CN109784603B - Method for solving flexible job shop scheduling based on mixed whale swarm algorithm - Google Patents

Abstract

The invention discloses a method for solving flexible job shop scheduling based on a mixed whale swarm algorithm, which comprises the steps of firstly defining a coding mode of flexible job shop scheduling as two-section random key coding, and then carrying out mapping conversion by adopting a conversion mechanism; defining an adaptability function to solve the shortest total processing time as an optimization target; initializing parameters in scheduling problems of a flexible job shop and whale population by adopting a whale swarm algorithm, wherein the initialization is divided into a sequencing scheme of a random generation procedure and a genetic variation mode adopting an improved genetic algorithm to generate a better machine allocation scheme corresponding to the procedure sequencing scheme, so as to generate a better initial population; calculating the fitness value of each scheduling scheme, and searching and reserving the best scheduling solution; and finally, outputting an optimal scheduling solution and a corresponding fitness function value thereof, namely the optimal scheduling solution is obtained, and the problems of low solving precision and low convergence speed in the existing flexible job shop scheduling problem are solved.

Description

A method for solving flexible job shop scheduling based on hybrid whale swarm algorithm

Technical field

The invention belongs to the technical field of flexible job shop scheduling, and specifically relates to a method for solving flexible job shop scheduling based on a hybrid whale swarm algorithm.

Background technique

As an extended form of the traditional job shop scheduling problem, the Flexible Job-shop Scheduling Problem (FJSP) adds the problem of selecting a flexible processing path for the workpiece. Its solution is more difficult and closer to the actual production. It has been It is proved to be a combinatorial optimization problem with NP-hard characteristics. Various intelligent algorithms are currently the main means to solve this problem, and have also become a research hotspot in the current field of production scheduling.

In the existing technology, FJSP associates the processing sequence with the start time. By learning the separation graph model, a tabu search algorithm with domain specificization and solution space diversification was designed, and the algorithm was verified through testing of standard examples. Advantages; an efficient ranking method for solving the minimum maximum completion time FJSP based on the discrete harmonious search algorithm, an improved hybrid gray wolf optimization algorithm, and three initialization populations through evolutionary population dynamics, reverse learning, and optimal individual mutation. On the other hand, the algorithm was improved and a standard example was simulated. The results show that the improved hybrid gray wolf optimization algorithm can effectively jump out of the local optimum and find a better solution, and the results are more robust. By applying the enclosing Boolean matrix in the polychromatic set theory to construct a multi-constraint model and a single-layer genetic coding method for the flexible job shop scheduling problem, the search range of the GA algorithm can be narrowed, and the space and time complexity of the chromosome can be effectively reduced. The speed and accuracy of the algorithm for solving FJSP problems are improved. The Hybrid Gray Wolf Optimization Algorithm (HGWO) uses a two-stage individual coding method to achieve continuous coding of discrete scheduling solutions, and uses a variable neighborhood search strategy to enhance the local search capability of the algorithm. At the same time, it introduces the crossover and mutation operations of the genetic algorithm to enhance In order to improve the diversity of the population, through simulation tests on standard examples of flexible job shop scheduling problems, the results show that this algorithm can effectively avoid premature maturity and improve the solution speed and accuracy. The whale optimization algorithm (WOA) improved by quantum computing theory was applied to solve the job shop scheduling problem, which proved the convergence and correctness of the algorithm. The job shop scheduling problem is a discrete scheduling problem, and the improved whale The algorithm mainly achieves solution and optimization by performing iterative operations on continuous whale individual position vectors. The existing technology does not provide a specific conversion process between the two. Therefore, applying new algorithms to solve production scheduling problems is still one of the hot topics in the manufacturing field.

WSA is a new metaheuristic algorithm inspired by the use of ultrasonic communication between individuals to guide whales during their hunting process. With its special iterative method, WSA has proven its superiority in solving combinatorial optimization problems. However, to apply it to solve discrete problems, we must first solve the mutual conversion problem between the problem solution and the individual position vector. Therefore, this paper solves the mutual conversion between the scheduling problem solution and the individual position vector by introducing the classic transformation mechanism. This algorithm is applied to solve the classic combinatorial optimization problem of FJSP, and the superiority of the algorithm is verified through practical solutions and comparative analysis, in order to explore a new way to solve FJSP.

Contents of the invention

The purpose of the present invention is to provide a method for solving flexible job shop scheduling based on the hybrid whale swarm algorithm, which solves the problems of low solution accuracy and slow convergence speed in flexible job shop scheduling problems existing in the prior art.

The technical solution adopted by the present invention is a method for solving flexible job shop scheduling based on the hybrid whale swarm algorithm, which is specifically implemented in accordance with the following steps:

Step 1. Define the coding method of flexible job shop scheduling as two-stage random key coding, and then use the conversion mechanism to perform mapping conversion;

Step 2. Define the fitness function to minimize the total processing time as the optimization goal;

Step 3. Use the whale swarm algorithm to initialize the parameters in the flexible job shop scheduling problem and the whale population. First, the upper and lower bounds ε of the individual position variable elements in the whale swarm algorithm, the sound source intensity ρ ₀ , the current iteration number t, and the maximum iteration The number M parameter is set; the population initialization is divided into two steps. The first step is to randomly generate the sorting scheme of the process, and the second step is to use the genetic mutation method of the improved genetic algorithm to generate a better machine allocation scheme corresponding to the process sorting scheme. Then produce a better initial population;

Step 4. Calculate the fitness value of each scheduling solution, find and retain the best scheduling solution S ^* ;

Step 5. Output the optimal scheduling solution S ^* and its corresponding fitness function value. S ^* is the optimal scheduling solution sought.

The present invention is also characterized in that,

The conversion mechanism in step 1 is specifically implemented according to the following steps:

a. Conversion of scheduling plan to individual location:

i) Machine selection: Convert the sequence number of the optional machine in the process into the element value of the individual position vector according to the following formula:

x(i)＝[2m/(s(i)-1)](n(i)-1)-m,s(i)≠1

Among them: x(i) represents the i-th element of the individual position vector; s(i) represents the number of machines that can be selected for the process corresponding to element i; n(i)∈[1,s(i)] represents the selected machine in The serial number in the optional machine set. If s(i)=1, then x(i) takes any value within [-m,m].

ii) Process sorting: First generate a set of random numbers within [-m, m], and arrange the ROV rules in ascending order to assign a unique ROV value to each random number so that each ROV value corresponds to a process, and then according to the process The ROV values are rearranged in the coding order, and the random number sequence corresponding to the rearranged ROV values is the value of each element in the individual position vector.

b. Conversion of individual position vectors to scheduling plans:

i) Machine selection: Perform the reverse operation of the formula in the conversion of the scheduling plan to individual positions in a, thereby obtaining the number of the machine:

ii) Process sorting: First, each element number corresponds to the process sequence and position element in a unique code, and then the position elements are arranged in ascending order according to the ROV rules. At this time, the ROV value corresponds to the element number to construct a process sequencing scheme;

Step 2 is specifically implemented according to the following steps:

Step 2.1. The problem model FJSP of flexible job shop scheduling is described as follows:

Suppose M is the number of processing equipment, N is the number of workpieces to be processed, P is the number of processes, and I is the set of all equipment; I _eg represents the set of available equipment for the gth process of workpiece e, J _e is the number of processes for workpiece e, x is the processing order of all workpieces, S _egk represents the start time of the gth process of workpiece e on equipment k; E _egk is the gth process of workpiece e on equipment k _{The processing end time of _} Indicates the final completion time of all workpieces;

When the j-th process of workpiece i and the g-th process of workpiece e are executed on the same equipment, if process j is processed before process g, Q _ijeg = 1, otherwise Q _ijeg = 0; if the g-th process of workpiece e If the process is processed on machine tool k, then X _egk = 1, otherwise X _egk = 0;

If there are S possible processing sequences for a certain FJSP, and the processing sequence with the shortest total operating time is required, first find the operating time corresponding to each processing sequence x (x∈{1,...,S}); obviously, the order The completion time of the last processing step in x is the final completion time of all workpieces:

MS=E _p (1)

Step 2.2. Let the objective function F(x) be:

F(x)=min( MS _x )=min((E _p ) _x ) (2)

Among them, x=1,...,S; _Qijeg =1;

STS _egk -E _e(g-1)n ≥0 (3)

e=1,...,N; g=1,...,J _e ;X _egk =1,X _e(g-1)n =1

S _egk -E _ijk ≥0 (4)

e=1,...,N; g=1,...,J _e ;X _ijk =1,X _egk =1,Q _ijeg =1.

Step 4 is specifically implemented according to the following steps:

Step 4.1, use formula (5) to perform iterative operations, as follows:

In the formula: and/> Refers to the position of the i-th element of X in the t step and t+1 step iteration respectively;

Refers to the position of the i-th element of Y in the t-step iteration;

d _X,Y refers to the distance between X and Y;

Indicates 0 to/> Random numbers generated between them, where the value of the attenuation coefficient η is related to the dimension, domain and peak distribution characteristics of the objective function;

Step 4.2: Use the conversion mechanism to convert the scheduling solution into a whale individual position vector, and retain the individual X ^* corresponding to S ^* ;

Step 4.3. Use formula (5) to perform an iterative operation to determine whether the termination condition of the algorithm is met. If not, set t=t+1 and repeat steps 4.3.1 to 4.3.7; if it is met, execute step 5. ,details as follows:

Step 4.3.1. Define the distance calculation method between two whales;

Step 4.3.2, for each whale Find the better and closest individual; if it does not exist, keep it unchanged;

Step 4.3.3. Find all fitness values greater than individuals, such as/>

Step 4.3.4. Calculate each fitness value greater than individual/> with/> The distance between D ₁ , D ₂ , D ₃ ;

Step 4.3.5. Sort D ₁ , D ₂ and D ₃ and select the individual corresponding to the minimum value. That is/> the better and closest individual;

Step 4.3.6, will and/> The value of and initialization parameters are brought into the iteration formula (5) to update each individual position vector/>

Step 4.3.7: Execute the conversion method of individual position vectors to scheduling plans, generate the scheduling plans corresponding to all individuals after the update, and return to step 4.2.

In step 4.1, ρ ₀ takes the empirical value 2.

In step 4.1, the initial value of the attenuation coefficient η is determined as follows:

First, let

Right now

d _max refers to the maximum possible distance between two whales in the search area,

Among them, n is the dimension of the objective function, with/> represent the lower limit and upper limit of the i-th variable respectively, therefore, η=-20·ln(0.25)/d _max .

The beneficial effect of the present invention is that, based on the hybrid whale swarm algorithm to solve the flexible job shop scheduling method, the mutual conversion between the FJSP scheduling solution and the whale individual position vector is realized through the conversion mechanism, and then the strong local and global search capabilities of WSA are utilized. As well as the advantages in maintaining population diversity, iterative solution is performed to improve the solution accuracy and speed of FJSP.

Description of the drawings

Figure 1 is a flow chart of the whale swarm algorithm for solving FJSP in a method of solving flexible job shop scheduling based on the hybrid whale swarm algorithm of the present invention;

Figure 2 is a WSA evolutionary convergence curve of Embodiment 1 of a method for solving flexible job shop scheduling based on the hybrid whale swarm algorithm of the present invention;

Figure 3 is a Gantt chart of the scheduling results in Embodiment 1 of a method for solving flexible job shop scheduling based on the hybrid whale swarm algorithm of the present invention;

Figure 4 is a WSA evolutionary convergence curve in Embodiment 2 of a method for solving flexible job shop scheduling based on the hybrid whale swarm algorithm of the present invention;

Figure 5 is a WSA evolutionary convergence curve in Embodiment 3 of a method for solving flexible job shop scheduling based on the hybrid whale swarm algorithm of the present invention.

Detailed ways

The present invention will be described in detail below with reference to the drawings and specific embodiments.

The present invention is a method for solving flexible job shop scheduling based on the hybrid whale swarm algorithm, as shown in Figure 1, which is specifically implemented in accordance with the following steps:

Step 1. Define the coding method of flexible job shop scheduling as two-stage random key coding, and then use the conversion mechanism to perform mapping conversion. The conversion mechanism is specifically implemented according to the following steps:

a. Conversion of scheduling plan to individual location:

i) Machine selection: Convert the sequence number of the optional machine in the process into the element value of the individual position vector according to the following formula:

x(i)＝[2m/(s(i)-1)](n(i)-1)-m,s(i)≠1

Among them: x(i) represents the i-th element of the individual position vector; s(i) represents the number of machines that can be selected for the process corresponding to element i; n(i)∈[1,s(i)] represents the selected machine in The serial number in the optional machine set. If s(i)=1, then x(i) takes any value within [-m,m].

ii) Process sorting: First generate a set of random numbers within [-m, m], and arrange the ROV rules in ascending order to assign a unique ROV value to each random number so that each ROV value corresponds to a process, and then according to the process The ROV values are rearranged in the coding order, and the random number sequence corresponding to the rearranged ROV values is the value of each element in the individual position vector.

b. Conversion of individual position vectors to scheduling plans:

i) Machine selection: Perform the reverse operation of the formula in the conversion of the scheduling plan to individual positions in a, thereby obtaining the number of the machine:

ii) Process sorting: First, each element number corresponds to the process sequence and position element in a unique code, and then the position elements are arranged in ascending order according to the ROV rules. At this time, the ROV value corresponds to the element number to construct a process sequencing scheme;

Step 2. Define the fitness function, with the optimization goal of minimizing the total processing time, and implement it according to the following steps:

Step 2.1. The problem model FJSP of flexible job shop scheduling is described as follows:

Suppose M is the number of processing equipment, N is the number of workpieces to be processed, P is the number of processes, and I is the set of all equipment; I _eg represents the set of available equipment for the gth process of workpiece e, J _e is the number of processes for workpiece e, x is the processing order of all workpieces, S _egk represents the start time of the gth process of workpiece e on equipment k; E _egk is the gth process of workpiece e on equipment k _{The processing end time of _} Indicates the final completion time of all workpieces;

When the j-th process of workpiece i and the g-th process of workpiece e are executed on the same equipment, if process j is processed before process g, Q _ijeg = 1, otherwise Q _ijeg = 0; if the g-th process of workpiece e If the process is processed on machine tool k, then X _egk = 1, otherwise X _egk = 0;

If there are S possible processing sequences for a certain FJSP, and the processing sequence with the shortest total operating time is required, first find the operating time corresponding to each processing sequence x (x∈{1,...,S}); obviously, the order The completion time of the last processing step in x is the final completion time of all workpieces:

MS=E _p (1)

Step 2.2. Let the objective function F(x) be:

F(x)=min( MS _x )=min((E _p ) _x ) (2)

Among them, x=1,...,S; _Qijeg =1;

STS _egk -E _e(g-1)n ≥0 (3)

e=1,...,N; g=1,...,J _e ;X _egk =1,X _e(g-1)n =1

S _egk -E _ijk ≥0 (4)

e=1,...,N; g=1,...,J _e ;X _ijk =1,X _egk =1,Q _ijeg =1;

Step 3. Use the whale swarm algorithm to initialize the parameters in the flexible job shop scheduling problem and the whale population. First, the upper and lower bounds ε of the individual position variable elements in the whale swarm algorithm, the sound source intensity ρ ₀ , the current iteration number t, and the maximum iteration The number M parameter is set; the population initialization is divided into two steps. The first step is to randomly generate the sorting scheme of the process, and the second step is to use the genetic mutation method of the improved genetic algorithm to generate a better machine allocation scheme corresponding to the process sorting scheme. Then produce a better initial population;

Step 4. Calculate the fitness value of each scheduling solution, find and retain the best scheduling solution S ^* , and implement it according to the following steps:

Step 4.1, use formula (5) to perform iterative operations, as follows:

In the formula: and/> Refers to the position of the i-th element of X in the t step and t+1 step iteration respectively;

Refers to the position of the i-th element of Y in the t-step iteration;

d _X,Y refers to the distance between X and Y;

Indicates 0 to/> Random numbers generated between them, where the value of the attenuation coefficient η is related to the dimension, domain and peak distribution characteristics of the objective function;

Step 4.2: Use the conversion mechanism to convert the scheduling solution into a whale individual position vector, and retain the individual X ^* corresponding to S ^* ;

Step 4.3. Use formula (5) to perform an iterative operation to determine whether the termination condition of the algorithm is met. If not, set t=t+1 and repeat steps 4.3.1 to 4.3.7; if it is met, execute step 5. ,details as follows:

Step 4.3.1. Define the distance calculation method between two whales;

Step 4.3.2, for each whale Find the better and closest individual; if it does not exist, keep it unchanged;

Step 4.3.3. Find all fitness values greater than individuals, such as/>

Step 4.3.4. Calculate each fitness value greater than individual/> with/> The distance between D ₁ , D ₂ , D ₃ ;

Step 4.3.5. Sort D ₁ , D ₂ and D ₃ and select the individual corresponding to the minimum value. That is/> the better and closest individual;

Step 4.3.6, will and/> The value of and initialization parameters are brought into the iteration formula (5) to update each individual position vector/>

Step 4.3.7. Execute the conversion method of individual position vectors to scheduling plans, generate the scheduling plans corresponding to all individuals after the update, and return to step 4.2;

In step 4.1, ρ ₀ takes the empirical value 2;

In step 4.1, the initial value of the attenuation coefficient η is determined as follows:

First, let

Right now

d _max refers to the maximum possible distance between two whales in the search area,

Among them, n is the dimension of the objective function, with/> represent the lower limit and upper limit of the i-th variable respectively, therefore, η=-20·ln(0.25)/d _max ;

Step 5. Output the optimal scheduling solution S ^* and its corresponding fitness function value. S ^* is the optimal scheduling solution sought.

This invention is a method for solving flexible job shop scheduling based on the hybrid whale swarm algorithm. In order to verify the feasibility of WSA in solving flexible job shop scheduling problems, this paper uses it to conduct simulation experiments on three embodiments respectively, and conducts simulation experiments with other intelligent algorithms. Comparative analysis. The simulation environment is: using MATILAB2016a, configured with a CPU frequency of 1.9GHz, AMD A10-7300 Radeon R6 processor, and Windows 10 operating system.

Example 1

Example 1 is a 3×6 FJSP. Its processing task information is shown in Table 1. The parameters of the whale group algorithm are set as follows: the population size is 50, the dimension of the individual whale position vector is 30, ρ ₀ is set to 2, and d _max = 32.86 , the attenuation coefficient η is set to 0, and the maximum number of iterations M is 400:

Table 1 Example 1 processing task information table

Machine allocation and process sequencing are encoded using a two-stage encoding based on random keys. The length of the individual position vector is 30, and each element takes an arbitrary value in [-3,3]. A computer is used to generate a 30-bit random numbers and stored in a certain order, as shown in Table 2, where OP _ij represents the jth process of workpiece i:

Table 2 Individual position vector diagram

For Embodiment 1, the WSA evolutionary convergence curve obtained by solving it is shown in Figure 2. It can be seen from Figure 2 that this whale group algorithm can quickly converge from 144 to 134 in 35 generations, and its corresponding scheduling result Gantt chart As shown in Figure 3.

Example 2

In order to further verify the feasibility and superiority of the algorithm, the literature [1] (Fu Weiping, Liu Dongmei, Lai Chunwei, Wang Wen. An improved genetic algorithm based on multi-color sets to solve multi-variety flexible scheduling problems [J]. Computer Integrated Manufacturing System, 2011 ,17(05):1004-1010.) to conduct simulation. For specific processing information, please refer to the literature [1]. The parameters of the whale group algorithm are set as follows: the population size is 100, and the dimension of the individual whale position vector in Example 1 is 108, ρ ₀ is set to 2, d _max =104.96, the attenuation coefficient η is set to 0, and the maximum number of iterations M is 200. The WSA evolutionary convergence curve is shown in Figure 4, and the optimal solution is 121 minutes. From the comparison of the WSA evolutionary convergence curve in Figure 4 and the evolution curve of Example 2, it can be seen that this whale swarm algorithm can quickly achieve the goal in 32 generations. It converges from 150 to 121. In comparison, its solution speed is obviously faster.

Example 3

In order to compare and verify the effect of the algorithm more comprehensively, the 8×8 example in the Kacem benchmark problem of flexible job scheduling is selected for solution and calculated 20 times. The parameters of the whale group algorithm are set as follows: the population size is 100, the dimension of the individual whale position vector is 54, ρ ₀ is set to 2, d _max = 117.57, the attenuation coefficient η is 0, the maximum number of iterations M is 200, WSA of Embodiment 3 The evolutionary convergence curve is shown in Figure 5. Table 3 compares the results obtained by the whale swarm algorithm with those of the evolutionary algorithm, master-slave genetic algorithm, multi-order genetic algorithm, and ant colony algorithm:

Table 3 Comparison of the solution results of various algorithms for the Kacem 8×8 benchmark problem

method name The optimal value Excellent value times mean convergence algebra Average convergence time/min evolutionary algorithm 15 1 Master-Slave Genetic Algorithm 19 2 200 41.8 multi-stage algorithm 15 Ant colony genetic algorithm 14 2 53 8.7 whale swarm algorithm 14 3 35 7.6

In view of the shortcomings in solving accuracy and speed of traditional algorithms when solving FJSP, the present invention uses a new meta-heuristic algorithm WSA to solve FJSP, and establishes a solution to the scheduling problem by using a two-stage encoding method and introducing a conversion mechanism. and the mutual mapping between the individual whale position vectors, and then use WSA to solve the FJSP. The results of the simulation example show that WSA has certain advantages in solving the FJSP. The special iteration formula of this algorithm finds the better and nearest iteration. This method has certain advantages in solving combinatorial optimization problems.

Claims (2)

1. The method for solving the flexible job shop scheduling based on the mixed whale swarm algorithm is characterized by comprising the following steps:

step 1, defining a coding mode of flexible job shop scheduling as two-section random key coding, and then mapping and converting by adopting a conversion mechanism;

the conversion mechanism in the step 1 is specifically implemented according to the following steps:

a. conversion of scheduling scheme to individual location:

i) Machine selection: converting sequence numbers in the process selectable machine set into individual position vector element values according to the following steps:

x(i)＝[2m/(s(i)-1)](n(i)-1)-m,s(i)≠1

wherein: x (i) represents the i-th element of the individual position vector; s (i) represents the number of machines which can be selected by the procedure corresponding to the element i; n (i) ∈ [1, s (i) ] represents the number of the selected machine in the set of selectable machines, and if s (i) =1, x (i) is arbitrarily valued within [ -m, m ];

ii) sequencing the working procedures: firstly, generating a group of random numbers in [ -m, m ], assigning a unique ROV value for each random number according to an ascending order ROV rule, enabling each ROV value to correspond to a procedure, rearranging the ROV values according to the coding sequence of the procedure, and enabling the random number sequence corresponding to the rearranged ROV values to be the value of each element in the individual position vector;

b. conversion of individual location vectors to scheduling scheme:

i) Machine selection: the inverse operation of the formula in the conversion of the scheduling scheme to individual positions in said a is performed, whereby the number of machines is obtained:

ii) sequencing the working procedures: firstly, enabling each element number to correspond to a unique sequence and position element in the code, then carrying out ascending arrangement on the position elements according to an ROV rule, and constructing a sequence ordering scheme by enabling the ROV value to correspond to the element number;

step 2, defining a fitness function to solve the shortest total processing time as an optimization target;

the step 2 is specifically implemented according to the following steps:

step 2.1, a problem model FJSP of flexible job shop scheduling is described as follows:

assuming that M is the number of processing devices, N is the number of workpieces to be processed, P is the number of procedures, and I is the set of all the devices; i _eg A set of available equipment representing the g-th pass of workpiece e,J _e the number of steps of the workpiece e, x is the processing sequence of all the workpieces, S _egk Indicating the start time of the processing of the g-th procedure of the workpiece e on the equipment k; e (E) _egk The end time of the processing of the g-th procedure of the workpiece e on the equipment k; t (T) _egk For the duration of the process of the g-th pass of the workpiece e on the apparatus k, and k.epsilon.I _eg Then there is E _egk ＝S _egk +T _egk ；E _p Indicating the finishing time of the final process; h represents the final finishing time of all the workpieces;

when the j-th process of the workpiece i and the g-th process of the workpiece e are performed on the same equipment, if the j-th process is processed before the g-th process, Q _ijeg =1, otherwise Q _ijeg =0; if the g-th step of the workpiece e is performed on the machine tool k, X _egk =1, otherwise X _egk ＝0；

If a certain FJSP has S possible machining sequences, a machining sequence with the shortest total working time is required, working time corresponding to each machining sequence x is firstly obtained, and x is { 1..once, S }; obviously, the finishing time of the last machining process in sequence x is the finishing time of all the workpieces:

H＝E _p (1)

step 2.2, setting an objective function F (x) as:

F(x)＝min(H)＝min((E _p ) _x ) (2)

wherein x=1, …, S; q (Q) _ijeg ＝1；

S.T. S _egk -E _e(g-1)n ≥0 (3)

e＝1，…，N；g＝1，…，J _e ；X _egk ＝1，X _e(g-1)n ＝1

S _egk -E _ijk ≥0 (4)

e＝1，…，N；g＝1，…，J _e ；X _ijk ＝1,X _egk ＝1,Q _ijeg ＝1；

Step 3, initializing parameters in scheduling problems of flexible job shops and whale population by adopting whale swarm algorithm, and firstly initializing upper and lower bounds epsilon and sound source intensity rho of individual position variable elements in whale swarm algorithm ₀ Setting parameters of the current iteration times t and the maximum iteration times M; the population initialization is carried out in two steps, wherein the first step is to randomly generate a sequencing scheme of a working procedure, and the second step is to generate a better machine allocation scheme corresponding to the working procedure sequencing scheme by adopting a genetic variation mode of an improved genetic algorithm, so as to generate a better initial population;

step 4, calculating the adaptability value of each scheduling scheme, and searching and reserving an optimal scheduling solution S ^* ；

The step 4 is specifically implemented according to the following steps:

step 4.1, performing iterative operation by using the formula (5), which is specifically as follows

Wherein:and->Respectively refers to the iterative positions of the ith element of X in the steps t and t+1;

refers to the position of the ith element of Y in t steps of iteration;

d _X,Y refers to the distance between X and Y;

representing 0 to->The random number generated in the process, wherein the value of the attenuation coefficient eta is related to the dimension, the definition domain and the peak distribution characteristic of the objective function;

in the step 4.1, the initial value of the attenuation coefficient η is determined according to the following method:

first, let the

I.e.

d _max Refers to the maximum distance possible between two whales in the search area,

where n is the dimension of the objective function,and->Represents the lower and upper limits of the ith variable, respectively, so η= -20·ln (0.25)/d _max ；

Step 4.2, converting the scheduling solution into whale individual position vectors by using a conversion mechanism, and reserving S ^* Corresponding individual X ^* ；

Step 4.3, performing iterative operation by using a formula (5), judging whether the termination condition of the algorithm is met, if not, making t=t+1, and repeatedly executing the steps 4.3.1-4.3.7; if yes, executing step 5, specifically as follows:

step 4.3.1, defining a distance calculating method between two whales;

step 4.3.2 for each whaleSearching for a preferred and nearest individual; if not, remain stationary;

step 4.3.3, finding all fitness values larger thanIndividuals of (E) such as->

Step 4.3.4, calculating each fitness value to be larger thanIs->And->Distance D between ₁ ,D ₂ ,D ₃ ；

Step 4.3.5, pair D ₁ ,D ₂ ,D ₃ Sorting, selecting the smallest one as D ₃ D is then ₃ Corresponding individualsNamely +.>Is "preferred and most recent" individuals;

step 4.3.6, willAnd->The values of (1) and the initialized parameters bring into the iterative formula (5) to update each individual position vector +.>

Step 4.3.7, executing a conversion mode from the individual position vector to the scheduling scheme, generating the scheduling schemes corresponding to all the individuals after updating, and returning to step 4.2;

step 5, outputting an optimal scheduling solution S ^* And the corresponding fitness function value, S ^* The optimal scheduling scheme is obtained.

2. The method for solving flexible job shop scheduling according to claim 1, wherein ρ in step 4.1 is ₀ Take the empirical value 2.