A Novel Parallel Assembly Sequence Planning Method for …downloads.hindawi.com/journals/mpe/2020/7848329.pdf · 2020-05-26 · A Novel Parallel Assembly Sequence Planning Method

Research ArticleA Novel Parallel Assembly Sequence Planning Method forComplex Products Based on PSOBC

Yanfang Yang12 Miao Yang1 Liang Shu 3 Shasha Li3 and Zhiping Liu12

1School of Logistic Engineering Wuhan University of Technology Wuhan China2Engineering Research Center of Port Logistics Technology and Equipment Ministry of Education Wuhan China3$e Low-Voltage Apparatus Technology Research Center of Zhejiang Wenzhou University Wenzhou China

Correspondence should be addressed to Liang Shu shuliangalbert163com

Received 27 December 2019 Revised 18 March 2020 Accepted 19 March 2020 Published 26 May 2020

Guest Editor Manjit Kaur

Copyright copy 2020 Yanfang Yang et alis 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

Parallel assembly sequence planning (PASP) greatly impacts on efficiency of assembly process In traditional methods large scaleof matrix calculation still limits efficiency of PASP for complex products A novel PASP method is proposed to address this issueTo avoid matrix calculation the synchronized assembly Petri net (SAPN) is firstly established to describe the precedence re-lationships Associated with the SAPN model the PASP process can be implemented via particle swarm optimization based onbacterial chemotaxis (PSOBC) Characterized by an attraction-repulsion phase PSOBC not only prevents premature convergenceto a high degree but also keeps a more rapid convergence rate than standard particle swarm optimization (PSO) algorithmFinally feasibility and effectiveness of the proposed method are verified via a case study With different assembly parallelismdegrees optimization results show that assembly efficiency of the solution calculated by PSOBC method is 90 42 and 31better than the standard PSO process

1 Introduction

ASP is one of the important parts of industrialmanufacturing which is able to provide fast and high ef-ficiency guidelines for equipment assembly High efficiencyof ASP directly determines enterprise profit However ASPis usually difficult due to the large number of equipmentcomponents (assembly operations) and complicated prior-itiesconstraints relations among components

Usually ASP can be treated as an NP-hard problem [1]which is generally solved to maximize the multiple benefitsduring the assembly process [2] ere are several challengesremaining to construct efficient and comprehensive ASPone of which is the modeling problem To optimize ASP theprecedence relationships among every individual compo-nent need to be investigated A number of methods havebeen developed to meet various requirements in the ASPoptimization ANDOR graph model was firstly developedby De Mello et al [3] and it has been widely employed inassembly sequence generation researches Precedence

matrix was built and embodied into Petri net to generate thereachability graph by Yang et al [4] Similar assembly di-rection matrix interference matrix and sequence-relationmatrix were introduced to generate optimal assembly se-quence for eccentric milling machine in [5] Such matrixmethod contributes to generation and acquisition of pro-duction information and it is usually straightforward andeasy to be implemented However due to the large numberof equipment components and the complicated constrainsthe size of precedence matrix is generally very largeComputational burden will limit the optimization processTo increase ASP efficiency some other advanced methodswere introduced and developed eg weighted precedencegraph [6] assembly tree [7] layout-based hierarchical graph[8] and part concatenation method [9ndash11] In thesemethods the weighted assembly graph reduces the com-plexity of ASP through considering qualitative and quan-titative constraints By using the assembly tree methodgeneration of feasible and stable sequences can be imple-mented through the recursive construction and stability

HindawiMathematical Problems in EngineeringVolume 2020 Article ID 7848329 11 pageshttpsdoiorg10115520207848329

criteria Part concatenation method is robust and able toaddress ASP problem by considering all necessary assemblypredicates However such models are still matrix based Notonly is enormous human calculation required to investigatecomplex products but also human error would occur with alarge probability To address this problem a SAPN model isdeveloped to describe the operations priorities and con-straints in assembly process Each transition is associatedwith the predecessors by using assembly synchronizerswhich illustrates the precedence relationshipse transitionstatus is updated in each assembly operation and theassembility can be identified by the status checking

ASP optimization could be implemented when the as-sembly model is developed Generally ASP can be catego-rized into two types the SASP and the PASP e assemblyoperations of SASP are constructed in series and are easy tobe implemented Since all the operations need to follow alinear sequence the efficiency is limited Assembly opera-tions of PASP can be performed in parallel and the efficiencycan be increased However it is still challenging to assemblecomplex equipment with PASP To increase efficiency andcorrectness the PASP needs to be optimally designedMoreover optimization of PASP needs to be subjected tostrict predetermined constraints

Heuristic algorithm is an effective way to solve opti-mization problem with constraints Typical methods in-clude genetic algorithm [12ndash14] PSO [15 16] ant colonyoptimization algorithm [17 18] and other evolutionaryalgorithms [19ndash21] GA is a nonlinear estimation methodthat generates solutions to optimization problems usingtechniques inspired by natural evolution [22] To tackleASP of complex products a chaotic PSO approach togenerate the optimal or near-optimal assembly sequencesof products was developed in [23] Generally heuristicsearching needs to be implemented with numerical itera-tions e coding rules and heuristic rules under pre-determined constraints are important to construct theefficient searching Like what has been discussed most ofthe ASP models and the coding rules are matrix based esearchingiteration performance is limited due to thematrix calculations We have developed the SAPNmodel todescribe the operations priorities and constraints in theassembly process Coding rules of PSOBC such as positionvelocity and diversity function are represented with indexvectors instead of matrix e consequent efficiency can beincreased

As for the heuristic rules design the prematurity phe-nomenon is the major problem that needs to be addressede searching result could be attracted to the local optima ifthe heuristic rules are designed improperly Inspired by thephenomenon of chemotaxis in bacteria colonies PSOBCwasproposed to improve the performance of standard PSO [24]Characterized by an attraction-repulsion phase PSOBC notonly prevents premature convergence to a high degree butalso keeps amore rapid convergence rate than standard PSOis algorithm has been successfully used in areas like CO2emission problem [25] and hydropower stations operations[26] e global and local search performance could be wellbalanced when the particle swarm diversity is properly

controlled Here we use a similar PSOBC framework to solvethe PASP problem based on the proposed SAPN model ediversity function is built up to avoid population prema-turity Feasibility and effectiveness of the proposed methodare also verified via a case study Optimization results havebeen compared with standard PSO method under differentDPAs

2 Establishment and Analysis of SAPN

21 Basic Petri Net

Definition 1 (Petri net) Let N (P T F) be the basic Petrinet which should satisfy the following three conditions

(1) PcupTneOslashandPcapTOslash(2) FsubePtimesTcupTtimes P(3) dom (F)cup cod (F) PcupT

Here ldquotimesrdquo is the Cartesian productdom(F) x |existy (x y) isin F1113864 1113865 and cod(F) y |existx (x1113864

y) isin F Condition (3) is defined to ensure that there is noisolated place transition or arc in the Petri net

22 Symbols in SAPN

Definition 2 (synchronizer) Let p (T1 T2 (a1 a2)) be asynchronizer where forallti1

isinT1 ti2isinT2 p tbull

i1 bullti2

w(ti1

p) a2 w(pti2) a1 T1 t11 t12 t1m11113864 1113865

T2 t21 t22 t2m21113864 1113865 1lea1lem1 |T1| and 1lea2lem2 |T2|e scheme of the defined synchronizer is shown in

Figure 1 e SAPN consists of a set of synchronizerscorresponding to different assembly status e synchro-nizers are used to illustrate the parallel and selective relationsof each assembly operation including the correspondingprecedence relationships

Definition 3 (SAPN) Let SAPN( P T FWM PT TM) bethe synchronized assembly Petri net consisting of assemblysynchronizers Places transitions arcs and other tuples areintegrated in the net Detailed explanation of these tuples isgiven as follows

P pj (j 1 2 m) represents the assembly statusand m is the amount of the placesT ti (i 1 2 n) represents the assembly oper-ations and n is the amount of the transitionsF fk (k 1 2 l) represents the directed arcsbetween transitions and places and l is the amount ofthe arcsW wk F⟶ 1 2 3 represents the weightfunctions associated with arcs e weight functionsdetermine the amount of the required tasks for a placeto transform into another one e weight functionsshould be assigned based on number of the arcs whichshare the same inputoutput place in the correspondingsynchronizer For example there are m1 arcs betweenT1 and p sharing the same output place p in Figure 1 sothe weight functions of these arcs should be both

2 Mathematical Problems in Engineering

assigned as m1 Also the weight functions of prsquos outputarcs should be both assigned asm2 according to similarrecursionM mi T⟶ 0 1 2 represents the markings as-sociated with transitions Here M is defined to repre-sent the transition status In the assembly process themarkingM is a piecewise constant vector and the valuesare dependent on the transition status based on thefollowing rules

mi

0 ti is fired

1 ti is fireable

2 ti is unfireablewhile it hasrsquot been fired

⎧⎪⎪⎨

⎪⎪⎩

(1)

PT pti21113966 1113967 T⟶ t i1

(i1 i2 1 2 n) repre-sents the precedence relationships among equipmentcomponentse element pti2

in PT is defined to collectthe preceding transitions of ti1

In others words if existp

(T1 T2 (a1 a2)) ti2isin T1 and ti1

isin T2 then the transi-tions in T1 are the predecessors of the transitions in T2TM tmiT represents operation times of thetransitions

23 Construction of SAPN is subsection introducesconstruction approach of SAPN through the following steps

Step 1 Generate transitions for each necessary as-sembly operationStep 2 Assign tms ms and pts to each correspondingtransitionStep 3 Generate the starting place the ending placeand the places between each transition and itspredecessorsStep 4 Integrate the duplicated input places of eachtransitionStep 5 Generate the directed arcs from predecessors tothe corresponding places and from the places to thecorresponding successorsStep 6 Assign the weight function to each arc

SAPN model can be constructed through the detailedsteps mentioned above Step 1 and Step 2 generate thefundamental transitions Generation of the corresponding

places is processed in Step 3 and Step 4 Since SAPN isoriented to the complex products the elements in SAPN areunavoidably large erefore the integration process in Step4 somewhat reduces the scale of SAPN model Step 5 andStep 6 generate and assign the arcs which link the elementsin each synchronizer

24 Analysis of Assembility A transition which is fireablemust satisfy the condition that its preceding transitions haveall been fired in the synchronizer Assembility of eachcomponent can be determined from the precedence rela-tionships and the status ie PT and M e necessary andsufficient condition of ti1

being fireable is given as below

mi1 1 (2)

Equation (2) can also be rewritten as

forallti2isin pti1

1113944 mi2 0andmi1

ne 0 (3)

It is seen from (3) that ti1is fireable when its preceding

transitions have all been fired namely mi2 0 Analysis of

assembility is necessary in the sequence optimization pro-cess to avoid violation of precedence relationships

3 Principles of Sequence Optimization

In this section PSOBC framework is developed to solve thePASP problem e algorithm alternates between phases ofattraction and repulsion Once the diversity of population istoo high the individuals will be congregated by attractionforce to explore better solution If the diversity of populationis too low the individuals will be dispersed by repulsionforce to ensure better convergence us the sequence op-timization can be efficiently performed

31 Problem Statement Here we use a simple example toillustrate the principles of PASP e parallel assemblyscheme is shown in Figure 2 It is assumed that there are 3manipulators and 13 individual parts of equipment elength of each rectangle in Figure 2 represents the assemblytime of the corresponding part ree manipulators areemployed and five assembly steps are involved in the pro-cess During the process time represents the operation timeof the whole assembly process To improve efficiency thegoal of sequence planning is set to minimize the assemblytime time DPA can be defined to describe the biggestnumber of components or parts to be assembled simulta-neously ie the amount of the manipulators According tothe definition the DPA of the example in Figure 2 is 3

To improve assembly performance time study is con-sidered to minimize assembly timee objective function isgiven in the following formulation

min(time) 1113944S

s1times (4)

In (4) times represents operation time of the subas-sembly s and time represents the whole assembly time

p

t11

t1m1 t2m2

t21

hellip hellip

a2 a1

a2 a1

Figure 1 Scheme of synchronizer

Mathematical Problems in Engineering 3

32 Coding Rules In this subsection coding rules for PASPproblem are proposed according to its features e rules in-clude particle position particle velocity fitness function anddiversity function Since matrix calculation is avoided calcu-lation process would be in high efficiency and effectiveness

Definition 4 (particle position) Let the particle position Xdbe a parallel sequence which is expressed by a vector asbelow

Xd xd1 xd2 xdi xdn1113858 1113859 (5)

Since xdi represents subassembly index of transition ti itis supposed to be an integer

Definition 5 (particle velocity) Let the particle velocity Vdbe the updating rule for the particles which is represented byanother vector as below

Vd vd1 vd2 vdi vdn1113858 1113859 (6)

Since vdi represents the updating rule for transition ti ofparticle Xd vdi is supposed to be a real number

Definition 6 (fitness function) Let the fitness function be theformulation to evaluate particle positions as expressed in

FT(X) Ct minus 1113944S

s1times + δpFp(s) + δlFl(s)1113960 1113961 (7)

In (7) Ct is a constant defined to convert minimizationto maximization Assembly time is considered in the fitnessfunction to optimize assembly efficiency while the twopenalty factorsfunctions are used to respectively investi-gate precedence relationships and parallel principles

To calculate the fitness function the first step is decodingX into parallel assembly sequence based on its definitionwhile the second step is calculating each times Fp (s) and Fl(s) times is the max operation time in subassemblys Fp (s)equals the number of unfireable transitions in subassembly sie the marking values of the transitions equal 2 accordingto analysis of assembility in the synchronizer (3) Aftercalculating each Fp (s) marking values of transitions in thesubassembly should be set as 0 and marking values of theunfired transitions in SDPN should be set as 1 if theirpreceding transitions have all been fired or the values re-main equal to 2 Fl (s) equals 0 if operation number ofsubassembly s is less than DPA or it equals DPA minusoperation number

Definition 7 (diversity of population) Let the diversity ofpopulation be the formulation shown in (8) to evaluate theaverage distance between all particles

Fd Pl( 1113857 1

|L| Pl

11138681113868111386811138681113868111386811138681113868

1113944

Pl| |

d1

1113944

n

i1xdi minus xi( 1113857

2

11139741113972

(8)

In (8) |Pl| is the population size while xi is the meansubassembly indices of ti in the population Diversityfunction is formulated to detect premature convergence and

then attractionrepulsion phase can be implemented toimprove algorithm performance

33 Algorithm Optimization Let Vg

d be the velocity ofparticle d in generation g and Xg

d be the position of particled in generation g e PSOBC algorithm should beimplemented with the following equations

Vg+1d ω times Vg

d + clr1 times Bi d minus Xg

d1113872 1113873 + cgr2 Bg d minus Xg

d1113872 1113873

(9)

Xg+1d Xg

d + Vg+1d (10)

Vg+1d ω times Vg

d minus clr1 times Wi d minus Xg

d1113872 1113873 minus cgr2 Wg d minus Xg

d1113872 1113873

(11)

In attraction phase (9) and (10) should be exerted tooptimize solutions with the principle of approaching localglobal best positions In repulsion phase (10) and (11)should be exerted to improve convergence with the principleof escaping from localglobal worst positions e calcu-lating rules of (9)sim(11) are proposed as below

Xd1plusmnXd2

V xd11plusmnxd21xd12plusmnxd22 xd1nplusmn xd2n1113960 1113961 (12)

k timesVd V kvd kvd kvd1113858 1113859 (13)

Vd1plusmnVd2

V vd11plusmn vd21 vd12plusmn vd22 vd1nplusmn vd2n1113960 1113961 (14)

Xd +Vd X xd1 + vd1 xd2 + vd2 xdi + vdi xdn + vdn1113858 1113859

(15)

Since xs are integers and vs are real numbers the ele-ments in X should be both rounded to integers after cal-culation using (15)

4 Case Study

In this paper a metallurgical reducer is taken as an example toverify the feasibility and the efficiency of the proposed methode DPAs are selected as 2 3 and 4 respectivelye explosivediagram of the reducer is shown in Figure 3 including 50components in total e main parts of the reducer are gearsboxes caps bears shafts lubricating rings and eccentric sleeves

41 Establishing SAPN Based on the reducer structure theplaces transitions arcs and other tuples in SAPN (P T FW M PT TM) are established in Section 23 Structurediagram of the SAPN is built as shown in Figure 4 In the netmodel P is the set of places defined to represent componentstatus p1 and p49 are the starting and ending places T is theset of transitions representing the assembly operationsDefinitions of the transitions in the SAPNmodel are given inTable 1 F is the set of arcs representing flow relationsbetween transition and place W is the set of weights as-sociated with the arcs e weight functions are assignedbased on the definition mentioned in Section 22 For


example weight functions of the arcs between p2 and t2 p5and t5 and p28 and t28 are assigned as 1 while weightfunction of the arcs between p29 and t41 is assigned as 3M isthe set of markings associated with transitions Precedencerelationships are illustrated in PT to analyze the componentassembility TM is the set of assembly times associated witheach assembly operation

It is seen from Table 1 that large size of matrices would begenerated if the matrix-based method was adopted formetallurgical reducer ASP design which would be complexand difficult to implement is problem can be addressedthrough the proposed PASP framework

42 Population Initialization Based on the heuristicmechanism of PSOBC population initialization is requiredfor the sequence optimization e stages of populationinitialization can be structured as a flow chart represented inFigure 5 Detailed explanation of the initialization procedureis given in the following steps

Step 1 Obtain the precedence relationships and the initialstatus of the components from PT and M Build IP (anempty set of particles position) as the initial populationStep 2 Build a new particle position PDS (an empty setof transitions)Step 3 Build a new set of all fireable transitions RTSbased on analysis of assembilityStep 4 Build a new assembly step DS (an empty set oftransitions) If nr (the amount of transitions in RTS)exceeds DPA randomly push 1simDPA assembly oper-ations intoDSOtherwise push all the operations formRTS into DS Update status and marking of thetransitions in DS and push the transitions from DSinto PDSStep 5 Remove the fired transitions in RTS based onPT and MStep 6 If nr is positive return to step 4 Otherwiseencode PDS into a particle position (an index vector)and push it into IP return to step 7

Step 7 If np (the amount of particles) exceeds thepopulation size nt output the initial population IP andend Otherwise return to step 3

In the C environment the population size nt is set as 50and the DPAs are set as 2 3 and 4 respectively e threeinitial populations are exported after running the initiali-zation codes According to the biologically inspired mech-anism of PSOBC the initial population is the origin for thebacterial evolution

e key procedure of PASP is the sequence optimi-zation process implemented to find the optimal solutionMain stages of heuristic optimization can be structured asanother flow chart represented in Figure 6 Detailed ex-planation of the procedure is illustrated in the followingsteps

43 Generation of the Optimal Sequences e key procedureof PDSP is the sequence optimization process implementedto find the optimal solution Main stages of heuristic op-timization can be structured as another flow chart repre-sented in Figure 6 Detailed explanation of the procedure isillustrated in the following steps

Step 1 Input the initial population from initializationprocess Set phase as ldquoattractionrdquoStep 2 Update the current best and worst particleposition for individuals and the whole populationStep 3 If the phase is ldquoattractionrdquo calculate the newvelocity and position for each particle using (9) and(10) Otherwise calculate using (10) and (11)Step 4 Calculate diversity function of the population Ifpremature convergence occurs set the phase asldquorepulsionrdquo and return to step 6 Otherwise return tostep 5Step 5 Calculate diversity function of the population Ifpremature convergence is escaped set the phase asldquoattractionrdquo and return to step 6

O1

O2

O3

time1

d1

d2

d3

d4

d5 d7

d8

d6 d9

d10 d12

d11

d13

time2 time3 time4 time5

time

Figure 2 Scheme of parallel assembly

1234

5

6

789

10

11

12131415

16

171819202122232425

2627

28

29303132

33

3435

36

3738

39404142

43

4445464748

49

50

Figure 3 Explosive diagram of the metallurgical reducer


Table 1 Detailed definitions of transitions in the SAPNT Targeted part No PT TM M T Targeted part No PT TM Mt1 Null Null empty 0 2 t26 Lubricating ring 3 41 t18 t23 18 2t2 Bear 12 18 t1 19 2 t27 Bear 4 3 t1 21 2t3 Bear 11 45 t1 16 2 t28 Bear 3 9 t1 21 2t4 Eccentric sleeve 5 44 t1 21 2 t29 Bear 2 47 t24 21 2t5 Spline 20 t1 18 2 t30 Lock 3 33 t1 12 2t6 Eccentric sleeve 4 24 t2 21 2 t31 Bear 1 39 t18 14 2t7 Bear 10 15 t1 13 2 t32 Lubricating ring 2 42 t26 15 2t8 Bear 9 17 t1 13 2 t33 Sleeve 3 2 t27 21 2t9 Gear 7 23 t1 13 2 t34 Sleeve 2 10 t1 38 2t10 Gear 6 22 t1 13 2 t35 Eccentric sleeve 2 8 t28 21 2t11 Bear 8 36 t1 21 2 t36 Lubricating ring 1 4 t1 15 2t12 Lubricating ring 4 27 t1 38 2 t37 Gear 1 7 t1 31 2t13 Sleeve 3 29 t3 t4 17 2 t38 Sleeve 1 6 t1 38 2t14 Inner splined sleeve 19 t5 28 2 t39 Eccentric sleeve 1 46 t29 23 2t15 End cap 4 13 t6 21 2 t40 Lock 2 34 t30 19 2t16 Gear 5 14 t7 t8 31 2 t41 Middle box 11 t31 t24 t39 21 2t17 Gear 4 25 t1 36 2 t42 Lock 1 32 t25 16 2t18 Gear 3 21 t9 t10 35 2 t43 Input shaft 43 t31 t32 42 2t19 Bear 7 38 t11 21 2 t44 Output shaft 1 5 t33 72 2t20 Blank cap 2 26 t12 09 2 t45 End cap 3 37 t33 09 1t21 Gear 2 30 t13 38 2 t46 End cap 3 37 t39 t32 t41 09 1t22 Bear 6 31 t1 19 2 t47 End cap 2 12 t1 09 1t23 Bear 5 40 t1 21 2 t48 End cap 1 35 t40 18 1t24 Intermediate shaft 16 t14simt19 95 2 t49 Upper box 49 t42 t43 41 1t25 Output shaft 2 28 t20simt22 121 2 t50 Oil collecting hood 1 t44 16 1

p1 0

0

0

0

t2

t1

p5

p6

p7

p8

p9

p10

p11

p12

p13

t5

t6

t7

t8

t9

t10

t11

t12

t13

p14

p15

p16

p17

p18

p19

p20

p21

t14 p24 p29t24

t29

t15

t16

t17

t18

t19

t20

p30 t30 p41 t40

p45p49

p44

t47

t46

t45

p46 t48

p48 t50

p47

t49

p40 t39

p42

p43

t41

t43

t44

t42

t21

p22 t22

p23 t23 p26 t26

p25 t25 p32

p31

t31

p33 t32

p34 t33

p35 t34

p36 t35

p37 t36

p38 t37

p39 t38

p27 t27

p28 t28

p2

p3

p4

t3

t4

Figure 4 SAPN model of the reducer


Step 6 If the termination condition is satisfied decodethe optimal solution and end Otherwise return to step2

In the optimization procedure the parameters are set asδp δl 5 itnmax 150 cl cg 2 dh 01 and dl 0001and ω will linearly decrease from 1 to 0 during the searchingprocess Here the parameters are set not only based on thefeatures of PASP problem and PSOBC algorithm but alsoreferring to researches in [25 26]

44 Discussions and Comparisons ree different simula-tions are conducted to verify the proposed method whenDPAs are settled as 2 3 and 4 respectively e proposedPSOBC simulations have been compared with standard PSOmethod in Figure 7 To keep consistency the simulationparameters including learning factor penalty factors andinertia factor are chosen as the same in both PSOBC andPSO for different DPAs In Figure 7(a) DPAs are bothsettled as 2 For PSO simulation the assembly time is 815seconds e assembly time of PSOBC method is 748seconds Compared to the standard PSO the efficiency isincreased by 90 Detailed explanation of 2-DPA assemblysequence is given in Table 2

In Figure 7(b) DPAs are both settled as 3 It is seen thatthe assembly time of PSO is 621 seconds while the assemblytime of PSOBC is 596 seconds Compared to the standardPSO the assembly efficiency is increased by 42 e

Output IP

Create PDS

Push all the able-to-disassemble transitions into RTS

nr gt dp

Randomly push 1~dptransitions from RTS

to DSDS = RTS

Push elements which are in DSinto TRS and update M

nt gt 0

Code PDS into index vector and push into IP

np = nt

np++

Input PT and M

Create IP

Create RTS

Create DS

Y N

Y

N

NY

Figure 5 Flow chart of population initialization

Update the best and the worst position

itn = itnmax

Input initial population

Set phase as attraction

Y N

Calculate with equation 9 and

equation 10

Calculate with equation10 and

equation 11

Calculate Fd (Pl)

Fd (Pl) lt dl

Set phase as repulsion

Fd (Pl) lt dh


itn++

Decode and output the optimal sequence

Is the phase attraction

Y N

N

Y

N

Y

Figure 6 Flow chart of sequence optimization


optimal sequence carried out by PSOBC method is given inTable 3 when DPA is set as 3

In Figure 7(c) DPAs are both settled as 4 roughstandard PSO method assembly time of the solution is 591secondse assembly time by using PSOBC is 573 secondse assembly efficiency is increased by 31 e optimizedsequence carried out by PSOBC is given in Table 4

Based on the optimization results and comparison thefollowing conclusions can be drawn

(1) Due to implementation of SAPN model the PASPproblem can be effectively and efficiently solvedfor complex products while huge matrix calcu-lation is avoided and possibility of human error isreduced

(2) Compared with the standard PSO algorithm pre-mature convergence can be effectively preventedthrough repulsion operation e solution of PASPproblem can be further optimized

70

80

90

100

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(a)

55

60

65

70

75

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(b)

55

60

65

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(c)

Figure 7 Optimization results and comparison (a) Optimization result with 2-DPD (b) Optimization result with 3-DPD (c) Optimizationresult with 4-DPD

Table 2 Parallel assembly sequence of 2-DPA

Subassembly index Subassembly Targeted parts1 (17 20) Bear 9 and spline2 (23 15) Gear 7 and bear 103 (22 36) Gear 6 and bear 84 (21 14) Gear 13 and gear 55 (40 18) Gear 7 and bear 126 (45 24) Bear 11 and eccentric sleeve 47 (38 13) Bear 7 and end cap 48 (19 27) Inner splined sleeve9 (41 44) Lubricating ring 3 and eccentric sleeve 510 (26 29) Blank cap 211 (31) Bear 612 (30 25) Gear 2 and gear 413 (28 16) Output shaft and intermediate shaft14 (32 3) Lock 1 and bear 415 (42 47) Lubricating ring 2 and bear 216 (39) Bear 117 (46 6) Eccentric sleeve 1 and sleeve 118 (33 9) Lock 3 and bear 319 (34 8) Lock 2 and eccentric sleeve 220 (7) Gear 121 (43 10) Input shaft and sleeve 222 (2) Sleeve 323 (11 4) Middle box and lubricating ring 124 (12 37) End cap 2 and end cap 325 (5 49) Output shaft 1 and upper box26 (1 35) Oil collecting hood27 (48) Blank cap 1


5 Conclusions

A novel PASP optimization method is developed forcomplex product in this paper A SAPN model is proposedto describe the precedence relationships in the assemblyprocess while huge matrix calculation is avoided andpossibility of human error is reduced To optimize PASPproblem a PSOBC method is developed along with theSAPN model Verifications show that assembly time of theoptima calculated using the proposed method is 90 42and 31 better than the standard PSO algorithm when theDPA is 2 3 and 4 respectively Assembly time of the threeoptimal sequences is 748 seconds 596 seconds and 573seconds separately Parallel assembly process can be opti-mally planned by using the proposed method

Nomenclature

SetsP Finite set of placesT Finite set of transitionsF Finite set of arcs

W Finite set of weightsM Finite set of markingspt Finite set of preceding transitionsPT Finite set of ptsTM Finite set of operation times

Indicesi i1 i2 Indices of transitionsj Index of placesk Index of arcsg Index of iterationss Index of subassembliesd d1 d2 Indices of particles

Symbolsp Placet Transitionf Arcw Weightm Markingtm Operation time of each transition


Subassembly index Subassembly Targeted parts1 (44 45 18) Eccentric sleeve 5 bear 11 and bear 122 (29 27 23) Sleeve 3 lubricating ring 4 and gear 73 (26 21 22) Blank cap 2 gear 3 and gear 64 (9 3 31) Bear 3 bear 4 and bear 65 (6 24 20) Sleeve 1 eccentric sleeve 4 and spline6 (15 17 19) Bear 10 bear 9 and inner splined sleeve7 (7 2 36) Gear 1 sleeve 3 and bear 88 (30 38 25) Gear 2 bear 7 and gear 49 (4 8) Lubricating ring 1 and eccentric sleeve 210 (14 10 13) Gear 5 sleeve 2 and end cap 411 (39 40) Bear 1 and bear 512 (5 28 16) Output shaft 113 (32 42 46) Lock 1 lubricating ring 2 and eccentric sleeve 114 (43 11 34) Input shaft middle box and lock 215 (49 1 37) Upper box oil collecting hood and end cap 316 (35 12 48) End cap 1 end cap 2 and blank cap 1


Subassembly index Subassembly Targeted parts1 (27 17) Lubricating ring 4 and bear 92 (26 15 20 18) Blank cap 2 bear 10 spline and bear 123 (19 36 24) Inner splined sleeve bear 8 and eccentric sleeve 44 (38 45 44 31) Bear 7 bear 11 eccentric sleeve 5 and bear 65 (23 29 4 22) Gear 7 sleeve 3 lubricating ring 1 and gear 66 (30 21 14) Gear 2 gear 3 and gear 57 (25 13) Gear 4 and end cap 48 (28 16) Output shaft 2 and intermediate shaft9 (7 3 47 40) Gear 1 bear 4 bear 2 and bear 510 (46 2 39 41) Eccentric sleeve 1 sleeve 3 bear 1 and lubricating ring 311 (9 42 32 11) Bear 3 lubricating ring 2 lock 1 and middle box12 (8 33 43) Eccentric sleeve 2 lock 3 and input shaft13 (34 6 10) Lock 2 sleeve 1 and sleeve 214 (49 37 5) Upper box end cap 3 and output shaft 115 (35 1 48 12) End cap 1 oil collecting hood blank cap 1 and end cap 2


O OperatorPl PopulationBid Bgd Current localglobal best positionsWid Wgd Current localglobal worst positions

Parametersδp Penalty factor for precedence relationships violationδl Penalty factor for parallel principles violationdh Upper bound of diversity functiondl Lower bound of diversity functionitn Current number of iterationsitnmax Max number of iterationsL Diagonal length of searching areaω Inertia factorcl Local learning factorcg Global learning factorr Random number with uniform distribution on

(0 1)DPA Degree of parallel assembly

Variablestime Operation time of assemblysubassemblyX Vector of particle positionV Vector of particle velocityFT (X) Fitness functionFp (s) Penalty function for violating precedence

relationshipsFl(s) Penalty function for violating parallel principlesFd (Pl) Diversity function

AbbreviationsSAPN Synchronized assembly Petri netASP Assembly sequence planningPASP Parallel assembly sequence planningSASP Sequential assembly sequence planningPSO Particle swarm optimizationPSOBC Particle swarm optimization based on bacterial

chemotaxis

Data Availability

No data were used to support this study

Conflicts of Interest

e authors declare that they have no conflicts of interest

Acknowledgments

is work was supported in part by the Zhejiang Key Re-search and Development Program under Grant 2017C01008the Analytical and Testing Technology Project of ZhejiangProvince under Grant 2018C37068 the Key TechnologyResearch Project of Wenzhou under Grants 2018ZG020 andZG2017002 and the Open Research Fund of the Low-Voltage Apparatus Technology Research Center of Zhejiangunder Grant 201711-01

References

[1] B Deepak M R G Bala Murali and B Biswal ldquoAssemblysequence planning using soft computing methods a reviewrdquoProceedings of the Institution of Mechanical Engineers Part EJournal of Process Mechanical Engineering vol 233 no 3pp 653ndash683 2019

[2] M R Bahubalendruni and B B Biswal ldquoA review on as-sembly sequence generation and its automationrdquo Proceedingsof the Institution of Mechanical Engineers Part C Journal ofMechanical Engineering Science vol 230 no 5 pp 824ndash8382016

[3] L S De Mello and A C Sanderson ldquoANDOR graph rep-resentation of assembly plansrdquo IEEE Transactions on Roboticsand Automation vol 6 no 2 pp 188ndash199 1990

[4] Y Yang P Yang J Li et al ldquoResearch on virtual hapticdisassembly platform considering disassembly processrdquoNeurocomputing vol 348 pp 74ndash81 2019

[5] Y-j Wu Y Cao and Q-f Wang ldquoAssembly sequenceplanning method based on particle swarm algorithmrdquo ClusterComputing vol 22 no S1 pp 835ndash846 2019

[6] YWang andD Tian ldquoA weighted assembly precedence graphfor assembly sequence planningrdquo$e International Journal ofAdvanced Manufacturing Technology vol 83 no 1ndash4pp 99ndash115 2016

[7] A Bedeoui R B Hadj M Hammadi et al ldquoAssembly se-quence plan generation of heavy machines based on thestability criterionrdquo $e International Journal of AdvancedManufacturing Technology 2019

[8] W Pan YWang and X-D Chen ldquoDomain knowledge basednon-linear assembly sequence planning for furniture prod-uctsrdquo Journal of Manufacturing Systems vol 49 pp 226ndash2442018

[9] M Bahubalendruni A Gulivindala S Varupala et alldquoOptimal Assembly Sequence generation through computa-tional approachrdquo Sadhana vol 44 no 8 p 174 2019

[10] M V A R Bahubalendruni A Gulivindala M KumarB B Biswal and L N Annepu ldquoA hybrid conjugated methodfor assembly sequence generation and explode view genera-tionrdquo Assembly Automation vol 39 no 1 pp 211ndash225 2019

[11] M R Bahubalendruni and B B Biswal ldquoAn intelligent ap-proach towards optimal assembly sequence generationrdquoProceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 232 no 4pp 531ndash541 2018

[12] L Zhang H Lv D Tan et al ldquoAdaptive quantum geneticalgorithm for task sequence planning of complex assemblysystemsrdquo Electronics Letters vol 54 no 14 pp 870ndash872 2018

[13] H S Wang C H Tu and K H Chen ldquoSupplier selection andproduction planning by using guided genetic algorithm anddynamic nondominated sorting genetic algorithm II ap-proachesrdquo Mathematical Problems in Engineering vol 2015pp 1ndash15 2015

[14] M Zhang L Wang Z Cui et al ldquoFast nondominated sortinggenetic algorithm II with levy distribution for network to-pology optimizationrdquoMathematical Problems in Engineeringvol 2020 Article ID 3094941 12 pages 2020

[15] X Hu Z Xu L Yang et al ldquoA novel assembly LineScheduling Algorithm based on CE-PSOrdquo MathematicalProblems in Engineering vol 2015 Article ID 685824 9 pages2015

[16] R Ab F Mohd A Tiwari et al ldquoIntegrated optimization ofmixed-model assembly sequence planning and line balancingusing Multi-Objective Discrete Particle Swarm


Optimizationrdquo Artificial Intelligence for Engineering DesignAnalysis and Manufacturing AIEDAM vol 33 no 3pp 332ndash345 2019

[17] Y Zhao W Li X Wang et al ldquoPath planning of slab librarycrane based on improved ant colony algorithmrdquo Mathe-matical Problems in Engineering vol 2019 Article ID7621464 16 pages 2019

[18] J Huo Z Wang F Chan et al ldquoAssembly line balancingbased on beam ant colony optimisationrdquo MathematicalProblems in Engineering vol 2018 Article ID 248143517 pages 2018

[19] J-F Tsai J G Carlsson D Ge Y-C Hu and J Shi ldquoIm-proved quantum-inspired evolutionary algorithm for engi-neering design optimizationrdquo Mathematical Problems inEngineering vol 2012 pp 1ndash7 2012

[20] W Hongbin D Jian and W Yueling ldquoHigh-order feedbackiterative learning control algorithm with forgetting factorrdquoMathematical Problems in Engineering vol 2015 Article ID826409 7 pages 2015

[21] M F F Ab Rashid W Hutabarat and A Tiwari ldquoMulti-objective discrete particle swarm optimisation algorithm forintegrated assembly sequence planning and assembly linebalancingrdquo Proceedings of the Institution of Mechanical En-gineers Part B Journal of Engineering Manufacture vol 232no 8 pp 1444ndash1459 2018

[22] L Shu M Dapino G Wu and D Chen ldquoFrequency-de-pendent sliding-mode control of Galfenol-driven unimorphactuator based on finite-element modelrdquo IEEE Transactionson Industrial Electronics vol 63 no 2 pp 1071ndash1082 2016

[23] Y Wang and J H Liu ldquoChaotic particle swarm optimizationfor assembly sequence planningrdquo Robotics and Computer-Integrated Manufacturing vol 26 no 2 pp 212ndash222 2010

[24] B Niu Y L Zhu X X He et al ldquoAn improved particle swarmoptimization based on bacterial chemotaxisrdquo Proceedings ofthe World Congress on Intelligent Control and Automation(WCICA) vol 1 pp 3193ndash3197 2006

[25] Y Cheng and Y Ren ldquoHeuristic search solvers for differentialgame model of CO2 emission in electricity marketrdquo Journal ofComputational Information Systems vol 8 no 19 pp 8151ndash8158 2012

[26] X Y Zhang C Li and Z Li ldquoOptimal reactive power dispatchbased on mixed bacterial chemotaxis algorithmrdquo AppliedMechanics and Materials vol 494-495 pp 1849ndash1852 2014


criteria Part concatenation method is robust and able toaddress ASP problem by considering all necessary assemblypredicates However such models are still matrix based Notonly is enormous human calculation required to investigatecomplex products but also human error would occur with alarge probability To address this problem a SAPN model isdeveloped to describe the operations priorities and con-straints in assembly process Each transition is associatedwith the predecessors by using assembly synchronizerswhich illustrates the precedence relationshipse transitionstatus is updated in each assembly operation and theassembility can be identified by the status checking

ASP optimization could be implemented when the as-sembly model is developed Generally ASP can be catego-rized into two types the SASP and the PASP e assemblyoperations of SASP are constructed in series and are easy tobe implemented Since all the operations need to follow alinear sequence the efficiency is limited Assembly opera-tions of PASP can be performed in parallel and the efficiencycan be increased However it is still challenging to assemblecomplex equipment with PASP To increase efficiency andcorrectness the PASP needs to be optimally designedMoreover optimization of PASP needs to be subjected tostrict predetermined constraints

Heuristic algorithm is an effective way to solve opti-mization problem with constraints Typical methods in-clude genetic algorithm [12ndash14] PSO [15 16] ant colonyoptimization algorithm [17 18] and other evolutionaryalgorithms [19ndash21] GA is a nonlinear estimation methodthat generates solutions to optimization problems usingtechniques inspired by natural evolution [22] To tackleASP of complex products a chaotic PSO approach togenerate the optimal or near-optimal assembly sequencesof products was developed in [23] Generally heuristicsearching needs to be implemented with numerical itera-tions e coding rules and heuristic rules under pre-determined constraints are important to construct theefficient searching Like what has been discussed most ofthe ASP models and the coding rules are matrix based esearchingiteration performance is limited due to thematrix calculations We have developed the SAPNmodel todescribe the operations priorities and constraints in theassembly process Coding rules of PSOBC such as positionvelocity and diversity function are represented with indexvectors instead of matrix e consequent efficiency can beincreased

As for the heuristic rules design the prematurity phe-nomenon is the major problem that needs to be addressede searching result could be attracted to the local optima ifthe heuristic rules are designed improperly Inspired by thephenomenon of chemotaxis in bacteria colonies PSOBCwasproposed to improve the performance of standard PSO [24]Characterized by an attraction-repulsion phase PSOBC notonly prevents premature convergence to a high degree butalso keeps amore rapid convergence rate than standard PSOis algorithm has been successfully used in areas like CO2emission problem [25] and hydropower stations operations[26] e global and local search performance could be wellbalanced when the particle swarm diversity is properly

controlled Here we use a similar PSOBC framework to solvethe PASP problem based on the proposed SAPN model ediversity function is built up to avoid population prema-turity Feasibility and effectiveness of the proposed methodare also verified via a case study Optimization results havebeen compared with standard PSO method under differentDPAs

2 Establishment and Analysis of SAPN

21 Basic Petri Net

Definition 1 (Petri net) Let N (P T F) be the basic Petrinet which should satisfy the following three conditions

(1) PcupTneOslashandPcapTOslash(2) FsubePtimesTcupTtimes P(3) dom (F)cup cod (F) PcupT

Here ldquotimesrdquo is the Cartesian productdom(F) x |existy (x y) isin F1113864 1113865 and cod(F) y |existx (x1113864

y) isin F Condition (3) is defined to ensure that there is noisolated place transition or arc in the Petri net

22 Symbols in SAPN

Definition 2 (synchronizer) Let p (T1 T2 (a1 a2)) be asynchronizer where forallti1

isinT1 ti2isinT2 p tbull

i1 bullti2

w(ti1

p) a2 w(pti2) a1 T1 t11 t12 t1m11113864 1113865

T2 t21 t22 t2m21113864 1113865 1lea1lem1 |T1| and 1lea2lem2 |T2|e scheme of the defined synchronizer is shown in

Figure 1 e SAPN consists of a set of synchronizerscorresponding to different assembly status e synchro-nizers are used to illustrate the parallel and selective relationsof each assembly operation including the correspondingprecedence relationships

Definition 3 (SAPN) Let SAPN( P T FWM PT TM) bethe synchronized assembly Petri net consisting of assemblysynchronizers Places transitions arcs and other tuples areintegrated in the net Detailed explanation of these tuples isgiven as follows

P pj (j 1 2 m) represents the assembly statusand m is the amount of the placesT ti (i 1 2 n) represents the assembly oper-ations and n is the amount of the transitionsF fk (k 1 2 l) represents the directed arcsbetween transitions and places and l is the amount ofthe arcsW wk F⟶ 1 2 3 represents the weightfunctions associated with arcs e weight functionsdetermine the amount of the required tasks for a placeto transform into another one e weight functionsshould be assigned based on number of the arcs whichshare the same inputoutput place in the correspondingsynchronizer For example there are m1 arcs betweenT1 and p sharing the same output place p in Figure 1 sothe weight functions of these arcs should be both



mi

0 ti is fired

1 ti is fireable


⎧⎪⎪⎨

⎪⎪⎩

(1)

PT pti21113966 1113967 T⟶ t i1












mi1 1 (2)


forallti2isin pti1

1113944 mi2 0andmi1

ne 0 (3)








min(time) 1113944S

s1times (4)


p

t11

t1m1 t2m2

t21

hellip hellip

a2 a1

a2 a1





Xd xd1 xd2 xdi xdn1113858 1113859 (5)



Vd vd1 vd2 vdi vdn1113858 1113859 (6)








Fd Pl( 1113857 1

|L| Pl

11138681113868111386811138681113868111386811138681113868

1113944

Pl| |

d1

1113944

n


2

11139741113972

(8)






Vg+1d ω times Vg


d1113872 1113873 + cgr2 Bg d minus Xg

d1113872 1113873

(9)

Xg+1d Xg

d + Vg+1d (10)

Vg+1d ω times Vg



d1113872 1113873

(11)


Xd1plusmnXd2



Vd1plusmnVd2



(15)


4 Case Study













O1

O2

O3

time1

d1

d2

d3

d4

d5 d7

d8

d6 d9

d10 d12

d11

d13


time


1234

5

6

789

10

11

12131415

16

171819202122232425

2627

28

29303132

33

3435

36

3738

39404142

43

4445464748

49

50




p1 0

0

0

0

t2

t1

p5

p6

p7

p8

p9

p10

p11

p12

p13

t5

t6

t7

t8

t9

t10

t11

t12

t13

p14

p15

p16

p17

p18

p19

p20

p21

t14 p24 p29t24

t29

t15

t16

t17

t18

t19

t20

p30 t30 p41 t40

p45p49

p44

t47

t46

t45

p46 t48

p48 t50

p47

t49

p40 t39

p42

p43

t41

t43

t44

t42

t21

p22 t22

p23 t23 p26 t26

p25 t25 p32

p31

t31

p33 t32

p34 t33

p35 t34

p36 t35

p37 t36

p38 t37

p39 t38

p27 t27

p28 t28

p2

p3

p4

t3

t4







Output IP

Create PDS


nr gt dp


to DSDS = RTS


nt gt 0


np = nt

np++

Input PT and M

Create IP

Create RTS

Create DS

Y N

Y

N

NY



itn = itnmax



Y N


equation 10


equation 11

Calculate Fd (Pl)

Fd (Pl) lt dl


Fd (Pl) lt dh


itn++



Y N

N

Y

N

Y








70

80

90

100

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(a)

55

60

65

70

75

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(b)

55

60

65

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(c)





5 Conclusions


Nomenclature
















chemotaxis

Data Availability




Acknowledgments


References































mi

0 ti is fired

1 ti is fireable


⎧⎪⎪⎨

⎪⎪⎩

(1)

PT pti21113966 1113967 T⟶ t i1












mi1 1 (2)


forallti2isin pti1

1113944 mi2 0andmi1

ne 0 (3)








min(time) 1113944S

s1times (4)


p

t11

t1m1 t2m2

t21

hellip hellip

a2 a1

a2 a1





Xd xd1 xd2 xdi xdn1113858 1113859 (5)



Vd vd1 vd2 vdi vdn1113858 1113859 (6)








Fd Pl( 1113857 1

|L| Pl

11138681113868111386811138681113868111386811138681113868

1113944

Pl| |

d1

1113944

n


2

11139741113972

(8)






Vg+1d ω times Vg


d1113872 1113873 + cgr2 Bg d minus Xg

d1113872 1113873

(9)

Xg+1d Xg

d + Vg+1d (10)

Vg+1d ω times Vg



d1113872 1113873

(11)


Xd1plusmnXd2



Vd1plusmnVd2



(15)


4 Case Study













O1

O2

O3

time1

d1

d2

d3

d4

d5 d7

d8

d6 d9

d10 d12

d11

d13


time


1234

5

6

789

10

11

12131415

16

171819202122232425

2627

28

29303132

33

3435

36

3738

39404142

43

4445464748

49

50




p1 0

0

0

0

t2

t1

p5

p6

p7

p8

p9

p10

p11

p12

p13

t5

t6

t7

t8

t9

t10

t11

t12

t13

p14

p15

p16

p17

p18

p19

p20

p21

t14 p24 p29t24

t29

t15

t16

t17

t18

t19

t20

p30 t30 p41 t40

p45p49

p44

t47

t46

t45

p46 t48

p48 t50

p47

t49

p40 t39

p42

p43

t41

t43

t44

t42

t21

p22 t22

p23 t23 p26 t26

p25 t25 p32

p31

t31

p33 t32

p34 t33

p35 t34

p36 t35

p37 t36

p38 t37

p39 t38

p27 t27

p28 t28

p2

p3

p4

t3

t4







Output IP

Create PDS


nr gt dp


to DSDS = RTS


nt gt 0


np = nt

np++

Input PT and M

Create IP

Create RTS

Create DS

Y N

Y

N

NY



itn = itnmax



Y N


equation 10


equation 11

Calculate Fd (Pl)

Fd (Pl) lt dl


Fd (Pl) lt dh


itn++



Y N

N

Y

N

Y








70

80

90

100

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(a)

55

60

65

70

75

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(b)

55

60

65

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(c)





5 Conclusions


Nomenclature
















chemotaxis

Data Availability




Acknowledgments


References
































Xd xd1 xd2 xdi xdn1113858 1113859 (5)



Vd vd1 vd2 vdi vdn1113858 1113859 (6)








Fd Pl( 1113857 1

|L| Pl

11138681113868111386811138681113868111386811138681113868

1113944

Pl| |

d1

1113944

n


2

11139741113972

(8)






Vg+1d ω times Vg


d1113872 1113873 + cgr2 Bg d minus Xg

d1113872 1113873

(9)

Xg+1d Xg

d + Vg+1d (10)

Vg+1d ω times Vg



d1113872 1113873

(11)


Xd1plusmnXd2



Vd1plusmnVd2



(15)


4 Case Study













O1

O2

O3

time1

d1

d2

d3

d4

d5 d7

d8

d6 d9

d10 d12

d11

d13


time


1234

5

6

789

10

11

12131415

16

171819202122232425

2627

28

29303132

33

3435

36

3738

39404142

43

4445464748

49

50




p1 0

0

0

0

t2

t1

p5

p6

p7

p8

p9

p10

p11

p12

p13

t5

t6

t7

t8

t9

t10

t11

t12

t13

p14

p15

p16

p17

p18

p19

p20

p21

t14 p24 p29t24

t29

t15

t16

t17

t18

t19

t20

p30 t30 p41 t40

p45p49

p44

t47

t46

t45

p46 t48

p48 t50

p47

t49

p40 t39

p42

p43

t41

t43

t44

t42

t21

p22 t22

p23 t23 p26 t26

p25 t25 p32

p31

t31

p33 t32

p34 t33

p35 t34

p36 t35

p37 t36

p38 t37

p39 t38

p27 t27

p28 t28

p2

p3

p4

t3

t4







Output IP

Create PDS


nr gt dp


to DSDS = RTS


nt gt 0


np = nt

np++

Input PT and M

Create IP

Create RTS

Create DS

Y N

Y

N

NY



itn = itnmax



Y N


equation 10


equation 11

Calculate Fd (Pl)

Fd (Pl) lt dl


Fd (Pl) lt dh


itn++



Y N

N

Y

N

Y








70

80

90

100

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(a)

55

60

65

70

75

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(b)

55

60

65

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(c)





5 Conclusions


Nomenclature
















chemotaxis

Data Availability




Acknowledgments


References







































O1

O2

O3

time1

d1

d2

d3

d4

d5 d7

d8

d6 d9

d10 d12

d11

d13


time


1234

5

6

789

10

11

12131415

16

171819202122232425

2627

28

29303132

33

3435

36

3738

39404142

43

4445464748

49

50




p1 0

0

0

0

t2

t1

p5

p6

p7

p8

p9

p10

p11

p12

p13

t5

t6

t7

t8

t9

t10

t11

t12

t13

p14

p15

p16

p17

p18

p19

p20

p21

t14 p24 p29t24

t29

t15

t16

t17

t18

t19

t20

p30 t30 p41 t40

p45p49

p44

t47

t46

t45

p46 t48

p48 t50

p47

t49

p40 t39

p42

p43

t41

t43

t44

t42

t21

p22 t22

p23 t23 p26 t26

p25 t25 p32

p31

t31

p33 t32

p34 t33

p35 t34

p36 t35

p37 t36

p38 t37

p39 t38

p27 t27

p28 t28

p2

p3

p4

t3

t4







Output IP

Create PDS


nr gt dp


to DSDS = RTS


nt gt 0


np = nt

np++

Input PT and M

Create IP

Create RTS

Create DS

Y N

Y

N

NY



itn = itnmax



Y N


equation 10


equation 11

Calculate Fd (Pl)

Fd (Pl) lt dl


Fd (Pl) lt dh


itn++



Y N

N

Y

N

Y








70

80

90

100

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(a)

55

60

65

70

75

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(b)

55

60

65

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(c)





5 Conclusions


Nomenclature
















chemotaxis

Data Availability




Acknowledgments


References































p1 0

0

0

0

t2

t1

p5

p6

p7

p8

p9

p10

p11

p12

p13

t5

t6

t7

t8

t9

t10

t11

t12

t13

p14

p15

p16

p17

p18

p19

p20

p21

t14 p24 p29t24

t29

t15

t16

t17

t18

t19

t20

p30 t30 p41 t40

p45p49

p44

t47

t46

t45

p46 t48

p48 t50

p47

t49

p40 t39

p42

p43

t41

t43

t44

t42

t21

p22 t22

p23 t23 p26 t26

p25 t25 p32

p31

t31

p33 t32

p34 t33

p35 t34

p36 t35

p37 t36

p38 t37

p39 t38

p27 t27

p28 t28

p2

p3

p4

t3

t4







Output IP

Create PDS


nr gt dp


to DSDS = RTS


nt gt 0


np = nt

np++

Input PT and M

Create IP

Create RTS

Create DS

Y N

Y

N

NY



itn = itnmax



Y N


equation 10


equation 11

Calculate Fd (Pl)

Fd (Pl) lt dl


Fd (Pl) lt dh


itn++



Y N

N

Y

N

Y








70

80

90

100

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(a)

55

60

65

70

75

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(b)

55

60

65

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(c)





5 Conclusions


Nomenclature
















chemotaxis

Data Availability




Acknowledgments


References


































Output IP

Create PDS


nr gt dp


to DSDS = RTS


nt gt 0


np = nt

np++

Input PT and M

Create IP

Create RTS

Create DS

Y N

Y

N

NY



itn = itnmax



Y N


equation 10


equation 11

Calculate Fd (Pl)

Fd (Pl) lt dl


Fd (Pl) lt dh


itn++



Y N

N

Y

N

Y








70

80

90

100

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(a)

55

60

65

70

75

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(b)

55

60

65

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(c)





5 Conclusions


Nomenclature
















chemotaxis

Data Availability




Acknowledgments


References



































70

80

90

100

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(a)

55

60

65

70

75

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(b)

55

60

65

0 30 60 90 120 150

Disa

ssem

bly

time

Generation

PSOPSOBC

(c)





5 Conclusions


Nomenclature
















chemotaxis

Data Availability




Acknowledgments


References






























5 Conclusions


Nomenclature
















chemotaxis

Data Availability




Acknowledgments


References




































chemotaxis

Data Availability




Acknowledgments


References










































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A Novel Parallel Assembly Sequence Planning Method for …downloads.hindawi.com/journals/mpe/2020/7848329.pdf · 2020-05-26 · A Novel Parallel Assembly Sequence Planning Method