optimum design of ship cabin equipment layout based on slp...

Research ArticleOptimum Design of Ship Cabin Equipment Layout Based onSLP Method and Genetic Algorithm

Jinghua Li1 Hui Guo 1 Shichao Zhang2 XiaoyuanWu2 and Liuling Shi2

1College of Shipbuilding Engineering Harbin Engineering University Harbin 150001 China2Shanghai Waigaoqiao Shipbuilding Co Ltd Shanghai 200137 China

Correspondence should be addressed to Hui Guo hui0625hrbeueducn

Received 30 August 2018 Revised 13 November 2018 Accepted 9 January 2019 Published 21 January 2019

Academic Editor Alessio Ishizaka

Copyright copy 2019 Jinghua Li et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The engine room is the heart of a ship and almost all of themain electromechanical equipment that supports the work on board canbe found here Finding a way to arrange the equipment in a small cabin space is an essential factor in the design and constructionof a ship However in existing research when an intelligent algorithm is used to optimize the design of a cabin the establishedmathematical model is not comprehensive and the solution has not been evaluatedThe optimal solution obtained is not feasible forthe actual design of a shipThis can lead to unnecessary redesign work which seriously affects design efficiency and increases designcosts In order to solve the above problems this paper innovatively refers to a Systematic Layout Planning (SLP) method (normallyapplied to the layout of plant equipment) to the cabin equipment layout issue The SLP method is used to quantitatively analyzethe adjacency and logistics relationship between devices and the mutual integration relationship between devices is obtained sothat a preliminary layout scheme can be retrieved The problem model is constructed by considering various factors such as thecomprehensive relationship between the equipment and the stability of the cabin and the corresponding objective function andconstraint function are established to further design the variables operators and steps of the genetic algorithmThe initial solutionobtained from the SLP method is used as part of an initial solution to the genetic algorithm and the genetic algorithm is usedto optimize the problem Finally the Analytic Hierarchy Process (AHP) is used to evaluate and optimize several groups of betterschemes obtained by running multiple genetic algorithms and select the better schemes The experimental design proves that theintegrated design method has certain feasibility and superiority

1 Introduction

The engine room is the heart of a ship A large amount ofmain equipment supporting the work on board is arranged inthe engine roomArranging all kinds of equipment effectivelyis an important problem [1] A reasonable layout can greatlyreduce the ship design and construction cycle and at thesame time reduce costs and ensure ship operation safetyWith research into and the application of an intelligentalgorithm the combination of an intelligent algorithm andequipment layout optimization has been attracting more andmore attention

The layout problem involves a lot of content which isessentially a complex multidisciplinary combination opti-mization problem [2] Its complexity is reflected in twoaspects on the one hand the model is complex Most

layout problems are derived from actual real-world problemsOften actual problems are difficult to describe accurately inmathematical language Usually assumptions can be used tosimplify actual problems and then describe them in math-ematical language to establish corresponding mathematicalmodels On the other hand these calculations are compli-cated The computational complexity of layout problems isa nondeterministic problem involving the complexity of thepolynomial that is an exact solution to the problem cannotbe found in a limited time Therefore most layout problemsoften end up not with an optimal solution but rather afeasible solution that satisfies the constraints and has a betterperformance [3]

In terms of the optimization of facility layouts theenlightenment period began in the early 20th century andwent on to the 1930s There was no systematic theory and

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 9492583 14 pageshttpsdoiorg10115520199492583

2 Mathematical Problems in Engineering

method due to the limitations of the times but it relied onthe subjective experience of human beings [4] From the1940s to the 1960s the issue of facility layout entered a periodof rapid development During this period researchers usedsystem theory cybernetics and other quantitative methodsto optimize facility layouts and achieved good results [5]Since the 1970s with the popularity of computers and theemergence of various intelligent algorithms the problem offacility layout optimization has once again entered anotherperiod of vigorous development [6] The use of intelligentalgorithms to optimize the layout of facilities has completelychanged past business processes and intelligent algorithmsare widely used to successfully solve the problem of facilitylayout design [7ndash17]

The optimization of the layout of ship cabin equipmentas a branch of facility layout optimization has also been ahot topic in recent years Over time many scholars haveconducted a lot of research and achieved certain results Atpresent in terms of the layout problem of cabin equipmentmost research has combined the layout problem with theintelligent algorithm optimized the algorithm and yielded abetter solution AI Olcer introduced amultiobjective geneticalgorithm and fuzzy multiattribute set in the layout designof a ro-ro shiprsquos cabin which improved the efficiency of thecabin layout [18] Y Yang proposed a facility layout optimiza-tion design method for cabin maintainability as a method toimprove the efficiency and quality of maintainability designand built a mathematical model for maintainability layoutoptimization An improved particle swarm optimizationalgorithm was also used to improve calculation efficiencyand accuracy and effectively solve themultipeak optimizationproblem [19] Wang Y L generated the initial value of thegenetic algorithm based on the energy method and thenused the improved genetic algorithm to design the shiprsquoscabin layout which solved the problem of layout evaluationto some extent [20 21] Literature [22ndash25] elaborates theapplication of genetic algorithm in equipment layout prob-lem in detail and proves that genetic algorithm has goodrobustness to this kind of problem through examples ZhengX L proposed a method based on game theory to promotethe multidisciplinary decision-making process involved inship cabin layout design and at the same time developed anoncooperative game strategy and determined the amountof equipment and furniture required for the correspondinglocation in order to achieve the highest possible performancein the cabin [26] The above research only used intelligentalgorithms to solve the problem yet it did not consider howto generate a more effective initial solution did not evaluatethe obtained solution and cannot determine whether thesolution was the most feasible solution

In 1961 Richard Joseph first proposed a systematic facilitylayout method [27] which enabled the plantrsquos designersto perform quantitative analyses based on objective datafrom previous qualitative analyses that had been based onsubjective experience The application of this method is notlimited to the industrial field but also used for various typesof facility layout optimization (among others) and it is stillused improved and innovated upon today The SystematicLayout Planning Method (SLP) is generally applied to the

layout design of workshop facilities [28ndash30] and researchon applying the SLP method to the layout of ship facilitiesis relatively rare Hu Yao et al optimized the complexmultiobjective combination optimization problem for theposition layout of the interior cabin of a volumetric ship andcomprehensively applied the SLP method to the heuristicmethod and the genetic algorithm (GA) in the intelligentalgorithm to solve the problem and optimize the solution[31] However it only lays out the location of differentcompartments and does not optimize the design of the cabininterior Moreover simply using the SLP method to solvethe layout of the cabin equipment only meets the adjacencyand logistics requirements of the equipment and does notguarantee that the scheme meets ship stability and otherperformance requirements

The Analytic Hierarchy Process (AHP) is a hierarchicalweighted decision analysis method proposed by the Pitts-burgh University Operations Researcher Professor Saaty inthe early 1970s when he was studying the subject of the USDepartment of Defense [32] Z Gao et al achieved goodresults by using the Analytic Hierarchy Process (AHP) andthe Pure Output Data Envelopment Analysis (DEA) method[33] When Wang Y L et al laid out the cabin of the shipintelligently the AHP method was used to evaluate severalschemes obtained by using the algorithmand finally theoptimal solution was selected [21] The above studies showthat it is feasible and effective to apply the AHP method toequipment layout optimization but there is still little researchinto the application of this method to the layout of ship cabinequipment

Based on the above research this paper creatively appliesthe SLP method to the cabin equipment layout optimizationproblem establishes a corresponding mathematical modeland uses a genetic algorithm to solve it Finally the AHPmethod is used to evaluate the multiple sets of solutionsobtained The rest of this paper is structured as followsSection 2 uses the SLP method to analyze the adjacencyrequirements of the cabin equipment and the personnelcirculation efficiency requirements and on this basis deter-mines a comprehensive interrelationship between differ-ent equipment thus determining the positional constraintsacross various pieces of equipment and using this as themethod of evaluation Section 3 comprehensively considersfactors such as cabin weight distance and stability and con-structs corresponding mathematical models and constraintsSection 4 details the operator design of the genetic algorithmand generates a cabin layout based on the genetic algorithmSection 5 is based on the AHP method and evaluates thescheme generated in the Section 4 to determine a better lay-out scheme Section 6 concludes the research and identifiesfuture work

2 SLP Analysis of SimplifiedLayout of Equipment

SLP is the earliest method used in the design of factory andworkshop equipment layout It uses the relationship betweenlogistics and nonlogistics of equipment as the main line and

Mathematical Problems in Engineering 3

Table 1 Definition of adjacent demand intensity level and coefficient

strength grade Strength coefficient Meaning1 1 The need for adjacency is very high2 08 Higher adjacency demand3 06 Contiguous demand4 04 Lower adjacency demand5 02 The need for adjacency is very low6 0 No adjacency requirements

adopts a set of expressive legend symbols and concise workforms and through the design process solves the problemof factory and workshop equipment location layout design[34ndash36] The cabin equipment layout problem is in the finalanalysis the layout optimization of the equipment so the SLPmethod can also be used in the layout optimization design ofthe cabin equipmentHowever regardless of the type of layoutdesign being used there are still some differences betweenthe two layout problems The main reason for this differenceis that the location layout design of the cabin equipment isdifferent from the layout design of the production systemThelayout design of the production system primarily considerslogistics transportation factors Relevant research shows thatabout 20 of the processing costs are used for materialtransportation and reasonable equipment layout can reducethe transportation cost of materials by at least 10-20 [37]However in terms of the layout of the cabin equipmentthe logistics relationship factor is not the primary factorand the influence of the adjacency relationship is greaterTherefore when using the SLP method it is necessary tomake adjustments to the constituent elements for the analysisof both the adjacent demand and the circulation relationshipAt the same time it is necessary to use the intensity coefficientto quantify the demand correlation between the constituentelements

21 Analysis of Adjacent Demand Intensity between Equip-ment According to certain layout criteria the adjacentdemand strength of each piece of equipment is determinedand expressed by strength grade and strength coefficient Theadjacency demand strength between devices is determinedaccording to certain arrangement criteria and is expressed byintensity level and intensity coefficient119886119895119896 isin [0 1] indicates the adjacent demand intensitybetween equipment 119895 and 119896 the larger 119886119895119896 indicates that theadjacency demand between 119895 and 119896 is stronger 119886119895119896 = 0indicates that devices 119895 and 119896 have no adjacency needs 119886119895119896 =1 indicates that devices 119895 and 119896 must be adjacent Matrix119860 = [119886119895119896]119899times119899 is a distribution matrix that represents adjacentdemand intensity in which only the adjacency relationshipbetween equipment 119895 and 119896 is considered while the adjacencyrelationship between 119896 and 119895 is not repeated and marked asempty At the same time there is no adjacency of equipmentin 119895 = 119896 The corresponding relationship between strengthgrade and strength coefficient is not present As shown inTable 1 [38] the strength factor of the table is determinedaccording to the demand relationship between the devices

Table 2 Device type

No Device name1 Host2 Alternator3 General use pump for cabin bottom4 Sewage comminution pump5 Fire Extinguisher6 Fuel tank7 Domestic sewage cabinet8 Staircase 19 Staircase 2

Table 3 Distribution of adjacent demand intensity in cabin

No 1 2 3 4 5 6 7 8 91 06 0 0 0 0 0 0 02 04 04 0 02 04 0 03 08 0 02 0 0 04 0 0 08 0 05 06 0 06 066 0 0 07 0 08 09

According to the equipment in Table 2 the layout criteriaare as follows the main engine and the generator should becloser and the distance from the other equipment is far apartthe total pump at the bottom of the cabin is not related tothe operation of other equipment the relative degree of thesewage pulverizing pump and the living sewage water tankis higher and should be placed in a relatively close positionfire extinguishers should be placed close to the fuel tank andstairs for easy access

The specific distribution matrix A of the relationshipdescribed above according to the analysis method in termsof the logistics and nonlogistics relationship using the SLPmethod is shown in Table 3

22 Analysis of the Intensity of Circulation Relationshipbetween Equipment In terms of cabin equipment layoutas well as needing to consider the adjacent requirementsof the operations between the various equipment it is also


Table 4 Definition of strength grade and coefficient of circulation relationship

strength grade Strength coefficient Meaning1 1 The demand for circulation is very high2 08 High demand for circulation3 06 General circulation demand4 04 Low demand for circulation5 02 The demand for circulation is very low6 0 Non circulation demand

Table 5 Distribution relationship intensity distribution in thecabin

No 1 2 3 4 5 6 7 8 91 08 0 0 0 0 0 0 02 06 06 0 02 04 0 03 08 0 0 02 0 04 0 0 08 0 05 06 0 06 066 0 0 07 0 08 09

necessary to consider the flow of personnel in order to ensurethe location of the equipment is convenient for personnelinstallation operation and evacuation

Using 119887119895119896 to express the strength of the circulationrelationship among personnel 119887119895119896 = 0 indicates no cir-culation relationship between 119895 and 119896 that is people donot operate equipment from 119895 to 119896 119887119895119896 =1 indicates that119895 and 119896 circulation is very high that is the frequency ofpersonnel operating from 119895 to 119896 is very high Matrix B is adistribution matrix representing the intensity of circulationrelationships distribution relationship grade and intensitycoefficient distribution as shown in Table 4 [38]

For the circulation relationship we mainly considerthe staff rsquos operation of the equipment and the circulationrelationship from the host to the stairs during evacuation Forthe logistics relationship the layout of the pipelines and cablesbetween the main engine and the generator is consideredmainly as well as the total use of the pump and the livingwater powder in the bottom of the cabin The intensity ofthe logistics relationship between the crushing pump and theliving water cabinet is also relatively small Referring to thecorresponding standard for ship engine roomdesign [39ndash42]the circulation relationship between the equipment is clearlydefined in the standard Referring to the corresponding regu-lations in the standard the personnel and logistics situation ofthe equipment in the cabin are determined The distributionmatrix B shown in Table 5 is obtained by selecting the flowintensity coefficient among the equipment

In Table 5 it should be noted that there is no flowrelationship between the equipment itself as there is no flowintensity grade between equipment 1 and equipment 1 theflow intensity between equipment 1 and equipment 2 is the

same as that between equipment 2 and equipment 1 only oneof them is counted

23 Analysis of the Comprehensive Relationship Strengthbetween Equipment The comprehensive inter-relationshiptable between devices is a combination of the adjacentdemand distribution and the circulation relationship analy-sis and the two relationships are integrated with each otherto produce a table Through a comprehensive analysis ofthe relationship between pieces of equipment the locationof each piece of equipment is reasonably planned and theequipment layout of the cabin is more reasonable

When the intensity of the adjacent demand and the inten-sity of the circulation relationship are all determined the sub-target can be weighted to be transformed into a multitargetstrength coefficient to obtain the comprehensive relationshipstrength of the SLP analysis and 119891119895119896 is expressed by the nextformula 119891119895119896 is determined by the following formula

119891119895119896 = 1199081119886119895119896 + 1199082119887119895119896 (1)

In the formula 119886119895119896 and 119887119895119896 represent two strength valuesbetween the cabins 119895 and 119896 and 1199081 and 1199082 are weightedcoefficients The relative importance of the adjacencyrelationship and the circulation relationship is determinedThe ratio of importance 1199081 1199082 (weighted value) shouldgenerally be in the range of 13sim31 When 1199081 1199082 lt13the layout is affected by the adjacency relationship and thekey planning objective is the circulation relationship whenthe equipment is arranged When 1199081 1199082 gt31 it indicatesthat the adjacency relationship between devices is dominant[43] When devices are deployed the devices occupyingan important proportion in the adjacency relationship areplanned Because the cabin equipment is in the actual shipoperation the adjacency relationship is more important andthe reasonable adjacency relationship can save a lot of cabinspace in order that the pipeline and circuit can be arrangedmore reasonably Referring to the value of this weight in[44] thus this article takes 1199081 1199082 =41 that is 1199081 = 08and 1199082 = 02 The distribution matrix F for synthesizing thestrength of correlation is shown in Table 6

The level of integrated correlation can be further deter-mined based on the data in Table 6 that is the levelof comprehensive correlation between devices in the SLPmethod The level is expressed as AE IOU and theintensity interval corresponding to the level in the exampleis [08 1] [06 08) [04 06) [02 04) [0 02) The rankdistribution is shown in Table 7


Table 6 Comprehensive relationship intensity distribution betweendevices

No 1 2 3 4 5 6 7 8 91 064 0 0 0 0 0 0 02 044 044 0 02 04 0 03 079 0 016 004 0 04 0 0 08 0 05 06 0 06 066 0 0 07 0 08 09

Table 7 Comprehensive relationship level distribution betweendevices

No 1 2 3 4 5 6 7 8 91 E U U U U U U U2 I I U O I U U3 E U U U U U4 U U A U U5 E U E E6 U U U7 U U8 U9

This paper introduces genetic algorithms for subsequentoptimization so the purpose of SLP analysis is to determinethe Positional relationship of a part of the equipment with rel-atively high comprehensive relationship strength In the def-inition of the SLP relational grade the highest level ldquoArdquo mustbe close The living sewage powder pump and the domesticsewage cabinet can be close together in order to facilitate thelaying of cables and pipes According to the requirements ofthe area of each item of equipment the position constraints ofthe algorithm are determined according to the optimizationof the algorithm and the preliminary arrangement of theequipment sequence (7 4 3 1 2 6 8 5 9) is obtained Theabove SLP analysis results can be used as part of the devicesequence in the initial solution of the genetic algorithm toaccelerate the convergence of the genetic algorithm

3 Mathematical Model and ConstraintConditions for Optimization of EngineRoom Layout

31 Establishment of Mathematical Model for Optimizationof Engine Room Layout The layout optimization problemcan also be regarded as a path planning problem The firstproblem is to establish an environment model of the cabinlayout In this paper the cabin equipment is simplified andabstracted as follows assume that the layout of the equipmentin the cabin space is rectangular and the placement of theinternal components of the equipment is also the optimal

placement A good layout of the interior equipment cancoordinate the operation relationship among the ship cabinequipment ensure the operation of the system improvethe circulation efficiency of the cabin crew and reduce thetime consumption of the circulation In engineering layoutproblems the basic layout forms include the single-linelayout multiline layout site layout ring layout and U-shapedlayout Most of the domestic related literatures refer to thesite layout and multiline layout as multiline layout The basiclayout is shown in Figure 1 [35] The layout position in theFigure is represented by two serial numbers the upper leftnumber is the layout position number and the lower rightnumber is the number of the facilities to be placed Combinedwith the cabin structure framework this paper has selectedthe multiline layout form

The topology model of the cabin and equipment is shownin Figure 2 In the picture the lower left corner of the cabinis the original point 119897119895 represents the length of the device119895 ℎ119892119896 represents the minimum horizontal spacing betweenequipment 119892 and 119896 ℎ1198950 represents the minimum horizontalspace between equipment 119895 and the cabin boundary and Δ 119895represents the net distance between equipment 119895 and device119895 minus 1 or the boundary The value range is [0 15] 119904 stands fordevice row spacing 1199040 represents the distance from the firstline device to the workshop boundary 119909119895 is the 119909 coordinatesof the center of device 119895 and 119910119895 is the 119910 coordinates of thecenter of device 11989532 Objective Function Thegoal of cabin layout optimizationis to properly place the equipment in a manner that ensuresthe stability of the shiprsquos structure and performance There-fore the objective function needs to meet the two objectivesof flow intensity and adjacent strength according to the SLPmethod In addition it is necessary to consider balance andcenter of gravity requirements equipment uniform arrange-ment and so on [39ndash42]

(1) Adjacent Intensity TargetThe higher the close relationshipbetweendevices the greater the flow intensity and the smallerthe distance between devices so the objective function fordefining adjacency strength is as follows

1198911 (119909) = 8sum119895=1

9sum119896=119895+1

119860 times 119863 (119909) (2)

The meanings of the letters in the above formula are asfollows

(1) 1198911(119909) is the sum of the product of the equipmentadjacency matrix A and the distance D between the devices

(2) D is the distance matrix between devices calculatedusing the following formula

119863119895119896 = 10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 + 10038161003816100381610038161003816119910119895 minus 11991011989610038161003816100381610038161003816 (3)

(2) Circulation Intensity Target The higher the degree ofclose relationship between devices is the greater the adjacentstrength and the smaller119889119894119895 is so the definition of the adjacentstrength objective function is as follows

1198912 (119909) = 8sum119895=1

9sum119896=119895+1

119861 times 119863 (119909) (4)


1 2 3 4 5

12 345

Single row layout

Multi-line layout

1

6

2

3

4 5

1 2 3

4 5

6

Site layout

1

6

9 82

3

4 5

7

1 2 3

6

98

4 5

7

Ring layout

1

3

2

4

6

5

8 7

1 2

6 5

8

7

3

4

U-shaped layout

14

3

2

5 01

86 7

3 2

87 9

9

10

5

4 1

6

a Layout locationnumber

b Number offacilities to be laidout

a

b

Figure 1 Classification of layout forms

Y

0 X

ℎj0

mj

xk

mk

ℎgk

mg

yj

s

s0

l

Δ Ｄ

Figure 2 Topology model between cabin and equipment

The meanings of the letters in the formula above are asfollows

(1) 1198912(119909) is the sum of the product of the equipmentcirculation strength matrix B and the distance D between thedevices


119863119895119896 = 10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 + 10038161003816100381610038161003816119910119895 minus 11991011989610038161003816100381610038161003816 (5)

(3) Ship Stability Requirements In order to improve thestability of the ship and ensure that the ship has a large heelwhen sailing ensure that the torque algebra and absolutevalue of the equipment for the midlongitudinal section are assmall as possible The distance between the center of gravityof the device and the longitudinal section is calculated as

1198913 (119909) =100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

119898119895 (119909119895 minus 1198972)10038161003816100381610038161003816100381610038161003816100381610038161003816 (6)

(4) Device ArrangedUniformly Auxiliary machines should bearranged as closely as possible to around the cabins mainlybecause if the auxiliary machines are arranged centrally onthe longitudinal line side of the shiprsquos nacelle there will bea free liquid level in the equipment when the equipment isworking normally This will cause the moment of inertia tobe unbalanced thus affecting the stability of the ship Thefollowing formula is used to control the equipment which hasbeen evenly arranged in the cabin

1198914 (119909) =100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

(119909119895 minus 1198972)10038161003816100381610038161003816100381610038161003816100381610038161003816 (7)

According to the mathematical model of the layoutprinciple it can be determined that the objective function ofthe cabin is

119865 (119909) = min4sum119890=1

119891119890 (119909) (8)


33 Constraint

(1) Equipment Must Not Overlap When the shiprsquos cabinequipment is arranged it should be ensured that there is nointerference between the equipment

10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 ge [(119897119895 + 119897119896)2 + ℎ119895119896]119911119895119901119911119896119901119895 119896 = 1 2 9

(9)

The formula for solving the horizontal axis of the deviceis

119909119895 = 119909119896 + (119897119896 + 119897119895)2 + ℎ119896119895 + Δ 119895= ℎ1198960 + Δ 119896 + (119897119895 + 2119897119896)2 + ℎ119896119895 + Δ 119895

(10)

The formula for solving the ordinate of the equipment is

119910119895 = (119896 minus 1) 119904 + 1199040if 119911119895119901 = 1 119895 = 1 2 9 119901 = 1 2 119903

119911119895119901 = 1 119863119890V119894119888119890 119895 119900119899 119897119894119899119890 1199010 119900119905ℎ119890119903

119895 = 1 2 9 119901 = 1 2 119903(11)

where 119903 is the total number of lines in the devicelayout

(2) During the calculation of the layout of the shiprsquoscabin equipment each device is required to appear only oncewhich is

119903sum119901=1

119911119895119901 = 1 119894 = 1 2 9 119909119895 119910119895 ge 0 Δ 119895 ge 0 (12)

(3) The weight of the mechanical equipment arranged onthe left and right sides should be kept as balanced as possibleto avoid the shiprsquos roll caused by the difference in weight onboth sides and 119908 is the cabin width

sum0le119909le1199082

119872119895 asymp sum1199082lt119909le119908

119872119896 (13)

In summary the mathematical model is established as

119865 (119909) = 8sum119895=1

9sum119896=119895+1

119860 times 119863 (119909) + 8sum119895=1

9sum119896=119895+1

119861 times 119863 (119909) +100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

119872119895 (119909119895 minus 1198712)10038161003816100381610038161003816100381610038161003816100381610038161003816 +

100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

(1199099 minus 1198712)10038161003816100381610038161003816100381610038161003816100381610038161003816

119904119905

10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 ge [(119897119895 + 119897119896)2 + ℎ119895119896]119911119895119901119911119896119901 119894 119895 = 1 2 9119903sum119901=1

119911119895119901 = 1 119895 = 1 2 9 119909119895 119910119895 ge 0 Δ 119895 ge 0 119895 = 1 2 9sum0le119909le1199082

119872119895 asymp sum1199082lt119909le119908

119872119896

(14)

By doing this according to the rules and design experi-ence of the cabin equipment layout the objective functionand constraints are determined and the mathematical modelof the cabin layout design is then established which is readyfor the next step whereby the genetic algorithm is used forintelligent optimization

4 Genetic Algorithm Design

In this paper the genetic algorithm is used to solve themodelThe genetic algorithm can be independent of the specific fieldof the problem and has strong robustness to this type of theproblem [45ndash50] Therefore the genetic algorithm can solvethe layout problem of the cabin equipment

According to the characteristics of the multiobjectiveoptimization model of cabin equipment this paper designsthe chromosome coding crossover mutation and algorithm

flow of the genetic algorithm The specific analysis is asfollows

41 Chromosome Coding Encoding extended transpositionexpressions using two lists of device symbols and net spacingare

[ 1198981 1198982 119898119899 Δ 1 Δ 2 Δ 119899] (15)

where 119898119899 represents the device serial number and Δ 119899represents the net spacing between device 119899 minus 1 and device119899 At the same time the automatic line-wrapping strategy isadopted that is when the sum of the lengths of the devicesin the same row and the actual mutual spacing exceeds themaximum lateral space length limit the last device of thebank automatically enters the next line


42 Initial Population The initial population is generatedrandomly In order to speed up the convergence processof the genetic algorithm the first device symbol sequencein the initial population can be replaced by the superiordevice symbol sequence obtained by the SLP method In thiscase the sequence of the cabin obtained by the SLP method(7 4 3 1 2 6 8 5 9) is used instead in order that the initialpopulation is formed

43 Fitness Function Because of the automatic line breakstrategy there is no device outside of the cabin area in theX-axis direction Therefore it is only necessary to determinewhether the last row exceeds the cabin area in the Y-axisdirection

119875119896 = 0 1199040 + (119898 minus 1) 119904 le 119867119879 119900119905ℎ119890119903 (16)

whereH is the width of the compartment is an unreason-able penalty and T is a positive large penalty value of 500

The fitness function is

119891119894119905 (V119896) = 1(119865 + 119875119896) (17)

In the formula 119865 is the objective function

44 Select The roulette selection mechanism is adopted -that is the probability of each individual being selectedis proportional to the fitness degree If the populationsize is M and the fitness of the individual 119894 is 119891119894119905(V119896)then the probability that the individual 119894 is selected is119875119894 = 119891119894119905(V119896)sum119898119894=1 119891119894119905(V119896) (119894 = 1 2 119872) - in otherwords the population is selected according to the proba-bility of obtaining a new population and the higher thefitness the greater the probability that the individual will beselected

45 Cross The crossover operation adopts the partial match-ing method of the two-point cross-binding repair programThe repair program can make the nonpopulation individ-uals in the cross become individual within the populationthus ensuring the smooth progress of the algorithm Thespecific implementation steps of the crossover method are asfollows

For parent one and parent two randomly find twonumbers from 1 to 9 as the intersection position

Father 1 (

Father 2 (

)7 82431956a aaaaaaaa

a aaaaaaaa )6 43179582

Exchange the parts between the two cross positions of theparent

Child 1 (

Child 2 (

a aaaaaaaa

a aaaaaaaa

)7 82179556

)6 43431982

After the crossover the same parent will have dupli-cate device numbers nonrepeating device numbers will beretained and conflicting device numbers will be mapped in

Table 8 Layout scheme

NO Layout Scheme1 [85][21][346][79]2 [7][43][296][185]3 [9][785][346][21]4 [8][59][347][621]5 [9][127][534][86]6 [96][851][732]

the corresponding order of the intermediate segments In thisexample the middle segment of Child 1 is (1198865 1198869 1198867 1198861) themiddle segment of Child 2 is (1198869 1198861 1198863 1198864) the conflictingdevice numbers of Child 1 are 1198865 and 1198867 and the missingparts are 1198863 and 1198864Therefore it is necessary to use the 1198863 and1198864 of the middle segment of Child 2 to fill the position Thecomplement order is complemented by the order of 1198863 and1198864 in (1198869 1198861 1198863 1198864) and Child 2 is also padded as describedabove Therefore the result is

Child 1 (

Child 2 (

a aaaaaaaa

a aaaaaaaa

)3 82179546

)6 75431982

46 Variation The mutation operation only operates on thenet spacing portion of the device assuming that the net spac-ing sequence for a given chromosome is Δ 1 Δ 2 Δ 119899Specify the mutated point Δ 119894 according to the probability ofmutation r is a given integer and [119880min 119880max] is the range ofvalues of the devicersquos net spacing Then within this intervalr net spacing can be generated randomly Δ1119894 Δ2119894 Δ119903119894Replacing the variation point Δ 119894 with Δ1119894 Δ2119894 Δ119903119894 rnew chromosomes can be produced The best one can beselected from the r new chromosome to replace the originalchromosome In this case 119903 = 10 [119880min 119880max] = [0 15]47 Decoding The layout adopts the automatic line-wrappingstrategy Therefore an array with the field name Layout isadded to the algorithm to store the sequence number of eachline of equipment after each device sequence is generated bythe iteration The resulting layout scheme is the data in thearray

48 Algorithm Flow Based on the above settings the GAalgorithm parameters are set as follows population size is 50crossover probability is 06 mutation probability is 01 andmaximum iteration number is 200The GA algorithm flow isshown in Figure 3

Using MATLAB software to optimize the solution theprogram can be run multiple times in order to obtain severalgroups of better solutions and select several sets of solutionsas the selection scheme as shown in Table 8

Because of the multi-line layout and the word-wrapstrategy each bracket represents a line and starts at the firstline


Start

Building a solutionmodel

Initial population

Is it less than the number

Computational fitness function

Yes

No

Select

Cross

Variation

Output result

End

Generating newpopulations

Figure 3 Algorithm flow

5 AHP-Based Cabin Layout Scheme Selection

51 The Basic Principle of AHP Method The Analytic Hier-archy Process (AHP) refers to a complex multi-objectivedecision-making problem as a system which decomposes thetarget into multiple goals or criteria and then decomposesthis into multiple levels of multiple indicators (or criteriaconstraints) The hierarchical single order (weight) and totalordering are calculated by using a qualitative index fuzzyquantization method which is used as the system method oftargeting (multi-indicator) and a multi-scheme optimizationdecision It is suitable for a target system with hierarchically-interlaced evaluation indicators and the target value is diffi-cult in order to quantitatively describe the decision problemOf course the biggest problem of analytic hierarchy process(AHP) is that it is difficult to guarantee the consistency ofthinking when there are many evaluation indicators at a

Building a hierarchicalmodel

Structural hierarchy judgment matrix

Hierarchical single sortconsistency test

Hierarchical total order consistency test

Meet theconditions

Meet the conditions

Determineweight

No

No

Yes

Yes

Figure 4 AHP Analysis flowchart

certain level (such as more than four) In this case the FuzzyAnalytic Hierarchy Process (FAHP) which combines theadvantages of the Fuzzy Method and the Analytic HierarchyProcess (AHP) can solve this problem well [51] Howeverthere are only three evaluation indicators in the criterionlevel of the problem studied in this paper so the nonfuzzyanalytic hierarchy process has been able to get a betterevaluation scheme When using the AHP method to modelproblems the following steps are generally required buildinga hierarchical model constructing a judgment (pairwisecomparison) matrix hierarchical single ordering and consis-tency checking hierarchical total ordering and consistencychecking [52]

The AHP analysis flowchart shown in Figure 4 is estab-lished and then the below six schemes are evaluated basedon this

52 Optimal Process

(1) Establish a Hierarchical Structure Model According tothe decision goal of this paper the target layer is definedas follows determine an optimal solution According to therelevant indicators for evaluating the location layout of the


Target layer TTDetermine the optimal layout

scheme

Z1Reasonable circulation route Z2Adjacent reasoning

P1PLan 1 P2PLan 2 P3PLan 3 P4PLan 4 P5PLan 5 P6PLan 6

Z3Cabin safety

Criteria layer Z

Solution layer P

Figure 5 Hierarchy diagram

cabin equipment the criterion layer is defined as followsthe reasonable degree of the circulation line (ie when theoperation route between the equipment in the scheme islowest and the evacuation path is the shortest the rationalityof the circulation route of the scheme is higher) adjacent tothe reasonable degree (ie the more the equipment must bein close proximity in the comprehensive correlation providedby the SLP method the more reasonable the proximity of thescheme is) the safety degree of the cabin (that is the layout ofthe scheme should be closer to the weight of the left and rightsides and the better the stability the higher the safety of thecabin) and the scheme layer is the six schemes for the layoutof the cabin equipment The hierarchical structure is shownin Figure 5

(2) Establish a Hierarchy of Judgment Matrices When deter-mining the weight between factors at each level if it is onlya qualitative result it is often difficult to be accepted byothers Themeaning of the judgment matrix is that the targetproblem is not compared with all the factors but the twoare compared with each other and the difficulties involved incomparing factors with different properties are compared asmuch as possible in order to improve accuracy For exampletaking the target layer in Figure 5 (determining the optimallayout scheme) as the standard it is more important to judgethe rationality of the circulation line of the criterion layer andthe reasonable degree of the adjacent level 119868119894119895 is the result ofcomparing the importance of element 119894 and element 119895 andthe importance degree is assigned according to Table 9 Thematrix formed by the comparison result of two pairs is calledthe judgment matrix The judgment matrix has the followingproperties

119868119894119895 = 1119868119895119894 (18)

Table 9 Proportion criteria table

Scaling Factor i ratio factor j1 Equally important3 Slightly important5 Stronger important7 Strongly important9 Extremely important2468 Intermediate value of two adjacent judgments

According to the scale value in Table 9 the criteria layercontains three criteria the reasonable degree of Z1 circulationline the reasonable degree of Z2 adjacency and the safetydegree of Z3 cabin The optimal layout scheme is determinedrelative to the target layer according to ship engine roomdesign specifications and references [42 53] and the twopoints are scored to obtain the judgment matrix of thecriterion layer for the target layer

119885119894119895 = [[[[[

1 12 152 1 135 3 1]]]]]

(19)

Similarly establish the decision matrix of the schemelayer for the criterion layer [42 53] 1198751119894119895 indicates the impor-tance of scheme 119894 and scheme 119895 relative to the rationality of thecriterion layer circulation line 1198752119894119895 indicates the importanceof scheme 119894 and scheme 119895 relative to the rationality of thecriterion layer 1198753119894119895 indicates the importance of scheme 119894


Table 10 Hierarchical single sort solution results

Judgment matrix 119885119894119895 1198751119894119895 1198752119894119895 1198753119894119895Maximum eigenvalue 3004 6489 6351 6146

Feature vector

0122 0272 0357 02520230 0228 0242 02560644 0184 0103 0144

0109 0103 00820130 0105 01710078 0091 0096

and scheme 119895 relative to the safety and reasonableness of thecriteria compartment

1198751119894119895 =

[[[[[[[[[[[[[[[[

1 1 2 3 2 41 1 2 3 1 212 12 1 2 3 213 13 12 1 2 112 1 13 12 1 314 12 12 1 13 1

]]]]]]]]]]]]]]]]

1198752119894119895 =

[[[[[[[[[[[[[[[[[

1 2 4 3 5 212 1 2 3 4 214 12 1 12 1 213 13 12 1 2 115 14 1 12 1 312 12 12 1 13 1

]]]]]]]]]]]]]]]]]

1198753119894119895 =

[[[[[[[[[[[[[[[[

1 1 3 3 1 21 1 2 3 2 213 12 1 2 1 213 13 12 1 12 11 12 1 2 1 212 12 12 1 12 1

]]]]]]]]]]]]]]]]

(20)

(3) Hierarchical Single Sort The eigenvector correspondingto the largest eigenvalue 120582max of the judgment matrix isnormalized (so that the sum of the elements in the vectoris equal to 1) and is denoted as 120596 The element of 120596 isthe ordering weight of the same level factor for the relativeimportance of a factor of the previous level factor Thisprocess is called hierarchical single orderingThe normalizedvector is set to 120596 the weight of each factor The solutionresults are shown in Table 10

Table 11 Average random consistency indicator RI standard value

n 1 2 3 4 5 6 7 8 9RI 0 0 058 089 112 124 132 141 145

Table 12 Judgment matrix CR value

Judgment matrix 119885119894119895 1198751119894119895 1198752119894119895 1198753119894119895CR 0003 0079 0057 0024

(4) Hierarchical Single Sort Consistency Test Whether itis possible or not to confirm the hierarchical ordering aconsistency check is required This so-called consistencycheck refers to determining the allowable inconsistency rangefor the matrix M Herein the unique nonzero eigenvalue ofthe n-order uniform matrix is n and the largest eigenvalue ofthe n-th order positive reciprocal matrix is M if and only ifM is a uniform matrix The definition consistency index 119862119868 is

119862119868 = 120582max minus 119899119899 minus 1 (21)

Considering that the deviation of consistency may becaused by one of any random reason when testing whetherthe judgment matrix has satisfactory consistency it is alsonecessary to compare the CI with the random consistencyindex RI to obtain the test coefficient CR and the formulais as follows

119862119877 = 119862119868119877119868 (22)

Generally if CRlt01 the judgmentmatrix is considered topass the consistency test otherwise there is no satisfactoryconsistency The random consistency index RI is related tothe order of the judgment matrix and the matrix orderis generally larger The probability of a uniform randomdeviation is also greater and the corresponding relationshipis shown in Table 11

Calculate the CR value of each judgment matrix accord-ing to the above formula as shown in Table 12

It can be seen from Table 12 that the CR value of eachjudgment matrix is less than 01 indicating that the judgmentmatrix established in this paper is correct

(5) The Total Order of the Hierarchy Calculating the weightof all factors at a certain level for the relative importanceof the highest level (total target) is called the total order ofthe hierarchy This process is carried out in order from thehighest level to the lowest level The weight of each factor atthe bottom is calculated according to the following formula

119882119894 = 119898sum119895=1

119887119895120596119894 (119894 = 1 2 119899) (23)

where119882119894 is the weight of the i-th factor 119875119894 of the solutionlayer to the target layer factor T m n is the number of targetlayer and criterion layer factors 119887119895 is the weight of the j-thfactor 119885119895 in the criterion layer to the target layer factor A 120596119894


Table 13 Hierarchical Total Ordering

Z layer 1198851 1198852 1198853 Z-layer total ordering of target layer PP layer 0122 0230 06441198751 0272 0357 0252 02771198752 0228 0242 0256 02481198753 0184 0103 0144 01381198754 0109 0103 0082 00901198755 0130 0105 0171 01501198756 0078 0091 0096 0092

Table 14 Calculate the required parameter values

Judgment matrix 1198851 1198852 1198853119862119868119895 0098 0070 0029119887119895 0122 0230 0644119877119868119895 1240 1240 1240

is the weight of the program layer factor to the criterion layerfactor 119885119895

According to the above steps the weight of each factor inthe target layer is as shown in Table 13

(6) Hierarchical Total Order Consistency Test First calculatethe CR value according to the following formula

119862119877 = (sum119898119895=1 119862119868119895119887119895)(sum119898119895=1 119877119868119895119887119895) (24)

Theparameter values required to solve the above equationcan be obtained as shown in Table 14

The data in Table 14 should be placed into the aboveformula in order to obtain the consistency ratio CR=0038 ofthe total order of the hierarchy which is less than 01[54]

Based on the above analysis according to the weightsof the six schemes in Table 13 the ranking of the six layoutschemes can be obtained as follows Scheme 1 gt Scheme 2 gtScheme 5gt Scheme 3gt Scheme6gt Scheme4Therefore afterthe AHP analysis Scheme 1 is the optimal solution among thesix layout schemes

6 Conclusions

In this paper the problem of the optimal design of shipcabin equipment layout is solved The SLP method is usedto analyze and determine the comprehensive relationshipbetween each item of equipment Circulation strength analy-sis is helpful for designers to choose the most effective layoutof machinery and equipment In addition to the analysisof circulation intensity it is also important to analyze theroute of the staff when they walk in the cabin during theirwork to facilitate the work of the staff These problems arenot considered in traditional cabin layout design At thesame time the genetic algorithm is used to solve the modelFinally the AHP method is used to evaluate and optimizethe scheme and a more suitable layout scheme is obtained

Compared with the simple use of intelligent algorithms theintegrated design method can more accurately quantitativeanalyze and express the relationship between each device anduse it to evaluate the solution produced by the algorithmwhich improves the accuracy of the feasible solution to someextent On the other hand there are relatively few studieson the application of the SLP method to the layout of cabinequipment This paper provides some ideas for using thismethod to optimize the layout of cabin equipment At thesame time the method of AHP is introduced into the eval-uation and selection of equipment layout schemeThe idea issimple and clear and there is no need to establish complexmathematical model It is very effective for multiobjectivesystem decision-making and the quantitative informationneeded after simplification is simple and easy to be acceptedby decision-makers By analyzing the subjective and fuzzyfactors the system error is reduced and the correctness of theselected layout scheme can be guaranteed to a greater extentOf course the comprehensive design method proposed inthis text still has some shortcomings in the expression andconstraints of the model Further research and discussion arerequired in order to further improve the effectiveness of theintegrated design method

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that they have no conflicts of interest

Authorsrsquo Contributions

Jinghua Li and Hui Guo contributed equally to this work

Acknowledgments

This research was funded by Ministry of Industry and Infor-mation Technology of the Peoplersquos Republic of China [Grantnumber 2016543] and National Natural Science Foundationof China [Grant number 51679059]


References

[1] S-Y Kim B-Y Moon and S-C Shin ldquoEvaluation criterion ofmachinery arrangement design in a ship engine roomrdquo Journalof Ship Production vol 25 no 3 pp 117ndash125 2009

[2] A Kusiak and S S Heragu ldquoThe facility layout problemrdquoEuropean Journal ofOperational Research vol 29 no 3 pp 229ndash251 1987

[3] S S Heragu and A Kusiak ldquoMachine layout problem in flexiblemanufacturing systemsrdquoOperations Research vol 36 no 2 pp258ndash268 1988

[4] HWiendahl P and P Nyhuis Facility Planning Springer BerlinHeidelberg 2014

[5] R D Meiler and K-Y Gau ldquoThe facility layout problemRecent and emerging trends and perspectivesrdquo Journal ofManufacturing Systems vol 15 no 5 pp 351ndash366 1996

[6] Z Liu and G Yao ldquoFacility Layout Design the Past the Presentand the Futurerdquo Journal of Jiangsu University of Science ampTechnology 2001

[7] J Balakrishnan and C H Cheng ldquoA note on ldquoa hybrid geneticalgorithm for the dynamic plant layout problemrdquordquo InternationalJournal of Production Economics vol 103 no 1 pp 87ndash89 2006

[8] M-JWangMH Hu andM-Y Ku ldquoA solution to the unequalarea facilities layout problem by genetic algorithmrdquo Computersin Industry vol 56 no 2 pp 207ndash220 2005

[9] A R McKendall and J Shang ldquoHybrid ant systems for thedynamic facility layout problemrdquo Computers amp OperationsResearch vol 33 no 3 pp 790ndash803 2006

[10] H Samarghandi P Taabayan and F F Jahantigh ldquoA particleswarm optimization for the single row facility layout problemrdquoComputers amp Industrial Engineering vol 58 no 4 pp 529ndash5342010

[11] S Kulturel-Konak and A Konak ldquoA new relaxed flexible baystructure representation and particle swarm optimization forthe unequal area facility layout problemrdquoEngineeringOptimiza-tion vol 43 no 12 pp 1263ndash1287 2011

[12] H Hosseini-Nasab and L Emami ldquoA hybrid particle swarmoptimisation for dynamic facility layout problemrdquo InternationalJournal of Production Research vol 51 no 14 pp 4325ndash43352013

[13] R Kothari and D Ghosh ldquoAn efficient genetic algorithm forsingle row facility layoutrdquoOptimization Letters vol 8 no 2 pp679ndash690 2014

[14] B Naderi and B Naderi A Hybrid Multi-Population GeneticAlgorithm for The Dynamic Facility Layout Problem ElsevierScience Publishers B V 2014

[15] N Banduka M Mladineo and M Eric ldquoDesigning a layoutusing Schmigallamethod combinedwith software tool vistablerdquoInternational Journal of Simulation Modelling vol 16 no 3 pp375ndash385 2017

[16] M Ficko and I Palcic ldquoDesigning a layout using the modifiedtriangle method and genetic algorithmsrdquo International Journalof Simulation Modelling vol 12 no 4 pp 237ndash251 2013

[17] Y J Xiao Y Zheng L M Zhang and Y H Kuo ldquoA combinedzone-LP and simulated annealing algorithm for unequal-areafacility layout problemrdquo Advances in Production Engineering ampManagement vol 11 no 4 pp 259ndash270 2016

[18] A I Olcer C Tuzcu and O Turan ldquoAn integrated multi-objective optimisation and fuzzy multi-attributive groupdecision-making technique for subdivision arrangement ofRo-Ro vesselsrdquo Applied Soft Computing vol 6 no 3 pp221ndash243 2006

[19] X Luo Y Yang Z Ge X Wen and F Guan ldquoMaintainability-based facility layout optimum design of ship cabinrdquo Interna-tional Journal of Production Research vol 53 no 3 pp 677ndash6942015

[20] Y LWang CWang andY Lin ldquoShip cabin layout optimizationdesign based on the improved genetic algorithm methodrdquoApplied Mechanics and Materials vol 300-301 pp 146ndash1492013

[21] Y Wang C Wang Z Ji and X Zhao ldquoA study on intelligentlayout design of ship cabinrdquo Ship Building of China vol 54 no3 pp 139ndash146 2013

[22] K Hauser and C H Chung ldquoGenetic algorithms for layoutoptimization in crossdocking operations of a manufacturingplantrdquo International Journal of Production Research vol 44 no21 pp 4663ndash4680 2006

[23] Z J Gang F E Min and L Z Min ldquoNon overlapped geneticalgorithm for layout problem with behavioral constraintsrdquoJournal of Dalian University of Technology vol 39 no 3 1999

[24] F Ozcelik and A A Islier ldquoGeneralisation of unidirectionalloop layout problem and solution by a genetic algorithmrdquoInternational Journal of Production Research vol 49 no 3 pp747ndash764 2011

[25] K Y Tam ldquoGenetic algorithms function optimizationand facility layout designrdquo European Journal of OperationalResearch vol 63 no 2 pp 322ndash346 1992

[26] Z X Liang L Yan and J Z Shang ldquoShip cabin layout designusing game theoryrdquo Journal of Marine Science and Technologyvol 13 no 4 pp 446ndash454 2008

[27] T-K Chien ldquoAn empirical study of facility layout using amodified SLP procedurerdquo Journal of Manufacturing TechnologyManagement vol 15 no 6 pp 455ndash465 2004

[28] D P van Donk and G Gaalman ldquoFood safety and hygienesystematic layout planning of food processesrdquo Chemical Engi-neering Research andDesign vol 82 no 11 pp 1485ndash1493 2004

[29] K-H Liu S-L Hwang M-H Hsieh S-F Max Liang andC-F Chuang ldquoSystematic layout planning in human-systeminterface An evaluation of alarmdisplayswith spatial proximityfor accidents diagnosis of advanced boiling water reactorrdquoInternational Journal of Industrial Ergonomics vol 51 pp 30ndash42 2016

[30] G B Benitez F S Fogliatto R B Cardoso F S Torres C SFaccin and J M Dora ldquoSystematic Layout Planning of a Radi-ology Reporting Area to Optimize Radiologistsrsquo PerformancerdquoJournal of Digital Imaging vol 31 no 2 pp 193ndash200 2018

[31] H U Yao Z Jiang Z Xiong et al ldquoThe Optimized LayoutDesign of Volume Type Ship Cabins Based on SLP and GArdquoChinese Journal of Ship Research vol 8 no 5 pp 19ndash26 2013

[32] E W L Cheng H Li and D C K Ho Analytic HierarchyProcess (AHP)[M] Encyclopedia of Biostatistics John Wiley ampSons Ltd 2016

[33] Z Gao K Yoshimoto and S Ohmori ldquoApplication of AHPDEA to facility layout selectionrdquo in Proceedings of the 3rdInternational Joint Conference on Computational Sciences andOptimization CSO 2010Theoretical Development and Engineer-ing Practice pp 252ndash254 China May 2010

[34] K Zhou Z Du B Liu R Zhang Y Wang and B WangldquoStudy on workshop layout of a motorcycle company based onsystematic layout planning (SLP)rdquo in Proceedings of the Interna-tional Conference on Image Processing and Pattern Recognitionin Industrial Engineering pp 1683ndash1688 International Societyfor Optics and Photonics Xirsquoan China 2010


[35] Q-L Lin H-C Liu D-J Wang and L Liu ldquoIntegratingsystematic layout planning with fuzzy constraint theory todesign and optimize the facility layout for operating theatre inhospitalsrdquo Journal of IntelligentManufacturing vol 26 no 1 pp87ndash95 2013

[36] S S Hosseini S A Mirzapour and K Y Wong ldquoImprovingmulti-floor facility layout problems using systematic layoutplanning and simulationrdquo Communications in Computer andInformation Science vol 409 pp 58ndash69 2013

[37] SXue Z PXuHHong et al ldquoApplicationof Systematic LayoutPlanning to Production Shop Design A Case Studyrdquo Journal ofIndustrial Engineering 2011

[38] Z-R Li L Qin and Z-Q Cao ldquoApplication of SLP method indesign of facilities layout in workshoprdquo Applied Mechanics andMaterials vol 190-191 pp 28ndash32 2012

[39] Z Zhihua Introduction to Marine Power Plant Harbin Engi-neering University Press 2002

[40] Z Shuwen Principle and Design of Marine Power PlantNational Defense Industry Press 1980

[41] L Jinming Principle and Design of Marine Power PlantNational Defense Industry Press 2014

[42] L Jianguang Guidelines for the Design of Marine and MarineEngineering PowerDevices HuazhongUniversity of Science andTechnology Press 2010

[43] K Q Zhou R J Zhang J A Liu et al ldquoApplication of SLP to theLayout Design ofWorkshop in aMotorcycle Companyrdquo Journalof Industrial Engineering 2011

[44] Y Zheng and B Zhan ldquoSLP-based layout optimization of logis-tics workshop facilities of huairsquoan courier postrdquo inProceedings ofthe 3rd International Conference on Transportation Informationand Safety ICTIS 2015 pp 848ndash851 China June 2015

[45] F Azadivar and J Wang ldquoFacility layout optimization usingsimulation and genetic algorithmsrdquo International Journal ofProduction Research vol 38 no 17 pp 4369ndash4383 2000

[46] T D Mavridou and P M Pardalos ldquoSimulated annealing andgenetic algorithms for the facility layout problem a surveyrdquoComputational Optimization and Applications vol 7 no 1 pp111ndash126 1997

[47] L Garcıa-Hernandez A Arauzo-Azofra H Pierreval andL Salas-Morera ldquoEncoding Structures and Operators Usedin Facility Layout Problems with Genetic Algorithmsrdquo inProceedings of the 2009 Ninth International Conference onIntelligent Systems Design and Applications pp 43ndash48 PisaItaly November 2009

[48] X Liu and X Li ldquoAn Improved Genetic Algorithms-basedApproach on Supply Chain-oriented Facility Layout SchedulingSystemrdquo in Proceedings of the World Congress on IntelligentControl amp Automation IEEE 2006

[49] R Pinto ldquoA Facility Layout Planner tool based on GeneticAlgorithmsrdquo in Proceedings of the Computational IntelligenceIEEE 2016

[50] R K Hasda R K Bhattacharjya and F Bennis ldquoModifiedgenetic algorithms for solving facility layout problemsrdquo Inter-national Journal on Interactive Design and Manufacturing vol11 no 3 pp 713ndash725 2017

[51] Z J Jun ldquoFuzzyAnalytical Hierarchy Processrdquo Fuzzy Systems ampMathematics vol 14 pp 80ndash88 2000

[52] R H Chiu L H Lin and S C Ting ldquoEvaluation of Green PortFactors and Performance A Fuzzy AHP AnalysisrdquoMathemati-cal Problems in Engineering vol 2014 no 5 Article ID 80297612 pages 2014

[53] Z Gao K Yoshimoto and S Ohmori ldquoApplication ofAHPDEA to facility layout selectionrdquo in Proceedings of theThird International Joint Conference on Computational Scienceamp Optimization IEEE Computer Society 2010

[54] A H P Morice I A Siegler and B G Bardy ldquoAction-perception patterns in virtual ball bouncing Combating systemlatency and tracking functional validityrdquo Journal ofNeuroscienceMethods vol 169 no 1 pp 255ndash266 2008

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of


Mathematical Problems in Engineering

Applied MathematicsJournal of


Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of



Engineering Mathematics

International Journal of


Operations ResearchAdvances in

Journal of


Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences


Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018


Decision SciencesAdvances in


AnalysisInternational Journal of


Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom


method due to the limitations of the times but it relied onthe subjective experience of human beings [4] From the1940s to the 1960s the issue of facility layout entered a periodof rapid development During this period researchers usedsystem theory cybernetics and other quantitative methodsto optimize facility layouts and achieved good results [5]Since the 1970s with the popularity of computers and theemergence of various intelligent algorithms the problem offacility layout optimization has once again entered anotherperiod of vigorous development [6] The use of intelligentalgorithms to optimize the layout of facilities has completelychanged past business processes and intelligent algorithmsare widely used to successfully solve the problem of facilitylayout design [7ndash17]

The optimization of the layout of ship cabin equipmentas a branch of facility layout optimization has also been ahot topic in recent years Over time many scholars haveconducted a lot of research and achieved certain results Atpresent in terms of the layout problem of cabin equipmentmost research has combined the layout problem with theintelligent algorithm optimized the algorithm and yielded abetter solution AI Olcer introduced amultiobjective geneticalgorithm and fuzzy multiattribute set in the layout designof a ro-ro shiprsquos cabin which improved the efficiency of thecabin layout [18] Y Yang proposed a facility layout optimiza-tion design method for cabin maintainability as a method toimprove the efficiency and quality of maintainability designand built a mathematical model for maintainability layoutoptimization An improved particle swarm optimizationalgorithm was also used to improve calculation efficiencyand accuracy and effectively solve themultipeak optimizationproblem [19] Wang Y L generated the initial value of thegenetic algorithm based on the energy method and thenused the improved genetic algorithm to design the shiprsquoscabin layout which solved the problem of layout evaluationto some extent [20 21] Literature [22ndash25] elaborates theapplication of genetic algorithm in equipment layout prob-lem in detail and proves that genetic algorithm has goodrobustness to this kind of problem through examples ZhengX L proposed a method based on game theory to promotethe multidisciplinary decision-making process involved inship cabin layout design and at the same time developed anoncooperative game strategy and determined the amountof equipment and furniture required for the correspondinglocation in order to achieve the highest possible performancein the cabin [26] The above research only used intelligentalgorithms to solve the problem yet it did not consider howto generate a more effective initial solution did not evaluatethe obtained solution and cannot determine whether thesolution was the most feasible solution

In 1961 Richard Joseph first proposed a systematic facilitylayout method [27] which enabled the plantrsquos designersto perform quantitative analyses based on objective datafrom previous qualitative analyses that had been based onsubjective experience The application of this method is notlimited to the industrial field but also used for various typesof facility layout optimization (among others) and it is stillused improved and innovated upon today The SystematicLayout Planning Method (SLP) is generally applied to the

layout design of workshop facilities [28ndash30] and researchon applying the SLP method to the layout of ship facilitiesis relatively rare Hu Yao et al optimized the complexmultiobjective combination optimization problem for theposition layout of the interior cabin of a volumetric ship andcomprehensively applied the SLP method to the heuristicmethod and the genetic algorithm (GA) in the intelligentalgorithm to solve the problem and optimize the solution[31] However it only lays out the location of differentcompartments and does not optimize the design of the cabininterior Moreover simply using the SLP method to solvethe layout of the cabin equipment only meets the adjacencyand logistics requirements of the equipment and does notguarantee that the scheme meets ship stability and otherperformance requirements

The Analytic Hierarchy Process (AHP) is a hierarchicalweighted decision analysis method proposed by the Pitts-burgh University Operations Researcher Professor Saaty inthe early 1970s when he was studying the subject of the USDepartment of Defense [32] Z Gao et al achieved goodresults by using the Analytic Hierarchy Process (AHP) andthe Pure Output Data Envelopment Analysis (DEA) method[33] When Wang Y L et al laid out the cabin of the shipintelligently the AHP method was used to evaluate severalschemes obtained by using the algorithmand finally theoptimal solution was selected [21] The above studies showthat it is feasible and effective to apply the AHP method toequipment layout optimization but there is still little researchinto the application of this method to the layout of ship cabinequipment

Based on the above research this paper creatively appliesthe SLP method to the cabin equipment layout optimizationproblem establishes a corresponding mathematical modeland uses a genetic algorithm to solve it Finally the AHPmethod is used to evaluate the multiple sets of solutionsobtained The rest of this paper is structured as followsSection 2 uses the SLP method to analyze the adjacencyrequirements of the cabin equipment and the personnelcirculation efficiency requirements and on this basis deter-mines a comprehensive interrelationship between differ-ent equipment thus determining the positional constraintsacross various pieces of equipment and using this as themethod of evaluation Section 3 comprehensively considersfactors such as cabin weight distance and stability and con-structs corresponding mathematical models and constraintsSection 4 details the operator design of the genetic algorithmand generates a cabin layout based on the genetic algorithmSection 5 is based on the AHP method and evaluates thescheme generated in the Section 4 to determine a better lay-out scheme Section 6 concludes the research and identifiesfuture work

2 SLP Analysis of SimplifiedLayout of Equipment

SLP is the earliest method used in the design of factory andworkshop equipment layout It uses the relationship betweenlogistics and nonlogistics of equipment as the main line and






Table 2 Device type



No 1 2 3 4 5 6 7 8 91 06 0 0 0 0 0 0 02 04 04 0 02 04 0 03 08 0 02 0 0 04 0 0 08 0 05 06 0 06 066 0 0 07 0 08 09








No 1 2 3 4 5 6 7 8 91 08 0 0 0 0 0 0 02 06 06 0 02 04 0 03 08 0 0 02 0 04 0 0 08 0 05 06 0 06 066 0 0 07 0 08 09








119891119895119896 = 1199081119886119895119896 + 1199082119887119895119896 (1)





No 1 2 3 4 5 6 7 8 91 064 0 0 0 0 0 0 02 044 044 0 02 04 0 03 079 0 016 004 0 04 0 0 08 0 05 06 0 06 066 0 0 07 0 08 09









1198911 (119909) = 8sum119895=1

9sum119896=119895+1

119860 times 119863 (119909) (2)




119863119895119896 = 10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 + 10038161003816100381610038161003816119910119895 minus 11991011989610038161003816100381610038161003816 (3)


1198912 (119909) = 8sum119895=1

9sum119896=119895+1

119861 times 119863 (119909) (4)


1 2 3 4 5

12 345

Single row layout

Multi-line layout

1

6

2

3

4 5

1 2 3

4 5

6

Site layout

1

6

9 82

3

4 5

7

1 2 3

6

98

4 5

7

Ring layout

1

3

2

4

6

5

8 7

1 2

6 5

8

7

3

4

U-shaped layout

14

3

2

5 01

86 7

3 2

87 9

9

10

5

4 1

6



a

b


Y

0 X

ℎj0

mj

xk

mk

ℎgk

mg

yj

s

s0

l

Δ Ｄ





119863119895119896 = 10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 + 10038161003816100381610038161003816119910119895 minus 11991011989610038161003816100381610038161003816 (5)


1198913 (119909) =100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

119898119895 (119909119895 minus 1198972)10038161003816100381610038161003816100381610038161003816100381610038161003816 (6)


1198914 (119909) =100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

(119909119895 minus 1198972)10038161003816100381610038161003816100381610038161003816100381610038161003816 (7)


119865 (119909) = min4sum119890=1

119891119890 (119909) (8)


33 Constraint


10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 ge [(119897119895 + 119897119896)2 + ℎ119895119896]119911119895119901119911119896119901119895 119896 = 1 2 9

(9)


119909119895 = 119909119896 + (119897119896 + 119897119895)2 + ℎ119896119895 + Δ 119895= ℎ1198960 + Δ 119896 + (119897119895 + 2119897119896)2 + ℎ119896119895 + Δ 119895

(10)


119910119895 = (119896 minus 1) 119904 + 1199040if 119911119895119901 = 1 119895 = 1 2 9 119901 = 1 2 119903

119911119895119901 = 1 119863119890V119894119888119890 119895 119900119899 119897119894119899119890 1199010 119900119905ℎ119890119903

119895 = 1 2 9 119901 = 1 2 119903(11)



119903sum119901=1

119911119895119901 = 1 119894 = 1 2 9 119909119895 119910119895 ge 0 Δ 119895 ge 0 (12)


sum0le119909le1199082

119872119895 asymp sum1199082lt119909le119908

119872119896 (13)


119865 (119909) = 8sum119895=1

9sum119896=119895+1

119860 times 119863 (119909) + 8sum119895=1

9sum119896=119895+1

119861 times 119863 (119909) +100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

119872119895 (119909119895 minus 1198712)10038161003816100381610038161003816100381610038161003816100381610038161003816 +

100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

(1199099 minus 1198712)10038161003816100381610038161003816100381610038161003816100381610038161003816

119904119905

10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 ge [(119897119895 + 119897119896)2 + ℎ119895119896]119911119895119901119911119896119901 119894 119895 = 1 2 9119903sum119901=1

119911119895119901 = 1 119895 = 1 2 9 119909119895 119910119895 ge 0 Δ 119895 ge 0 119895 = 1 2 9sum0le119909le1199082

119872119895 asymp sum1199082lt119909le119908

119872119896

(14)







[ 1198981 1198982 119898119899 Δ 1 Δ 2 Δ 119899] (15)





119875119896 = 0 1199040 + (119898 minus 1) 119904 le 119867119879 119900119905ℎ119890119903 (16)



119891119894119905 (V119896) = 1(119865 + 119875119896) (17)





Father 1 (

Father 2 (




Child 1 (

Child 2 (

a aaaaaaaa

a aaaaaaaa

)7 82179556

)6 43431982



NO Layout Scheme1 [85][21][346][79]2 [7][43][296][185]3 [9][785][346][21]4 [8][59][347][621]5 [9][127][534][86]6 [96][851][732]


Child 1 (

Child 2 (

a aaaaaaaa

a aaaaaaaa

)3 82179546

)6 75431982






Start


Initial population



Yes

No

Select

Cross

Variation

Output result

End









Meet theconditions

Meet the conditions

Determineweight

No

No

Yes

Yes




52 Optimal Process




scheme



Z3Cabin safety

Criteria layer Z

Solution layer P




119868119894119895 = 1119868119895119894 (18)




119885119894119895 = [[[[[

1 12 152 1 135 3 1]]]]]

(19)





Feature vector

0122 0272 0357 02520230 0228 0242 02560644 0184 0103 0144

0109 0103 00820130 0105 01710078 0091 0096


1198751119894119895 =

[[[[[[[[[[[[[[[[

1 1 2 3 2 41 1 2 3 1 212 12 1 2 3 213 13 12 1 2 112 1 13 12 1 314 12 12 1 13 1

]]]]]]]]]]]]]]]]

1198752119894119895 =

[[[[[[[[[[[[[[[[[

1 2 4 3 5 212 1 2 3 4 214 12 1 12 1 213 13 12 1 2 115 14 1 12 1 312 12 12 1 13 1

]]]]]]]]]]]]]]]]]

1198753119894119895 =

[[[[[[[[[[[[[[[[

1 1 3 3 1 21 1 2 3 2 213 12 1 2 1 213 13 12 1 12 11 12 1 2 1 212 12 12 1 12 1

]]]]]]]]]]]]]]]]

(20)



n 1 2 3 4 5 6 7 8 9RI 0 0 058 089 112 124 132 141 145




119862119868 = 120582max minus 119899119899 minus 1 (21)


119862119877 = 119862119868119877119868 (22)





119882119894 = 119898sum119895=1

119887119895120596119894 (119894 = 1 2 119899) (23)






Judgment matrix 1198851 1198852 1198853119862119868119895 0098 0070 0029119887119895 0122 0230 0644119877119868119895 1240 1240 1240




119862119877 = (sum119898119895=1 119862119868119895119887119895)(sum119898119895=1 119877119868119895119887119895) (24)




6 Conclusions



Data Availability






Acknowledgments



References































































Journal of












Journal of







Volume 2018






Volume 2018













Table 2 Device type



No 1 2 3 4 5 6 7 8 91 06 0 0 0 0 0 0 02 04 04 0 02 04 0 03 08 0 02 0 0 04 0 0 08 0 05 06 0 06 066 0 0 07 0 08 09








No 1 2 3 4 5 6 7 8 91 08 0 0 0 0 0 0 02 06 06 0 02 04 0 03 08 0 0 02 0 04 0 0 08 0 05 06 0 06 066 0 0 07 0 08 09








119891119895119896 = 1199081119886119895119896 + 1199082119887119895119896 (1)





No 1 2 3 4 5 6 7 8 91 064 0 0 0 0 0 0 02 044 044 0 02 04 0 03 079 0 016 004 0 04 0 0 08 0 05 06 0 06 066 0 0 07 0 08 09









1198911 (119909) = 8sum119895=1

9sum119896=119895+1

119860 times 119863 (119909) (2)




119863119895119896 = 10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 + 10038161003816100381610038161003816119910119895 minus 11991011989610038161003816100381610038161003816 (3)


1198912 (119909) = 8sum119895=1

9sum119896=119895+1

119861 times 119863 (119909) (4)


1 2 3 4 5

12 345

Single row layout

Multi-line layout

1

6

2

3

4 5

1 2 3

4 5

6

Site layout

1

6

9 82

3

4 5

7

1 2 3

6

98

4 5

7

Ring layout

1

3

2

4

6

5

8 7

1 2

6 5

8

7

3

4

U-shaped layout

14

3

2

5 01

86 7

3 2

87 9

9

10

5

4 1

6



a

b


Y

0 X

ℎj0

mj

xk

mk

ℎgk

mg

yj

s

s0

l

Δ Ｄ





119863119895119896 = 10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 + 10038161003816100381610038161003816119910119895 minus 11991011989610038161003816100381610038161003816 (5)


1198913 (119909) =100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

119898119895 (119909119895 minus 1198972)10038161003816100381610038161003816100381610038161003816100381610038161003816 (6)


1198914 (119909) =100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

(119909119895 minus 1198972)10038161003816100381610038161003816100381610038161003816100381610038161003816 (7)


119865 (119909) = min4sum119890=1

119891119890 (119909) (8)


33 Constraint


10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 ge [(119897119895 + 119897119896)2 + ℎ119895119896]119911119895119901119911119896119901119895 119896 = 1 2 9

(9)


119909119895 = 119909119896 + (119897119896 + 119897119895)2 + ℎ119896119895 + Δ 119895= ℎ1198960 + Δ 119896 + (119897119895 + 2119897119896)2 + ℎ119896119895 + Δ 119895

(10)


119910119895 = (119896 minus 1) 119904 + 1199040if 119911119895119901 = 1 119895 = 1 2 9 119901 = 1 2 119903

119911119895119901 = 1 119863119890V119894119888119890 119895 119900119899 119897119894119899119890 1199010 119900119905ℎ119890119903

119895 = 1 2 9 119901 = 1 2 119903(11)



119903sum119901=1

119911119895119901 = 1 119894 = 1 2 9 119909119895 119910119895 ge 0 Δ 119895 ge 0 (12)


sum0le119909le1199082

119872119895 asymp sum1199082lt119909le119908

119872119896 (13)


119865 (119909) = 8sum119895=1

9sum119896=119895+1

119860 times 119863 (119909) + 8sum119895=1

9sum119896=119895+1

119861 times 119863 (119909) +100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

119872119895 (119909119895 minus 1198712)10038161003816100381610038161003816100381610038161003816100381610038161003816 +

100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

(1199099 minus 1198712)10038161003816100381610038161003816100381610038161003816100381610038161003816

119904119905

10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 ge [(119897119895 + 119897119896)2 + ℎ119895119896]119911119895119901119911119896119901 119894 119895 = 1 2 9119903sum119901=1

119911119895119901 = 1 119895 = 1 2 9 119909119895 119910119895 ge 0 Δ 119895 ge 0 119895 = 1 2 9sum0le119909le1199082

119872119895 asymp sum1199082lt119909le119908

119872119896

(14)







[ 1198981 1198982 119898119899 Δ 1 Δ 2 Δ 119899] (15)





119875119896 = 0 1199040 + (119898 minus 1) 119904 le 119867119879 119900119905ℎ119890119903 (16)



119891119894119905 (V119896) = 1(119865 + 119875119896) (17)





Father 1 (

Father 2 (




Child 1 (

Child 2 (

a aaaaaaaa

a aaaaaaaa

)7 82179556

)6 43431982



NO Layout Scheme1 [85][21][346][79]2 [7][43][296][185]3 [9][785][346][21]4 [8][59][347][621]5 [9][127][534][86]6 [96][851][732]


Child 1 (

Child 2 (

a aaaaaaaa

a aaaaaaaa

)3 82179546

)6 75431982






Start


Initial population



Yes

No

Select

Cross

Variation

Output result

End









Meet theconditions

Meet the conditions

Determineweight

No

No

Yes

Yes




52 Optimal Process




scheme



Z3Cabin safety

Criteria layer Z

Solution layer P




119868119894119895 = 1119868119895119894 (18)




119885119894119895 = [[[[[

1 12 152 1 135 3 1]]]]]

(19)





Feature vector

0122 0272 0357 02520230 0228 0242 02560644 0184 0103 0144

0109 0103 00820130 0105 01710078 0091 0096


1198751119894119895 =

[[[[[[[[[[[[[[[[

1 1 2 3 2 41 1 2 3 1 212 12 1 2 3 213 13 12 1 2 112 1 13 12 1 314 12 12 1 13 1

]]]]]]]]]]]]]]]]

1198752119894119895 =

[[[[[[[[[[[[[[[[[

1 2 4 3 5 212 1 2 3 4 214 12 1 12 1 213 13 12 1 2 115 14 1 12 1 312 12 12 1 13 1

]]]]]]]]]]]]]]]]]

1198753119894119895 =

[[[[[[[[[[[[[[[[

1 1 3 3 1 21 1 2 3 2 213 12 1 2 1 213 13 12 1 12 11 12 1 2 1 212 12 12 1 12 1

]]]]]]]]]]]]]]]]

(20)



n 1 2 3 4 5 6 7 8 9RI 0 0 058 089 112 124 132 141 145




119862119868 = 120582max minus 119899119899 minus 1 (21)


119862119877 = 119862119868119877119868 (22)





119882119894 = 119898sum119895=1

119887119895120596119894 (119894 = 1 2 119899) (23)






Judgment matrix 1198851 1198852 1198853119862119868119895 0098 0070 0029119887119895 0122 0230 0644119877119868119895 1240 1240 1240




119862119877 = (sum119898119895=1 119862119868119895119887119895)(sum119898119895=1 119877119868119895119887119895) (24)




6 Conclusions



Data Availability






Acknowledgments



References































































Journal of












Journal of







Volume 2018






Volume 2018












No 1 2 3 4 5 6 7 8 91 08 0 0 0 0 0 0 02 06 06 0 02 04 0 03 08 0 0 02 0 04 0 0 08 0 05 06 0 06 066 0 0 07 0 08 09








119891119895119896 = 1199081119886119895119896 + 1199082119887119895119896 (1)





No 1 2 3 4 5 6 7 8 91 064 0 0 0 0 0 0 02 044 044 0 02 04 0 03 079 0 016 004 0 04 0 0 08 0 05 06 0 06 066 0 0 07 0 08 09









1198911 (119909) = 8sum119895=1

9sum119896=119895+1

119860 times 119863 (119909) (2)




119863119895119896 = 10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 + 10038161003816100381610038161003816119910119895 minus 11991011989610038161003816100381610038161003816 (3)


1198912 (119909) = 8sum119895=1

9sum119896=119895+1

119861 times 119863 (119909) (4)


1 2 3 4 5

12 345

Single row layout

Multi-line layout

1

6

2

3

4 5

1 2 3

4 5

6

Site layout

1

6

9 82

3

4 5

7

1 2 3

6

98

4 5

7

Ring layout

1

3

2

4

6

5

8 7

1 2

6 5

8

7

3

4

U-shaped layout

14

3

2

5 01

86 7

3 2

87 9

9

10

5

4 1

6



a

b


Y

0 X

ℎj0

mj

xk

mk

ℎgk

mg

yj

s

s0

l

Δ Ｄ





119863119895119896 = 10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 + 10038161003816100381610038161003816119910119895 minus 11991011989610038161003816100381610038161003816 (5)


1198913 (119909) =100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

119898119895 (119909119895 minus 1198972)10038161003816100381610038161003816100381610038161003816100381610038161003816 (6)


1198914 (119909) =100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

(119909119895 minus 1198972)10038161003816100381610038161003816100381610038161003816100381610038161003816 (7)


119865 (119909) = min4sum119890=1

119891119890 (119909) (8)


33 Constraint


10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 ge [(119897119895 + 119897119896)2 + ℎ119895119896]119911119895119901119911119896119901119895 119896 = 1 2 9

(9)


119909119895 = 119909119896 + (119897119896 + 119897119895)2 + ℎ119896119895 + Δ 119895= ℎ1198960 + Δ 119896 + (119897119895 + 2119897119896)2 + ℎ119896119895 + Δ 119895

(10)


119910119895 = (119896 minus 1) 119904 + 1199040if 119911119895119901 = 1 119895 = 1 2 9 119901 = 1 2 119903

119911119895119901 = 1 119863119890V119894119888119890 119895 119900119899 119897119894119899119890 1199010 119900119905ℎ119890119903

119895 = 1 2 9 119901 = 1 2 119903(11)



119903sum119901=1

119911119895119901 = 1 119894 = 1 2 9 119909119895 119910119895 ge 0 Δ 119895 ge 0 (12)


sum0le119909le1199082

119872119895 asymp sum1199082lt119909le119908

119872119896 (13)


119865 (119909) = 8sum119895=1

9sum119896=119895+1

119860 times 119863 (119909) + 8sum119895=1

9sum119896=119895+1

119861 times 119863 (119909) +100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

119872119895 (119909119895 minus 1198712)10038161003816100381610038161003816100381610038161003816100381610038161003816 +

100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

(1199099 minus 1198712)10038161003816100381610038161003816100381610038161003816100381610038161003816

119904119905

10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 ge [(119897119895 + 119897119896)2 + ℎ119895119896]119911119895119901119911119896119901 119894 119895 = 1 2 9119903sum119901=1

119911119895119901 = 1 119895 = 1 2 9 119909119895 119910119895 ge 0 Δ 119895 ge 0 119895 = 1 2 9sum0le119909le1199082

119872119895 asymp sum1199082lt119909le119908

119872119896

(14)







[ 1198981 1198982 119898119899 Δ 1 Δ 2 Δ 119899] (15)





119875119896 = 0 1199040 + (119898 minus 1) 119904 le 119867119879 119900119905ℎ119890119903 (16)



119891119894119905 (V119896) = 1(119865 + 119875119896) (17)





Father 1 (

Father 2 (




Child 1 (

Child 2 (

a aaaaaaaa

a aaaaaaaa

)7 82179556

)6 43431982



NO Layout Scheme1 [85][21][346][79]2 [7][43][296][185]3 [9][785][346][21]4 [8][59][347][621]5 [9][127][534][86]6 [96][851][732]


Child 1 (

Child 2 (

a aaaaaaaa

a aaaaaaaa

)3 82179546

)6 75431982






Start


Initial population



Yes

No

Select

Cross

Variation

Output result

End









Meet theconditions

Meet the conditions

Determineweight

No

No

Yes

Yes




52 Optimal Process




scheme



Z3Cabin safety

Criteria layer Z

Solution layer P




119868119894119895 = 1119868119895119894 (18)




119885119894119895 = [[[[[

1 12 152 1 135 3 1]]]]]

(19)





Feature vector

0122 0272 0357 02520230 0228 0242 02560644 0184 0103 0144

0109 0103 00820130 0105 01710078 0091 0096


1198751119894119895 =

[[[[[[[[[[[[[[[[

1 1 2 3 2 41 1 2 3 1 212 12 1 2 3 213 13 12 1 2 112 1 13 12 1 314 12 12 1 13 1

]]]]]]]]]]]]]]]]

1198752119894119895 =

[[[[[[[[[[[[[[[[[

1 2 4 3 5 212 1 2 3 4 214 12 1 12 1 213 13 12 1 2 115 14 1 12 1 312 12 12 1 13 1

]]]]]]]]]]]]]]]]]

1198753119894119895 =

[[[[[[[[[[[[[[[[

1 1 3 3 1 21 1 2 3 2 213 12 1 2 1 213 13 12 1 12 11 12 1 2 1 212 12 12 1 12 1

]]]]]]]]]]]]]]]]

(20)



n 1 2 3 4 5 6 7 8 9RI 0 0 058 089 112 124 132 141 145




119862119868 = 120582max minus 119899119899 minus 1 (21)


119862119877 = 119862119868119877119868 (22)





119882119894 = 119898sum119895=1

119887119895120596119894 (119894 = 1 2 119899) (23)






Judgment matrix 1198851 1198852 1198853119862119868119895 0098 0070 0029119887119895 0122 0230 0644119877119868119895 1240 1240 1240




119862119877 = (sum119898119895=1 119862119868119895119887119895)(sum119898119895=1 119877119868119895119887119895) (24)




6 Conclusions



Data Availability






Acknowledgments



References































































Journal of












Journal of







Volume 2018






Volume 2018










No 1 2 3 4 5 6 7 8 91 064 0 0 0 0 0 0 02 044 044 0 02 04 0 03 079 0 016 004 0 04 0 0 08 0 05 06 0 06 066 0 0 07 0 08 09









1198911 (119909) = 8sum119895=1

9sum119896=119895+1

119860 times 119863 (119909) (2)




119863119895119896 = 10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 + 10038161003816100381610038161003816119910119895 minus 11991011989610038161003816100381610038161003816 (3)


1198912 (119909) = 8sum119895=1

9sum119896=119895+1

119861 times 119863 (119909) (4)


1 2 3 4 5

12 345

Single row layout

Multi-line layout

1

6

2

3

4 5

1 2 3

4 5

6

Site layout

1

6

9 82

3

4 5

7

1 2 3

6

98

4 5

7

Ring layout

1

3

2

4

6

5

8 7

1 2

6 5

8

7

3

4

U-shaped layout

14

3

2

5 01

86 7

3 2

87 9

9

10

5

4 1

6



a

b


Y

0 X

ℎj0

mj

xk

mk

ℎgk

mg

yj

s

s0

l

Δ Ｄ





119863119895119896 = 10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 + 10038161003816100381610038161003816119910119895 minus 11991011989610038161003816100381610038161003816 (5)


1198913 (119909) =100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

119898119895 (119909119895 minus 1198972)10038161003816100381610038161003816100381610038161003816100381610038161003816 (6)


1198914 (119909) =100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

(119909119895 minus 1198972)10038161003816100381610038161003816100381610038161003816100381610038161003816 (7)


119865 (119909) = min4sum119890=1

119891119890 (119909) (8)


33 Constraint


10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 ge [(119897119895 + 119897119896)2 + ℎ119895119896]119911119895119901119911119896119901119895 119896 = 1 2 9

(9)


119909119895 = 119909119896 + (119897119896 + 119897119895)2 + ℎ119896119895 + Δ 119895= ℎ1198960 + Δ 119896 + (119897119895 + 2119897119896)2 + ℎ119896119895 + Δ 119895

(10)


119910119895 = (119896 minus 1) 119904 + 1199040if 119911119895119901 = 1 119895 = 1 2 9 119901 = 1 2 119903

119911119895119901 = 1 119863119890V119894119888119890 119895 119900119899 119897119894119899119890 1199010 119900119905ℎ119890119903

119895 = 1 2 9 119901 = 1 2 119903(11)



119903sum119901=1

119911119895119901 = 1 119894 = 1 2 9 119909119895 119910119895 ge 0 Δ 119895 ge 0 (12)


sum0le119909le1199082

119872119895 asymp sum1199082lt119909le119908

119872119896 (13)


119865 (119909) = 8sum119895=1

9sum119896=119895+1

119860 times 119863 (119909) + 8sum119895=1

9sum119896=119895+1

119861 times 119863 (119909) +100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

119872119895 (119909119895 minus 1198712)10038161003816100381610038161003816100381610038161003816100381610038161003816 +

100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

(1199099 minus 1198712)10038161003816100381610038161003816100381610038161003816100381610038161003816

119904119905

10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 ge [(119897119895 + 119897119896)2 + ℎ119895119896]119911119895119901119911119896119901 119894 119895 = 1 2 9119903sum119901=1

119911119895119901 = 1 119895 = 1 2 9 119909119895 119910119895 ge 0 Δ 119895 ge 0 119895 = 1 2 9sum0le119909le1199082

119872119895 asymp sum1199082lt119909le119908

119872119896

(14)







[ 1198981 1198982 119898119899 Δ 1 Δ 2 Δ 119899] (15)





119875119896 = 0 1199040 + (119898 minus 1) 119904 le 119867119879 119900119905ℎ119890119903 (16)



119891119894119905 (V119896) = 1(119865 + 119875119896) (17)





Father 1 (

Father 2 (




Child 1 (

Child 2 (

a aaaaaaaa

a aaaaaaaa

)7 82179556

)6 43431982



NO Layout Scheme1 [85][21][346][79]2 [7][43][296][185]3 [9][785][346][21]4 [8][59][347][621]5 [9][127][534][86]6 [96][851][732]


Child 1 (

Child 2 (

a aaaaaaaa

a aaaaaaaa

)3 82179546

)6 75431982






Start


Initial population



Yes

No

Select

Cross

Variation

Output result

End









Meet theconditions

Meet the conditions

Determineweight

No

No

Yes

Yes




52 Optimal Process




scheme



Z3Cabin safety

Criteria layer Z

Solution layer P




119868119894119895 = 1119868119895119894 (18)




119885119894119895 = [[[[[

1 12 152 1 135 3 1]]]]]

(19)





Feature vector

0122 0272 0357 02520230 0228 0242 02560644 0184 0103 0144

0109 0103 00820130 0105 01710078 0091 0096


1198751119894119895 =

[[[[[[[[[[[[[[[[

1 1 2 3 2 41 1 2 3 1 212 12 1 2 3 213 13 12 1 2 112 1 13 12 1 314 12 12 1 13 1

]]]]]]]]]]]]]]]]

1198752119894119895 =

[[[[[[[[[[[[[[[[[

1 2 4 3 5 212 1 2 3 4 214 12 1 12 1 213 13 12 1 2 115 14 1 12 1 312 12 12 1 13 1

]]]]]]]]]]]]]]]]]

1198753119894119895 =

[[[[[[[[[[[[[[[[

1 1 3 3 1 21 1 2 3 2 213 12 1 2 1 213 13 12 1 12 11 12 1 2 1 212 12 12 1 12 1

]]]]]]]]]]]]]]]]

(20)



n 1 2 3 4 5 6 7 8 9RI 0 0 058 089 112 124 132 141 145




119862119868 = 120582max minus 119899119899 minus 1 (21)


119862119877 = 119862119868119877119868 (22)





119882119894 = 119898sum119895=1

119887119895120596119894 (119894 = 1 2 119899) (23)






Judgment matrix 1198851 1198852 1198853119862119868119895 0098 0070 0029119887119895 0122 0230 0644119877119868119895 1240 1240 1240




119862119877 = (sum119898119895=1 119862119868119895119887119895)(sum119898119895=1 119877119868119895119887119895) (24)




6 Conclusions



Data Availability






Acknowledgments



References































































Journal of












Journal of







Volume 2018






Volume 2018









1 2 3 4 5

12 345

Single row layout

Multi-line layout

1

6

2

3

4 5

1 2 3

4 5

6

Site layout

1

6

9 82

3

4 5

7

1 2 3

6

98

4 5

7

Ring layout

1

3

2

4

6

5

8 7

1 2

6 5

8

7

3

4

U-shaped layout

14

3

2

5 01

86 7

3 2

87 9

9

10

5

4 1

6



a

b


Y

0 X

ℎj0

mj

xk

mk

ℎgk

mg

yj

s

s0

l

Δ Ｄ





119863119895119896 = 10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 + 10038161003816100381610038161003816119910119895 minus 11991011989610038161003816100381610038161003816 (5)


1198913 (119909) =100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

119898119895 (119909119895 minus 1198972)10038161003816100381610038161003816100381610038161003816100381610038161003816 (6)


1198914 (119909) =100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

(119909119895 minus 1198972)10038161003816100381610038161003816100381610038161003816100381610038161003816 (7)


119865 (119909) = min4sum119890=1

119891119890 (119909) (8)


33 Constraint


10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 ge [(119897119895 + 119897119896)2 + ℎ119895119896]119911119895119901119911119896119901119895 119896 = 1 2 9

(9)


119909119895 = 119909119896 + (119897119896 + 119897119895)2 + ℎ119896119895 + Δ 119895= ℎ1198960 + Δ 119896 + (119897119895 + 2119897119896)2 + ℎ119896119895 + Δ 119895

(10)


119910119895 = (119896 minus 1) 119904 + 1199040if 119911119895119901 = 1 119895 = 1 2 9 119901 = 1 2 119903

119911119895119901 = 1 119863119890V119894119888119890 119895 119900119899 119897119894119899119890 1199010 119900119905ℎ119890119903

119895 = 1 2 9 119901 = 1 2 119903(11)



119903sum119901=1

119911119895119901 = 1 119894 = 1 2 9 119909119895 119910119895 ge 0 Δ 119895 ge 0 (12)


sum0le119909le1199082

119872119895 asymp sum1199082lt119909le119908

119872119896 (13)


119865 (119909) = 8sum119895=1

9sum119896=119895+1

119860 times 119863 (119909) + 8sum119895=1

9sum119896=119895+1

119861 times 119863 (119909) +100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

119872119895 (119909119895 minus 1198712)10038161003816100381610038161003816100381610038161003816100381610038161003816 +

100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

(1199099 minus 1198712)10038161003816100381610038161003816100381610038161003816100381610038161003816

119904119905

10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 ge [(119897119895 + 119897119896)2 + ℎ119895119896]119911119895119901119911119896119901 119894 119895 = 1 2 9119903sum119901=1

119911119895119901 = 1 119895 = 1 2 9 119909119895 119910119895 ge 0 Δ 119895 ge 0 119895 = 1 2 9sum0le119909le1199082

119872119895 asymp sum1199082lt119909le119908

119872119896

(14)







[ 1198981 1198982 119898119899 Δ 1 Δ 2 Δ 119899] (15)





119875119896 = 0 1199040 + (119898 minus 1) 119904 le 119867119879 119900119905ℎ119890119903 (16)



119891119894119905 (V119896) = 1(119865 + 119875119896) (17)





Father 1 (

Father 2 (




Child 1 (

Child 2 (

a aaaaaaaa

a aaaaaaaa

)7 82179556

)6 43431982



NO Layout Scheme1 [85][21][346][79]2 [7][43][296][185]3 [9][785][346][21]4 [8][59][347][621]5 [9][127][534][86]6 [96][851][732]


Child 1 (

Child 2 (

a aaaaaaaa

a aaaaaaaa

)3 82179546

)6 75431982






Start


Initial population



Yes

No

Select

Cross

Variation

Output result

End









Meet theconditions

Meet the conditions

Determineweight

No

No

Yes

Yes




52 Optimal Process




scheme



Z3Cabin safety

Criteria layer Z

Solution layer P




119868119894119895 = 1119868119895119894 (18)




119885119894119895 = [[[[[

1 12 152 1 135 3 1]]]]]

(19)





Feature vector

0122 0272 0357 02520230 0228 0242 02560644 0184 0103 0144

0109 0103 00820130 0105 01710078 0091 0096


1198751119894119895 =

[[[[[[[[[[[[[[[[

1 1 2 3 2 41 1 2 3 1 212 12 1 2 3 213 13 12 1 2 112 1 13 12 1 314 12 12 1 13 1

]]]]]]]]]]]]]]]]

1198752119894119895 =

[[[[[[[[[[[[[[[[[

1 2 4 3 5 212 1 2 3 4 214 12 1 12 1 213 13 12 1 2 115 14 1 12 1 312 12 12 1 13 1

]]]]]]]]]]]]]]]]]

1198753119894119895 =

[[[[[[[[[[[[[[[[

1 1 3 3 1 21 1 2 3 2 213 12 1 2 1 213 13 12 1 12 11 12 1 2 1 212 12 12 1 12 1

]]]]]]]]]]]]]]]]

(20)



n 1 2 3 4 5 6 7 8 9RI 0 0 058 089 112 124 132 141 145




119862119868 = 120582max minus 119899119899 minus 1 (21)


119862119877 = 119862119868119877119868 (22)





119882119894 = 119898sum119895=1

119887119895120596119894 (119894 = 1 2 119899) (23)






Judgment matrix 1198851 1198852 1198853119862119868119895 0098 0070 0029119887119895 0122 0230 0644119877119868119895 1240 1240 1240




119862119877 = (sum119898119895=1 119862119868119895119887119895)(sum119898119895=1 119877119868119895119887119895) (24)




6 Conclusions



Data Availability






Acknowledgments



References































































Journal of












Journal of







Volume 2018






Volume 2018









33 Constraint


10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 ge [(119897119895 + 119897119896)2 + ℎ119895119896]119911119895119901119911119896119901119895 119896 = 1 2 9

(9)


119909119895 = 119909119896 + (119897119896 + 119897119895)2 + ℎ119896119895 + Δ 119895= ℎ1198960 + Δ 119896 + (119897119895 + 2119897119896)2 + ℎ119896119895 + Δ 119895

(10)


119910119895 = (119896 minus 1) 119904 + 1199040if 119911119895119901 = 1 119895 = 1 2 9 119901 = 1 2 119903

119911119895119901 = 1 119863119890V119894119888119890 119895 119900119899 119897119894119899119890 1199010 119900119905ℎ119890119903

119895 = 1 2 9 119901 = 1 2 119903(11)



119903sum119901=1

119911119895119901 = 1 119894 = 1 2 9 119909119895 119910119895 ge 0 Δ 119895 ge 0 (12)


sum0le119909le1199082

119872119895 asymp sum1199082lt119909le119908

119872119896 (13)


119865 (119909) = 8sum119895=1

9sum119896=119895+1

119860 times 119863 (119909) + 8sum119895=1

9sum119896=119895+1

119861 times 119863 (119909) +100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

119872119895 (119909119895 minus 1198712)10038161003816100381610038161003816100381610038161003816100381610038161003816 +

100381610038161003816100381610038161003816100381610038161003816100381610038169sum119895=1

(1199099 minus 1198712)10038161003816100381610038161003816100381610038161003816100381610038161003816

119904119905

10038161003816100381610038161003816119909119895 minus 11990911989610038161003816100381610038161003816 ge [(119897119895 + 119897119896)2 + ℎ119895119896]119911119895119901119911119896119901 119894 119895 = 1 2 9119903sum119901=1

119911119895119901 = 1 119895 = 1 2 9 119909119895 119910119895 ge 0 Δ 119895 ge 0 119895 = 1 2 9sum0le119909le1199082

119872119895 asymp sum1199082lt119909le119908

119872119896

(14)







[ 1198981 1198982 119898119899 Δ 1 Δ 2 Δ 119899] (15)





119875119896 = 0 1199040 + (119898 minus 1) 119904 le 119867119879 119900119905ℎ119890119903 (16)



119891119894119905 (V119896) = 1(119865 + 119875119896) (17)





Father 1 (

Father 2 (




Child 1 (

Child 2 (

a aaaaaaaa

a aaaaaaaa

)7 82179556

)6 43431982



NO Layout Scheme1 [85][21][346][79]2 [7][43][296][185]3 [9][785][346][21]4 [8][59][347][621]5 [9][127][534][86]6 [96][851][732]


Child 1 (

Child 2 (

a aaaaaaaa

a aaaaaaaa

)3 82179546

)6 75431982






Start


Initial population



Yes

No

Select

Cross

Variation

Output result

End









Meet theconditions

Meet the conditions

Determineweight

No

No

Yes

Yes




52 Optimal Process




scheme



Z3Cabin safety

Criteria layer Z

Solution layer P




119868119894119895 = 1119868119895119894 (18)




119885119894119895 = [[[[[

1 12 152 1 135 3 1]]]]]

(19)





Feature vector

0122 0272 0357 02520230 0228 0242 02560644 0184 0103 0144

0109 0103 00820130 0105 01710078 0091 0096


1198751119894119895 =

[[[[[[[[[[[[[[[[

1 1 2 3 2 41 1 2 3 1 212 12 1 2 3 213 13 12 1 2 112 1 13 12 1 314 12 12 1 13 1

]]]]]]]]]]]]]]]]

1198752119894119895 =

[[[[[[[[[[[[[[[[[

1 2 4 3 5 212 1 2 3 4 214 12 1 12 1 213 13 12 1 2 115 14 1 12 1 312 12 12 1 13 1

]]]]]]]]]]]]]]]]]

1198753119894119895 =

[[[[[[[[[[[[[[[[

1 1 3 3 1 21 1 2 3 2 213 12 1 2 1 213 13 12 1 12 11 12 1 2 1 212 12 12 1 12 1

]]]]]]]]]]]]]]]]

(20)



n 1 2 3 4 5 6 7 8 9RI 0 0 058 089 112 124 132 141 145




119862119868 = 120582max minus 119899119899 minus 1 (21)


119862119877 = 119862119868119877119868 (22)





119882119894 = 119898sum119895=1

119887119895120596119894 (119894 = 1 2 119899) (23)






Judgment matrix 1198851 1198852 1198853119862119868119895 0098 0070 0029119887119895 0122 0230 0644119877119868119895 1240 1240 1240




119862119877 = (sum119898119895=1 119862119868119895119887119895)(sum119898119895=1 119877119868119895119887119895) (24)




6 Conclusions



Data Availability






Acknowledgments



References































































Journal of












Journal of







Volume 2018






Volume 2018











119875119896 = 0 1199040 + (119898 minus 1) 119904 le 119867119879 119900119905ℎ119890119903 (16)



119891119894119905 (V119896) = 1(119865 + 119875119896) (17)





Father 1 (

Father 2 (




Child 1 (

Child 2 (

a aaaaaaaa

a aaaaaaaa

)7 82179556

)6 43431982



NO Layout Scheme1 [85][21][346][79]2 [7][43][296][185]3 [9][785][346][21]4 [8][59][347][621]5 [9][127][534][86]6 [96][851][732]


Child 1 (

Child 2 (

a aaaaaaaa

a aaaaaaaa

)3 82179546

)6 75431982






Start


Initial population



Yes

No

Select

Cross

Variation

Output result

End









Meet theconditions

Meet the conditions

Determineweight

No

No

Yes

Yes




52 Optimal Process




scheme



Z3Cabin safety

Criteria layer Z

Solution layer P




119868119894119895 = 1119868119895119894 (18)




119885119894119895 = [[[[[

1 12 152 1 135 3 1]]]]]

(19)





Feature vector

0122 0272 0357 02520230 0228 0242 02560644 0184 0103 0144

0109 0103 00820130 0105 01710078 0091 0096


1198751119894119895 =

[[[[[[[[[[[[[[[[

1 1 2 3 2 41 1 2 3 1 212 12 1 2 3 213 13 12 1 2 112 1 13 12 1 314 12 12 1 13 1

]]]]]]]]]]]]]]]]

1198752119894119895 =

[[[[[[[[[[[[[[[[[

1 2 4 3 5 212 1 2 3 4 214 12 1 12 1 213 13 12 1 2 115 14 1 12 1 312 12 12 1 13 1

]]]]]]]]]]]]]]]]]

1198753119894119895 =

[[[[[[[[[[[[[[[[

1 1 3 3 1 21 1 2 3 2 213 12 1 2 1 213 13 12 1 12 11 12 1 2 1 212 12 12 1 12 1

]]]]]]]]]]]]]]]]

(20)



n 1 2 3 4 5 6 7 8 9RI 0 0 058 089 112 124 132 141 145




119862119868 = 120582max minus 119899119899 minus 1 (21)


119862119877 = 119862119868119877119868 (22)





119882119894 = 119898sum119895=1

119887119895120596119894 (119894 = 1 2 119899) (23)






Judgment matrix 1198851 1198852 1198853119862119868119895 0098 0070 0029119887119895 0122 0230 0644119877119868119895 1240 1240 1240




119862119877 = (sum119898119895=1 119862119868119895119887119895)(sum119898119895=1 119877119868119895119887119895) (24)




6 Conclusions



Data Availability






Acknowledgments



References































































Journal of












Journal of







Volume 2018






Volume 2018









Start


Initial population



Yes

No

Select

Cross

Variation

Output result

End









Meet theconditions

Meet the conditions

Determineweight

No

No

Yes

Yes




52 Optimal Process




scheme



Z3Cabin safety

Criteria layer Z

Solution layer P




119868119894119895 = 1119868119895119894 (18)




119885119894119895 = [[[[[

1 12 152 1 135 3 1]]]]]

(19)





Feature vector

0122 0272 0357 02520230 0228 0242 02560644 0184 0103 0144

0109 0103 00820130 0105 01710078 0091 0096


1198751119894119895 =

[[[[[[[[[[[[[[[[

1 1 2 3 2 41 1 2 3 1 212 12 1 2 3 213 13 12 1 2 112 1 13 12 1 314 12 12 1 13 1

]]]]]]]]]]]]]]]]

1198752119894119895 =

[[[[[[[[[[[[[[[[[

1 2 4 3 5 212 1 2 3 4 214 12 1 12 1 213 13 12 1 2 115 14 1 12 1 312 12 12 1 13 1

]]]]]]]]]]]]]]]]]

1198753119894119895 =

[[[[[[[[[[[[[[[[

1 1 3 3 1 21 1 2 3 2 213 12 1 2 1 213 13 12 1 12 11 12 1 2 1 212 12 12 1 12 1

]]]]]]]]]]]]]]]]

(20)



n 1 2 3 4 5 6 7 8 9RI 0 0 058 089 112 124 132 141 145




119862119868 = 120582max minus 119899119899 minus 1 (21)


119862119877 = 119862119868119877119868 (22)





119882119894 = 119898sum119895=1

119887119895120596119894 (119894 = 1 2 119899) (23)






Judgment matrix 1198851 1198852 1198853119862119868119895 0098 0070 0029119887119895 0122 0230 0644119877119868119895 1240 1240 1240




119862119877 = (sum119898119895=1 119862119868119895119887119895)(sum119898119895=1 119877119868119895119887119895) (24)




6 Conclusions



Data Availability






Acknowledgments



References































































Journal of












Journal of







Volume 2018






Volume 2018










scheme



Z3Cabin safety

Criteria layer Z

Solution layer P




119868119894119895 = 1119868119895119894 (18)




119885119894119895 = [[[[[

1 12 152 1 135 3 1]]]]]

(19)





Feature vector

0122 0272 0357 02520230 0228 0242 02560644 0184 0103 0144

0109 0103 00820130 0105 01710078 0091 0096


1198751119894119895 =

[[[[[[[[[[[[[[[[

1 1 2 3 2 41 1 2 3 1 212 12 1 2 3 213 13 12 1 2 112 1 13 12 1 314 12 12 1 13 1

]]]]]]]]]]]]]]]]

1198752119894119895 =

[[[[[[[[[[[[[[[[[

1 2 4 3 5 212 1 2 3 4 214 12 1 12 1 213 13 12 1 2 115 14 1 12 1 312 12 12 1 13 1

]]]]]]]]]]]]]]]]]

1198753119894119895 =

[[[[[[[[[[[[[[[[

1 1 3 3 1 21 1 2 3 2 213 12 1 2 1 213 13 12 1 12 11 12 1 2 1 212 12 12 1 12 1

]]]]]]]]]]]]]]]]

(20)



n 1 2 3 4 5 6 7 8 9RI 0 0 058 089 112 124 132 141 145




119862119868 = 120582max minus 119899119899 minus 1 (21)


119862119877 = 119862119868119877119868 (22)





119882119894 = 119898sum119895=1

119887119895120596119894 (119894 = 1 2 119899) (23)






Judgment matrix 1198851 1198852 1198853119862119868119895 0098 0070 0029119887119895 0122 0230 0644119877119868119895 1240 1240 1240




119862119877 = (sum119898119895=1 119862119868119895119887119895)(sum119898119895=1 119877119868119895119887119895) (24)




6 Conclusions



Data Availability






Acknowledgments



References































































Journal of












Journal of







Volume 2018






Volume 2018











Feature vector

0122 0272 0357 02520230 0228 0242 02560644 0184 0103 0144

0109 0103 00820130 0105 01710078 0091 0096


1198751119894119895 =

[[[[[[[[[[[[[[[[

1 1 2 3 2 41 1 2 3 1 212 12 1 2 3 213 13 12 1 2 112 1 13 12 1 314 12 12 1 13 1

]]]]]]]]]]]]]]]]

1198752119894119895 =

[[[[[[[[[[[[[[[[[

1 2 4 3 5 212 1 2 3 4 214 12 1 12 1 213 13 12 1 2 115 14 1 12 1 312 12 12 1 13 1

]]]]]]]]]]]]]]]]]

1198753119894119895 =

[[[[[[[[[[[[[[[[

1 1 3 3 1 21 1 2 3 2 213 12 1 2 1 213 13 12 1 12 11 12 1 2 1 212 12 12 1 12 1

]]]]]]]]]]]]]]]]

(20)



n 1 2 3 4 5 6 7 8 9RI 0 0 058 089 112 124 132 141 145




119862119868 = 120582max minus 119899119899 minus 1 (21)


119862119877 = 119862119868119877119868 (22)





119882119894 = 119898sum119895=1

119887119895120596119894 (119894 = 1 2 119899) (23)






Judgment matrix 1198851 1198852 1198853119862119868119895 0098 0070 0029119887119895 0122 0230 0644119877119868119895 1240 1240 1240




119862119877 = (sum119898119895=1 119862119868119895119887119895)(sum119898119895=1 119877119868119895119887119895) (24)




6 Conclusions



Data Availability






Acknowledgments



References































































Journal of












Journal of







Volume 2018






Volume 2018












Judgment matrix 1198851 1198852 1198853119862119868119895 0098 0070 0029119887119895 0122 0230 0644119877119868119895 1240 1240 1240




119862119877 = (sum119898119895=1 119862119868119895119887119895)(sum119898119895=1 119877119868119895119887119895) (24)




6 Conclusions



Data Availability






Acknowledgments



References































































Journal of












Journal of







Volume 2018






Volume 2018









References































































Journal of












Journal of







Volume 2018






Volume 2018




































Journal of












Journal of







Volume 2018






Volume 2018








optimum design of ship cabin equipment layout based on slp...

Documents