CN112257297A - Welding shop comprehensive scheduling method based on improved firework algorithm - Google Patents

Welding shop comprehensive scheduling method based on improved firework algorithm Download PDF

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CN112257297A
CN112257297A CN202011359791.9A CN202011359791A CN112257297A CN 112257297 A CN112257297 A CN 112257297A CN 202011359791 A CN202011359791 A CN 202011359791A CN 112257297 A CN112257297 A CN 112257297A
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张剑
袁铭辉
谭光鑫
郑婷娟
李俊达
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Southwest Jiaotong University
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Abstract

The invention discloses a welding shop comprehensive scheduling method based on an improved firework algorithm, which specifically comprises the following steps: firstly, establishing a mathematical model of comprehensive scheduling of a welding workshop, and establishing the comprehensive scheduling model of the welding workshop based on an optimization target of minimum maximum completion time of complex welding products processed in the workshop and reasonable machine load and considering constraint conditions of tight front and back of product procedures, machine resource occupation and the like; and finally, solving by adopting an improved firework algorithm, wherein in the solving process, firework explosion and mutation operators which can meet the processing sequence of the working procedures are designed, and the condition that illegal solutions cannot be generated in the whole solving process is ensured. The invention fully considers the influence of complex product process constraint and different processing machine types on the actual welding shop scheduling, so that the scheduling solution scheme is more reasonable; the invention ensures different search purposes, ensures the diversity of the population, ensures that the solving process is not easy to fall into local optimum, and has more superiority than genetic algorithms and other heuristic methods.

Description

Welding shop comprehensive scheduling method based on improved firework algorithm
Technical Field
The invention belongs to the technical field of multi-objective optimization of welding workshops, and particularly relates to a welding workshop comprehensive scheduling method based on an improved firework algorithm.
Background
The welding Shop integrated Scheduling WSISP (welding Shop integrated Scheduling Problem) is widely existed in enterprises of aerospace, automobile manufacturing, ship manufacturing and the like, and is taken as one of key material forming and processing technologies in modern manufacturing industry.
At present, only a few students partially research the problems of a welding workshop, but the researched welding scheduling problem is only suitable for a simple flowing water welding workshop, so that the difference between an established mathematical model and an actual model is very large, and the internal parallelizable relation of a process tree in the manufacturing process is definitely split if the traditional scheduling algorithm is still adopted for processing the welding scheduling of complex welding products (welding products with processing and assembly at the same time) with a tree-shaped process structure, so that the production period is prolonged.
In the actual welding production of assembly manufacturing, a plurality of machines are needed to be cooperatively processed in a single process, when two large components are welded and assembled, one main welding machine is needed, and other auxiliary equipment is needed to fix the main welding machine and the auxiliary equipment, so that one process of the welding process needs a plurality of pieces of equipment to complete, and under the condition that a welding area is enough to accommodate a plurality of main welding equipment, the number of the main welding machines can be increased to shorten the time of the welding process, so that the parallel processing of various types of machines exists in the welding process, one is that the common welding machine which reduces the processing time by increasing the number of the main welding machines is parallel, and the other is that the auxiliary equipment fixes the welding components or performs other functions in parallel. Due to the fact that the complexity of the problem is high, the research results are reported recently, and therefore the research for the comprehensive scheduling problem of the complex welding products is of great significance.
The Firework Algorithm (FA) is a swarm intelligence optimization Algorithm and has good optimization performance.
Disclosure of Invention
In order to overcome the defects of complex welding product comprehensive scheduling in the prior art, the invention provides a welding workshop comprehensive scheduling method based on an improved firework algorithm.
The invention discloses a welding shop comprehensive scheduling method based on an improved firework algorithm, which specifically comprises the following steps:
step 1: and establishing a mathematical model for comprehensive scheduling optimization of the welding workshop.
The method comprises the following steps of establishing a mathematical model for comprehensive scheduling optimization of a welding workshop by taking the minimum maximum completion time and reasonable machine load of a workshop processed complex welding product as optimization targets, wherein the objective function is as follows:
Figure BDA0002803678250000011
constraint conditions are as follows:
Figure BDA0002803678250000021
Figure BDA0002803678250000022
Figure BDA0002803678250000023
Figure BDA0002803678250000024
Figure BDA0002803678250000025
Figure BDA0002803678250000026
Figure BDA0002803678250000027
Figure BDA0002803678250000028
Figure BDA0002803678250000029
Figure BDA00028036782500000210
in the formula, the process set included in the product a is i ═ {1,2,3, …, n }, and the processing time is ti＝{t1,…,tnN represents the total number of working procedures, and N is a working procedure set on a processing technology tree; n is a radical ofableIn order to increase the process set of the number of main welding machines,
Figure BDA00028036782500000211
Figure BDA00028036782500000216
is a standard process set; n is a radical ofmutRepresents a multi-device process set; i represents the ith procedure on the product, i belongs to N; n is a radical ofinvRepresenting a virtual work process set; t is tiThe processing time of the ith procedure; siRepresenting the starting processing time of the ith procedure; ciEnd of the i-th step, CmaxMaximum completion time; j is a function ofrAn index indicating the j-th process on the machine r; j. the design is a squareiA process immediately before the same equipment representing the ith process; fiThe process immediately before the ith process is shown,
Figure BDA00028036782500000212
representing the starting processing time of the process immediately before the ith process;
Figure BDA00028036782500000213
representing the i-th stepStarting processing time of a process immediately before equipment; u. ofiA penalty factor, u, representing an increase in the number of main welding machinesiE (0, 1); m is the total number of machines; m represents a machine set M ═ {1, …, M }; miThe method is characterized in that the method is a set of machines with selectable same types of the ith procedure, namely a main welding machine, i belongs to N;
Figure BDA00028036782500000214
representing a needed cooperative auxiliary machine set of the ith process, i belongs to N; r is the index of the main welding machine, rassIndexing for auxiliary machines; y isi，rA variable of 0,1, y if the ith process is performed on the main welding machine r i，r1, otherwise yi，r＝0，r∈Mi；
Figure BDA00028036782500000215
Is a 0,1 variable; if the ith process is in the auxiliary machine rassUpper working procedure
Figure BDA0002803678250000031
Otherwise
Figure BDA0002803678250000032
Figure BDA0002803678250000033
Representing the maximum allowable number of the same type parallel processors of the ith procedure; l isiRepresenting the actual number of main welding machines of the ith pass,
Figure BDA0002803678250000034
Figure BDA0002803678250000035
representing the number of co-operating auxiliary machines used during the working of step i, wherein
Figure BDA0002803678250000036
Figure BDA0002803678250000037
Adding a main welding machine on the representative procedure iThe processing time after the treatment of the processor meets the requirements
Figure BDA0002803678250000038
Figure BDA0002803678250000039
The process immediately before the process representing the process i increases the processing time after the treatment of the main welding machine;
Figure BDA00028036782500000310
the process immediately before the equipment representing the process i increases the processing time after the treatment of the main welding machine; miRepresents the set of main welding machines,
Figure BDA00028036782500000311
Figure BDA00028036782500000312
represents a set of auxiliary machines that are to be operated,
Figure BDA00028036782500000313
parallel processor M with no difference included in MiAnd co-operating parallel processors
Figure BDA00028036782500000314
At a certain process time tnAt least one machine tool r is needed, r belongs to M, and the main welding machine set is MiThe main functional machine in the ith process is integrated with auxiliary machines
Figure BDA00028036782500000315
The set of auxiliary machines required for the ith pass, i.e. in addition to the calling of the main welding machine M in the ith passiIn addition, corresponding auxiliary machines are required to assist the process.
In the formula (1) f1Is to minimize the maximum completion time C of the processed productmax，f2The machine load is minimized, and the formula (2) is to restrict the number of auxiliary machines of the standard process, and the value is 0; the formula (3) is the number of auxiliary machines belonging to the multi-equipment processCarrying out constraint, wherein the value of the constraint is a positive integer; the formula (4) is used for restricting the number of main welding machines in all the working procedures, and the value of the formula is a positive integer; equation (5) is a constraint on the number of parallel processors invested in each workstation; equation (6) is a 0-1 variable constraint; the formula (7) is to distinguish the number of machines of the parallel process and the non-parallel process; formula (8) indicates that each process of the product task can only occur once in the scheduled position; formula (9) xi,kIs a 0-1 variable constraint; the formula (10) shows that the processing starting time of each process is required to be more than or equal to the larger value of the sum of the process time immediately before all the same equipment and the current process preparation time and the sum of the process time immediately before all the processes and the current process preparation time; equation (11) represents the maximum completion time CmaxThe time for completing the last working procedure of the product.
Step 2: and (5) optimizing and solving by using an improved firework algorithm.
The firework group is X ═ X1,X2,X3,…,XKK is the index of the serial number of the group fireworks, K is the total number of the individual fireworks, K is {1,2, …, K }, and X iskIs the position of the kth fireworks, fkRepresenting the fitness value of fireworks k, Sn being the number of exploding sparks of fireworks, SekRepresenting the number of explosion sparks of the fireworks k, and adopting rounding operation in actual calculation; a is the basic explosive radius, RkIs the explosion radius of the fireworks k, fmaxFor the optimal fitness value, f, in the current algebraic firework setminThe epsilon is a particularly small constant which is the worst fitness value in the current algebraic firework set and avoids the denominator being 0; the firework k is arranged at the radius R according to the fitness valuekInternally generated SekAn explosion spark Xnew. Expressed as:
Figure BDA00028036782500000316
Figure BDA0002803678250000041
the specific operation process of the improved firework algorithm is as follows:
s1: initializing a firework population with K individuals, calculating the fitness values of the K fireworks, and calculating the explosion radius and the explosion number according to the fitness values.
S2: generating explosion sparks, and reserving K individuals according to an individual elite reservation strategy; randomly selecting individuals to generate GK Gaussian variation sparks, wherein GK is 10% of K.
Explosion spark: note XkIs the solution of fireworks k, and the fitness is f (X)k)，newXkFor candidate fireworks solutions generated by explosion, Explode is the explosion function, XkObtaining candidate firework solution newX after explosion function ExplodekThe definition of the improved explosion operator is given by equation (14):
newXk＝Explode(Xk,min(Rk,Rfb)) (14)
random selection of XkThe distance between the immediately-before process and the immediately-before process is RfbControlling the explosion distance between the immediately preceding process and the immediately subsequent process, wherein the new solution obtained by each normal explosion of explosion is a legal solution meeting the constraint, and the explosion offset distance is R calculated by the formula (13)kAnd RfbThe smaller value of (a).
Gaussian variation: mutXkCandidate firework solution generated by Gaussian variation, Gs is Gaussian variation function, and firework XkThe Gaussian variation is not influenced by the fitness value fkCalculated explosion range RkConstraint of XkObtaining candidate firework solution mutX after passing through Gaussian variation function GskThe definition of which is given by formula (15):
mutXk＝Gs(Xk,Rfb) (15)
s3: and mixing the explosion sparks and the Gaussian variation sparks to obtain a population, keeping 50% of individuals with the first fitness, and using a roulette strategy on the individuals with the residual population capacity to obtain a next-generation firework population.
S4: judging whether the explosion frequency of the fireworks reaches the set maximum explosion number or not, if so, ending the explosion process and outputting fireworks with the optimal value; otherwise, the iteration is continued.
Compared with the prior art, the invention has the following advantages and effects:
(1) aiming at the problem of comprehensive scheduling of various machines of different types in a complex welding product, a complex welding comprehensive scheduling model is provided, a process tree of comprehensive scheduling is used as the input of process information of the complex welding product for scheduling solution, whether processes are parallel or not between the processes and between the processes, namely the processes before and after the processes, and the other processes are required to be considered in the scheduling process, meanwhile, the cooperative processing of a main welding machine and an auxiliary machine exists in the welding process, and the scheduling result is often ignored in the actual welding scheduling process, so that the scheduling result is inconsistent with the actual field. The invention fully considers the influence of complex product process constraint and different processing machine types on the actual welding workshop scheduling, and establishes a complex welding comprehensive scheduling mathematical model, so that the scheduling solving scheme is more reasonable.
(2) The improved firework algorithm is adopted to solve the comprehensive scheduling problem of the welding workshop, so that illegal solutions cannot be generated in the whole processes of initialization solution set and iterative explosion, different search purposes are guaranteed due to the designed explosion operator and Gaussian variation, the diversity of population is guaranteed, the solving process is not easy to fall into local optimum, and the method has better superiority than genetic algorithm and other heuristic methods.
Drawings
FIG. 1 is a flow chart of an improved fireworks algorithm of the present invention;
FIG. 2 is a flow chart of the individual fireworks initialization;
FIG. 3 is a schematic diagram of an explosion operator;
FIG. 4 is a schematic representation of Gaussian variation;
FIG. 5 is a diagram of complex product A weld process tree information;
FIG. 6 is a Gantt chart of the scheduling result obtained by the improved firework algorithm.
Detailed Description
The present invention will be described in further detail with reference to the accompanying drawings and examples.
The invention discloses a welding shop comprehensive scheduling method based on an improved firework algorithm, which specifically comprises the following steps:
step 1: and establishing a mathematical model for comprehensive scheduling optimization of the welding workshop.
The comprehensive Scheduling WSISP (welding Shop integrated Scheduling System) of a welding Shop is a Scheduling problem for processing welding processing and welding assembly of welding products together, and the welding assembly process of each workpiece is regarded as a special processing process. The processing process of the complex welding product is that according to the obtained tree-shaped structure of the restriction relation of the constraint relation before tightening or after tightening among all working procedures of the product, the nodes on the processing technology tree correspond to the working procedures of the product, the working procedure i has the working procedure before tightening and the working procedure after tightening, the working procedure j pointed by the arrow is the working procedure after tightening of the working procedure i, and when the last working procedure of the root node of the technology tree is finished, the product is processed.
The premise assumptions for WSISP problem establishment include:
each process must wait until all of its processes have been completed and the processes must be completed before the equipment.
Each device can only process one procedure at any time, and the processing process can not be interrupted.
The process set allows a multi-device process in which a plurality of related devices are collectively processed, and a plurality of related devices in which a process can be simultaneously and cooperatively processed in the process are referred to as a cooperative device.
Some procedures can call a plurality of same machines to process a certain procedure of the workpiece simultaneously and parallelly so as to reduce the processing time of the procedure, the processing time is in inverse proportion to the number of the machines, and the total number of the machines which meet the characteristics is limited.
The WSISP scheduling problem is described as follows: product A contains n multiple parallel processing tasks pi＝{p1,…,pnThe process set is i ═ 1,2,3, …, n, and the processing time is ti＝{t1,…,tnN represents the total number of working procedures, and N is a working procedure set on a processing technology tree; n is a radical ofableIn order to increase the process set of the number of main welding machines,
Figure BDA0002803678250000061
Noneis a standard process set; n is a radical ofmutRepresents a multi-device process set; i represents the ith procedure on the product, i belongs to N; n is a radical ofinvRepresenting a virtual work process set; t is tiThe processing time of the ith procedure; siRepresenting the starting processing time of the ith procedure; ciEnd of the i-th step, CmaxMaximum completion time; j is a function ofrAn index indicating the j-th process on the machine r; j. the design is a squareiA process immediately before the same equipment representing the ith process; fiA step immediately before the ith step; u. ofiA penalty factor, u, representing an increase in the number of main welding machinesiE (0, 1); m is the total number of machines; m represents a machine set M ═ {1, …, M }; miThe method is characterized in that the method is a set of machines with selectable same types of the ith procedure, namely a main welding machine, i belongs to N;
Figure BDA0002803678250000062
representing a needed cooperative auxiliary machine set of the ith process, i belongs to N; r is the index of the main welding machine, rassIndexing for auxiliary machines; y isi，rA variable of 0,1, y if the ith process is performed on the main welding machine r i，r1, otherwise yi，r＝0，r∈Mi；
Figure BDA0002803678250000063
Is a 0,1 variable; if the ith process is in the auxiliary machine rassUpper working procedure
Figure BDA0002803678250000064
Otherwise
Figure BDA0002803678250000065
Figure BDA0002803678250000066
Representing the maximum allowable number of the same type parallel processors of the ith procedure; l isiRepresenting the actual number of main welding machines of the ith pass,
Figure BDA0002803678250000067
Figure BDA0002803678250000068
representing the number of co-operating auxiliary machines used during the working of step i, wherein
Figure BDA0002803678250000069
Figure BDA00028036782500000610
Represents the increase of the processing time after the treatment of the main welding machine and meets the requirements
Figure BDA00028036782500000611
Figure BDA00028036782500000612
The process immediately before the process representing the process i increases the processing time after the treatment of the main welding machine;
Figure BDA00028036782500000613
the process immediately before the equipment representing the process i increases the processing time after the treatment of the main welding machine; (ii) a MiRepresents the set of main welding machines,
Figure BDA00028036782500000614
Figure BDA00028036782500000615
represents a set of auxiliary machines that are to be operated,
Figure BDA00028036782500000616
parallel processor M with no difference included in MiAnd co-operating parallel processors
Figure BDA00028036782500000617
At a certain process time tnAt least one machine tool r is needed, r belongs to M, and the main welding machine set is MiThe main functional machine in the ith procedure is shortened by increasing the number of machines of the same type under the constraint of meeting the number of machines of the same type, the size of a working space and the proceduresThe purpose of the sequence processing time. The auxiliary machine is composed of
Figure BDA00028036782500000618
The set of auxiliary machines required for the ith pass, i.e. in addition to the calling of the main welding machine M in the ith passiBesides, corresponding auxiliary machines are needed to assist the process, such as clamping, positioning and the like.
The method comprises the following steps of establishing a mathematical model for comprehensive scheduling optimization of a welding workshop by taking the minimum maximum completion time and reasonable machine load of a workshop processed complex welding product as optimization targets, wherein the objective function is as follows:
Figure BDA00028036782500000619
constraint conditions are as follows:
Figure BDA00028036782500000620
Figure BDA00028036782500000621
Figure BDA0002803678250000071
Figure BDA0002803678250000072
Figure BDA0002803678250000073
Figure BDA0002803678250000074
Figure BDA0002803678250000075
Figure BDA0002803678250000076
Figure BDA0002803678250000077
Figure BDA0002803678250000078
in the formula (1) f1Is to minimize the maximum completion time C of the processed productmax，f2The machine load is minimized, and the formula (2) is to restrict the number of auxiliary machines of the standard process, and the value is 0; the formula (3) is used for restricting the number of auxiliary machines belonging to the multi-equipment process, and the value of the formula is a positive integer; the formula (4) is used for restricting the number of main welding machines in all the working procedures, and the value of the formula is a positive integer; equation (5) is a constraint on the number of parallel processors invested in each workstation; equation (6) is a 0-1 variable constraint; the formula (7) is to distinguish the number of machines of the parallel process and the non-parallel process; formula (8) indicates that each process of the product task can only occur once in the scheduled position; formula (9) xi,kIs a 0-1 variable constraint; the formula (10) shows that the processing starting time of each process is required to be more than or equal to the larger value of the sum of the process time immediately before all the same equipment and the current process preparation time and the sum of the process time immediately before all the processes and the current process preparation time; equation (11) represents the maximum completion time CmaxThe time for completing the last working procedure of the product.
Step 2: and (5) optimizing and solving by using an improved firework algorithm.
The firework group is X ═ X1,X2,X3,…,XKK is the index of the serial number of the group fireworks, K is the total number of the individual fireworks, K is {1,2, …, K }, and X iskIs the position of the kth fireworks, fkFitness value representing fireworks kSn is the number of explosion sparks of fireworks and SekRepresenting the number of explosion sparks of the fireworks k, and adopting rounding operation in actual calculation; a is the basic explosive radius, RkIs the explosion radius of the fireworks k, fmaxFor the optimal fitness value, f, in the current algebraic firework setminThe epsilon is a particularly small constant which is the worst fitness value in the current algebraic firework set and avoids the denominator being 0; the firework k is arranged at the radius R according to the fitness valuekInternally generated SekAn explosion spark Xnew. Expressed as:
Figure BDA0002803678250000081
Figure BDA0002803678250000082
the improved firework algorithm is shown in fig. 1, and specifically comprises the following steps:
s1: initializing a firework population with K individuals, calculating the fitness values of the K fireworks, and calculating the explosion radius and the explosion number according to the fitness values, wherein an initialization flow chart is shown in an attached figure 2.
S2: generating explosion sparks, and reserving K individuals according to an individual elite reservation strategy; randomly selecting individuals to generate GK Gaussian variation sparks, wherein GK is 10% of K.
Explosion spark: note XkIs the solution of fireworks k, and the fitness is f (X)k)，newXkFor candidate fireworks solutions generated by explosion, Explode is the explosion function, XkObtaining candidate firework solution newX after explosion function ExplodekThe definition of the improved explosion operator is given by equation (14):
newXk＝Explode(Xk,min(Rk,Rfb)) (14)
the definition of equation (14) is shown in FIG. 3, where X is chosen randomlykThe distance between the immediately-before process and the immediately-before process is RfbControlling the explosion distance between the immediately preceding process and the immediately subsequent process, wherein the new solution obtained by each normal explosion of explosion is a legal solution meeting the constraint, and the explosion offset distance is R calculated by the formula (13)kAnd RfbThe smaller value of (a).
Gaussian variation: mutXkCandidate firework solution generated by Gaussian variation, Gs is Gaussian variation function, and firework XkThe Gaussian variation is not influenced by the fitness value fkCalculated explosion range RkConstraint of XkObtaining candidate firework solution mutX after passing through Gaussian variation function GskThe definition of which is given by formula (15):
mutXk＝Gs(Xk,Rfb) (15)
the definition of equation (15) is shown in FIG. 4: z is a radical offAt the location of the immediately preceding process, zbAnd performing Gaussian position transformation on the selected dimension to obtain a Gaussian variation result under the condition that the selected dimension is the position of the process immediately before and after the process is constrained.
S3: and mixing the explosion sparks and the Gaussian variation sparks to obtain a population, keeping 50% of individuals with the first fitness, and using a roulette strategy on the individuals with the residual population capacity to obtain a next-generation firework population.
S4: judging whether the explosion frequency of the fireworks reaches the set maximum explosion number or not, if so, ending the explosion process and outputting fireworks with the optimal value; otherwise, the iteration is continued.
Example (b):
a manufacturing enterprise prepares to produce a complex welding product A, process tree information is shown in figure 5 and consists of 40 working procedures, machine information is shown in table 1 and comprises 8 machine models, wherein 1-5 machines are main welding machines, the upper limit of the number of machines and the load coefficient of the machines can be read out from the table, and 6-8 machines are auxiliary machines. The second column of machine types 1 represents the main welding machine, 0 represents the auxiliary machine, and the number represents the upper limit number of machines.
TABLE 1 machine information Table
Figure BDA0002803678250000091
The product is scheduled by adopting a genetic algorithm and an improved firework algorithm, each algorithm independently runs for 20 times, makespan and machine load of each test are taken and calculated to obtain a total target value, and the comparison result is shown in table 2:
TABLE 2 comparison of algorithms
Figure BDA0002803678250000092
Figure BDA0002803678250000101
The experimental result shows that compared with the genetic algorithm, in the comprehensive welding scheduling solution, the minimum value of the maximum makespan obtained by the improved firework algorithm is 146, the minimum value is smaller than 150 times obtained by the genetic algorithm, the average value of 20 times obtained by the improved firework algorithm is also lower than the result obtained by the genetic algorithm, and the superiority and effectiveness of the improved firework algorithm are verified. Coding of the optimal firework individual: [ 312121413826539153671629142011253261712232210193728271839243035343338401223221221121122321332213133423223221143 ] corresponding to the Gantt chart as shown in FIG. 6, the ordinate of the Gantt chart is { M1, M2, M3, M4, M5, A3, A7, A8}, which represents that from machine 1 to machine 5 are main welding machines, from machine 6 to machine 8 are auxiliary machines, and the abscissa represents the finishing time. The first scheduled process in the sequence of processes is 31, the corresponding number of main welding machines is 1, and the ordinate on the gantt chart is the position of M4; the process J32 was simultaneously processed by the main welding machine M5 and the auxiliary machine a8, and the rectangular block on the main welding machine M5 was divided into three rectangular blocks by black stripes, representing that the number of the main welding machines M5 occupied by the process was 3; similarly, the 37 th scheduled process is the process 33, which corresponds to the number of main welding machines being 1, and is cooperatively processed by the main welding machine M3 and the auxiliary machines a7 and a 8. In the gantt chart, the maximum completion time is 146, and the machine load is 30.4; as can be seen comprehensively, the comprehensive scheduling model and the solving method of the welding shop provided by the invention are superior to those of the prior art.

Claims (1)

1. A welding shop comprehensive scheduling method based on an improved firework algorithm is characterized by comprising the following steps:
step 1: establishing a mathematical model for comprehensive scheduling optimization of a welding workshop;
the method comprises the following steps of establishing a mathematical model for comprehensive scheduling optimization of a welding workshop by taking the minimum maximum completion time and reasonable machine load of a workshop processed complex welding product as optimization targets, wherein the objective function is as follows:
Figure FDA0002803678240000011
constraint conditions are as follows:
Figure FDA0002803678240000012
Figure FDA0002803678240000013
Figure FDA0002803678240000014
Figure FDA0002803678240000015
Figure FDA0002803678240000016
Figure FDA0002803678240000017
Figure FDA0002803678240000018
Figure FDA0002803678240000019
Figure FDA00028036782400000110
Figure FDA00028036782400000111
in the formula, the process set included in the product a is i ═ {1,2,3, …, n }, and the processing time is ti＝{t1,…,tnN represents the total number of working procedures, and N is a working procedure set on a processing technology tree; n is a radical ofableIn order to increase the process set of the number of main welding machines,
Figure FDA00028036782400000112
Noneis a standard process set; n is a radical ofmutRepresents a multi-device process set; i represents the ith procedure on the product, i belongs to N; n is a radical ofinvRepresenting a virtual work process set; t is tiThe processing time of the ith procedure; siRepresenting the starting processing time of the ith procedure; ciEnd of the i-th step, CmaxMaximum completion time; j is a function ofrAn index indicating the j-th process on the machine r; j. the design is a squareiA process immediately before the same equipment representing the ith process; fiThe process immediately before the ith process is shown,
Figure FDA00028036782400000113
representing the starting processing time of the process immediately before the ith process;
Figure FDA0002803678240000021
representing the starting processing time of the process immediately before the equipment of the ith process; u. ofiA penalty factor, u, representing an increase in the number of main welding machinesiE (0, 1); m is the total number of machines; m represents a machine set M ═ {1, …, M }; miThe method is characterized in that the method is a set of machines with selectable same types of the ith procedure, namely a main welding machine, i belongs to N;
Figure FDA0002803678240000022
representing a needed cooperative auxiliary machine set of the ith process, i belongs to N; r is the index of the main welding machine, rassIndexing for auxiliary machines; y isi，rA variable of 0,1, y if the ith process is performed on the main welding machine ri，r1, otherwise yi，r＝0，r∈Mi；
Figure FDA0002803678240000023
Is a 0,1 variable; if the ith process is in the auxiliary machine rassUpper working procedure
Figure FDA0002803678240000024
Otherwise
Figure FDA0002803678240000025
Figure FDA0002803678240000026
Figure FDA0002803678240000027
Representing the maximum allowable number of the same type parallel processors of the ith procedure; l isiRepresenting the actual number of main welding machines of the ith pass,
Figure FDA0002803678240000028
Figure FDA0002803678240000029
representing the working process in step iNumber of co-operating auxiliary machines used in the middle or upper part, wherein
Figure FDA00028036782400000210
Figure FDA00028036782400000211
The processing time after the treatment of the main welding machine is increased in the representative procedure i, and the requirements are met
Figure FDA00028036782400000212
Figure FDA00028036782400000213
The process immediately before the process representing the process i increases the processing time after the treatment of the main welding machine;
Figure FDA00028036782400000214
the process immediately before the equipment representing the process i increases the processing time after the treatment of the main welding machine; (ii) a MiRepresents the set of main welding machines,
Figure FDA00028036782400000215
Figure FDA00028036782400000216
represents a set of auxiliary machines that are to be operated,
Figure FDA00028036782400000217
parallel processor M with no difference included in MiAnd co-operating parallel processors
Figure FDA00028036782400000218
At a certain process time tnAt least one machine tool r is needed, r belongs to M, and the main welding machine set is MiThe main functional machine in the ith process is integrated with auxiliary machines
Figure FDA00028036782400000219
The set of auxiliary machines required for the ith pass, i.e. in addition to the calling of the main welding machine M in the ith passiBesides, corresponding auxiliary machines are needed to assist the process;
in the formula (1) f1Is to minimize the maximum completion time C of the processed productmax，f2The machine load is minimized, and the formula (2) is to restrict the number of auxiliary machines of the standard process, and the value is 0; the formula (3) is used for restricting the number of auxiliary machines belonging to the multi-equipment process, and the value of the formula is a positive integer; the formula (4) is used for restricting the number of main welding machines in all the working procedures, and the value of the formula is a positive integer; equation (5) is a constraint on the number of parallel processors invested in each workstation; equation (6) is a 0-1 variable constraint; the formula (7) is to distinguish the number of machines of the parallel process and the non-parallel process; formula (8) indicates that each process of the product task can only occur once in the scheduled position; equation (9) is a 0-1 variable constraint; the formula (10) shows that the processing starting time of each process is required to be more than or equal to the larger value of the sum of the process time immediately before all the same equipment and the current process preparation time and the sum of the process time immediately before all the processes and the current process preparation time; equation (11) represents the maximum completion time CmaxThe time for completing the last procedure of the product;
step 2: optimizing and solving by using an improved firework algorithm;
the firework group is X ═ X1,X2,X3,…,XKK is the index of the serial number of the group fireworks, K is the total number of the individual fireworks, K is {1,2, …, K }, and X iskIs the position of the kth fireworks, fkRepresenting the fitness value of fireworks k, Sn being the number of exploding sparks of fireworks, SekRepresenting the number of explosion sparks of the fireworks k, and adopting rounding operation in actual calculation; a is the basic explosive radius, RkIs the explosion radius of the fireworks k, fmaxFor the optimal fitness value, f, in the current algebraic firework setminThe epsilon is a particularly small constant which is the worst fitness value in the current algebraic firework set and avoids the denominator being 0; the firework k is arranged at the radius R according to the fitness valuekInternally generated SekAn explosion spark Xnew(ii) a Expressed as:
Figure FDA0002803678240000031
Figure FDA0002803678240000032
the specific operation process of the improved firework algorithm is as follows:
s1: initializing a firework population with K individuals, calculating the fitness values of the K fireworks, and calculating the explosion radius and the explosion number according to the fitness values;
s2: generating explosion sparks, and reserving K individuals according to an individual elite reservation strategy; randomly selecting individuals to generate GK Gaussian variation sparks, wherein GK is 10% of K;
explosion spark: note XkIs the solution of fireworks k, and the fitness is f (X)k)，newXkFor candidate fireworks solutions generated by explosion, Explode is the explosion function, XkObtaining candidate firework solution newX after explosion function ExplodekThe definition of the improved explosion operator is given by equation (14):
newXk＝Explode(Xk,min(Rk,Rfb)) (14)
random selection of XkThe distance between the immediately-before process and the immediately-before process is RfbControlling the explosion distance between the immediately preceding process and the immediately subsequent process, wherein the new solution obtained by each normal explosion of explosion is a legal solution meeting the constraint, and the explosion offset distance is R calculated by the formula (13)kAnd RfbThe smaller value of (d);
gaussian variation: mutXkCandidate firework solution generated by Gaussian variation, Gs is Gaussian variation function, and firework XkThe Gaussian variation is not influenced by the fitness value fkCalculated explosion range RkConstraint of XkAfter passing through a Gaussian variation function GsObtaining candidate firework solution mutXkThe definition of which is given by formula (15):
mutXk＝Gs(Xk,Rfb) (15)
s3: mixing the explosion sparks and the Gaussian variation sparks to obtain a population, reserving 50% of individuals with the first fitness, and using a roulette strategy on the individuals with the residual population capacity to obtain a next-generation firework population;
s4: judging whether the explosion frequency of the fireworks reaches the set maximum explosion number or not, if so, ending the explosion process and outputting fireworks with the optimal value; otherwise, the iteration is continued.
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