CN109784603B - Method for solving flexible job shop scheduling based on mixed whale swarm algorithm - Google Patents
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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
Technical Field
The invention belongs to the technical field of flexible job shop scheduling, and particularly relates to a method for solving flexible job shop scheduling based on a mixed whale swarm algorithm.
Background
The Flexible Job shop scheduling problem (FJSP) is used as an extension form of the traditional Job shop scheduling problem, so that the selection problem of a Flexible processing path of a workpiece is increased, the difficulty of solving the Flexible processing path is higher, the Flexible Job shop scheduling problem is closer to the actual production, and the Flexible Job shop scheduling problem has proved to be a combined optimization problem with NP difficult characteristics. For solving the problem, various intelligent algorithms are currently the main means of the intelligent algorithms, and are also becoming research hotspots in the current production scheduling field.
In the prior art, FJSP related to processing sequence and starting time designs a tabu search algorithm for field materialization and space diversification through learning of a separation graph model, and verifies the superiority of the algorithm through testing of standard calculation examples; the method for solving the high-efficiency sequencing of the minimized finishing time FJSP based on the discrete harmony search algorithm improves the mixed wolf optimization algorithm, improves the algorithm by three aspects of population dynamic evolution, population initialization by reverse learning, optimal individual variation and the like, simulates a standard example, and shows that the improved mixed wolf optimization algorithm can effectively jump out of a local optimal value, find a better solution and has stronger result robustness. The multi-constraint model and the single-layer genetic coding mode of the flexible job shop scheduling problem are constructed by applying the girth Boolean matrix in the multicolor set theory, so that the searching range of the GA algorithm is reduced, the space and time complexity of the chromosome is effectively reduced, and the speed and the precision of solving the FJSP problem by the algorithm are further improved. A hybrid gray wolf optimization algorithm (HGWO) adopts a two-section individual coding mode to realize continuous coding of discrete scheduling solutions, and utilizes a variable neighborhood search strategy to strengthen local search capability of the algorithm, meanwhile, cross and mutation operations of a genetic algorithm are introduced to strengthen diversity of a population, and simulation tests of standard examples of scheduling problems of a flexible job shop show that the algorithm can effectively avoid premature and improve solving speed and precision. The improved Whale Optimization Algorithm (WOA) based on the quantum computing theory is applied to solve a job shop scheduling problem, convergence and correctness of the algorithm are proved, the job shop scheduling problem belongs to a discrete scheduling problem, the improved whale algorithm mainly realizes solving and optimizing by carrying out iterative operation on continuous whale individual position vectors, and a specific conversion process between the two is not given in the prior art. Therefore, the application of new algorithms to solve the production scheduling problem remains one of the hot spots of research in the manufacturing field.
WSA is a novel meta-heuristic algorithm which is developed by the guidance of ultrasonic mutual communication among individuals in the process of whale predation. WSA proves the superiority of solving the combined optimization problem in a special iteration mode. However, to solve the discrete problem, firstly, the problem solution and the inter-conversion problem between the individual position vectors are solved, so that the algorithm is applied to solve the FJSP classical combination optimization problem on the basis of introducing a classical conversion mechanism to solve the inter-conversion problem between the scheduling problem solution and the individual position vectors, and the superiority of the algorithm is verified through embodiment solution and comparative analysis, so that a new way for solving the FJSP is explored.
Disclosure of Invention
The invention aims to provide a method for solving flexible job shop scheduling based on a mixed whale swarm algorithm, which solves the problems of low solving precision and low convergence speed in the flexible job shop scheduling problem in the prior art.
The technical scheme adopted by the invention is that the method for solving the scheduling of the flexible job shop based on the mixed whale swarm algorithm is implemented according to 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;
step 2, defining a fitness function to solve the shortest total processing time as an optimization target;
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 corresponding to the working procedure sequencing scheme by adopting a genetic variation mode of an improved genetic algorithmThe distribution scheme is adopted, so that a better initial population is generated;
step 4, calculating the adaptability value of each scheduling scheme, and searching and reserving the best scheduling solution S * ；
Step 5, outputting an optimal scheduling solution S * And the corresponding fitness function value, S * The optimal scheduling scheme is obtained.
The present invention is also characterized in that,
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 within the set of selectable machines. If s (i) =1, x (i) takes any value 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;
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; MS 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 processing sequences, processing sequences with the shortest total operation time are required, and the operation time corresponding to each processing sequence x (x is { 1.,. The S }) is firstly obtained; obviously, the finishing time of the last machining process in sequence x is the finishing time of all the workpieces:
MS＝E p (1)
step 2.2, setting an objective function F (x) as:
F(x)＝min( x MS)＝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 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;
step 4.2, converting the scheduling solution into whale by using a conversion mechanismIndividual position vectors, and preserve 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 individual corresponding to the minimum valueNamely +.>Is the most preferred and recent individual of (a);
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 the 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.
ρ in step 4.1 0 Take the empirical value 2.
In 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 。
The method for solving the flexible job shop scheduling based on the mixed whale swarm algorithm has the advantages that the method realizes the mutual conversion between the FJSP scheduling solution and the whale individual position vector through a conversion mechanism, and further performs iterative solution by utilizing the strong local and global searching capability of WSA and the advantages in the aspect of maintaining population diversity so as to improve the solving precision and speed of FJSP.
Drawings
FIG. 1 is a flow chart of solving FJSP by using a whale swarm algorithm in a method for solving flexible job shop scheduling based on a mixed whale swarm algorithm;
FIG. 2 is a graph showing the convergence of WSA evolution of example 1 in a method for solving a flexible job shop schedule based on a hybrid whale swarm algorithm of the present invention;
FIG. 3 is a Gantt chart of scheduling results of example 1 in a method for solving flexible job shop scheduling based on a hybrid whale swarm algorithm of the present invention;
FIG. 4 is a graph showing the convergence of WSA evolution of example 2 of a method for solving a flexible job shop schedule based on a hybrid whale swarm algorithm of the present invention;
FIG. 5 is an example 3WSA evolution convergence curve of a method of the present invention for solving flexible job shop schedules based on a hybrid whale swarm algorithm.
Detailed Description
The invention will be described in detail below with reference to the drawings and the detailed description.
The invention discloses a method for solving flexible job shop scheduling based on a mixed whale swarm algorithm, which is implemented according to the following steps as shown in figure 1:
step 1, 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, wherein the conversion mechanism is implemented specifically 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 within the set of selectable machines. If s (i) =1, x (i) takes any value 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, and specifically implementing the method 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 The g-th procedure of the workpiece e is shown in the equipment kThe starting time of the upper machining; 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; MS 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 processing sequences, processing sequences with the shortest total operation time are required, and the operation time corresponding to each processing sequence x (x is { 1.,. The S }) is firstly obtained; obviously, the finishing time of the last machining process in sequence x is the finishing time of all the workpieces:
MS＝E p (1)
step 2.2, setting an objective function F (x) as:
F(x)＝min( x MS)＝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 Current iteration numbert, setting a maximum iteration number M parameter; 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 the best scheduling solution S * The method is 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;
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 individual corresponding to the minimum valueNamely +.>Is the most preferred and recent individual of (a);
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;
ρ in step 4.1 0 Taking an experience value of 2;
in 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 5, outputting an optimal scheduling solution S * And the corresponding fitness function value, S * I.e. the optimal adjustmentDegree scheme.
In order to verify the feasibility of WSA for solving the scheduling problem of the flexible job shop, the simulation experiment is respectively carried out on three embodiments by using the method, and the comparison analysis is carried out on other intelligent algorithms. The simulation environment is as follows: MATILAB2016a is used in the environment of CPU main frequency 1.9GHz, AMD A10-7300 Radeon R6 processor, windows 10 operating system.
Example 1
Example 1 is a 3 x 6 FJSP with processing task information as shown in table 1, parameters set for whale swarm algorithm as follows: population size 50, whale individual position vector dimension 30, ρ 0 Set to 2, d max =32.86, the attenuation coefficient η is taken to be 0, and the maximum number of iterations M is 400:
table 1 example 1 processing task information table
Encoding machine allocation and sequence ordering using random key based two-segment encoding, the length of individual position vector is 30, and each element is [ -3,3]The random number of 30 bits is generated by a computer and stored in a certain order as shown in Table 2, wherein OP ij The j-th step of representing the workpiece i:
TABLE 2 Individual position vector diagrams
For example 1, the WSA evolution convergence curve obtained by solving the method is shown in fig. 2, and fig. 2 shows that the whale shoal algorithm can converge from 144 to 134 faster at the 35 th generation, and the corresponding scheduling result Gantt chart is shown in fig. 3.
Example 2
To further verify the feasibility and superiority of the algorithm, document [1 ] was chosen](Fu Weiping, liu)Winter plum, laichun Wang improved genetic algorithm based on polychromatic set to solve the flexible scheduling problem of multiple varieties [ J ]]Simulation was performed in example 2 of a computer integrated manufacturing system 2011,17 (05): 1004-1010., see document [1 ] for specific process information]Parameters of whale swarm algorithm were set as follows: population size 100, whale individual position vector dimension 108, ρ in example 1 0 Set to 2, d max = 104.96, the attenuation coefficient η is taken to be 0, and the maximum number of iterations M is 200. The WSA evolutionary convergence curve is shown in fig. 4, the optimal solution is 121 minutes, and comparison between the WSA evolutionary convergence curve of fig. 4 and the evolutionary curve of example 2 shows that the whale swarm algorithm can converge from 150 to 121 quickly at 32 generations, and compared with the solution, the solution speed is obviously higher.
Example 3
To more fully compare and verify the algorithm effect, 8×8 embodiments in the Kacem reference problem of flexible job scheduling are selected for solving, and 20 times of calculation are performed. Parameters of whale swarm algorithm were set as follows: the population scale is 100, the whale individual position vector dimension is 54, ρ 0 Set to 2, d max = 117.57, the attenuation coefficient η is taken to be 0, the maximum iteration number M is 200, and the WSA evolution convergence curve of example 3 is shown in fig. 5. Table 3 compares the results obtained with the results obtained by the solution of the evolutionary algorithm, the master-slave genetic algorithm, the multi-order genetic algorithm, the ant colony algorithm:
table 3 comparison of the results of the solutions of the algorithms of the Kacem8×8 reference problem
| Method name | Optimum value | Figure of merit | Average 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 shoal algorithm | 14 | 3 | 35 | 7.6 |
Aiming at the defects of solving precision and speed of the traditional algorithm in solving the FJSP, the invention utilizes a new meta-heuristic algorithm WSA to solve the FJSP, establishes the mutual mapping between a scheduling problem solution and whale individual position vectors by adopting a two-section coding mode and introducing a conversion mechanism, then solves the FJSP by utilizing the WSA, and a simulation embodiment result shows that the WSA has certain superiority in solving the FJSP.
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.
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