CN110889534A - Product principle arrangement method based on improved NSGA-II - Google Patents

Product principle arrangement method based on improved NSGA-II Download PDF

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CN110889534A
CN110889534A CN201911006109.5A CN201911006109A CN110889534A CN 110889534 A CN110889534 A CN 110889534A CN 201911006109 A CN201911006109 A CN 201911006109A CN 110889534 A CN110889534 A CN 110889534A
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刘峰
汪孝胜
陶凌峰
李茂盛
高磊
洪鹏
吕向峰
刘哲
金磊
张斌
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Abstract

A product principle arrangement method based on improved NSGA-II. Relates to the technical field of automatic production lines, in particular to a product principle arrangement method. The method is characterized in that NSGA-II (non-dominated Sorting Genetic Algorithm II, NSGA-II) is improved, three distribution modes are adopted to distribute operation procedures, insert sequence cross operation, single-point variation operation and initialize population strategies, and therefore the optimized design of product principle arrangement is achieved. The method comprises the following steps: 1) modeling; 2) initializing a population; 3) selecting a championship game; 4) genetic manipulation; 5) selecting elite; 6) and judging whether the stop criterion is met. Simulation experiments prove that the improved algorithm can obtain a high-quality non-dominated solution set, the construction cost of a production line is reduced, and the workload of each station is balanced, so that the optimal design of product principle arrangement is realized.

Description

Product principle arrangement method based on improved NSGA-II
Technical Field
The invention relates to the technical field of automatic production lines, in particular to a product principle arrangement method.
Background
The product principle arrangement is suitable for production modes with few varieties and large batch, the essence is assembly line arrangement, and the core problem is assembly line balance. The assembly line balancing means that the operation procedures are distributed to different operation stations under the condition of meeting certain constraint conditions, so that the station number and the production rhythm are reasonable, and the production load of each operation station is balanced as much as possible. Aiming at assembly line balance, the high-quality operation procedure distribution scheme can effectively reduce the construction cost of the production line, reduce the actual production takt time and balance the load among operation stations. The problem is a combined optimization problem, belongs to an NP difficult problem, and a good optimization result is difficult to obtain by a general mathematical means.
For the multi-objective optimization problem, the prior art often adopts a pareto optimization method, a hierarchical optimization method and a weighted combination method. When the methods are used for processing the multi-objective optimization problem, a better optimization result is difficult to obtain due to the limitations of the methods.
Disclosure of Invention
Aiming at the problems, the invention provides an improved NSGA-II (non-dominated Sorting Genetic Algorithm II, NSGA-II) product principle arrangement method which adopts three distribution modes to distribute operation procedures, embed order cross operation, single point variation operation and initialize population strategy, thereby realizing the optimized design of product principle arrangement and being based on the improved NSGA-II.
The technical scheme of the invention is as follows: the method comprises the following steps:
1) and modeling:
1.1) establishing a mathematical model of the objective function and determining constraint conditions of the mathematical model;
1.2) adopting a Pareto optimization method to process a multi-objective optimization problem;
1.3), carrying out discretization treatment on NSGA-II, coding by adopting a procedure sequence coding mode, and simultaneously decoding by applying three distribution modes;
2) initializing a population:
2.1), initializing population control parameters including population number SN and cross probability pcAnd the probability of variation pm；
2.2) applying three initialization population criteria to generate an initialized population, and storing non-dominant solutions in the population into an external Pareto solution set;
3) and selecting the championship game: selecting individuals in the population by adopting a championship method, wherein the championship scale is 3, and forming a genetic operation pool with the size being half of the population number;
4) and genetic manipulation: performing cross operation and mutation operation on individuals in the genetic operation pool; first, a random number p between 0 and 1 is generated, if p<pcThen choose the individual to do the order-interleaving operation, if p<pmSelecting individuals to perform single-point mutation operation;
5) selecting elite: combining the individuals in the genetic operation pool in the step 3) with the individuals generated by the genetic operation in the step 4), performing non-dominant sorting and crowded distance sorting on the combined population, screening to keep the population size the same as a set value, and storing non-dominant solutions in an external Pareto solution set;
6) judging whether the stop criterion is met; if yes, exporting an external Pareto storage set; if not, go to step 3).
The variable symbols of the mathematical model in step 1.1) are shown in the following table:
TABLE 1 variable symbol definitions
Figure BDA0002242824950000021
The constraint conditions in step 1.1) are as follows:
Figure BDA0002242824950000022
Figure BDA0002242824950000023
Figure BDA0002242824950000031
Figure BDA0002242824950000032
Figure BDA0002242824950000033
Figure BDA0002242824950000034
xik0 or
Figure BDA0002242824950000035
Wherein, the formula (1) is an optimization target of minimizing the number of stations; the formula (2) and the formula (3) are respectively a smooth index optimization target for maximizing the balance efficiency of the assembly line and minimizing the load of each station; the expression (4) indicates that any operation process can be distributed into only one station and cannot be simultaneously distributed into a plurality of stations; equation (5) represents arbitrary operation processes i and j, and if process i is a process immediately before process j, process i cannot be assigned to a process after the station where process j is locatedWithin a bit; the formula (6) shows that the sum of all the distributed operation process time in any station can not exceed the given beat time; equation (7) represents the control variable xikWhen operation i is assigned to the kth station, x ik1, otherwise, xik＝0。
Pareto domination in step 1.2) denotes the hypothesis X1And X2Are two solutions different from each other, and any solution has r optimization objective function values, if the following conditions are satisfied, X is indicated1Dominating X2；
fi(X1)≤fi(X2),i＝1,2,...,r(8)
Figure BDA0002242824950000036
Pareto optimal solution indicates if X1Dominating X2Then, X is described1R optimized objective function values are all superior to X2(ii) a Therefore, the solution that is not dominated by any solution is called Pareto optimal solution;
the Pareto external storage set indicates that all Pareto optimal solutions are recorded in one set; the set is updated synchronously as the algorithm iterates, in order to keep the solutions in the set all Pareto optimal solutions.
Step 1.3) adopts a procedure sequence coding mode to represent a feasible solution, namely, coding is a data string of 1 to n (total number of operation procedures) which meets procedure priority relation constraint; each number in the data string represents an operation procedure, and the total number of the operation procedures is the total length of the data string;
the three distribution modes in the step 1.3) are respectively as follows: forward allocation, reverse allocation, and bi-directional allocation.
The three initialization population criteria in step 2.2) are respectively: the method includes the steps of allocating a large number of subsequent steps, allocating a step with the longest working time of the step, and randomly selecting one step for allocation.
If the sequence crossing method is adopted in the crossing operation in the step 4), the method comprises the following operation steps:
a4.1), randomly selecting two individuals from the population, and randomly generating a cross point;
a4.2), the offspring individual 1 copies the code on the left side of the cross point of the parent individual 1;
a4.3), deleting the code contained by the parent individual 1 in the parent individual 2;
a4.4), copying the rest codes in the parent individual 2 to the right area of the cross point of the child individual 1 in the order from left to right to generate a child individual 1;
A45) the same steps are repeated to generate the filial generation individuals 2.
If the variation operation in the step 4) adopts a single point variation method, the method comprises the following operation steps:
b4.1) randomly generating a variation point in the parent individuals;
b4.2) determining the position which can be inserted in the parent individual (the process priority relation constraint needs to be met);
b4.3) inserting the process of the position of the variation point into a determined position, and moving other processes backwards in sequence after the position;
b4.4) completing mutation operation to generate offspring individuals.
In step 5), the non-dominated sorting is mainly divided into two parts: the first part is to calculate the set of solutions that each solution i in the population can dominate and the number of solutions i in the population that can dominate; the second part is to divide all individuals into different layers, and the solutions within each layer cannot dominate each other.
Compared with the prior art, the invention has the following remarkable advantages: the invention designs a product principle layout optimization method based on improved NSGA-II, which improves NSGA-II, adopts three distribution modes to distribute operation procedures, embeds sequence cross operation, single-point variation operation and initializes a population strategy, and further increases the diversity of the population; simulation experiments prove that the improved algorithm can obtain a high-quality non-dominated solution set, the construction cost of a production line is reduced, and the workload of each station is balanced, so that the optimal design of product principle arrangement is realized.
Drawings
FIG. 1 is a flow diagram of the modified NAGA-II of the present invention;
FIG. 2 is an exemplary illustration of operation process priorities in accordance with the present invention;
FIG. 3 is a diagram illustrating a forward allocation manner of the present invention;
FIG. 4 is an exemplary diagram of a reverse allocation scheme of the present invention;
FIG. 5(a) is a diagram illustrating a second bi-directional allocation scheme of the present invention;
FIG. 5(b) is a diagram illustrating a third bi-directional allocation scheme according to the present invention;
FIG. 6 is a schematic diagram of an order-crossing method of the present invention;
FIG. 7 is a schematic diagram of a single point mutation method according to the present invention;
FIG. 8 is a diagram of a process priority relationship for an aerospace product assembly line, in accordance with an embodiment of the present invention;
FIG. 9(a) is a Gantt chart of a first process step allocation scheme according to an embodiment of the present invention;
FIG. 9(b) is a Gantt chart of a second process step allocation scheme according to an embodiment of the present invention;
FIG. 9(c) is a Gantt chart of a third process step allocation scheme according to an embodiment of the present invention;
FIG. 9(d) is a Gantt chart of a fourth process step allocation scheme according to an embodiment of the present invention;
FIG. 10 is an assembly line balancing efficiency iteration graph of an embodiment of the present invention;
FIG. 11 is a graph of a smoothing exponential iteration for each process load for an embodiment of the present invention.
Detailed Description
The invention is shown in figures 1-4, 5(a) -5(b), 6-8, 9(a) -9(d), 10-11, and is arranged as follows:
1) and modeling: the principle arrangement of the product is essentially assembly line arrangement, and the core problem is assembly line balance;
1.1) establishing a mathematical model of the objective function and determining constraint conditions of the mathematical model;
1.2) adopting a Pareto optimization method to process a multi-objective optimization problem;
1.3), carrying out discretization treatment on NSGA-II, coding by adopting a procedure sequence coding mode, and simultaneously decoding by applying three distribution modes;
2) initializing a population:
2.1), initializing population control parameters including population number SN and cross probability pcAnd the probability of variation pm；
2.2) applying three initialization population criteria to averagely divide all individuals (solutions) in the population into three parts, wherein each part uses one initialization criterion to generate an initialized population, and storing non-dominant solutions in the population into an external Pareto solution set;
3) and selecting the championship game: selecting individuals in the population by adopting a championship method, wherein the championship scale is 3, and forming a genetic operation pool with the size being half of the population number;
4) and genetic manipulation: performing cross operation and mutation operation on individuals in the genetic operation pool; first, a random number p between 0 and 1 is generated, if p<pcThen choose the individual to do the order-interleaving operation, if p<pmSelecting individuals to perform single-point mutation operation;
5) selecting elite: combining the individuals in the genetic operation pool in the step 3) with the individuals generated by the genetic operation in the step 4), performing non-dominant sorting and crowded distance sorting on the combined population, screening to keep the population size the same as a set value, and storing non-dominant solutions in an external Pareto solution set;
6) judging whether a stopping criterion is met (namely, the algorithm stops when the total iteration times reach the set iteration times during running); if yes, exporting an external Pareto storage set which already contains the arrangement flow of the assembly line; if not, go to step 3). The final external Pareto solution set is a plurality of optimal solutions, namely, the final external Pareto solution set is an arrangement process of a plurality of assembly lines, and which one is specifically used in the final actual use can be automatically determined by a human.
The variable symbols of the mathematical model in step 1.1) are shown in the following table:
TABLE 1 variable symbol definitions
Figure BDA0002242824950000061
The three optimization targets are respectively the minimum number of stations, the maximum assembly line balance efficiency and the minimum smooth index of each station load; therefore, the constraints in step 1.1) are as follows:
Figure BDA0002242824950000062
Figure BDA0002242824950000063
Figure BDA0002242824950000064
Figure BDA0002242824950000071
Figure BDA0002242824950000072
Figure BDA0002242824950000073
xik0 or
Figure BDA0002242824950000074
The formula (1) is an optimization target of minimizing the number of stations, and the construction cost of an assembly line can be reduced, resources are saved, and the utilization rate of equipment is improved due to fewer stations; the formula (2) and the formula (3) are respectively an optimization target for maximizing assembly line balance efficiency and a smooth index optimization target for minimizing the load of each station, and if the work load among the stations is relatively balanced, the actual production takt time can be effectively reduced; the expression (4) indicates that any operation process can be distributed into only one station and cannot be simultaneously distributed into a plurality of stations; equation (5) represents arbitrary operation processes i and j, and if process i is a process immediately before process j, process i cannot be assigned to a station subsequent to the station where process j is locatedInternal; the formula (6) shows that the sum of all the distributed operation process time in any station can not exceed the given beat time; equation (7) represents the control variable xikWhen operation i is assigned to the kth station, xik1, otherwise, xik＝0。
Pareto domination in step 1.2) denotes the hypothesis X1And X2Are two solutions different from each other, and any solution has r optimization objective function values, if the following conditions are satisfied, X is indicated1Dominating X2；
fi(X1)≤fi(X2),i＝1,2,...,r(8)
Figure BDA0002242824950000075
Pareto optimal solution indicates if X1Dominating X2Then, X is described1R optimized objective function values are all superior to X2(ii) a Therefore, the solution that is not dominated by any solution is called Pareto optimal solution;
the Pareto external storage set indicates that all Pareto optimal solutions are recorded in one set; the set is updated synchronously as the algorithm iterates, in order to keep the solutions in the set all Pareto optimal solutions.
Step 1.3) adopts a procedure sequence coding mode to represent a feasible solution, namely, coding is a data string of 1 to n (total number of operation procedures) which meets procedure priority relation constraint; each number in the data string represents an operation procedure, and the total number of the operation procedures is the total length of the data string; fig. 2 shows an example of the operation process priority relationship, in which the number of the operation process is indicated in the circle, the number above the circle indicates the operation time of the operation process, and the tact time is 11.
In step 1.3), in order to extract the potential of the feasible solution and break through the existing limitations, three distribution modes are adopted to distribute the same feasible solution, wherein the three distribution modes are respectively as follows: forward allocation, reverse allocation, and bi-directional allocation.
Forward direction distribution: in the forward distribution mode, the operation steps in the feasible solution are distributed into the workstations one by one in sequence, namely, from left to right. And distributing the current operation process into the station, and if the total load in the station does not exceed the given takt time, continuing to distribute the next operation process. Once the total load in a station exceeds the actual tempo, the current operating procedure is assigned to the next station. In this manner, all of the operational procedures within the feasible solution are allocated to completion until completed. An example of a forward allocation is shown in fig. 3. The results of the assignment of the operating procedures and the values corresponding to the three optimization objectives are given in the figure.
Reverse distribution: in the reverse distribution mode, the operation steps in the feasible solution are also sequentially distributed into the work stations one by one, and are carried out in a right-to-left mode unlike the forward distribution mode. The rest of the operation is the same as the forward allocation. An example of a reverse allocation is shown in fig. 4.
Bidirectional allocation: the bidirectional allocation mode is that when the working procedure in the feasible solution is allocated, the working procedure is respectively allocated from the left side and the right side of the solution at the same time, and the allocation mode is the same as the forward allocation mode and the reverse allocation mode. And when the two distribution modes finish the distribution of one station, comparing the working procedures in the two stations. If the working procedures in the two stations are not intersected, the two modes simultaneously carry out the distribution of the working procedure in the next station. And if the working procedures in the two working stations have intersection, taking out the same working procedure in the two working stations from the front working station. And (3) adopting an optimized distribution mode for the same procedures, namely, distributing the same procedures into two stations again, wherein the distributed procedures in the two stations are different, so as to ensure that the solution obtained after distribution is a feasible solution. The aim of the re-allocation is to optimize the allocated optimal target value. The selection of the optimal value adopts a Pareto optimization method.
For example, when the feasible solutions are distributed by bidirectional distribution, the processes in the second stations of the front-side and back-side distribution modes are respectively 234 and 3456, and the repeated processes of the two distribution modes are 34, which results in ① {1}, {2}, { 3456 } and { 789 }, ② {1}, { 23 }, { 456 } and { 789 }, ③ {1}, { 234 }, { 56 } and { 789 }.
In step 2.2), all individuals (solutions) in the population are averagely divided into three parts, each part uses an initialization criterion, and the three initialization population criteria are respectively as follows: the method includes the steps of allocating a large number of subsequent steps, allocating a step with the longest working time of the step, and randomly selecting one step for allocation.
If the interleaving operation in step 4) adopts an order interleaving method, as shown in fig. 6 specifically, the method includes the following operation steps:
a4.1), randomly selecting two individuals from the population, and randomly generating a cross point;
a4.2), the offspring individual 1 copies the code on the left side of the cross point of the parent individual 1;
a4.3), deleting the code contained by the parent individual 1 in the parent individual 2;
a4.4), copying the rest codes in the parent individual 2 to the right area of the cross point of the child individual 1 in the order from left to right to generate a child individual 1;
A45) the same steps are repeated to generate the filial generation individuals 2.
If the mutation operation in step 4) adopts a single point mutation method, as shown in fig. 7, the method comprises the following operation steps:
b4.1) randomly generating a variation point in the parent individuals;
b4.2) determining the position which can be inserted in the parent individual (the process priority relation constraint needs to be met);
b4.3) inserting the process of the position of the variation point into a determined position, and moving other processes backwards in sequence after the position;
b4.4) completing mutation operation to generate offspring individuals.
In step 5), the non-dominated sorting is mainly divided into two parts: the first part is to calculate the set of solutions that each solution i in the population can dominate and the number of solutions i in the population that can dominate; the second part is to divide all individuals into different layers, and the solutions within each layer cannot dominate each other. After the non-dominated sorting is completed, the quality of the non-dominated solution in each layer needs to be measured, and therefore, the concept of the crowding distance is introduced to evaluate the quality of the solution. Generally, the crowding density of other solutions around a solution with a large crowding distance is small, and an individual with a large crowding distance is generally selected in order to maintain the diversity and distribution of solutions in a population.
The invention is further illustrated by the following examples and figures.
The invention solves an assembly line example of certain aerospace products by using the improved NSGA-II, and further verifies the effectiveness of the algorithm. The priority relationship of the process and the operation time of the process of an aerospace product assembly line are shown in FIG. 8 and Table 2, and the production tact time of the product is 52.5. The code for all algorithms was written on the MATLAB R2012a platform and run on a Core-i5(3.2GHz) personal computer. The parameters of the algorithm are set as follows: SN of 100, pcIs 0.8, pm0.2 and 50 iterations.
TABLE 2 operating time of an aerospace product assembly line procedure
Figure BDA0002242824950000091
Figure BDA0002242824950000101
For this example problem, the improved algorithm runs independently 10 times under the same test environment. The result obtained after algorithm optimization is as follows: the number of work stations m is 4, the assembly line balance efficiency E is 0.9239, and the smoothness index Wv of the load of each work station is 0.0474. At this objective function value, the modified algorithm results in four different process recipes, as shown in fig. 9(a) to 9 (d). Algorithm during operation, a smooth exponential iterative process of population assembly line balancing efficiency and load at each workstation is shown in fig. 10-11.
From the calculation results, the improved algorithm adopts three distribution modes to distribute the working procedures, so that the quality of feasible solutions is improved, and a high-quality solution effect can be obtained. In the iterative process, the algorithm is converged quickly and obtains a global optimal solution, and the algorithm is proved to have strong searching capability. In conclusion, when the improved algorithm is used for solving the multi-target assembly line balance problem, a competitive solving effect can be obtained, and the practicability and the effectiveness of the algorithm are proved.

Claims (7)

1. A product principle arrangement method based on improved NSGA-II is characterized by comprising the following steps:
1) and modeling:
1.1) establishing a mathematical model of the objective function and determining constraint conditions of the mathematical model;
1.2) adopting a Pareto optimization method to process a multi-objective optimization problem;
1.3), carrying out discretization treatment on NSGA-II, coding by adopting a procedure sequence coding mode, and simultaneously decoding by applying three distribution modes;
2) initializing a population:
2.1), initializing population control parameters including population number SN and cross probability pcAnd the probability of variation pm；
2.2) applying three initialization population criteria to generate an initialized population, and storing non-dominant solutions in the population into an external Pareto solution set;
3) and selecting the championship game: selecting individuals in the population by adopting a championship method, wherein the championship scale is 3, and forming a genetic operation pool with the size being half of the population number;
4) and genetic manipulation: performing cross operation and mutation operation on individuals in the genetic operation pool; first, a random number p between 0 and 1 is generated, if p<pcThen choose the individual to do the order-interleaving operation, if p<pmSelecting individuals to perform single-point mutation operation;
5) selecting elite: combining the individuals in the genetic operation pool in the step 3) with the individuals generated by the genetic operation in the step 4), performing non-dominant sorting and crowded distance sorting on the combined population, screening to keep the population size the same as a set value, and storing non-dominant solutions in an external Pareto solution set;
6) judging whether the stop criterion is met; if yes, exporting an external Pareto storage set; if not, go to step 3).
2. A method for arranging product principles based on modified NSGA-II according to claim 1, wherein the variable symbols of the mathematical model in step 1.1) are shown in the following table:
TABLE 1 variable symbol definitions
Figure FDA0002242824940000011
Figure FDA0002242824940000021
The constraint conditions in step 1.1) are as follows:
Figure FDA0002242824940000022
Figure FDA0002242824940000023
Figure FDA0002242824940000024
Figure FDA0002242824940000025
Figure FDA0002242824940000026
Figure FDA0002242824940000027
Figure FDA0002242824940000028
wherein, the formula (1) is an optimization target of minimizing the number of stations; the formula (2) and the formula (3) are respectively a smooth index optimization target for maximizing the balance efficiency of the assembly line and minimizing the load of each station; the expression (4) indicates that any operation process can be distributed into only one station and cannot be simultaneously distributed into a plurality of stations; equation (5) represents any operation processes i and j, and if the process i is a process immediately before the process j, the process i cannot be distributed into a station behind a station where the process j is located; the formula (6) shows that the sum of all the distributed operation process time in any station can not exceed the given beat time; equation (7) represents the control variable xikWhen operation i is assigned to the kth station, xik1, otherwise, xik＝0。
3. The method for improving NSGA-II based product principles layout, according to claim 1, is characterized in that Pareto domination in step 1.2) represents the assumption of X1And X2Are two solutions different from each other, and any solution has r optimization objective function values, if the following conditions are satisfied, X is indicated1Dominating X2；
fi(X1)≤fi(X2),i＝1,2,...,r (8)
Figure FDA0002242824940000031
Pareto optimal solution indicates if X1Dominating X2Then, X is described1R optimized objective function values are all superior to X2(ii) a Therefore, the solution that is not dominated by any solution is called Pareto optimal solution;
the Pareto external storage set indicates that all Pareto optimal solutions are recorded in one set; the set is updated synchronously as the algorithm iterates, in order to keep the solutions in the set all Pareto optimal solutions.
4. The method for arranging the product principles based on the improved NSGA-II is characterized in that a feasible solution is represented in a process sequence coding mode in the step 1.3), namely a data string from 1 to n (the total number of operation processes) meeting process priority relation constraints is coded; each number in the data string represents an operation procedure, and the total number of the operation procedures is the total length of the data string;
the three distribution modes in the step 1.3) are respectively as follows: forward allocation, reverse allocation, and bi-directional allocation.
5. The NSGA-II based product rule placement method according to claim 1, wherein the three initial population criteria in step 2.2) are: the method includes the steps of allocating a large number of subsequent steps, allocating a step with the longest working time of the step, and randomly selecting one step for allocation.
6. The NSGA-II based product rule placement method according to claim 1, wherein if the interleaving operation in step 4) is an order interleaving method, the method comprises the following steps:
a4.1), randomly selecting two individuals from the population, and randomly generating a cross point;
a4.2), the offspring individual 1 copies the code on the left side of the cross point of the parent individual 1;
a4.3), deleting the code contained by the parent individual 1 in the parent individual 2;
a4.4), copying the rest codes in the parent individual 2 to the right area of the cross point of the child individual 1 in the order from left to right to generate a child individual 1;
A45) the same steps are repeated to generate the filial generation individuals 2.
If the variation operation in the step 4) adopts a single point variation method, the method comprises the following operation steps:
b4.1) randomly generating a variation point in the parent individuals;
b4.2) determining the position which can be inserted in the parent individual (the process priority relation constraint needs to be met);
b4.3) inserting the process of the position of the variation point into a determined position, and moving other processes backwards in sequence after the position;
b4.4) completing mutation operation to generate offspring individuals.
7. The method for arranging the product principles based on the improved NSGA-II according to the claim 1, wherein in the step 5), the non-dominated sorting is mainly divided into two parts: the first part is to calculate the set of solutions that each solution i in the population can dominate and the number of solutions i in the population that can dominate; the second part is to divide all individuals into different layers, and the solutions within each layer cannot dominate each other.
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