CN110516821B - Ship maintenance resource allocation optimization method and device - Google Patents

Ship maintenance resource allocation optimization method and device Download PDF

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CN110516821B
CN110516821B CN201910657480.1A CN201910657480A CN110516821B CN 110516821 B CN110516821 B CN 110516821B CN 201910657480 A CN201910657480 A CN 201910657480A CN 110516821 B CN110516821 B CN 110516821B
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魏来
解锋
冯源
黄登斌
叶晓慧
彭丹
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Naval University of Engineering PLA
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    • GPHYSICS
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Abstract

The invention relates to a method and a device for optimizing ship maintenance resource allocation, wherein a ship state evaluation index system is established based on a fuzzy analytic hierarchy process, the index system comprises an index layer and a sub-index layer, and the comprehensive importance of each element of the sub-index layer in the index system to the ship state is calculated; and optimizing the coefficient of a pre-established ship maintenance resource demand prediction model by using the current data of each element of the sub-index layer in the evaluation index system and the comprehensive importance of each element to the ship state. The invention uses fuzzy mathematic principle to describe qualitative or uncertain factors in accurate mathematic language, and evaluates the repair state of the factors to obtain an optimized configuration model.

Description

Ship maintenance resource allocation optimization method and device
Technical Field
The invention relates to the field of ship maintenance, in particular to a method and a device for optimizing ship maintenance resource allocation.
Background
The ship maintenance resources refer to materials necessary for implementing maintenance work and other conditions for completing the maintenance work, and mainly comprise personnel, materials, environment, regulations and the like. The optimal allocation problem of the ship maintenance resources is a complex and systematic mathematical problem, and relates to various problems such as optimal overall allocation of the maintenance resources, optimal allocation of the resources by considering a ship maintenance unit as a whole, optimal allocation of single resources, optimal allocation of multi-dimensional resources and the like.
Most of the current ship maintenance resource allocation is based on a single analysis method, namely, the optimization allocation is developed only by taking single maintenance resources as an optimization target, and important influence factors such as the technical state parameters, the navigation mileage, the comprehensive repair capability of a dock and the like of a ship are not fully and systematically considered during the allocation, and the maintenance resource allocation data cannot be dynamically adjusted according to the actual ship repair requirement, so that the maintenance resource allocation is unscientific and unreasonable, and unnecessary waste is generated.
Disclosure of Invention
Aiming at the technical problems in the prior art, the invention provides a method and a device for optimizing the configuration of ship maintenance resources.
The technical scheme for solving the technical problems is as follows:
in a first aspect, the present invention provides a method for optimizing a ship maintenance resource allocation, including the following steps:
establishing a ship state evaluation index system based on a fuzzy analytic hierarchy process, wherein the index system comprises an index layer and a sub-index layer, and calculating the comprehensive importance of each element of the sub-index layer in the index system to the ship state;
optimizing the coefficient of a pre-established ship maintenance resource demand prediction model by using the current data of each element of the sub-index layer in the evaluation index system and the comprehensive importance of each element to the ship state;
the model for forecasting the ship maintenance resource demand is shown as the following formula,
ZYSLi=αJIBIE1SLi+βJIBIE2SLi+γJIBIE3SLi
in the formula, ZYSLiRepresenting the required quantity of the ith type of resources in the maintenance process;
JIBIE1SLi、JIBIE2SLi、JIBIE3SLirespectively representing the average values of general, more serious and serious historical statistics of the dock repair level corresponding to the ith type of resource; alpha, beta and gamma are weighting coefficients.
On the basis of the technical scheme, the method further comprises a coefficient optimization result evaluation method, and the method comprises the following steps:
calculating the membership value of each element of the sub-index layer to the ship state;
and calculating the evaluation value of the coefficient optimization result by using the comprehensive importance of each element to the ship state and the membership value of each element.
Further, the calculating the membership value of each element of the sub-index layer to the ship state includes:
defining a membership coefficient according to expert experience;
calculating a correction coefficient of the ship maintenance resource demand prediction model corresponding to each element of the sub-index layer by using the current data of each element of the sub-index layer in the evaluation index system;
and calculating the membership value of each element of the sub-index layer to the ship state according to the membership coefficient and the correction coefficient corresponding to each element of the sub-index layer.
In a second aspect, the present invention further provides a device for optimizing ship maintenance resource allocation, including:
the evaluation index system establishing module is used for establishing a ship state evaluation index system based on a fuzzy analytic hierarchy process, wherein the index system comprises an index layer and a sub-index layer, and the comprehensive importance of each element of the sub-index layer in the index system to the ship state is calculated;
and the coefficient optimization module is used for optimizing the coefficient of the pre-established ship maintenance resource demand prediction model by utilizing the current data of each element of the sub-index layer in the evaluation index system and the comprehensive importance of each element to the ship state.
The invention has the beneficial effects that: the weighting coefficient adopted by the optimal configuration of the ship maintenance resources is a fixed value, but the actual ship technical state is certainly dynamic, so the ship state must be evaluated, the more accurate ship state weighting coefficient can better reflect the ship to-be-repaired state, and the more targeted guidance of the configuration of the dock maintenance resources is realized. Therefore, the maintenance resource optimization configuration research mainly optimizes the ship state weighting coefficients, and the ship state weighting coefficients [ alpha, beta and gamma ] are determined by adopting a weight configuration method based on a fuzzy hierarchy analysis method, so that the optimized ship state weighting coefficients can more accurately reflect the ship maintenance state.
The invention adopts a ship technical state evaluation system based on a fuzzy hierarchical analysis method, establishes a hierarchical structure of an object to be evaluated on the basis of comprehensively analyzing parameters such as a ship technical state, sailing mileage and the like, adopts a dynamic coefficient weighted model according to the attributes of evaluation indexes, and evaluates the repair state of the object by describing qualitative or uncertain factors in an accurate mathematical language by using a fuzzy mathematical principle to obtain an optimized configuration model.
Drawings
Fig. 1 is a flowchart of a method for optimizing a ship maintenance resource allocation according to an embodiment of the present invention;
fig. 2 is a structural diagram of a ship maintenance resource allocation optimization apparatus according to a second embodiment of the present invention.
Detailed Description
The principles and features of this invention are described below in conjunction with the following drawings, which are set forth by way of illustration only and are not intended to limit the scope of the invention.
The weighting coefficient adopted by the optimized configuration of the ship maintenance resources is a fixed value, but the actual ship technical state is certainly dynamic, so the ship state must be evaluated, the more accurate ship state weighting coefficient can better reflect the ship to-be-repaired state, and the more targeted guidance of the configuration of the dock maintenance resources is realized. Therefore, the maintenance resource optimization configuration research mainly optimizes the ship state weighting coefficients, and the ship state weighting coefficients [ alpha, beta and gamma ] are determined by adopting a weight configuration method based on a fuzzy hierarchy analysis method, so that the optimized ship state weighting coefficients can more accurately reflect the ship maintenance state.
The invention adopts a ship technical state evaluation system based on a fuzzy hierarchical analysis method, establishes a hierarchical structure of an object to be evaluated on the basis of comprehensively analyzing parameters such as a ship technical state, sailing mileage and the like, adopts a dynamic coefficient weighted model according to the attributes of evaluation indexes, and evaluates the repair state of the object by describing qualitative or uncertain factors in an accurate mathematical language by using a fuzzy mathematical principle to obtain an optimized configuration model.
Example one
As shown in fig. 1, a method for optimizing a ship maintenance resource configuration according to an embodiment of the present invention includes the following steps:
and 100, establishing a ship state evaluation index system based on a fuzzy analytic hierarchy process, wherein the index system comprises an index layer and a sub-index layer, and calculating the comprehensive importance of each element of the sub-index layer in the index system to the ship state.
The model for forecasting the ship maintenance resource demand is shown as the following formula,
ZYSLi=αJIBIE1SLi+βJIBIE2SLi+γJIBIE3SLi
in the formula, ZYSLiRepresenting the required quantity of the ith type of resources in the maintenance process;
JIBIE1SLi、JIBIE2SLi、JIBIE3SLirespectively representing the average values of general, more serious and serious historical statistics of the dock repair level corresponding to the ith type of resource; alpha, beta and gamma are weighting coefficients.
And carrying out hierarchical division on the factors according to the types of the factors in the evaluation system to form a hierarchical structure model for evaluating the ship state. The ship state evaluation structure model is a completely independent hierarchical structure and is characterized in that indexes at the same level are independent and completely different from indexes at the next level.
Determining a ship maintenance state evaluation index system and a hierarchical relation structure through investigation of a certain dock maintenance unit, setting two index parameters of 'navigation condition' and 'ship condition' in a first layer, wherein the lower-level index of the navigation condition is 'navigation time after last maintenance', 'navigation mileage after last maintenance' and 'interval time after last maintenance'; the "hull state" is set here as an evaluation of the ship's technical state by the ship's length. The hierarchical relationship structure is shown in table 1. The hierarchical relationship can be continuously corrected and supplemented and perfected subsequently.
TABLE 1 hierarchical relationship structure of certain ship maintenance state evaluation index system
Figure BDA0002137282320000051
After the hierarchical structure is established, the element membership between the upper layer and the lower layer is determined, and the index system is divided into two layers as shown in table 1:
the index layer only lists two elements of ship navigation conditions and ship body states; but it is not excluded that other category elements are added depending on the type of vessel.
The sub-index layer also lists only four elements of the navigation time after the last maintenance, the navigation process after the last maintenance, the interval time after the last maintenance and the ship body state captain evaluation value;
the navigation time after the last maintenance, the navigation flow after the last maintenance and the interval time after the last maintenance belong to the ship course condition, and the ship state captain evaluation value belongs to the ship state.
After the hierarchical structure is established, calculating the comprehensive importance w of each level of index according to the influence importance of the lower level index on the upper level evaluation criterionj
wj=ai*bj
Wherein, aiValue representing index layer element i, bjRepresenting the value of sub-index layer element j.
The sum of the importance of each level of the index is 1. Here, it is assumed that the importance of the two first-layer indexes of "voyage situation" and "hull situation" is (0.6, 0.4), respectively (the importance value is generally evaluated by a repair specialist), and the importance of the indexes of "voyage situation" three sub-indexes of "voyage time after last maintenance", "voyage mileage after last maintenance", and "interval time after last maintenance" is (0.4,0.4,0.2), respectively (the importance value is generally evaluated by a repair specialist). Comprehensive importance w of indexes of each leveljThe calculation results are shown in table 2.
TABLE 2 comprehensive importance of certain ship maintenance state evaluation index
Figure BDA0002137282320000052
Figure BDA0002137282320000061
And 200, optimizing the coefficient of a pre-established ship maintenance resource demand prediction model by using the current data of each element of the sub-index layer in the evaluation index system and the comprehensive importance of each element to the ship state.
Firstly, calculating a correction coefficient of the ship maintenance resource demand prediction model corresponding to each element of the sub-index layer by using the current data of each element of the sub-index layer in the evaluation index system;
specifically, the method for calculating the correction coefficient of the ship maintenance resource demand prediction model corresponding to each element of the sub-index layer by using the current data of each element of the sub-index layer in the evaluation index system comprises the following substeps:
step 201, discussing the relationship between the ship maintenance resource demand prediction model weight coefficients [ α, β, γ ], and firstly establishing a constraint condition: and (2) initializing the ship maintenance resource demand prediction model coefficient, wherein alpha + beta + gamma is 1, and making alpha, beta and gamma take the average value of the sum of the weight coefficients to be 0.33.
Step 202, performing normalization processing on the current data of each element of the sub-index layer, and then discussing values of alpha and gamma.
1. The normalization processing is carried out aiming at the 'voyage time after the last maintenance',
Figure BDA0002137282320000062
Figure BDA0002137282320000063
representing normalized "last repair after voyage time";
Thtrepresenting "last repair after voyage time";
Thtmaxindicating "maximum time to voyage after repair".
In the [ alpha, beta, gamma ] coefficients, gamma represents the severe state of the dock maintenance level of the ship to be maintained, so the actual value of each element of the sub-index layer is in direct proportion to gamma.
When in use
Figure BDA0002137282320000071
When the temperature of the water is higher than the set temperature,
Figure BDA0002137282320000072
when in use
Figure BDA0002137282320000073
When the temperature of the water is higher than the set temperature,
Figure BDA0002137282320000074
since the initialized β is 0.33, when determining the corrected weight coefficient, α and γ must be multiplied by 2/3, that is, the correction coefficient corresponding to the last-maintained voyage time is
Figure BDA0002137282320000075
2. Firstly, the normalization processing is carried out aiming at the sailing mileage after the last maintenance,
Figure BDA0002137282320000076
Figure BDA0002137282320000077
representing normalized "voyage mileage after last repair";
Thl"last-repair after-voyage mileage";
Thlmaxand represents "longest after-repair voyage mileage".
In the [ alpha, beta, gamma ] coefficients, gamma represents the severe state of the dock maintenance level of the ship to be maintained, so the actual value of each element of the sub-index layer is in direct proportion to gamma.
When in use
Figure BDA0002137282320000078
When the temperature of the water is higher than the set temperature,
Figure BDA0002137282320000079
when in use
Figure BDA00021372823200000710
When the temperature of the water is higher than the set temperature,
Figure BDA00021372823200000711
since the initialized β is 0.33, when determining the corrected weighting coefficients, α and γ must be multiplied by 2/3, that is, the correction coefficient corresponding to the mileage after the last maintenance is equal to
Figure BDA0002137282320000081
3. The normalization processing is carried out aiming at the interval time after the last maintenance,
Figure BDA0002137282320000082
Figure BDA0002137282320000083
representing normalized "voyage mileage after last repair";
Tjt"last-repair after-voyage mileage";
Tjtmaxand represents "longest after-repair voyage mileage".
In the [ alpha, beta, gamma ] coefficients, gamma represents the severe state of the dock maintenance level of the ship to be maintained, so the actual value of each element of the sub-index layer is in direct proportion to gamma.
When in use
Figure BDA0002137282320000084
When the temperature of the water is higher than the set temperature,
Figure BDA0002137282320000085
when in use
Figure BDA0002137282320000086
When the temperature of the water is higher than the set temperature,
Figure BDA0002137282320000087
since the initialized β is 0.33, α and γ must be multiplied by 2/3 when determining the corrected weight coefficient, that is, the correction coefficient corresponding to the interval time after the last maintenance is equal to
Figure BDA0002137282320000088
4. Aiming at the 'ship state ship length evaluation value', the value interval of the 'ship state ship length evaluation value' is [0, 1], so that normalization processing is not needed. Assuming that the 'hull state ship length evaluation value' is P, the 'hull state ship length evaluation value' is in direct proportion to alpha, namely the higher the score is, the better the ship state is, and the higher the weight coefficient alpha is.
When P is more than or equal to 0.5,
Figure BDA0002137282320000089
when the P is less than 0.5,
Figure BDA0002137282320000091
since the initialized β is 0.33, α and γ must be multiplied by 2 when determining the corrected weight coefficient/3, namely the correction coefficient corresponding to the ship body state ship length evaluation value is
Figure BDA0002137282320000092
After the correction coefficients corresponding to the elements of the sub-index layer are obtained, the comprehensive importance w of the elements to the ship state is combinediAnd calculating an optimization coefficient of the ship maintenance resource demand prediction model.
Specifically, the correction coefficients corresponding to each element of the sub-index layer are combined into a coefficient matrix, as follows:
Figure BDA0002137282320000093
simultaneously according to the comprehensive importance w of each element to the state of the shipiForm the composite importance vector W ═ W1,w2,w3,w4) And calculating an optimization coefficient of a ship maintenance resource demand prediction model by using the comprehensive importance vector and the coefficient matrix, wherein the optimization coefficient is represented by the following formula:
Figure BDA0002137282320000094
on the basis of the method, the embodiment of the invention also comprises a coefficient optimization result evaluation method, which comprises the following steps:
step 301, calculating membership values of each element of the sub-index layer to the ship state;
specifically, a membership coefficient is defined according to expert experience; and then calculating the membership value of each element of the sub-index layer to the ship state according to the membership coefficient and the correction coefficient corresponding to each element of the sub-index layer.
Defining a membership coefficient Z ═ Z (Z)1,z2,z3),z1,z2,z3And the coefficients alpha, beta and gamma respectively correspond to the ship maintenance resource demand prediction model. According to expert experience, Z is (0.6,0.8,1), and then each element pair of the sub-index layer is calculated according to the following formulaMembership value U (b) in ship statej):
Figure BDA0002137282320000101
And step 302, calculating a coefficient optimization result evaluation value Q according to the following formula by using the comprehensive importance of each element to the ship state and the membership value of each element.
Q=[U(b1),U(b2),U(b3),U(b4)](w1,w2,w3,w4)T
When describing the matching degree of the predicted value and the real result, seven grades of good, medium, poor and poor are generally adopted, which correspond to 1.0, 0.9, 0.7, 0.5, 0.3, 0.1 and 0 respectively, and the worst value and the optimal value of the predicted value are 0 and 1.0 respectively. And comparing seven grade values according to the evaluation value Q of the coefficient optimization result, and converting the qualitative representation of the index into quantitative description.
Example two
As shown in fig. 2, a ship maintenance resource allocation optimizing apparatus provided in an embodiment of the present invention includes:
the evaluation index system establishing module is used for establishing a ship state evaluation index system based on a fuzzy analytic hierarchy process, wherein the index system comprises an index layer and a sub-index layer, and the comprehensive importance of each element of the sub-index layer in the index system to the ship state is calculated;
and the coefficient optimization module is used for optimizing the coefficient of the pre-established ship maintenance resource demand prediction model by utilizing the current data of each element of the sub-index layer in the evaluation index system and the comprehensive importance of each element to the ship state.
The above description is only for the purpose of illustrating the preferred embodiments of the present invention and is not to be construed as limiting the invention, and any modifications, equivalents, improvements and the like that fall within the spirit and principle of the present invention are intended to be included therein.

Claims (5)

1. A ship maintenance resource allocation optimization method is characterized by comprising the following steps:
establishing a ship state evaluation index system based on a fuzzy analytic hierarchy process, wherein the index system comprises an index layer and a sub-index layer, and calculating the comprehensive importance of each element of the sub-index layer in the index system to the ship state;
optimizing the coefficient of a pre-established ship maintenance resource demand prediction model by using the current data of each element of the sub-index layer in the evaluation index system and the comprehensive importance of each element to the ship state;
the model for forecasting the ship maintenance resource demand is shown as the following formula,
Figure 174122DEST_PATH_IMAGE001
in the formula (I), the compound is shown in the specification,
Figure 553282DEST_PATH_IMAGE002
indicating the first in the maintenance process
Figure 854951DEST_PATH_IMAGE003
The required number of class resources;
Figure 215525DEST_PATH_IMAGE004
Figure 664961DEST_PATH_IMAGE005
Figure 34762DEST_PATH_IMAGE006
respectively represent the first
Figure 925358DEST_PATH_IMAGE008
The generic resources correspond to the average historical statistics of general, more serious and serious dock repair levels;
Figure 535462DEST_PATH_IMAGE009
Figure 144298DEST_PATH_IMAGE010
Figure 317790DEST_PATH_IMAGE011
is a weighting coefficient;
the comprehensive importance of each index in the calculation index system to the ship state comprises the following steps:
assigning values to each element of the index layer according to the importance of each element of the index layer on the ship state evaluation;
assigning values to each element in the sub-index layer according to the importance of each element in the sub-index layer to each element in the index layer;
calculating the comprehensive importance of each element in the sub-index layer relative to the state of the ship by using the following formula;
Figure 390788DEST_PATH_IMAGE012
wherein the content of the first and second substances,
Figure 358744DEST_PATH_IMAGE013
representing index layer elements
Figure 454876DEST_PATH_IMAGE014
The value of (a) is,
Figure 976600DEST_PATH_IMAGE015
representing sub-index layer elements
Figure 841788DEST_PATH_IMAGE016
A value of (d);
the method for optimizing the coefficient of the pre-established ship maintenance resource demand prediction model by using the current data of each element of the sub-index layer in the evaluation index system and the comprehensive importance of each element to the ship state comprises the following steps:
establishing a constraint condition:
Figure 980645DEST_PATH_IMAGE017
initializing the ship maintenance resource demand prediction model coefficients and order
Figure 626390DEST_PATH_IMAGE009
Figure 876106DEST_PATH_IMAGE018
Figure 674429DEST_PATH_IMAGE019
The average value of the sum of the weight coefficients is 0.33;
normalizing the current data of each element of the sub-index layer;
correcting coefficients of the ship maintenance resource demand prediction model corresponding to each element of the operator index layer are calculated according to the relation among the coefficients of the prediction model, each coefficient value and each element data value after normalization processing;
and calculating the optimization coefficient of the ship maintenance resource demand prediction model by a linear weighting method by combining the comprehensive importance of each element to the ship state and utilizing the correction coefficient corresponding to each element of the sub-index layer.
2. The method of claim 1, wherein the vessel condition evaluation index system comprises:
the index layer at least comprises two elements of ship navigation condition and ship body state;
the sub-index layer at least comprises four elements of navigation time after last maintenance, navigation flow after last maintenance, interval time after last maintenance and ship hull state captain evaluation value;
the index layer and the sub-index layer have a membership relationship, the navigation time after the last maintenance, the navigation flow after the last maintenance and the interval time after the last maintenance belong to the ship course condition, and the ship state captain evaluation value belongs to the ship state.
3. The method of claim 1, further comprising a coefficient optimization result evaluation method comprising:
calculating the membership value of each element of the sub-index layer to the ship state;
and calculating the evaluation value of the coefficient optimization result by using the comprehensive importance of each element to the ship state and the membership value of each element.
4. The method of claim 3, wherein calculating the membership value of each element of the sub-index layer to the ship state comprises:
defining a membership coefficient according to expert experience;
calculating a correction coefficient of the ship maintenance resource demand prediction model corresponding to each element of the sub-index layer by using the current data of each element of the sub-index layer in the evaluation index system;
and calculating the membership value of each element of the sub-index layer to the ship state according to the membership coefficient and the correction coefficient corresponding to each element of the sub-index layer.
5. A ship maintenance resource allocation optimizing device is characterized by comprising:
the evaluation index system establishing module is used for establishing a ship state evaluation index system based on a fuzzy analytic hierarchy process, wherein the index system comprises an index layer and a sub-index layer, and the comprehensive importance of each element of the sub-index layer in the index system to the ship state is calculated;
the coefficient optimization module is used for optimizing the coefficient of a pre-established ship maintenance resource demand prediction model by utilizing the current data of each element of the sub-index layer in the evaluation index system and the comprehensive importance of each element to the ship state;
the model for forecasting the ship maintenance resource demand is shown as the following formula,
Figure 984187DEST_PATH_IMAGE001
in the formula (I), the compound is shown in the specification,
Figure 54911DEST_PATH_IMAGE002
indicating the first in the maintenance process
Figure 170635DEST_PATH_IMAGE003
The required number of class resources;
Figure 10415DEST_PATH_IMAGE004
Figure 491075DEST_PATH_IMAGE005
Figure 596565DEST_PATH_IMAGE006
respectively represent the first
Figure 453663DEST_PATH_IMAGE014
The generic resources correspond to the average historical statistics of general, more serious and serious dock repair levels;
Figure 147949DEST_PATH_IMAGE009
Figure 861827DEST_PATH_IMAGE010
Figure 907144DEST_PATH_IMAGE011
is a weighting coefficient;
the comprehensive importance of each index in the calculation index system to the ship state comprises the following steps:
assigning values to each element of the index layer according to the importance of each element of the index layer on the ship state evaluation;
assigning values to each element in the sub-index layer according to the importance of each element in the sub-index layer to each element in the index layer;
calculating the comprehensive importance of each element in the sub-index layer relative to the state of the ship by using the following formula;
Figure 302353DEST_PATH_IMAGE012
wherein the content of the first and second substances,
Figure 664195DEST_PATH_IMAGE013
representing index layer elements
Figure 752237DEST_PATH_IMAGE014
The value of (a) is,
Figure 19270DEST_PATH_IMAGE015
representing sub-index layer elements
Figure 280488DEST_PATH_IMAGE016
A value of (d);
the coefficient optimization module is specifically configured to:
establishing a constraint condition:
Figure 683787DEST_PATH_IMAGE020
initializing the ship maintenance resource demand prediction model coefficients and order
Figure 942730DEST_PATH_IMAGE009
Figure 530617DEST_PATH_IMAGE010
Figure 267628DEST_PATH_IMAGE011
The average value of the sum of the weight coefficients is 0.33;
normalizing the current data of each element of the sub-index layer;
correcting coefficients of the ship maintenance resource demand prediction model corresponding to each element of the operator index layer are calculated according to the relation among the coefficients of the prediction model, each coefficient value and each element data value after normalization processing;
and calculating the optimization coefficient of the ship maintenance resource demand prediction model by a linear weighting method by combining the comprehensive importance of each element to the ship state and utilizing the correction coefficient corresponding to each element of the sub-index layer.
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