CN101957945A - Method and device for optimizing goods loading of container - Google Patents
Method and device for optimizing goods loading of container Download PDFInfo
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- CN101957945A CN101957945A CN2010102589845A CN201010258984A CN101957945A CN 101957945 A CN101957945 A CN 101957945A CN 2010102589845 A CN2010102589845 A CN 2010102589845A CN 201010258984 A CN201010258984 A CN 201010258984A CN 101957945 A CN101957945 A CN 101957945A
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Abstract
The invention discloses a method and a device for optimizing the goods loading of a container. According to the size and bearing power of the container, and related information about the lengths, widths, heights and weights of various case goods, pre-loading is performed, and search is optimized by an algorithm to find an optimal loading scheme, thereby improving the utilization rate of resources and the loading efficiency.
Description
Technical field
The present invention relates to port traffic management and Computer Applied Technology field, relate in particular to a kind of container cargo and load optimization method, device.
Background technology
Along with development of global economy, world's encased conveying already presents quick rising tendency, encased conveying in transport by sea in occupation of more and more important ratio.At present, the goods of world commerce 90% is born by sea-freight, and most goods all adopt encased conveying.According to statistics, 1996 to 2006, the average annual amount of increase of world's container cargo was 10%, and the average annual amount of increase of world's trade of goods is 5%.In China, the foreign trade logistics capacity more than 90% will rely on the harbour to realize, and main means of transportation is exactly encased conveying.The handling capacity crisis at harbour has been brought up in global container trade sustainable growth, and this has become a major challenge to harbor industry.
The harbour except facilities and equipment is carried out the investment construction, more need consider to adopt effective production management method to impel the existing facilities and equipment performance in harbour largest production efficient in the face of ever-increasing cargo handling capacity.The port traffic resource makes full use of with the production efficiency raising has become the major issue that many harbours mainly face and need to solve, even becomes one of main bottleneck that influences port development.Set up integrated service system of port traffic efficiently and technical method (container cargo loading optimization method) and seem necessary and urgent, it can bring directly, quick and considerable economic.
Current container cargo stowage is random loading, perhaps loads by rule of thumb, causes the waste of container space and bearing capacity.In fact, cargo loading is a kind of combinatorial optimization problem, has many loading patterns, has huge optimization space.
Therefore, it is very necessary to design a kind of container cargo loading optimization method and device thereof, is one of port traffic management and Computer Applied Technology field problem anxious to be solved at present.
Summary of the invention
The embodiment of the invention provides container cargo to load optimization method, device, by length, re-correlation information according to container size and bearing capacity and various case goods, carry out preloaded, search for algorithm optimization, find out optimum loading pattern, thereby improve resource utilization and efficiency of loading.
The embodiment of the invention provides following technical scheme:
A kind of container cargo loads optimization method, and step comprises:
Parameter configuration is carried out in the size of step 1, the volume to every case goods, weight, can make a profit profit and container and load-bearing restriction.
Step 2, in loading process, every case goods that the candidate is loaded produces a first generation initial solution at random.
Step 4, utilize iterative algorithm, the parameter configuration by the reorganization iteration is optimized first generation initial solution, obtains second generation feasible solution.
Step 5, pass through
Calculate the adaptive value of second generation feasible solution, wherein, p represents the profit of every case goods, and x represents the carrying case number.
Step 6, according to maximum iteration time or (with) candidate solution is set at the predetermined minimum adaptive value of optimum loading pattern, when satisfying the iteration stopping condition, the output array.
Preferably, in above-mentioned steps two, all goods that in loading process the candidate loaded produce initial solution, i.e. an array at random
...,
...,
| k, i, j, s, t ∈ n}.In the element of array G1, it is individual identical to have m, and it represents to have in this loading pattern m identical goods, wherein, and the length of container (l), wide (b), high (h).
Preferably, in above-mentioned steps two, it is as follows further to comprise recurrence equation:
Wherein, k is an iteration algebraically; v
Id(k) be d dimension component (is-symbol coding or natural number coding, the then v that this algorithm is used of the k time iteration candidate solution i search speed vector
Id(k) is-symbol or natural number); x
Id(k) be the d dimension component of the k time iteration candidate solution i vector, v
Id(k) also is-symbol or natural number; p
Id(k) be the d dimension component that the individual best candidate of candidate solution i is separated; p
Gd(k) be the d dimension component of the best loading pattern of colony, obviously v '
Id(k), p
Id(k) and p
Gd(k) all is-symbol or natural number;
Be the commutating operator symbol, represent two candidate solutions or speed to carry out swap operation.
Preferably, in above-mentioned steps four, utilize iterative algorithm, the parameter configuration by the reorganization iteration is optimized first generation initial solution, i.e. array
...,
...,
| k, i, j, s, t ∈ n} is that starting point is loaded with the initial point, first goods D
sLoad mode be h
s, l
s, b
sThe limit respectively with Y, X, the Z axle is parallel; Follow lade D
i, its load mode is l
i, h
i, b
iThe limit respectively with D
sThe h of goods
s, l
s, b
sThe limit is parallel; And the like, promptly withdraw from up to the length that exceeds container (l), wide (b), high (h) any one restriction.
Preferably, in above-mentioned steps four, utilize iterative algorithm, by the reorganization iteration parameter configuration further comprise commutating operator parameter configuration and mutation operator parameter configuration.
A kind of container cargo loads optimization means, loads optimization means at above-mentioned container cargo and comprises parameter configuration module, first generation initial solution generation module, first generation initial solution adaptive value computing module, second generation feasible solution generation module, second generation feasible solution adaptive value computing module and cycle calculations module.
Preferably, the above-mentioned parameter configuration module comprises volume, weight to every case goods, the size of can make a profit profit and container and load-bearing restriction carry out parameter configuration.
Preferably, in the above-mentioned first generation initial solution generation module, all goods that in loading process the candidate loaded produce initial solution, i.e. an array at random
...,
| k, i, j, s, t ∈ n}.In the element of array G1, it is individual identical to have m, and it represents to have in this loading pattern m identical goods.
Preferably, above-mentioned first generation initial solution adaptive value computing module passes through
Calculate the adaptive value of first generation initial solution.
Preferably, above-mentioned second generation feasible solution generation module utilizes iterative algorithm, and the parameter configuration by the reorganization iteration is optimized first generation initial solution, i.e. array
...,
...,
| k, i, j, s, t ∈ n} is that starting point is loaded with the initial point, first goods D
sLoad mode be h
s, l
s, b
sThe limit respectively with Y, X, the Z axle is parallel; Follow lade D
i, its load mode is l
i, h
i, b
iThe limit respectively with D
sThe h of goods
s, l
s, b
sThe limit is parallel; And the like, promptly withdraw from up to any one restriction of the length that exceeds container.
Preferably, above-mentioned second generation feasible solution generation module further comprises commutating operator parameter configuration module and mutation operator parameter configuration module.
Preferably, above-mentioned second generation feasible solution adaptive value computing module passes through
Calculate the adaptive value of second generation feasible solution.
Preferably, above-mentioned cycle calculations module according to maximum iteration time or (with) candidate solution is set at the predetermined minimum adaptive value of optimum loading pattern, when satisfying the iteration stopping condition, the output array.
Preferably, further to comprise recurrence equation as follows for above-mentioned described first generation initial solution generation module:
Wherein, k is an iteration algebraically; v
Id(k) be d dimension component (is-symbol coding or natural number coding, the then v that this algorithm is used of the k time iteration candidate solution i search speed vector
Id(k) is-symbol or natural number); x
Id(k) be the d dimension component of the k time iteration candidate solution i vector, v
Id(k) also is-symbol or natural number; p
Id(k) be the d dimension component that the individual best candidate of candidate solution i is separated; p
Gd(k) be the d dimension component of the best loading pattern of colony, obviously v '
Id(k), p
Id(k) and p
Gd(k) all is-symbol or natural number;
Be the commutating operator symbol, represent two candidate solutions or speed to carry out swap operation.
A kind of container cargo provided by the invention loads optimization method, device, by length, re-correlation information according to container size and bearing capacity and various case goods, carry out preloaded, search for algorithm optimization, find out optimum loading pattern, thereby improve resource utilization and efficiency of loading.
Description of drawings
In order to be illustrated more clearly in the embodiment of the invention or technical scheme of the prior art, will do to introduce simply to the accompanying drawing of required use in embodiment or the description of the Prior Art below.Apparently, the accompanying drawing in describing below only is some embodiments of the present invention, for those of ordinary skills, under the prerequisite of not paying creative work, can also obtain other accompanying drawing according to these accompanying drawings.
Fig. 1 is that the container cargo that the embodiment of the invention provides loads the optimization method process flow diagram;
Fig. 2 is that the container cargo that the embodiment of the invention provides loads the optimization means synoptic diagram;
Fig. 2 a is the second generation feasible solution generation module inner structure synoptic diagram that the embodiment of the invention provides;
The container cargo load mode synoptic diagram that Fig. 3 provides for the embodiment of the invention;
The swap operation synoptic diagram that Fig. 4 provides for the embodiment of the invention;
The mutation operation synoptic diagram that Fig. 5 provides for the embodiment of the invention.
Embodiment
The embodiment of the invention provides a kind of container cargo to load optimization method, device.By length, re-correlation information according to container size and bearing capacity and various case goods, carry out preloaded, with the algorithm optimization search, find out optimum loading pattern, thereby improve resource utilization and efficiency of loading.For making purpose of the present invention, technical scheme and advantage clearer, the embodiment that develops simultaneously with reference to the accompanying drawings, the present invention is described in more detail.
The embodiment of the invention provides a kind of container cargo to load optimization method, and as shown in Figure 1, concrete steps comprise:
Parameter configuration is carried out in the size of step 1, the volume to every case goods, weight, can make a profit profit and container and load-bearing restriction.
Step 2, in loading process, every case goods that the candidate is loaded produces a first generation initial solution at random.
Step 4, utilize iterative algorithm, the parameter configuration by the reorganization iteration is optimized first generation initial solution, obtains second generation feasible solution.
Step 6, root maximum iteration time or (with) candidate solution is set at the predetermined minimum adaptive value of optimum loading pattern, when satisfying the iteration stopping condition, the output array.
In addition, the embodiment of the invention also provides a kind of container cargo to load optimization means.As shown in Figure 2, a kind of container cargo that provides for the embodiment of the invention loads the optimization means synoptic diagram.
A kind of container cargo loads optimization means, comprises parameter configuration module 11, first generation initial solution generation module 22, first generation initial solution adaptive value computing module 33, second generation feasible solution generation module 44, second generation feasible solution adaptive value computing module 55 and cycle calculations module 66.
Container cargo loading problem is meant puts the goods of different size in the container of one constant volume, to obtain best benefit (as the profit value maximization).Container cargo loading problem is a combinatorial optimization problem with complicated constraint condition, belongs to the NP-hard problem in theory.When container carries out cargo loading, generally all be according to from inside to outside, from top to bottom, from left to right (principle is identical from right to left) loads successively, till filling.As shown in Figure 3.
Particularly, the size of the volume of every case goods, weight, can make a profit profit and delivery instrument (as container) and load-bearing restriction are set up the cargo loading model below as table 1, provide optimum container cargo loading pattern, generate profit value maximization.
Table 1
Particularly, x
1, x
2..., x
nRepresent goods D respectively
1, D
2..., D
nThe carrying case number.As shown in Equation (1), can not overcharge after the goods optimum of loading is put the length of restriction.Just cargo loading the time, horizontal stroke or vertical pendulum are put the different utilization ratios that directly influence the space.Different disposing way optimizing is exactly Combinatorial Optimization in fact.
First generation initial solution generation module 22: in loading process, every case goods that the candidate is loaded produces a first generation initial solution at random.
Particularly, suppose in the target search space of a D dimension (the D case is treated lade) that have m candidate solution (loading pattern) to form a population, wherein i candidate solution is expressed as a D dimensional vector
I=1,2 ..., m, promptly i candidate solution in the search volume of D dimension is
With
Be respectively the search speed and the acceleration of i candidate solution,
Be the optimum loading pattern pbest that i candidate solution searches up to now,
Be the optimum loading pattern gbest that whole candidate solution group searches up to now.The recurrence equation of this new algorithm is as follows:
In formula (2), (3), (4) and (5), k is an iteration algebraically; v
Id(k) be d dimension component (is-symbol coding or natural number coding, the then v that this algorithm is used of the k time iteration candidate solution i search speed vector
Id(k) is-symbol or natural number); x
Id(k) be the d dimension component of the k time iteration candidate solution i vector, v
Id(k) also is-symbol or natural number; p
Id(k) be the d dimension component that the individual best candidate of candidate solution i is separated; p
Gd(k) be the d dimension component of the best loading pattern of colony, obviously v '
Id(k), p
Id(k) and p
Gd(k) all is-symbol or natural number;
Be the commutating operator symbol, represent two candidate solutions or speed to carry out swap operation; Stopping criterion for iteration according to particular problem generally elect as maximum iteration time or (with) the predetermined minimum adaptive value of the optimum loading pattern that searches up to now of candidate solution.
Particularly, all goods that in loading process the candidate loaded produce an initial solution at random.It is array
...,
...,
| k, i, j, s, t ∈ n}.In the element of array G1, it is individual identical to have m, and it represents to have in this loading pattern m identical goods; The limit that rotational symmetry equates can be regarded same limit as; Initial solution G1 represents, is that starting point is loaded with the initial point, first goods D
iLoad mode be l
i, b
i, h
iThe limit respectively with Y, X, the Z axle is parallel; Follow lade D
j, its load mode is l
j, b
j, h
jThe limit respectively with D
kThe l of goods
i, b
i, h
iThe limit is parallel; And the like, promptly withdraw from operation up to the length that exceeds container (L), wide (B), high (H) any one parameter limit.Might as well suppose to be loaded into goods D
kThe time, exceeded the long restriction of container.
First generation initial solution adaptive value computing module 33:
Second generation feasible solution generation module 44: utilize iterative algorithm, the parameter configuration by the reorganization iteration is optimized first generation initial solution, obtains second generation feasible solution.
Particularly, utilize iterative algorithm, the parameter configuration by the reorganization iteration is optimized first generation initial solution, i.e. array
...,
...,
| k, i, j, s, t ∈ n}.Separating G2 and represent, is that starting point is loaded with the initial point, first goods D
sLoad mode be h
s, l
s, b
sThe limit respectively with Y, X, the Z axle is parallel; Follow lade D
i, its load mode is l
i, h
i, b
iThe limit respectively with D
sThe h of goods
s, l
s, b
sThe limit is parallel; And the like, promptly withdraw from up to any one restriction of the length that exceeds container.Might as well suppose to be loaded into goods D
tThe time, exceeded the long restriction of container.When reorganization, iteration, the element of array is " alternately " mutually, and the element interior element is mutual " exchange " also.
Particularly, second generation feasible solution generation module 44 comprises following two modules:
Commutating operator parameter configuration module 441:
One section exchange (C1): 1. and 2. select two exchange spots at random in the candidate solution length at parent, with parent 1. and 2. the lade correspondence in the exchange spot copy to half son's generation, then with its exchange.The type swap operation is shown in Fig. 4 (a).
Two sections exchanges (C2): 1. and 2. select two pairs of exchange spots at random in the candidate solution length at parent, with parent 1. and 2. the lade correspondence in the exchange spot copy to half son's generation, then with its exchange.The type swap operation is shown in Fig. 4 (b).
Three sections exchanges (C3) are similar with the step of two sections exchanges, and concrete operations are shown in Fig. 5 (c).
Mutation operator parameter configuration module 442:
One section move to be inserted (M1): select one section goods to be installed at random at parent, move to the insertion point of selecting at random in the length of parent again after.One section is moved the insertion operation shown in Fig. 5 (c).
Two sections mobile insert (M2) are similar to one section mobile insertion, shown in Fig. 5 (d).
Select container upset (M3) at random: in parent candidate solution length, select several containers to be installed at random, select overturn point then respectively at random, with its overturn point and the long exchange of container.The type mutation operator parameter configuration step is shown in Fig. 5 (e).
Select one section big upset of container (M4) at random: in parent candidate solution length, select one section container to be installed at random, the length of all containers in this section is become height, length and width.The type mutation operator parameter configuration step is shown in Fig. 5 (f).
Select at random that two sections big upsets of container (M5) are similar selects one section big upset of container at random.
Produce new lade sequence (M6) at random.As produce in the new population the new candidate solution, produce new cargo loading sequence at random and get final product.
Second generation feasible solution adaptive value computing module 55:
Particularly, hypothesis is loaded into goods D in first generation initial solution optimal module
tThe time, the restriction that has exceeded container long (L).So we can pass through formula
First generation initial solution is calculated, to obtain the maximized profit value that container cargo loads.
Cycle calculations module 66:
Particularly, suppose in the target search space of a D dimension (the D case is treated lade) that have m candidate solution (loading pattern) to form a population, wherein i candidate solution is expressed as a D dimensional vector
I=1,2 ..., m, promptly i candidate solution in the search volume of D dimension is
With
Be respectively the search speed and the acceleration of i candidate solution,
Be the optimum loading pattern pbest that i candidate solution searches up to now,
Be the optimum loading pattern gbest that whole candidate solution group searches up to now.The recurrence equation of this new algorithm is as follows:
In formula (2), (3), (4) and (5), k is an iteration algebraically; v
Id(k) be d dimension component (is-symbol coding or natural number coding, the then v that this algorithm is used of the k time iteration candidate solution i search speed vector
Id(k) is-symbol or natural number); x
Id(k) be the d dimension component of the k time iteration candidate solution i vector, v
Id(k) also is-symbol or natural number; p
Id(k) be the d dimension component that the individual best candidate of candidate solution i is separated; p
Gd(k) be the d dimension component of the best loading pattern of colony, obviously v '
Id(k), p
Id(k) and p
Gd(k) all is-symbol or natural number;
Be the commutating operator symbol, represent two candidate solutions or speed to carry out swap operation, as Fig. 4, shown in Figure 5; Stopping criterion for iteration according to particular problem generally elect as maximum iteration time or (with) the predetermined minimum adaptive value of the optimum loading pattern that searches up to now of candidate solution.When the iteration stopping condition satisfies, export array, be loading pattern.
One of ordinary skill in the art will appreciate that and realize that all or part of step that the foregoing description method is carried is to instruct relevant hardware to finish by program, described program can be stored in a kind of computer-readable recording medium, this program comprises one of step or its combination of method embodiment when carrying out.
In addition, each functional unit in each embodiment of the present invention can be integrated in the processing module, also can be that the independent physics in each unit exists, and also can be integrated in the module two or more unit.Above-mentioned integrated module both can adopt the form of hardware to realize, also can adopt the form of software function module to realize.If described integrated module realizes with the form of software function module and during as independently production marketing or use, also can be stored in the computer read/write memory medium.
In sum, this paper provides the embodiment of the invention to provide a kind of container cargo to load optimization method, device.By length, re-correlation information according to container size and bearing capacity and various case goods, carry out preloaded, with the algorithm optimization search, find out optimum loading pattern, thereby improve resource utilization and efficiency of loading.
More than to a kind of container cargo provided by the present invention load optimization method, device is described in detail, used specific case herein principle of the present invention and embodiment are set forth, the explanation of above embodiment just is used for helping to understand the solution of the present invention; Simultaneously, for one of ordinary skill in the art, according to thought of the present invention, the part that all can change in specific embodiments and applications, in sum, this description should not be construed as limitation of the present invention.
Claims (14)
1. a container cargo loads optimization method, it is characterized in that, described container cargo loads optimization method and comprises:
Parameter configuration is carried out in the size of step 1, the volume to every case goods, weight, can make a profit profit and container and load-bearing restriction;
Step 2, in loading process, every case goods that the candidate is loaded produces a first generation initial solution at random
Step 3, pass through
Calculate the adaptive value of first generation initial solution, wherein, p represents the profit of every case goods, and x represents the carrying case number;
Step 4, utilize iterative algorithm, the parameter configuration by the reorganization iteration is optimized first generation initial solution, obtains second generation feasible solution;
Step 5, pass through
Calculate the adaptive value of second generation feasible solution, wherein, p represents the profit of every case goods, and x represents the carrying case number;
Step 6, according to maximum iteration time or (with) candidate solution is set at the predetermined minimum adaptive value of optimum loading pattern, when satisfying the iteration stopping condition, the output array.
2. container cargo according to claim 1 loads optimization method, it is characterized in that, in described step 2, all goods that in loading process the candidate loaded produce initial solution, i.e. an array at random
...,
...,
| k, i, j, s, t ∈ n}.In the element of array G1, it is individual identical to have m, and it represents to have in this loading pattern m identical goods, wherein, and the length of container (l), wide (b), high (h).
3. container cargo according to claim 1 loads optimization method, it is characterized in that in described step 2, it is as follows further to comprise recurrence equation:
Wherein, k is an iteration algebraically; v
Id(k) be d dimension component (is-symbol coding or natural number coding, the then v that this algorithm is used of the k time iteration candidate solution i search speed vector
Id(k) is-symbol or natural number); x
Id(k) be the d dimension component of the k time iteration candidate solution i vector, v
Id(k) also is-symbol or natural number; p
Id(k) be the d dimension component that the individual best candidate of candidate solution i is separated; p
Gd(k) be the d dimension component of the best loading pattern of colony, obviously v '
Id(k), p
Id(k) and p
Gd(k) all is-symbol or natural number;
Be the commutating operator symbol, represent two candidate solutions or speed to carry out swap operation.
4. container cargo according to claim 1 loads optimization method, it is characterized in that, in described step 4, utilizes iterative algorithm, and the parameter configuration by the reorganization iteration is optimized first generation initial solution, i.e. array
...,
...,
| k, i, j, s, t ∈ n} is that starting point is loaded with the initial point, first goods D
sLoad mode be h
s, l
s, b
sThe limit respectively with Y, X, the Z axle is parallel; Follow lade D
i, its load mode is l
i, h
i, b
iThe limit respectively with D
sThe h of goods
s, l
s, b
sThe limit is parallel; And the like, promptly withdraw from up to the length that exceeds container (l), wide (b), high (h) any one restriction.
5. container cargo according to claim 4 loads optimization method, it is characterized in that, in described step 4, utilizes iterative algorithm, and the parameter configuration by the reorganization iteration further comprises commutating operator parameter configuration and mutation operator parameter configuration.
6. a container cargo loads optimization means, it is characterized in that described container cargo loads optimization means and comprises parameter configuration module, the first initial solution generation module, the first initial solution adaptive value computing module, the second initial solution generation module, the second initial solution adaptive value computing module and cycle calculations module.
7. container cargo according to claim 6 loads optimization means, it is characterized in that, described parameter configuration module comprises carries out parameter configuration to the size of the volume of every case goods, weight, can make a profit profit and container and load-bearing restriction.
8. container cargo according to claim 6 loads optimization means, it is characterized in that, in the described first generation initial solution generation module, all goods that in loading process the candidate loaded produce initial solution, i.e. an array at random
...,
...,
| k, i, j, s, t ∈ n}.In the element of array Gl, it is individual identical to have m, and it represents to have in this loading pattern m identical goods.
10. container cargo according to claim 6 loads optimization means, it is characterized in that described second generation feasible solution generation module utilizes iterative algorithm, and the parameter configuration by the reorganization iteration is optimized first generation initial solution, i.e. array
...,
...,
| k, i, j, s, t ∈ n} is that starting point is loaded with the initial point, first goods D
sLoad mode be h
s, l
s, b
sThe limit respectively with Y, X, the Z axle is parallel; Follow lade D
i, its load mode is l
i, h
i, b
iThe limit respectively with D
sThe h of goods
s, l
s, b
sThe limit is parallel; And the like, promptly withdraw from up to any one restriction of the length that exceeds container.
11. container cargo according to claim 10 loads optimization means, it is characterized in that the utilization of described second generation feasible solution generation module further comprises commutating operator parameter configuration module and mutation operator parameter configuration module.
12. container cargo according to claim 6 loads optimization means, it is characterized in that described second generation feasible solution adaptive value computing module passes through
Calculate the adaptive value of second generation feasible solution.
13. container cargo according to claim 6 loads optimization means, it is characterized in that, the cycle calculations module according to maximum iteration time or (with) candidate solution is set at the predetermined minimum adaptive value of optimum loading pattern, when satisfying the iteration stopping condition, the output array.
14. container cargo according to claim 6 loads optimization means, it is characterized in that it is as follows that described first generation initial solution generation module further comprises recurrence equation:
Wherein, k is an iteration algebraically; v
Id(k) be d dimension component (is-symbol coding or natural number coding, the then v that this algorithm is used of the k time iteration candidate solution i search speed vector
Id(k) is-symbol or natural number); x
Id(k) be the d dimension component of the k time iteration candidate solution i vector, v
Id(k) also is-symbol or natural number; p
Id(k) be the d dimension component that the individual best candidate of candidate solution i is separated; p
Gd(k) be the d dimension component of the best loading pattern of colony, obviously v '
Id(k), p
Id(k) and p
Gd(k) all is-symbol or natural number;
Be the commutating operator symbol, represent two candidate solutions or speed to carry out swap operation.
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