.48，f₂＝312.24，f₃＝306.24，f₄＝964.00，f₅＝721.92，f₆＝1294.44，f₇＝1316.76，f₈＝1321.32，f₉＝1676.12，f₁₀＝2070.20，f₁₁＝1683.16，f₁₂＝2271.88，f₁₃＝2281.16，f₁₄＝2291.16，f₁₅2293.76, and its workflow response time rs₁₀2302.16; due to rs₁₀2302.16 is less than

Go to step C3;

step C3 is executed: an individual ch₁₀The gene sequence corresponding to each virtual machine j in the database

Rearranging according to the task completion time from large to small, wherein j is 1, … and 6, and forming reverse individuals

Step C4 is executed: to pair

Decoding by adopting a serial reverse individual decoding method based on an insertion mode to obtain reverse completion time of all tasks:

and its inverse fitness value

Due to the fact that

Less than rs₁₀When the result is 2302.16, go to step C5;

step C5 is executed: make the opposite direction single body

The gene sequence corresponding to each virtual machine j in the database

Rearranging according to the reverse completion time of the task from large to small, wherein j is 1, … and 6 to form individuals

ch₁₀＝{(1,2,5),(3),(),(6,7,11,10,12), (8,9), (4,13,14,15) }, go to step C2;

step C2 is executed: individual ch adopting serial individual decoding method based on insertion mode₁₀Decoding is carried out, and the completion time of all tasks is obtained: f. of₁＝300.24，f₂＝612.48，f₃＝306.24，f₄＝651.76，f₅＝721.92，f₆＝988.20，f₇＝1010.52，f₈＝1015.08，f₉＝1369.88，f₁₀＝1763.96，f₁₁＝1376.92，f₁₂＝1965.64，f₁₃＝1974.92，f₁₄＝1984.92，f₁₅1987.52, and its workflow response time rs₁₀1995.92; due to rs₁₀1995.92 equal to

Go to step C6

Step C6 is executed: output individual ch₁₀{ (1,2,5), (3), (), (6,7,11,10,12), (8,9), (4,13,14,15) } and its fitness value, i.e., workflow response time rs₁₀1995.92, the operation ends;

with the above mentioned individual ch₁₀As an example, the serial individual decoding method based on the insertion mode is implemented as follows:

step G1 is executed: let ready times rt of all tasks_i0,1, …, 15; make available time period list of virtual machine vatl_j＝{[0,M]1,2,3,4,5,6, where M is a number close to infinity; order list VTI of task number assigned to virtual machine₁＝(2,1,5)，VTI₂＝(3)，VTI₃＝()，VTI₄＝(6,7,11,10,12)，VTI₅＝(8,9)，VTI₆＝(4,13,14,15)；

Step G2 is executed: due to VTI₁Has a first element of 2, and task 2 has no parent task, so the slave VTI₁The first element 2 is taken out of (2,1, 5);

step G3 is executed: assigning task 2 to virtual machine 1 based on the insertion pattern; namely: step G3.1 is performed: meterComputing the execution time of task 2

Step G3.2 is performed: in vatl₁Find out an idle time period [0, M ] from morning to evening]Satisfy M-0 ≥ et₂312.24 and M-312.24 ≧ rt₂0; step G3.3 is performed: calculate start time and completion time for task 2:

s₂＝max{v₁,rt₂}＝max{0,0}＝0，f₂＝s₂+et₂0+ 312.24-312.24; update ready time of subtask of task 2: rt is an integer of₅＝max{rt₅,f₂}＝max{0,312.24}＝312.24，rt₁₀312.24; step G3.4 is performed: list of time slots available in virtual machine, vatl₁Deletion of [0, M]Insertion interval length greater than 0 [312.24, M]Then, vall₁＝{[312.24,M]}；

Step G4 is executed: due to VTI₁,…,VTI₆If not, go to step G2;

step G2 is executed: due to VTI₁Has a first element of 1 and task 1 has no parent task, so the slave VTI₁The first element 1 is taken out of (1, 5);

step G3 is executed: assigning task 1 to virtual machine 1 based on the insertion pattern; namely: step G3.1 is performed: computing the execution time of task 1

Step G3.2 is performed: in vatl₁Find an idle period of time from morning to evening [312.24, M]The requirements that M-312.24 is more than or equal to 300.24 and M-300.24 is more than or equal to 0 are met; step G3.3 is performed: calculate start time and completion time for task 1: s₁＝max{v₁,rt₁}＝max{312.24,0}＝312.24，f₁612.48, update the ready time for the subtask of task 1: rt is an integer of₄＝max{rt₄,f₁}＝max{0,612.48}＝612.48，rt₅＝max{rt₅,f₁}＝612.48，rt₆＝612.48，rt₉612.48; step G3.4 is performed: list of time slots available in virtual machine, vatl₁Deletion [312.24, M]Insertion interval length greater than 0 [612.48, M]Then, vall₁＝{[612.48,M]}；

Step G4 is executed: due to VTI₁,…,VTI₆If not, go to step G2;

step G2 is executed: since VTI is now₁Has a first element of 5 and the parent task of task 5 has all been scheduled, thus the slave VTI₁Taking out a first element 5 from the formula (5);

step G3 is executed: assigning task 5 to virtual machine 1 based on the insertion pattern; namely: step G3.1 is performed: computing the execution time of task 1

Step G3.2 is performed: in vatl₁Find an idle period of time from morning to evening [612.48, M]The requirements that M-612.48 is more than or equal to 109.44 and M-109.44 is more than or equal to 612.48 are met; step G3.3 is performed: calculate start time and completion time of task 5: s₅＝max{v₁,rt₅}＝max{612.48,612.48}＝612.48，f₅721.92, the ready time rt of the subtask of task 5 is updated₇721.92; step G3.4 is performed: list of time slots available in virtual machine, vatl₁Deletion [612.48, M]Insertion interval length greater than 0 [721.92, M]Then, vall₁＝{[721.92,M]}；

Step G4 is executed: due to VTI₁,…,VTI₆If not, go to step G2;

step G2 is executed: since VTI is now₁Become () and VTI₂Has a first element of 3, and task 3 has no parent task, so the slave VTI₂Taking out a first element 3 from the group (3);

step G3 is executed: assigning task 3 to virtual machine 2 based on the insertion pattern; namely: step G3.1 is performed: computing the execution time et of task 3₃306.24; step G3.2 is performed: in vatl₂Find out an idle time period [0, M ] from morning to evening]M-0 is equal to or more than 306.24 and M-306.24 is equal to or more than 0; step G3.3 is performed: calculate the start time s of task 3₃＝max{v₂,rt₃Max {0,0}, 0, completion time f₃＝306.24，Update ready time of subtask of task 3: rt is an integer of₆＝max{rt₆,f₃}＝max{612.48,306.24}＝612.48，rt₁₁＝max{rt₁₁,f₃306.24; step G3.4 is performed: list of time slots available in virtual machine, vatl₂Deletion of [0, M]When the length of the insertion section is greater than 0, vall₂＝{[306.24,M]}；

Step G4 is executed: due to VTI₁,…,VTI₆If not, go to step G2;

……

so that steps G2 to G4 are repeatedly executed until VTI₁,…,VTI₆Go to step G5 if all are empty;

step G5 is executed: obtain start and completion times for all tasks: s₁＝312.24，s₂＝0.00，s₃＝0.00，s₄＝612.48，s₅＝612.48，s₆＝612.48，s₇＝1294.44，s₈＝1316.76，s₉＝1321.32，s₁₀＝1683.16，s₁₁＝1321.32，s₁₂＝2070.20，s₁₃＝2271.88，s₁₄＝2281.16，s₁₅＝2291.16；f₁＝612.48，f₂＝312.24，f₃＝306.24，f₄＝964.00，f₅＝721.92，f₆＝1294.44，f₇＝1316.76，f₈＝1321.32，f₉＝1676.12，f₁₀＝2070.20，f₁₁＝1683.16，f₁₂＝2271.88，f₁₃＝2281.16，f₁₄＝2291.16，f₁₅2293.76, its fitness value, i.e. workflow response time, is calculated: due to the fact that

And SFL₁₅＝{f_15-1Get the results

Finishing the operation;

with the above-mentioned individuals

For example, the serial reverse individual decoding method based on the insertion mode is implemented as follows:

step H1 is performed: due to the fact that

And SFL₁₅＝{f_15-1Let the reverse ready time of the task:

make virtual machine available a time period list vatl₁＝{[0,M]}，vatl₂＝{[0,M]}，……，vatl₆＝{[0,M]And ordering a task number list VTI allocated to the virtual machine₁＝(5,1,2)，VTI₂＝(3)，VTI₃＝()，VTI₄＝(12,10,11,7,6)，VTI₅＝(9,8)，VTI₆(15,14,13,4) where M is a number close to infinity;

step H2 is performed: due to VTI₁、VTI₂、VTI₃、VTI₄、VTI₅Tasks that are either empty or numbered with their first element have subtasks that have not yet been scheduled, while the VTI₆Is 15, and task 15 has no subtasks, so the slave VTI₆First element 15, VTI, is taken out of (15,14,13,4)₆＝(14,13,4)；

Step H3 is performed: assigning tasks 15 to virtual machines 6 based on the insertion pattern; namely: step H3.1 is performed: computing the execution time of a task 15

Step H3.2 is performed: in vatl₆Find out an idle time interval [0, M ] from morning to evening]Satisfy M-0 ≥ et₁₅2.60 and

step H3.3 is performed: calculating the reverse start time of task 15

Reverse completion time

Updating the reverse ready time of the parent task of task 15

Step H3.4 is performed: list of time slots available in virtual machine, vatl₆Deletion of [0, M]Insertion interval length greater than 0 [0,8.4 ]]And [11, M]；

Step H4 is performed: due to VTI₁,…,VTI₆If not, go to step H2;

step H2 is performed: due to VTI₁、VTI₂、VTI₃、VTI₄、VTI₅Tasks that are either empty or numbered with their first element have subtasks that have not yet been scheduled, while the VTI₆Is 14 and task 14 has no subtasks, so the slave VTI₆First element 14, VTI, out of (14,13,4)₆＝(13,4)；

Step H3 is performed: assigning the task 14 to the virtual machine 6 based on the insertion pattern; namely: step H3.1 is performed: computing the execution time of a task 14

Step H3.2 is performed: in vatl₆Find out an idle period [11, M ] from morning to evening]M-11 is not less than 10 and

step H3.3 is performed: calculating the reverse start time of task 14

Reverse completion time

Updating the reverse ready time of the parent task of task 14

Step H3.4 is performed: in deficiencyList of available time slots of virtual machine vatl₆Deletion in [11, M]Insertion interval length greater than 0 [21, M ]]；

Step H4 is performed: due to VTI₁,…,VTI₆If not, go to step H2;

……

so that the steps H2 to H4 are repeatedly executed until VTI₁,…,VTI₆Go to step H5 if empty;

step H5 is performed: obtaining reverse start times and reverse completion times for all tasks:

calculating reverse workflow response times

Finishing the operation;

similarly, other individuals in the contemporary population become:

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

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

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

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

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

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

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

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

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

the fitness values, i.e. the workflow response times, are respectively: rs₁＝1098.16，rs₂＝5667.80，rs₃＝5432.56，rs₄＝5024.20，rs₅＝5075.36，rs₆＝4288.64，rs₇＝2833.72，rs₈＝2195.68，rs₉＝2873.16；

All individuals in the contemporary population are improved by the LDI method, for example, ch in the contemporary population₂Specific implementation procedures modified by the LDI method are as follows:

step D1 is executed: order a task list VTN assigned to a virtual machine₁＝{t₄,t₁₅}，VTN₂＝{t₅,t₇,t₁₀,t₉,t₁₄}，VTN₃＝{t₃}，VTN₄＝{t₂,t₁,t₁₁,t₁₂}，VTN₅＝{t₆}，VTN₆＝{t₈,t₁₃}; the load of each virtual machine is calculated,

the load ld of other virtual machines can be obtained in the same way₂＝374.4，ld₃＝66，ld₄＝194.4，ld₅＝38，ld₆＝8；

Step D2 is executed: find the least loaded virtual machine, which is vm₆Due to ld₆＝8>0, so go to step D3;

step D3 is executed:

go to step D5;

step D5 is executed: due to ST₆＝{t₁₁,t₁₄If not, from ST₆Find a task with the highest load of the virtual machine, which is t₁₄Go to step D6;

step D6 is executed: due to (g)_6,1,g_6,2) If (8,13) the ancestral task number of task 14 exists, but the descendant task number of task 14 does not exist, and therefore (g) is (g)_6,1,g_6,2) Look for its last ancestor task number from front to back in (8,13), which is g_6,213 at g_6,2Then randomly find a position to insert 14, g_6,314; task list (g) of virtual machine 2 in which task 14 originally exists_2,1,g_2,2,g_2,3,g_2,4,g_2,5) Deleting 14 from (5,7,10,9,14) to form new individuals

Using FBI&Method for treating new individuals

Performing decoding improvement to obtain workflow response time

Due to the fact that

Thus replacing the original individual with a modified individual, i.e.

Go to step D7;

step D7 is executed: the LDI operation is finished;

similarly, other individuals in the contemporary population become:

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

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

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

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

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

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

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

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

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

the fitness values, i.e. the workflow response times, are respectively: rs₁＝1098.16，rs₃＝5432.56，rs₄＝2061.80，rs₅＝4912.16，rs₆＝4192.64，rs₇＝2305.96，rs₈＝2189.68，rs₉＝2873.16，rs₁₀＝1887.12；

And 5, executing the step: judging whether a termination condition is met, if so, turning to a step 10 after the evolution is finished, otherwise, turning to a step 6;

the termination condition is that continuous iteration GG is 20 generations of optimal individuals has no improvement;

the population generated by the above process is an improved population after initialization, and currently only evolution and iteration are performed for one generation, and the termination condition is not met, so that the process goes to step 6.

And 6, executing the step: carrying out N times of parameterized uniform cross operations based on preference and two-dimensional topological sorting on the contemporary population to form a new population;

preference rate p_b＝0.7；

The specific implementation process of carrying out the preference and two-dimensional topological ordering based parameterized uniform crossing operation on the contemporary population is as follows:

executeStep E1: two different individuals were randomly selected as parents from the contemporary population using a rank-based betting round method, namely: step E1.1 is performed: sequencing the individuals in the current generation population from good to bad, namely, the fitness value is from small to large, and obtaining a sequencing value corresponding to each individual: rk₁＝1，rk₂＝6，rk₃＝10，rk₄＝3，rk₅＝9，rk₆＝8，rk₇＝5，rk₈＝4，rk₉＝7，rk₁₀2; step E1.2 is performed: let R be 1+1/N be 1.1, calculate individual ch₁Probability of being selected

Same principle A₂＝0.092，A₃＝0.063，A₄＝0.122，A₅＝0.069，A₆＝0.076，A₇＝0.101，A₈＝0.111，A₉＝0.083，A₁₀0.135; step E1.3 is performed: calculating cumulative probability

Step E1.4 is performed: randomly generating a random number λ of [0,1 ]₁It is 0.812 because

Thus selecting ch₉(ii) a Step E1.5 is performed: randomly generating a random number λ of [0,1 ]₂It is 0.397, due to

And 4 ≠ 9, so ch is selected₄Go to step E1.6; step E1.6 is performed: obtaining two different individuals ch₉And ch₄As a father, the individual selection operation ends; and due to rs₉＝2873.16>rs₄2061.80 due to its root causeThis selects ch₄As the father 1, ch₉As father 2, namely: ch (channel)^p1＝ch₄＝{(6,9),(10),(3),(1,4,11,12),(2,7),(5,8,13,14,15)}，ch^p2＝ch₉{ (4), (7), (9), (2,6,8,10,12), (3,14), (1,5,11,13,15) }; order to

Order to

δ＝1；

Step E2 is executed: if δ 1 ≦ I ≦ 15, go to step E3;

step E3 is executed: generating a random number between [0,1), which is 0.03, since 0.03. ltoreq. p_bIf 0.7, go to step E4;

step E4 is executed: due to the fact that

In (1),

the condition that the task numbered as its first element has no parent task or that its parent task has been scheduled is satisfied, and thus

Randomly select one of which is

Thus, from

Taking out the first element 2, adding it to

At the tail, i.e.

And from

In deletion 2, i.e.

Go to step E2, if δ + 1+ 2;

step E2 is executed: since δ 2 ≦ I ≦ 15, go to step E3;

step E3 is executed: generate a random number between [0,1), which is 0.94, since 0.94>p_bIf 0.7, go to step E5;

step E5 is executed: due to the fact that

In (1),

the condition that the task numbered as its first element has no parent task or that its parent task has been scheduled is satisfied, and thus

Randomly select one of which is

Thus, from

Taking out the first element 1, and adding it to

At the tail, i.e.

And from

In deletion of 1, i.e.

Go to step E2, δ +1 — 2+1 — 3;

step E2 is executed: if δ is 3 ≦ I ≦ 15, go to step E3;

step E3 is executed: generating a random number between [0,1), which is 0.17, since 0.17 ≦ p_bIf 0.7, go to step E4;

step E4 is executed: due to the fact that

In (1),

the condition that the task numbered as its first element has no parent task or that its parent task has been scheduled is satisfied, and thus

Randomly select one of which is

Thus, from

Taking out the first element 3, adding it to

At the tail, i.e.

And from

Deletion 3 in (i)

Go to step E2, δ +1 — 3+1 — 4;

step E2 is executed: since δ 4 ≦ I ≦ 15, go to step E3;

step E3 is executed: generating a random number between [0,1), which is 0.57, since 0.57 ≦ p_bIf 0.7, go to step E4;

step E4 is executed: due to the fact that

In (1),

the condition that the task numbered as its first element has no parent task or that its parent task has been scheduled is satisfied, and thus

Randomly select one of which is

Thus, from

Taking out the first element 4, adding it to

At the tail, i.e.

And from

In deletion 4, i.e.

Go to step E2, δ + 1+ 4+ 1+ 5;

step E2 is executed: if δ is 5 ≦ I ≦ 15, go to step E3;

step E3 is executed: generating a random number between [0,1), which is 0.34, since 0.34 ≦ p_bIf 0.7, go to step E4;

step E4 is executed: due to the fact that

In (1),

the condition that the task numbered as its first element has no parent task or that its parent task has been scheduled is satisfied, and thus

Randomly select one of which is

Thus, from

Taking out the first element 5, adding it to

At the tail, i.e.

And from

Deletion of 5, i.e.

Go to step E2, δ + 1+ 5+ 1+ 6;

……

the steps E2 to E5 are repeated until δ ═ 16> I ═ 15, and the process goes to step E6;

step E6 is executed: output sub-body ch^c{ (9), (7,10), (3), (4,6,11,12), (2,14), (1,5,8,13,15) }, the operation ends;

thus obtaining ch'₁＝ch^c＝{(9),(7,10),(3),(4,6,11,12),(2,14),(1,5,8,13,15)}；

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

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

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

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

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

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

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

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

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

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

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

And 7, executing the step: performing mutation operation on each individual in the new population;

taking the rate of variation p_m＝0.2；

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

for ch'₁＝{(9),(7,10),(3),(4,6,11,12),(2,14),(1,5,8,13,15)}；

Step F1 is performed: randomly generating a random number λ of [0,1), which is 0.54, since λ is 0.54>p_mIf 0.2, go to step F5;

step F5 is performed: finishing individual variation operation;

for ch'₂＝{(4,15),(5,10),(11),(9,12),(2,1,3,6,7),(8,13,14)}；

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

step F2 is performed: randomly select a task number, which is 2, from the individuals ch'₂Deletion of the Gene having a deletion number of 2, i.e., deletion of Gene g_5,1Then ch'₂＝{(4,15),(5,10),(11),(9,12),(1,3,6,7),(8,13,14)}；

Step F3 is performed: from virtual machine set VM₂＝{vm₁,vm₃,vm₄,vm₅Randomly selecting a virtual machine which is a virtual machine 1 again;

step F4 is performed: due to (g)_1,1,g_1,2) Since the ancestral task number of task 2 does not exist in (4,15) and the descendant task number of task 2 exists in (g)_1,1,g_1,2) Looking for the first descendant task number from front to back in (4,15), which is g_1,215 at g_1,2Previously randomly finding a position which is g_1,2Insertion of 2, i.e. g_1,2＝2，g_1,3＝15；

Step F5 is performed: finishing individual variation operation;

thus, the mutated individuals become: ch'₂＝{(4,2,15),(5,10),(11),(9,12),(1,3,6,7),(8,13,14)}；

Similarly, for other individuals in the new population, the variation becomes:

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

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

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

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

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

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

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

ch′₁₀＝{(2,6,9),(),(3),(1,4,11,10,12),(7),(5,8,13,14,15)}。

and step 8 is executed: improving all individuals in the new population by adopting FBI & D and LDI methods and calculating the fitness value of the individuals;

all individuals in the new population become, after improvement by the FBI & D method:

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

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

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

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

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

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

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

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

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

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

its fitness value, i.e. workflow response time: rs'₁＝3491.08，rs′₂＝2193.72，rs′₃＝1249.32，rs′₄＝2173.32，rs′₅＝2329.24，rs′₆＝2189.68，rs′₇＝1907.4，rs′₈＝1919.64，rs′₉＝2515.56，rs′₁₀＝2055.00；

All individuals in the new population become, after improvement by the LDI method:

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

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

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

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

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

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

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

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

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

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

its fitness value, i.e. workflow response time: rs'₁＝3491.08，rs′₂＝2117.72，rs′₃＝1178.60，rs′₄＝1949.84，rs′₅＝1893.64，rs′₆＝2189.68，rs′₇＝1907.4，rs′₈＝1899.72，rs′₉＝2034.64，rs′₁₀＝2055.00；

And step 9 is executed: selecting N different individuals from the current generation population and the new population from the superior to the inferior to form a new current generation population, and turning to the step 5;

selecting from good to bad according to the fitness valueCh of generation group₁、ch₄、ch₁₀And ch 'of the New population'₃、ch′₄、ch′₅、ch′₇、ch′₈、ch′₉、ch′₁₀Forming the next generation of population, i.e. GP ═ ch₁,ch₄,ch₁₀,ch′₃,ch′₄,ch′₅,ch′₇,ch′₈,ch′₉,ch′₁₀}；

Let CP be GP, then all individuals of the contemporary population are:

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

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

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

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

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

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

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

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

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

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

the fitness values, i.e. the workflow response times, are respectively: rs₁＝1098.16，rs₂＝2061.80，rs₃＝1887.12，rs₄＝1178.60，rs₅＝1949.84，rs₆＝1893.64，rs₇＝1907.40，rs₈＝1899.72，rs₉＝2034.64，rs₁₀＝2055.00；

Go to step 5.

……

Thus, the steps 5 to 9 are continuously and repeatedly executed until the optimal individual continuously iterates for 20 generations without improvement, and the contemporary population becomes:

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

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

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

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

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

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

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

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

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

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

the fitness values, i.e. the workflow response times, are respectively: rs₁＝1104.80，rs₂＝1103.60，rs₃＝1113.52，rs₄＝1107.44，rs₅＝1108.40，rs₆＝1109.6，rs₇＝1110.68，rs₈＝1099.36，rs₉＝1100.44，rs₁₀＝1018.08；

Go to step 10.

Executing the step 10: outputting the optimal individuals in the contemporary population, wherein the corresponding scheduling scheme is an optimal scheme;

the best individual in the contemporary population is ch₁₀{ (), (), (3,6), (2,10,11,12), (4,7,8,9), (1,5,13,14,15) }, workflow response time is: rs₁₀1018.08, the corresponding schedule optimization scheme is shown in table 3.

Scheduling order	Task numbering	Starting time	Execution time	End time	Virtual machine numbering
						1	3	0.00	182.16	182.16	3
2	2	0.00	185.16	185.16	4
						3	1	0.00	129.12	129.12	6
4	6	182.16	269.16	451.32	3
						5	4	129.12	193.12	322.24	5
6	5	185.16	254.32	439.48	6
						7	7	451.32	11.76	463.08	5
8	8	463.08	2.96	466.04	5
						9	10	466.04	60.64	526.68	4
10	11	526.68	259.44	786.12	4
						11	9	466.04	196.40	662.44	5
12	12	786.12	201.68	987.80	4
						13	13	987.80	9.28	997.08	6
14	14	997.08	10.00	1007.08	6
						15	15	1007.08	2.60	1009.68	6

TABLE 3

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 (4)

1. A cloud workflow scheduling optimization method based on a two-dimensional coding genetic algorithm is characterized by comprising the following steps: the method comprises the following steps: step 1: formalizing a scheduling problem, and acquiring information required by scheduling optimization;

get task set T ═ T₁,t₂,…,t_IWhere I is the number of tasks, t_iRepresenting a task i, namely a task with the number i;

acquiring a time sequence relation between tasks: parent task set PR of task i_iSubstask set SC for task i_iWherein I is 1,2 …, I;

acquiring task related parameters: length t of task i_iLength, i.e. the number of instructions that need to be consumed by a virtual machine when task i is processed, list t of input files that need to be processed when task i is processed_iIFL, output File List t generated after task i is processed_i-OFL, and the size of the file in the file list, I ═ 1,2 …, I; task i is task i⁺The requirements of the parent task of (2) are: there is a file that is the output file of task i and is also task i⁺The input file of (a), namely:

obtaining a virtual machine set VM ═ VM in a cloud computing environment₁,vm₂,…,vm_JWhere J is the number of virtual machines, vm_jRepresents virtual machine j, i.e., the virtual machine numbered j;

acquiring related parameters of the virtual machine: computing power vm of virtual machine j_jPs, bandwidth vm of virtual machine j_jBw, wherein J ═ 1,2 …, J;

acquiring a support relationship between the task and the virtual machine: task set T that virtual machine j can handle_jWherein J is 1,2 …, J; virtual machine set VM capable of processing task i_iWherein I is 1,2 …, I;

step 2: calculating the level value level of the task;

for a starting task i without a parent task, the hierarchy value is:

level_i＝1 (1)

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

and step 3: initializing a contemporary population;

initializing a current generation population, wherein the current generation population comprises 1 individual generated by an individual generation method based on HEFT-baseline and the remaining N-1 different individuals generated by an individual random generation method based on a hierarchy, and N is the population scale;

the individuals adopt two-dimensional integer coding, and the method comprises the following steps:

wherein g is_j,iA number indicating a task assigned to virtual machine j and scheduled i in virtual machine j;

is a sequence of task numbers assigned to virtual machine j and satisfies a task timing constraint, I_jRepresenting the number of tasks assigned to virtual machine j;

the individual generating method based on HEFT-baseline comprises the following steps:

step A1: make virtual machine available a time period list vatl_j＝{[0,M]}, variable k_j1, J-1, 2, …, J, where M is a near infinite number; enabling a task set UT to be processed to be T; let ready times rt of all tasks_i＝0，i＝1,…,I；

Step A2: finding out the task with the minimum level value in the UT, taking out the task with the maximum number of subtasks from the task, and if the task with the maximum number of subtasks is more than one, randomly taking one of the tasks without setting t as t_i；

Step A3: make available virtual machine set AVM_i＝VM_iCalculating t_iAre respectively allocated to AVM_iT after each virtual machine in (1)_iCompletion time of (d):

step A3.1: slave AVM_iGet one virtual machine out of it, set to vm_j；

Step A3.2: calculating t_iTo vm_jExecution time after processing

Step A3.3: in vatl_jFinding out an idle time period [ v ] from morning to evening_j,υ_j]Satisfy upsilon_j-ν_j≥et_i,jAnd upsilon_j-et_i,j≥rt_i；

Step A3.4: calculating t_iTo vm_jStart time after treatment s_i,j＝max{ν_j,rt_iH, completion time f_i,j＝s_i,j+et_i,j；

Step A3.5: if AVM_iIf not, go to step A3.1, otherwise go to step A4;

step A4: finding earliest achievable t in virtual machine order_iVirtual machine of (2) is not set to vm_j(ii) a Order to

k_j＝k_j+ 1; handle t_iTo vm_j：

Step A4.1: let t_iStart time s of_i＝s_i,j，t_iCompletion time f_i＝f_i,j；

Step A4.2: updating t_iReady time of subtask of (2)

Step A4.3: list of time slots available in virtual machine, vatl_jDeletion of [ v ]_j,υ_j]V, with insertion interval length greater than 0_j,s_i]And [ f_i,υ_j]；

Step A5: if UT is not empty, go to step A2, otherwise go to step A6;

step A6: outputting an individual based on HEFT-baseline

Finishing the operation;

wherein:

ω_i,j: is vm_jTreatment t_iThe time of (a) is,

is a handle t_iTo vm_jThe processing needs to obtain the file transfer time of the input file from other virtual machines,

is to treat

The virtual machine of (1);

τ_i,j: is a handle t_iTo vm_jThe process requires obtaining the file transfer time of the input file from the shared database,

the individual random generation method based on the hierarchy comprises the following steps:

step B1: let UT be T, let k be variable_j＝1，j＝1,2,…,J；

Step B2: randomly fetch a task with the minimum hierarchy value from UT, and set t as t_i；

Step B3: slave VM_iRandomly selecting one virtual machine in the virtual network, and setting the virtual machine not to be vm_j；

Step B4: order to

k_j＝k_j+1；

Step B5: if UT is not empty, go to step B2; otherwise go to step B6;

step B6: outputting an individual

Finishing the operation;

and 4, step 4: improving all individuals in the contemporary population by adopting FBI & D and LDI methods and calculating the fitness value of the individuals;

the fitness value is workflow response time rs, and the calculation method is as follows:

wherein: rf (radio frequency)_iIs the response time of the task i,

SFL_iis the set of output files that task i outputs to the shared database, i.e.

j is the virtual machine number of processing task i, i.e. satisfies

J of (1);

the smaller the individual fitness value is, the better the individual is;

the FBI & D method comprises the steps of:

step C1: make reverse workflow response time

Wherein M is a number approaching infinity;

step C2: decoding the individual ch by adopting a serial individual decoding method based on an insertion mode to obtain the completion time f of all tasks₁,…,f_IAnd its workflow response time rs; if rs is less than

Go to step C3, otherwise, go to step C6;

step C3: corresponding gene sequence of each virtual machine j in individual ch

Rearranging according to the task completion time from large to small, J is 1, …, J, forming reverse individuals

Step C4: to pair

Decoding by adopting a serial reverse individual decoding method based on an insertion mode to obtain reverse completion time of all tasks

And its reverse workflow response time

If it is

If the value is less than rs, go to step C5, otherwise, go to step C6;

step C5: make the opposite direction single body

The gene sequence corresponding to each virtual machine j in the database

Rearranging according to the reverse completion time of the task from large to small, wherein J is 1, …, J, forming an individual ch, and turning to the step C2;

step C6: outputting the individual ch and the fitness value thereof, namely the workflow response time rs, and finishing the operation;

the LDI method comprises the following steps:

step D1: ordering task lists assigned to virtual machines

Calculating each virtual machine load

Step D2: finding out a virtual machine j' with the minimum load; if ld is_j′>0, go to step D3, otherwise go to step D4;

step D3: order task set

Go to step D5;

step D4: order task set ST_j′＝T_j′Go to step D5;

step D5: if ST_j′If not, from ST_j′Sequentially fetching a task i 'with the highest load of the virtual machine j' to go to step D6; otherwise go to step D7;

step D6: in that

Searching the last ancestor task number and the first descendant task number of the task i 'from front to back, and if both the ancestor task number and the first descendant task number are found, randomly searching a position between the last ancestor task number and the first descendant task number to insert the task i'; if the descendant task number does not exist and the ancestor task number exists, randomly finding a position behind the last ancestor task number to insert i'; if the descendant task number exists and the ancestor task number does not exist, randomly finding a position before the first descendant task number and inserting i'; if the descendant task number and the ancestor task number do not exist, randomly finding a position to insert i'; in that

Deleting i' to form new individual, using FBI&D, the method carries out decoding improvement on the new individual, if the new individual is improved, the improved individual is used for replacing the original individual, and the step D7 is carried out; otherwise go to step D5;

step D7: the LDI operation is finished;

the definition of the descendant task and the ancestor task is described as follows: if there is a task sequence

Satisfy the requirement of

Is that

Where 1 is not more than k<n is then

Is that

The task of the ancestor of (c),

is that

The descendant task of (2);

and 5: judging whether a termination condition is met, if so, turning to a step 10 after the evolution is finished, otherwise, turning to a step 6;

the termination condition is that the optimal individual is not improved after iteration to a designated generation TG or continuous iteration GG generation;

step 6: carrying out N times of parameterized uniform cross operations based on preference and two-dimensional topological sorting on the contemporary population to form a new population;

the parameterized uniform crossover operation based on preference and two-dimensional topological ordering comprises the following steps:

step E1: randomly selecting two different individuals from the contemporary population as parents by adopting a ranking-based round-robin method, and setting the individuals as parents

And ch^p1Has a fitness value of ch or less^p2Wherein: task number list

Task number list

J is 1, …, J, order the task number list

Making a cycle control variable delta equal to 1;

step E2: if δ ≦ I, go to step E3, otherwise go to step E6;

step E3: generating a random number λ ∈ [0,1) if λ<p_bGo to step E4, otherwise go to step E5;

step E4: from ch^p1Randomly selecting one task which is not empty and is numbered as the first element of the task, wherein the parent task does not exist in the task or the parent task is scheduled

From

Take out the first element i' and add it to

And from ch^p2If i', δ is δ +1, go to step E2;

step E5: from ch^p2Randomly selecting one task which is not empty and is numbered as the first element of the task, wherein the parent task does not exist in the task or the parent task is scheduled

From

Take out the first element i' and add it to

And from ch^p1Deleting i ", δ ═ δ +1, go to step E2;

step E6: output sub-body

Finishing the operation;

wherein: p is a radical of_bE (0.5,1) is the preference rate;

and 7: performing mutation operation on each individual in the new population;

the mutation operation comprises the following steps:

step F1: randomly generating 1 random number lambda epsilon [0,1) if lambda is less than or equal to p_mGo to step F2, otherwise go to step F5;

step F2: randomly selecting a task number i, and deleting a gene with the value of i in an individual;

step F3: slave VM_iRandomly selecting a virtual machine again, and setting the virtual machine as a virtual machine j;

step F4: in that

Searching the last ancestor task number and the first descendant task number of the task i from front to back, and if both the last ancestor task number and the first descendant task number are found, randomly finding a position between the last ancestor task number and the first descendant task number to insert the i; if the descendant task number does not exist and the ancestor task number exists, randomly finding a position behind the last ancestor task number to insert i; if the descendant task number exists and the ancestor task number does not exist, randomly finding a position before the first descendant task number to insert i; if the descendant task number and the ancestor task number do not exist, randomly finding a position to insert i;

step F5: finishing individual variation operation;

wherein: p is a radical of_mE [0,1) is the mutation rate;

and 8: improving all individuals in the new population by adopting FBI & D and LDI methods and calculating the fitness value of the individuals;

and step 9: selecting N different individuals from the current generation population and the new population from the superior to the inferior to form a new current generation population, and turning to the step 5;

step 10: and outputting the optimal individuals in the contemporary population, wherein the corresponding scheduling scheme is an optimal scheme.

2. The cloud workflow scheduling optimization method based on the two-dimensional coding genetic algorithm according to claim 1, characterized in that: the serial individual decoding method based on the insertion mode in the step C2 specifically includes the following steps:

step G1: let ready times rt of all tasks_i0, I-1, …, I; make all virtual machines available a time period list vatl_j＝{[0,M]And the task number list allocated to the virtual machine

Wherein M is a number approaching infinity;

step G2: selecting a VTI which is not empty and has no parent task or has the scheduled parent task in the first element according to the numbering sequence of the virtual machine_jFrom VTI_jTaking out the first element i;

step G3: assigning task i to virtual machine j based on the insertion pattern;

step G3.1: computing the execution time of task i

Step G3.2: in vatl_jFinding out an idle time period [ v ] from morning to evening_j,υ_j]Satisfies upsilon_j-ν_j≥et_iAnd upsilon_j-et_i≥rt_i；

Step G3.3: calculating the start time and the completion time of task i: s_i＝max{ν_j,rt_i}，f_i＝s_i+et_i(ii) a Updating the Ready time of a subtask of task i

Step G3.4: in vatl_jDeletion of [ v ]_j,υ_j]V, with insertion interval length greater than 0_j,s_i]And [ f_i,υ_j]；

Step G4: if VTI₁,…,VTI_JIf not, go to step G2, otherwise, go to step G5;

step G5: obtain start and completion times for all tasks: s_i、f_iI1, …, I, calculating a fitness value; the operation is ended.

3. The cloud workflow scheduling optimization method based on the two-dimensional coding genetic algorithm according to claim 1, characterized in that: the specific steps of the serial reverse individual decoding method based on the insertion mode in step C4 are as follows:

step H1: make reverse ready time of all tasks

Wherein: i is 1, …, and I, j is the virtual machine number that processes task I, i.e., satisfies

J of (1); make virtual machine available a time period list vatl_j＝{[0,M]And the task number list allocated to the virtual machine

Wherein M is a number approaching infinity;

step H2: selecting a VTI which is not empty and has no subtask numbered first element or has all subtasks scheduled according to the numbering sequence of the virtual machine_jFrom VTI_jTaking out the first element i;

step H3: assigning task i to virtual machine j based on insertion patterns:

step H3.1: computing the execution time of task i

Step H3.2: in vatl_jFinding out an idle time period [ v ] from morning to evening_j,υ_j]Satisfy upsilon_j-ν_j≥et_iAnd

step H3.3: calculating a reverse start time for task i

Reverse completion time

Updating the Ready time of the parent task of task i

Step H3.4: list of time slots available in virtual machine, vatl_jDeletion of [ v ]_j,υ_j]With an insertion interval length greater than 0

And

step H4: if VTI₁,…,VTI_JIf not, go to step H2, otherwise, go to step H5;

step H5: obtaining reverse start times and reverse completion times for all tasks:

calculating reverse workflow response times

The operation is ended.

4. The cloud workflow scheduling optimization method based on the two-dimensional coding genetic algorithm according to claim 1, characterized in that: the specific steps of randomly selecting two different individuals from the contemporary population as parents based on the ranking round-robin in the step E1 are as follows:

step E1.1: sequencing the individuals in the contemporary population from good to bad, namely, the fitness value is from small to large, and obtaining 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 E1.2: calculating the probability that the individual n is selected

Wherein R is>1 is the discrimination coefficient;

step E1.3: calculating cumulative probability

Step E1.4: generating a random number lambda₁E [0,1) if

Then the individual n is selected;

step E1.5: generating a random number lambda₂E [0,1) if

And n '≠ n, then individual n' is selected, go to step E1.6, otherwise go to step E1.5;

step E1.6: two different individuals n and n' are obtained as parents and the individual selection operation ends.

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