mathematical model construction of the production workshop

Research ArticleMathematical Model Construction of the Production WorkshopBased on the Complex Network and Markov Theory

Gang Guo 12 and Fengjing Shao 1

1Institute of Complexity Science Qingdao University Qingdao 266071 China2Qingdao Open University Qingdao 266199 China

Correspondence should be addressed to Fengjing Shao sfjqdueducn

Received 10 April 2021 Revised 7 May 2021 Accepted 25 May 2021 Published 8 June 2021

Academic Editor Libor Pekar

Copyright copy 2021 Gang Guo and Fengjing Shao -is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

Because of the advantages of the complex network in describing the interaction between nodes the complex network theory isintroduced into the production process of the modern workshop in this paper According to the characteristics of the workshopbased on extracted key nodes the complex network model of the workshop is constructed to realize the mathematical descriptionof the production process of the workshop Aiming at the multidisturbance factors in the production process of the workshop thekey disturbance factors are predicted based on the Markov method and the propagation dynamics model close to the actualproduction of the workshop is established Finally the bottleneck prediction model of the workshop under the disturbanceenvironment is established -e simulation results show that the proposed prediction model is in good agreement with the actualdata and the coincidence rate is as high as 937

1 Introduction

Production workshop is the basic unit of manufacturingenterprises which consists of equipment personnel de-partments and various workpieces In the productionworkshop the workpiece is processed to the final productaccording to the established process route and the process isaccompanied by many uncertain factors -ese disturbancefactors lead to the generation transfer and disappearance ofthe bottleneck in the workshop and the bottleneck unitrestricts the effective output of the production system-erefore it is necessary to study the dynamic prediction ofthe bottleneck in the workshop under the disturbance en-vironment for improving themanagement of the productionsystem

In the manufacturing process how to establish a rea-sonable and accurate mathematical model to identify andpredict bottlenecks can provide a theoretical basis for en-terprises to make production plans and management Wuet al [1] thought that from the point of view of complexnetwork statistical parameters the cooperation supply and

demand information transmission and other aspects be-tween production enterprises show self-similar character-istics Hao et al [2] used simulation software to simulate andanalyze the production process of the production system-e research showed that the evolution process of theproduction system was extremely sensitive to the systemparameters Sun and Bin [3] analyzed that all kinds ofmanufacturing workshop cell network mouldmanufacturing network and all kinds of supply chainnetwork show small-world characteristics

In recent years more and more researchers have appliedthe complex network theory to the manufacturing industryin order to make new breakthroughs in complex productdevelopment supply chain optimization and enterpriseproduction management and optimization Ma et al [4]proposed the concept of collaborative manufacturing chainbased on the complex network theory for the effectiveutilization of resources in the complex environment andfinally realized the optimal utilization of resources Cao et al[5] modelled the production process of the product with themodular network analyzed the evolution of the product

HindawiMathematical Problems in EngineeringVolume 2021 Article ID 1912442 10 pageshttpsdoiorg10115520211912442

module by using the brittleness theory of the complexnetwork and improved the antirisk ability of the modularproduct network Funke and Becker [6] analyzed the ap-plication of the complex network theory in themanufacturing system and provided a new method fornetwork modelling of the manufacturing system Giret et al[7] applied the complex network theory to the service-oriented manufacturing network and analyzed the networkstructure and statistical parameters of the service-orientedmanufacturing network

Nowadays artificial intelligent algorithms had beenapplied in the construction for different applications Keaet al [8] proposed a newmodel to estimate the disc cutter lifeby integrating a group method of data handling- (GMDH-)type neural network with a genetic algorithm Wei et al [9]proposed an analytical method for estimating the horizontaltransition probability matrix which is one of the importantinput parameters for the coupled Markov chain model Shenet al [10] presented an automatic pumping-recharge systemto maintain groundwater balance during dewatering Linet al [11] presented an approach for eutrophication eval-uation based on the technique for order preference Zhanget al [12] proposed an artificial intelligence model to predictground settlement Ssl et al [13] proposed an intelligentframework for predicting the advancing speed during earthpressure balance

-e definition and classification of bottlenecks are dif-ferent according to different research objects differentmethods and different observation angles Wang et al [14]regarded the queuing number of parts in the processing areaas the bottleneck identification index and the bottleneck isthe one with the largest queuing number Samouei et al [15]studied the bottleneck in the workshop from various aspectsof the equipment parameters and identified the bottleneckunit by building a time-based mathematical model Azadehand Shoja [16] proposed a network model of bottleneckidentification which converted the main units of theworkshop into a network model to identify the bottlenecklocation from the perspective of the system Rui and Cheng[17] studied the workshop of the assembly line productionmode constructed a mathematical model based on thebalance rate of the production line and identified thebottleneck in real time Based on the TOC constraint theoryBin and Sun [18] combined with the intelligent algorithmand improved some processes of the previous genetic al-gorithm so as to optimize the bottleneck identificationmethod According to the linear programming methodBrucker et al [19] constructed a programming model for theobjective function and constraints of the productionworkshop to identify the bottleneck unit of the productionsystem Masoud et al [20] used the model graph and net-work model construction algorithm to identify the real-timebottleneck in the production workshop -urer and Ste-venson [21] used simulation software to simulate the mainproduction factors of the production system and comparedthe simulation data with the theoretical model and work-shop data to identify the real-time bottleneck Sweeney et al[22] carried out orthogonal experiments on the factors thataffect the output target of the system in the production

workshop compared with the mobile bottleneck generatorand verified the superiority of the orthogonal experimentbottleneck identification method

-e existing bottleneck research methods are limited to aspecific parameter index such as equipment station per-sonnel and materials which lack systematic thinking -ebottleneck identification method in this paper is based onthe complex network theory the Markov model is simpleand it can be used to describe complex random phenomenaMarkov process is used to analyze the disturbance factorsand the state transition probability and steady distribution ofthe Markov chain are obtained it provides a way for thequantitative analysis of the disturbance factors in the pro-duction workshop From the perspective of the complexnetwork all production factors are considered to identifyand predict the bottleneck And then the accuracy andeffectiveness of the proposed method are verified by theactual data of the workshop and simulation software

2 Disturbance Factorsrsquo Analysis of theProduction Workshop Based on theMarkov Model

In this paper Markov model is used to evaluate and predictthe intensity of disturbance factors -e method has a goodeffect on process state prediction and it can be used forproduction site state prediction However it is not suitablefor medium- and long-term system prediction From theperspective of the production process the disturbancefactors of the workshop are analyzed the disturbance factorintensity matrix is constructed the relationship between thematrix and disturbance factor intensity is simulated and theMarkov chain prediction model is established to predict thechange of disturbance factor intensity so as to determine thekey disturbance factors and their occurrence probability-eflowchart indicating the bottleneck prediction of the pro-duction workshop is shown in Figure 1

In each link of the workshop there will be a variety ofdisturbance factors such as demand change emergencyorder insertion equipment failure machining accuracy andpersonnel absence At the same time there will be a lot ofinaccurate information such as material arrival timemanual clamping time and auxiliary processing time -eexistence of these disturbance factors will make the work-shop conditions change dynamically and seriously affect thenormal production activities

-e disturbance in the workshop mainly refers to thefactors that affect the effective output of production unitssuch as equipment personnel process and departmentDisturbance factors can be divided into the following fourcategories internal production environment externalenvironment monitoring technology and human factorsDisturbance factors caused by the internal productionenvironment although the production in themanufacturing workshop has been oriented to stan-dardization and precision the microdifferences andrandomness in time cannot be eliminated -ese uncer-tainties are mostly related to the internal production

2 Mathematical Problems in Engineering

environment such as equipment random failure andequipment accuracy error Disturbance factors caused bythe external environment they are materials not trans-ported in place according to regulations order changesetc -e disturbance factors caused by the external en-vironment will lead to the cancellation of existing pro-duction tasks or the change of production schedule whichis a kind of disturbance factors with great influence on thedetermination of results -e disturbance factors causedby monitoring technology include detection methoddetection time and detection environment -e influenceof such disturbance factors is relatively weak Disturbancefactors caused by human factors mainly included workersabsence workersrsquo proficiency and workersrsquo mood -esedisturbance factors affect the quality and productionschedule of the product directly

-e intensity of the disturbance factor is a quantitativedescription of the disturbance factor -e expression ofdisturbance factor intensity is different in different fields-e intensity of the disturbance factor reflects an inherentdynamic characteristic of the disturbance factor which isused to describe the direct influence of the disturbancefactor on the workshop network -e influence of differentdisturbance factors on the workshop is different Chinet al [23] classified the disturbance factors into four typesexternal disturbance internal disturbance monitoringdisturbance and human disturbance -e bottleneckinfluencing factor matrix is shown in Table 1

-ere are various uncertain disturbance factors in theprocess of production and processing -e occurrence ofdisturbance factors may lead to the generation transferincrease or decrease of bottlenecks in the productionworkshop In this paper the intensity of the disturbancefactor is defined to express the comprehensive action degreeof various disturbance factors in the process of productionand processing -e intensity of the disturbance factor isexpressed as follows

I 1113944n

i1widi i 1 2 n (1)

where wi represents the importance of each disturbancefactor and di represents the measurement indexes of theintensity of each human disturbance factor to the distur-bance factor

Because the intensity of disturbance factors in theproduction workshop is a comprehensive quantitative indexof various disturbance factors and the intensity of distur-bance factors is a fuzzy quantity the intensity of disturbancefactors in the production workshop can be measured by thefuzzy evaluation method at all levels analytic hierarchyprocess expert scoring and other comprehensive methods-e specific steps are as follows

(i) Step 1 the set of disturbance factors in the work-shop is established D d1 d2 d3 d41113864 1113865 is the dis-turbance factorsrsquo set Analytic hierarchy process

Complex network model of the productionprocess in the workshop

Complex network

Node degreeAverage

degree valueClusteringcoefficient

Analysis of disturbance factors in the workshop

Disturbance factorsrsquodivision

Disturbance factorsrsquoassessment

Markov analysis of disturbance factors

Key disturbancefactors


Nonkeydisturbance factors

N

Y

x1(T)

x2(T)

Time T

Node stateprediction

model

x1(T + T0)

x2(T + T0)

Time T + T0

Prediction of bottleneck nodes in the workshop

Bottleneckprediction

model

Bottleneckdetermination

Nonbottleneckunit set

Bottleneck unitset

N

Y

Number ofbottlenecks gt1


Bottleneckpolymorphismdetermination

Y

N

Main bottleneck

Secondarybottleneck

Bottleneck determination mechanism

Figure 1 -e flowchart indicating the bottleneck prediction of the production workshop

Mathematical Problems in Engineering 3

(AHP) is used to measure the disturbance intensityW w1 w2 w3 w41113864 1113865 1113936

4i1 wi 1 wi ge 0 -e

weight of fuzzy subsets in D iswi wi1 wi1 wim1113864 1113865 1113936

mi1 wim 1 wim ge 0

(ii) Step 2 the workshop influence evaluation set isestablished By using the expert scoring method themanufacturing workshop bottleneck research ex-perts score the intensity of disturbance factors in theproduction workshop and grade different distur-bance factors Each disturbance factor is dividedinto five levels according to the influence level andthe level from low to high represents the influencedegree It is assumed that the evaluation set isC c1 c2 c3 c4 c51113864 1113865 ci represents the influence ofeach disturbance factor on the intensity of thedisturbance factor in the workshop

(iii) Step 3 the evaluation of the disturbance factor todisturbance factor intensity is obtained by expertscoring fuzzy function f D⟶ F(V) is estab-lished F(V) is the fuzzy complete set of subset Vand di⟶ f(di) (di1 di2 di3 di4 di5) is thefeedback of disturbance factor di in the workshop tothe expert scoring system and evaluation set It isassumed that the feedback vector of disturbancefactors to bottleneck set V is Ri ri1 ri2 ri51113864 1113865after fuzzy transformation the following formulacan be obtained

Bi wi middot Ri b1 b2 b3 b4 b5( 1113857 (2)

-e obtained values are imported into the upper layer bythe analytic hierarchy process and then evaluated Finallythe overall disturbance factor intensity is calculated as

D W middot R D1 D2 D3 D4 D5( 1113857 (3)

-e bottleneck of the production workshop is producedin the production activities and it is the result of thecomprehensive effect of various disturbance factors in theproduction activities -e intensity of disturbance factorscaused by human disturbance factors is divided into statesMarkov model is used to predict the divided states Finallythe probability of each state is obtained to predict the keydisturbance factors

In this paper Markov chain is used to model the dis-turbance intensity caused by disturbance factors in theworkshop S s1 s2 sn1113864 1113865 represents the state set it is thedivision of the disturbance factor intensity and the proba-bility measure of the disturbance factor intensity under thecomprehensive action of each disturbance factor in theproduction workshop S can be preliminarily determinedaccording to the historical data of disturbance factors in theproduction workshop -e state of disturbance factors isdefined in Table 2

-e initial probability of the state set is determined byfitting the curve with the historical data of disturbance factorsin the workshop-e probability of the initial state is expressedby vector a

a a1 a2 ak1113858 1113859 (4)

where ak represents the occurrence probability of state kIn the Markov process xk is the state in time step tk

xi(k) P(xk i) the probability in matrix Pij representsthe probability that the state is i at this moment and j at thenext moment -e matrix expression is as follows

P

p11 middot middot middot p1n

⋮ ⋱ ⋮

pn1 middot middot middot pnn

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (5)

where pij is defined as follows

pij Nij

1113936nj1 Nij

(6)

and Nij is the statistical historical data it represents thenumber of times that the node state changes from i to j in theproduction activity of the workshop

According to the Markov theory the probability of thestate of disturbance intensity in the workshop can be cal-culated as follows

1113954p a middot p (7)

3 Bottleneck Prediction of the ProductionWorkshop Based on the ComplexNetworkunder theDisturbanceEnvironment

In a workshopNworkpieces are produced and processed onM sets of equipment and workpieces of the same specifi-cation have their own process routes It is assumed that thesame equipment can only process one workpiece at the sametime and the processing sequence is first come first process-ere are time constraints and process constraints in theprocessing of workpieces Different equipment processesdifferent workpieces with different times and processes -evariables are defined as follows

-e workpiece sequence is represented byJ Jj|j 1 2 N1113966 1113967 Jj represents the j-th workpiece theequipment sequence is represented byQ Qi|i 1 2 M1113864 1113865 Qi represents the i-th equipmentand the process sequence is represented by Pp then theprocess matrix of N workpieces produced and processed onM sets of equipment is defined as

Pp

Pp1(1) middot middot middot Pp1(n)

⋮ ⋱ ⋮

Ppj(1) middot middot middot Ppj(n)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (8)

where n represents the number of processes and Pij(n)

represents the n-th process of the j-th workpiece producedon the i-th equipment

Machining time series of the workpiece is represented byTi which is defined as follows


Ti

Ti1(1) middot middot middot Ti1(n)

⋮ ⋱ ⋮

Tij(1) middot middot middot Tij(n)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (9)

where Tij(n) represents the machining time of the n-thprocess of the j-th workpiece on the i-th equipment

Each resource node (such as workshop departmentequipment personnel and tools) involved in the productionprocess of a production workshop is regarded as a networknode -e possible process routes and logistics paths be-tween nodes are regarded as the connecting edges in thenetwork -e direction of the connecting edges betweennodes is determined by the priority relationship of processesas the weight on the connected edge of the network thedevice load is used to measure the closeness of the rela-tionship between nodes -erefore each production processconstitutes a complex multitask weighted directed networkmodel An example of a workshop network model is shownin Figure 2

Figure 2 represents a workshop production processwhich consists of two workshops node P represents theresources (equipment personnel departments etc) in themanufacturing system the direction of the edge representsthe flow direction of the logistics process and the node stateis represented by the load of the node -us the disturbancefactors can be described as follows

Pp Pp + ΔPp

xi(t) xi(t) + Δxi(t)

ϑ ϑ + Δϑ

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)

where Pp represents changes in disturbance factors ΔPp

represents the process matrix change increment Δxi(t)

represents the fluctuation of the node state caused by dis-turbance factors and Δϑ represents the influence of dis-turbance factors on network coupling strength

Considering the interaction between nodes and thedynamic characteristics of nodes in the whole network a

coupled map lattice node state prediction model with thenetwork scale of n is constructed in this paper

xi(t + 1) (1 minus ϑ)f xi(t)( 1113857 +ϑ1113936

Nj1jne i aij + rij1113872 1113873

k(i)

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868 (11)

where xi(t) represents the state value of node i at time txi(t + 1) represents the calculated value of the next time stepof the node k(i) represents the degree of node i in thecomplex network aij represents the network connectionmatrix ϑ represents the coupling strength between nodesfunction f represents the dynamic behaviour of nodes andrij represents the fluctuation of the connection matrix afterdisturbance

In this paper from the perspective of the complexsystem considering the network characteristics such as nodedegree value and clustering coefficient the bottleneckjudgment standard is given Finally according to thejudgment standard the bottleneck classification is imple-mented such as primary and secondary bottlenecks andnonbottlenecks [24] Suppose the bottleneck criterion is τthe calculation formula of τ is as follows

τ αC + βK + cL (12)

where C represents the network clustering coefficient K

represents the node degree L represents the node load andα β and c represent weights and α + β + c 1

-erefore the bottleneck identification formula in thenetwork is defined as follows

BSN Xi(t)ge τ1113864 1113865

BSnonminusN 0ltXi(t)lt τ1113864 1113865(13)

where BSN represents the bottleneck sequence and BSnonminusN

represents the nonbottleneck sequenceIn order to analyze the fluctuation and influence of

disturbance factors on the production process of theworkshop the load change rate Fi is defined to describe thefluctuation caused by disturbance factors

Table 1 -e bottleneck influencing factor matrix

Disturbance factorsDisturbance intensity level

Level 1 Level 2 Level 3 Level 4 Level 50-01 01ndash03 03ndash05 05ndash08 08ndash10

External disturbance factors

Order change lowast

Material supply lowast

Production plan lowast

Policy changes lowast

Internal disturbance factors

Equipment failure lowast

Route change lowast

Machining accuracy lowast

Dispatcher change lowast

Human factors Personnel absence lowast

Proficiency lowast

Monitoring factorsMonitoring method lowast

Monitoring technology lowast

Monitoring environment lowast


Fi 1113944T

t1limΔt⟶0

xi(t + 1) minus xi(t)1113868111386811138681113868

1113868111386811138681113868

Δt (14)

where T represents the total simulation time

4 Simulation Results and Discussion

41 Analysis of Disturbance Factors In the experiment theactual production workshop data of an automobilemanufacturing enterprise [25] are taken as an exampleMarkov process is used to analyze the disturbance factorsFirstly the analytic hierarchy process is used to analyzemany disturbance factors and determine their respectiveweights as shown in Table 3

After classifying and subdividing the disturbance factorsand calculating the weight the occurrence frequency is fittedaccording to the historical data of the disturbance factors inthe production workshop -e state probability transitionmatrix is determined according to Table 3 in which the stateprobability transition diagram and probability matrix are asfollows

P

0382 0387 0288 0 0

0522 0330 0120 0120 0

0450 0270 0350 0120 0

0 1 0 0 0

0 0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(15)

-e production process parameters of the workshop areshown in Table 4

According to Table 4 the node state transition frequencymatrix in the process of production activities is calculated

N

6 6 4 0 0

6 3 2 1 0

3 2 3 0 0

0 2 0 0 0

0 0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)

According to the expert scoring method the initial stateprobabilities of ten disturbance factors to the disturbancefactor intensity are determined as follows

a 04 03 02 01 001113858 1113859 (17)

In this way the prediction probability of each node stateSi will be calculated as follows

1113954p a middot p 0382 0380 0201 0027 01113858 1113859 (18)

It can be seen that in the production process in eachstate of 23 production process indicators the key distur-bance factors are equipment failure material supply orderchange and so on and the influence of the occurrenceprobability of these disturbance factors accounted for 862

42 Bottleneck Prediction Model Simulation Aiming at anauto parts production workshop the data of the productionprocess are collected in hours by sensors on the productionline According to the working characteristics resource flowprocess constraints information flow and personnel allo-cation of the production workshop the data are collectedand the key nodes are extracted and the complex networkmodel is constructed to realize the network mathematicaldescription of the production process of the productionworkshop -e specific production data of 10 sets ofequipment and 6 workpiecesrsquo sequence are shown in Table 5

-e layout of the workshop is shown in Figure 3After the initial state of each node is set the subsequent

state of each node is calculated according to the predictionmodel -e node state values without disturbance are shownin Table 6

Table 6 shows that the state values of each node arebasically stable without disturbance and there is no bot-tleneck unit

When the disturbance factor is added the node statevalues of the workshop after 3 hours of disturbance areshown in Table 7

According to the criterion of bottleneck judgment [26]the state value of each node is counted to judge the bot-tleneck -e results are shown in Table 8

As shown in Tables 7 and 8 before the disturbance thestate value of the workshop nodes fluctuates little and thereis no bottleneck unit After the disturbance factor is addedthe state value of each node begins to be affected -roughsimulation calculation it is found that equipment P6 is theprimary bottleneck and equipment P5 is the secondarybottleneck -rough the analysis of Table 8 it is found thatthe degree value and clustering coefficient of the bottleneckunit are larger than other nodes and the bottleneck unitappears on the equipment directly affected by orderinsertion

In order to verify the rationality and correctness of themodel the production process of the workshop is simulatedand analyzed-rough the simulation analysis when there isno disturbance factor the production activities of the

Table 2 State of disturbance factors

Frequency of disturbance factors StateFgt 7 s54leFlt 7 s42leFlt 4 s30leFlt 2 s2Fge 0 s1


workshop are normal and there is no bottleneck With theinput of the disturbance factor the process matrix changesresulting in the changes of the network topology clusteringcoefficient node degree value node state and processmatrix Accuracy index is used to verify the accuracy of theprediction model and it is defined as follows

accuracy (TP + TN)

(TP + FP + TN + FN) (19)

where TP represents true positive FP represents falsepositive FN represents false negative and TN representstrue negative

-e fluctuation caused by disturbance is positivelycorrelated with the degree value but not with betweennessand clustering coefficient After the disturbance the pro-posed prediction method in this paper is in good agreementwith the actual data and the coincidence rate is as high as937 -e coincidence rate with other prediction methods

P1

P2

P

P9 P10

P3

P4

P5 P6

P8

P7

Operator 1

Workshop 1

Operator n

Equipment 1

Equipment n

Workshop 1

Operator 1

Operator n

Equipment 1

Equipment 1

Production resources

Production department

Quality check

Figure 2 An example of a workshop production network model

Table 3 Disturbance factors and their weights in the workshop

Symbol a1 a11 a12 a13 a14 a15Disturbancefactors

Externaldisturbance Order change Material supply Demand change Policy change Environment

changeWeight 05637 01947 04628 01889 00725 00733Symbol a2 a21 a22 a23 a24Disturbancefactors

Internaldisturbance

Equipmentfailure Process change Machining accuracy Workshop

schedulingWeight 02631 02751 00643 05502 01236Symbol a3 a31 a32 a33Disturbancefactors Human factor Personnel

absence Skill level Quality defects

Weight 01202 00927 05201 03896Symbol a4 a41 a42 a43Disturbancefactors Monitoring Monitoring

methodMonitoringtechnology

Monitoringenvironment

Weight 00565 06386 01037 02579


Table 4 -e production process parameters of the workshop

Ai Di Fi Si

A1 (order information) 02652 1 S1A2 (material purchasing plan) 03275 0 S1A3 (demand risk analysis) 01861 1 S1A4 (initial data) 03785 1 S2A5 (material arrival time) 03952 0 S2A6 (manual clamping time) 03562 1 S1A7 (auxiliary processing time) 03283 0 S1A8 (industry policy changes) 03183 1 S2A9 (market environment changes) 05902 3 S3A10 (equipment failure rate) 03392 0 S2A11 (aging degree of equipment) 03287 1 S1A12 (importance of equipment) 03916 1 S3A13 (route layout) 03285 1 S2A14 (process constraints) 03852 0 S2A15 (accuracy of machining different parts) 03907 1 S2A16 (selection of the production scheduling mode) 05589 2 S3A17 (scheduling objectives) 03205 0 S1A18 (staff leave rate) 05960 3 S3A19 (processing pass rate) 02298 2 S3A20 (time required for monitoring) 03805 5 S4A21 (monitoring environmental impacts) 03607 3 S4A22 (production line balance rate) 02385 1 S2A23 (change of delivery date) 03762 2 S3

P1P2

P3

P4P8

P5

P6

P7P9

P10

Equipment

Buffer

Figure 3 -e layout of the workshop

Table 5 Original data of workpiece production

Workpiece Process routearrival rate per unit timeprocessing rate per unit timeJ1 P135120 P240110 P34090 P435100 P535130 mdashJ2 P1035100 P940100 P83090 P74080 P640110 mdashJ3 P1025100 P33580 P63090 mdash mdash mdashJ4 P130110 P830120 P53080 mdash mdash mdashJ5 P145120 P445150 P640160 mdash mdash mdashJ6 P1045110 P255100 P855120 P645130 P555110 mdash


[27 28] is also high it shows that the theoretical analysis ofthe disturbance factors in the proposed prediction model isconsistent with the simulation results which verifies thecorrectness and rationality of the proposed model

5 Conclusions

In this paper a complex network is introduced into theproduction process of the production workshop Accordingto the characteristics of the production workshop such asthe working characteristics resource flow process con-straints information flow and personnel allocation the dataare collected and the key nodes are extracted and then thecomplex network model is constructed to realize the net-work mathematical description of the production process ofthe production workshop -e disturbance factors are de-scribed mathematically and the mechanism of the distur-bance factors in the complex network system is constructed

so as to identify the bottleneck position of the productionworkshop under the disturbance factors

Data Availability

-e basic data used in this article are downloaded from theonline public dataset weighted tardiness (httppeoplebrunelacuksimmastjjbjebinfohtm)

Conflicts of Interest

-e authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

-is work was supported by a grant from the NationalNatural Science Foundation of China (no 91130035)

Table 6 Node state values without disturbance

Time (d)Node state values

P1 P2 P3 P4 P5 P6 P7 P8 P9 P101 0875 0400 0875 0625 0538 0763 0429 0542 0365 06252 0677 0706 0687 0809 0925 0736 0659 0980 0937 08363 0675 0833 0738 0639 0206 0755 0482 0196 0286 05694 0565 0686 0737 0579 0198 0859 0783 0837 0691 08965 0763 0902 0849 0782 0390 0491 0509 0359 0425 07396 0803 0285 0729 0832 0849 0729 0492 0401 0839 06857 0586 0339 0605 0572 0449 0839 0837 0938 0729 08178 0806 0839 0358 0609 0362 0937 0885 0829 0582 0606

Table 7 Node state values after 3 hours of disturbance


P1 P2 P3 P4 P5 P6 P7 P8 P9 P101 0875 0400 0875 0625 0538 0763 0429 0542 0365 06252 0677 0706 0687 0809 0925 0736 0659 0980 0937 08363 0875 0952 0896 1000 0984 1255 0482 1000 0325 10004 0865 0977 0904 0892 0795 1389 0886 0949 0892 09165 0763 0902 0849 0782 1286 0695 0616 0479 0495 07396 0933 0858 0786 0897 1497 0881 0797 0771 0739 06577 0266 0217 0605 0572 1930 0780 0633 0814 0796 08178 0766 0583 0658 0509 3362 0837 0775 0626 0594 0316

Table 8 Statistics of workshop network parameters

Node Node degree Clustering coefficient State fluctuation rate Node average state value Bottleneck nodeP1 3 0667 0196 0722 mdashP2 4 0333 0175 0672 mdashP3 4 0167 0189 0736 mdashP4 5 0500 0072 0741 mdashP5 3 0333 1020 1106 Secondary bottleneckP6 6 0667 1108 1528 Primary bottleneckP7 2 0167 0207 0532 mdashP8 3 0200 0433 0602 mdashP9 4 0500 0051 0622 mdashP10 3 0333 0176 0752 mdash


References

[1] B Wu P Liu and X Xu ldquoAn evolutionary analysis of low-carbon strategies based on the government-enterprise game inthe complex network contextrdquo Journal of Cleaner Productionvol 141 pp 168ndash179 2017

[2] X Hao H An H Qi and X Gao ldquoEvolution of the exergyflow network embodied in the global fossil energy trade basedon complex networkrdquo Applied Energy vol 162 pp 1515ndash1522 2016

[3] G Sun and S Bin ldquoRouter-level internet topology evolutionmodel based on multi-subnet composited complex networkmodelrdquo Journal of Internet Technology vol 18 no 6pp 1275ndash1283 2017

[4] F Ma H Xue K F Yuen et al ldquoAssessing the vulnerability oflogistics service supply chain based on complex networkrdquoSustainability vol 12 no 5 p 1991 2020

[5] H Cao H Zhang and M Liu ldquoGlobal modular productionnetwork from system perspectiverdquo Procedia Engineeringvol 23 pp 786ndash791 2011

[6] T Funke and T Becker ldquoComplex networks of material flowin manufacturing and logistics modeling analysis andprediction using stochastic block modelsrdquo Journal ofManufacturing Systems vol 56 pp 296ndash311 2020

[7] A Giret E Garcia and V Botti ldquoAn engineering frameworkfor service-oriented intelligent manufacturing systemsrdquoComputers in Industry vol 81 pp 116ndash127 2016

[8] B Kea B Slsa and C Az ldquoPrediction of disc cutter life duringshield tunneling with AI via the incorporation of a geneticalgorithm into a GMDH-type neural networkrdquo Engineeringvol 7 no 2 pp 238ndash251 2021

[9] C Wei C Az and D Slsc ldquoAn analytical method for esti-mating horizontal transition probability matrix of coupledMarkov chain for simulating geological uncertaintyrdquo Com-puters and Geotechnics vol 129 p 2021

[10] S L Shen H M Lyu and A Zhou ldquoAutomatic control ofgroundwater balance to combat dewatering during con-struction of a metro systemrdquo Automation in Constructionvol 123 no 5 103536

[11] S-S Lin S-L Shen A Zhou and Y-S Xu ldquoApproach basedon TOPSIS and Monte Carlo simulation methods to evaluatelake eutrophication levelsrdquo Water Research vol 187p 116437 2020

[12] K Zhang H-M Lyu S-L Shen A Zhou and Z-Y YinldquoEvolutionary hybrid neural network approach to predictshield tunneling-induced ground settlementsrdquo Tunnellingand Underground Space Technology vol 106 p 103594 2020

[13] A Ssl B Slsa and Z B Ning ldquoModelling the performance ofEPB shield tunnelling using machine and deep learning al-gorithmsrdquo Geoscience Frontiers vol 12 no 5 Article ID101177 2021

[14] J-Q Wang J Chen Y Zhang and G Q Huang ldquoSchedule-based execution bottleneck identification in a job shoprdquoComputers amp Industrial Engineering vol 98 pp 308ndash3222016

[15] P Samouei P Fattahi and J Ashayeri ldquoBottleneck easing-based assignment of work and product mixture determina-tion fuzzy assembly line balancing approachrdquo AppliedMathematical Modelling vol 40 no 7 pp 4323ndash4340 2016

[16] A Azadeh and B M Shoja ldquoA neural network meta-modelfor identification of optimal combination of priority dis-patching rules and makespan in a deterministic job shopscheduling problemrdquo International Journal of AdvancedManufacturing Technology vol 67 no 8 pp 1549ndash1561 2013

[17] Z Rui and W Cheng ldquoBottleneck machine identificationmethod based on constraint transformation for job shopscheduling with genetic algorithmrdquo Information Sciencesvol 188 pp 236ndash252 2012

[18] S Bin and G Sun ldquoOptimal energy resources allocationmethod of wireless sensor networks for intelligent railwaysystemsrdquo Sensors vol 20 no 2 p 482 2020

[19] P Brucker E K Burke and S Groenemeyer ldquoA mixed in-teger programming model for the cyclic job-shop problemwith transportationrdquo Discrete Applied Mathematics vol 160no 13 pp 1924ndash1935 2012

[20] M Masoud E Kozan and G Kent ldquoA job-shop schedulingapproach for optimising sugarcane rail operationsrdquo FlexibleServices and Manufacturing Journal vol 23 no 2 pp 181ndash206 2011

[21] M -urer and M Stevenson ldquoOn the beat of the drumimproving the flow shop performance of the Drum-Buffer-Rope scheduling mechanismrdquo International Journal of Pro-duction Research vol 56 no 9 pp 3294ndash3305 2018

[22] K D Sweeney D C Sweeney and J F Campbell ldquo-eperformance of priority dispatching rules in a complex jobshop a study on the Upper Mississippi Riverrdquo InternationalJournal of Production Economics vol 216 pp 154ndash172 2019

[23] K-S Chin D-W Tang J-B Yang S Y Wong and HWangldquoAssessing new product development project risk by Bayesiannetwork with a systematic probability generation method-ologyrdquo Expert Systems with Applications vol 36 no 6pp 9879ndash9890 2009

[24] S Bin G Sun N Cao et al ldquoCollaborative filtering rec-ommendation algorithm based on multi-relationship socialnetworkrdquo Computers Materials amp Continua vol 60 no 2pp 659ndash674 2019

[25] G Sun C-C Chen and S Bin ldquoStudy of cascading failure inmultisubnet composite complex networksrdquo Symmetryvol 13 no 3 pp 523ndash537 2021

[26] G Tian S Zhou G Sun and C-C Chen ldquoA novel intelligentrecommendation algorithm based on mass diffusionrdquo Dis-crete Dynamics in Nature and Society vol 2020 no 11pp 1ndash9 Article ID 4568171 2020

[27] W Fang Y Guo and W Liao ldquoA Parallel Gated RecurrentUnits (P-GRUs) network for the shifting lateness bottleneckprediction in make-to-order production systemrdquo Computersamp Industrial Engineering vol 140 no 2 pp 1ndash12 2020

[28] A Yazdanbakhsh B -waites H EsmaeilzadehG Pekhimenko O Mutlu and T C Mowry ldquoMitigating thememory bottleneck with approximate load value predictionrdquoIEEE Design amp Test vol 33 no 1 pp 32ndash42 2016


module by using the brittleness theory of the complexnetwork and improved the antirisk ability of the modularproduct network Funke and Becker [6] analyzed the ap-plication of the complex network theory in themanufacturing system and provided a new method fornetwork modelling of the manufacturing system Giret et al[7] applied the complex network theory to the service-oriented manufacturing network and analyzed the networkstructure and statistical parameters of the service-orientedmanufacturing network

Nowadays artificial intelligent algorithms had beenapplied in the construction for different applications Keaet al [8] proposed a newmodel to estimate the disc cutter lifeby integrating a group method of data handling- (GMDH-)type neural network with a genetic algorithm Wei et al [9]proposed an analytical method for estimating the horizontaltransition probability matrix which is one of the importantinput parameters for the coupled Markov chain model Shenet al [10] presented an automatic pumping-recharge systemto maintain groundwater balance during dewatering Linet al [11] presented an approach for eutrophication eval-uation based on the technique for order preference Zhanget al [12] proposed an artificial intelligence model to predictground settlement Ssl et al [13] proposed an intelligentframework for predicting the advancing speed during earthpressure balance

-e definition and classification of bottlenecks are dif-ferent according to different research objects differentmethods and different observation angles Wang et al [14]regarded the queuing number of parts in the processing areaas the bottleneck identification index and the bottleneck isthe one with the largest queuing number Samouei et al [15]studied the bottleneck in the workshop from various aspectsof the equipment parameters and identified the bottleneckunit by building a time-based mathematical model Azadehand Shoja [16] proposed a network model of bottleneckidentification which converted the main units of theworkshop into a network model to identify the bottlenecklocation from the perspective of the system Rui and Cheng[17] studied the workshop of the assembly line productionmode constructed a mathematical model based on thebalance rate of the production line and identified thebottleneck in real time Based on the TOC constraint theoryBin and Sun [18] combined with the intelligent algorithmand improved some processes of the previous genetic al-gorithm so as to optimize the bottleneck identificationmethod According to the linear programming methodBrucker et al [19] constructed a programming model for theobjective function and constraints of the productionworkshop to identify the bottleneck unit of the productionsystem Masoud et al [20] used the model graph and net-work model construction algorithm to identify the real-timebottleneck in the production workshop -urer and Ste-venson [21] used simulation software to simulate the mainproduction factors of the production system and comparedthe simulation data with the theoretical model and work-shop data to identify the real-time bottleneck Sweeney et al[22] carried out orthogonal experiments on the factors thataffect the output target of the system in the production

workshop compared with the mobile bottleneck generatorand verified the superiority of the orthogonal experimentbottleneck identification method

-e existing bottleneck research methods are limited to aspecific parameter index such as equipment station per-sonnel and materials which lack systematic thinking -ebottleneck identification method in this paper is based onthe complex network theory the Markov model is simpleand it can be used to describe complex random phenomenaMarkov process is used to analyze the disturbance factorsand the state transition probability and steady distribution ofthe Markov chain are obtained it provides a way for thequantitative analysis of the disturbance factors in the pro-duction workshop From the perspective of the complexnetwork all production factors are considered to identifyand predict the bottleneck And then the accuracy andeffectiveness of the proposed method are verified by theactual data of the workshop and simulation software

2 Disturbance Factorsrsquo Analysis of theProduction Workshop Based on theMarkov Model

In this paper Markov model is used to evaluate and predictthe intensity of disturbance factors -e method has a goodeffect on process state prediction and it can be used forproduction site state prediction However it is not suitablefor medium- and long-term system prediction From theperspective of the production process the disturbancefactors of the workshop are analyzed the disturbance factorintensity matrix is constructed the relationship between thematrix and disturbance factor intensity is simulated and theMarkov chain prediction model is established to predict thechange of disturbance factor intensity so as to determine thekey disturbance factors and their occurrence probability-eflowchart indicating the bottleneck prediction of the pro-duction workshop is shown in Figure 1

In each link of the workshop there will be a variety ofdisturbance factors such as demand change emergencyorder insertion equipment failure machining accuracy andpersonnel absence At the same time there will be a lot ofinaccurate information such as material arrival timemanual clamping time and auxiliary processing time -eexistence of these disturbance factors will make the work-shop conditions change dynamically and seriously affect thenormal production activities

-e disturbance in the workshop mainly refers to thefactors that affect the effective output of production unitssuch as equipment personnel process and departmentDisturbance factors can be divided into the following fourcategories internal production environment externalenvironment monitoring technology and human factorsDisturbance factors caused by the internal productionenvironment although the production in themanufacturing workshop has been oriented to stan-dardization and precision the microdifferences andrandomness in time cannot be eliminated -ese uncer-tainties are mostly related to the internal production





I 1113944n

i1widi i 1 2 n (1)





Complex network

Node degreeAverage









N

Y

x1(T)

x2(T)

Time T


model

x1(T + T0)

x2(T + T0)

Time T + T0



model



Bottleneck unitset

N

Y




Y

N

Main bottleneck

Secondarybottleneck





4i1 wi 1 wi ge 0 -e


mi1 wim 1 wim ge 0





D W middot R D1 D2 D3 D4 D5( 1113857 (3)




a a1 a2 ak1113858 1113859 (4)



P


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (5)


pij Nij

1113936nj1 Nij

(6)



1113954p a middot p (7)




Pp


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (8)





Ti


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (9)




Pp Pp + ΔPp


ϑ ϑ + Δϑ

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)






xi(t + 1) (1 minus ϑ)f xi(t)( 1113857 +ϑ1113936

Nj1jne i aij + rij1113872 1113873

k(i)

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868 (11)



τ αC + βK + cL (12)




BSN Xi(t)ge τ1113864 1113865









Order change lowast






Route change lowast




Proficiency lowast





Fi 1113944T

t1limΔt⟶0

xi(t + 1) minus xi(t)1113868111386811138681113868

1113868111386811138681113868

Δt (14)





P

0382 0387 0288 0 0

0522 0330 0120 0120 0

0450 0270 0350 0120 0

0 1 0 0 0

0 0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(15)



N

6 6 4 0 0

6 3 2 1 0

3 2 3 0 0

0 2 0 0 0

0 0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)


a 04 03 02 01 001113858 1113859 (17)


1113954p a middot p 0382 0380 0201 0027 01113858 1113859 (18)














accuracy (TP + TN)

(TP + FP + TN + FN) (19)



P1

P2

P

P9 P10

P3

P4

P5 P6

P8

P7

Operator 1

Workshop 1

Operator n

Equipment 1

Equipment n

Workshop 1

Operator 1

Operator n

Equipment 1

Equipment 1



Quality check






Internaldisturbance







Weight 00565 06386 01037 02579



Ai Di Fi Si


P1P2

P3

P4P8

P5

P6

P7P9

P10

Equipment

Buffer






5 Conclusions



Data Availability




Acknowledgments




P1 P2 P3 P4 P5 P6 P7 P8 P9 P101 0875 0400 0875 0625 0538 0763 0429 0542 0365 06252 0677 0706 0687 0809 0925 0736 0659 0980 0937 08363 0675 0833 0738 0639 0206 0755 0482 0196 0286 05694 0565 0686 0737 0579 0198 0859 0783 0837 0691 08965 0763 0902 0849 0782 0390 0491 0509 0359 0425 07396 0803 0285 0729 0832 0849 0729 0492 0401 0839 06857 0586 0339 0605 0572 0449 0839 0837 0938 0729 08178 0806 0839 0358 0609 0362 0937 0885 0829 0582 0606



P1 P2 P3 P4 P5 P6 P7 P8 P9 P101 0875 0400 0875 0625 0538 0763 0429 0542 0365 06252 0677 0706 0687 0809 0925 0736 0659 0980 0937 08363 0875 0952 0896 1000 0984 1255 0482 1000 0325 10004 0865 0977 0904 0892 0795 1389 0886 0949 0892 09165 0763 0902 0849 0782 1286 0695 0616 0479 0495 07396 0933 0858 0786 0897 1497 0881 0797 0771 0739 06577 0266 0217 0605 0572 1930 0780 0633 0814 0796 08178 0766 0583 0658 0509 3362 0837 0775 0626 0594 0316




References

































I 1113944n

i1widi i 1 2 n (1)





Complex network

Node degreeAverage









N

Y

x1(T)

x2(T)

Time T


model

x1(T + T0)

x2(T + T0)

Time T + T0



model



Bottleneck unitset

N

Y




Y

N

Main bottleneck

Secondarybottleneck





4i1 wi 1 wi ge 0 -e


mi1 wim 1 wim ge 0





D W middot R D1 D2 D3 D4 D5( 1113857 (3)




a a1 a2 ak1113858 1113859 (4)



P


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (5)


pij Nij

1113936nj1 Nij

(6)



1113954p a middot p (7)




Pp


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (8)





Ti


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (9)




Pp Pp + ΔPp


ϑ ϑ + Δϑ

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)






xi(t + 1) (1 minus ϑ)f xi(t)( 1113857 +ϑ1113936

Nj1jne i aij + rij1113872 1113873

k(i)

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868 (11)



τ αC + βK + cL (12)




BSN Xi(t)ge τ1113864 1113865









Order change lowast






Route change lowast




Proficiency lowast





Fi 1113944T

t1limΔt⟶0

xi(t + 1) minus xi(t)1113868111386811138681113868

1113868111386811138681113868

Δt (14)





P

0382 0387 0288 0 0

0522 0330 0120 0120 0

0450 0270 0350 0120 0

0 1 0 0 0

0 0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(15)



N

6 6 4 0 0

6 3 2 1 0

3 2 3 0 0

0 2 0 0 0

0 0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)


a 04 03 02 01 001113858 1113859 (17)


1113954p a middot p 0382 0380 0201 0027 01113858 1113859 (18)














accuracy (TP + TN)

(TP + FP + TN + FN) (19)



P1

P2

P

P9 P10

P3

P4

P5 P6

P8

P7

Operator 1

Workshop 1

Operator n

Equipment 1

Equipment n

Workshop 1

Operator 1

Operator n

Equipment 1

Equipment 1



Quality check






Internaldisturbance







Weight 00565 06386 01037 02579



Ai Di Fi Si


P1P2

P3

P4P8

P5

P6

P7P9

P10

Equipment

Buffer






5 Conclusions



Data Availability




Acknowledgments




P1 P2 P3 P4 P5 P6 P7 P8 P9 P101 0875 0400 0875 0625 0538 0763 0429 0542 0365 06252 0677 0706 0687 0809 0925 0736 0659 0980 0937 08363 0675 0833 0738 0639 0206 0755 0482 0196 0286 05694 0565 0686 0737 0579 0198 0859 0783 0837 0691 08965 0763 0902 0849 0782 0390 0491 0509 0359 0425 07396 0803 0285 0729 0832 0849 0729 0492 0401 0839 06857 0586 0339 0605 0572 0449 0839 0837 0938 0729 08178 0806 0839 0358 0609 0362 0937 0885 0829 0582 0606



P1 P2 P3 P4 P5 P6 P7 P8 P9 P101 0875 0400 0875 0625 0538 0763 0429 0542 0365 06252 0677 0706 0687 0809 0925 0736 0659 0980 0937 08363 0875 0952 0896 1000 0984 1255 0482 1000 0325 10004 0865 0977 0904 0892 0795 1389 0886 0949 0892 09165 0763 0902 0849 0782 1286 0695 0616 0479 0495 07396 0933 0858 0786 0897 1497 0881 0797 0771 0739 06577 0266 0217 0605 0572 1930 0780 0633 0814 0796 08178 0766 0583 0658 0509 3362 0837 0775 0626 0594 0316




References































4i1 wi 1 wi ge 0 -e


mi1 wim 1 wim ge 0





D W middot R D1 D2 D3 D4 D5( 1113857 (3)




a a1 a2 ak1113858 1113859 (4)



P


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (5)


pij Nij

1113936nj1 Nij

(6)



1113954p a middot p (7)




Pp


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (8)





Ti


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (9)




Pp Pp + ΔPp


ϑ ϑ + Δϑ

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)






xi(t + 1) (1 minus ϑ)f xi(t)( 1113857 +ϑ1113936

Nj1jne i aij + rij1113872 1113873

k(i)

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868 (11)



τ αC + βK + cL (12)




BSN Xi(t)ge τ1113864 1113865









Order change lowast






Route change lowast




Proficiency lowast





Fi 1113944T

t1limΔt⟶0

xi(t + 1) minus xi(t)1113868111386811138681113868

1113868111386811138681113868

Δt (14)





P

0382 0387 0288 0 0

0522 0330 0120 0120 0

0450 0270 0350 0120 0

0 1 0 0 0

0 0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(15)



N

6 6 4 0 0

6 3 2 1 0

3 2 3 0 0

0 2 0 0 0

0 0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)


a 04 03 02 01 001113858 1113859 (17)


1113954p a middot p 0382 0380 0201 0027 01113858 1113859 (18)














accuracy (TP + TN)

(TP + FP + TN + FN) (19)



P1

P2

P

P9 P10

P3

P4

P5 P6

P8

P7

Operator 1

Workshop 1

Operator n

Equipment 1

Equipment n

Workshop 1

Operator 1

Operator n

Equipment 1

Equipment 1



Quality check






Internaldisturbance







Weight 00565 06386 01037 02579



Ai Di Fi Si


P1P2

P3

P4P8

P5

P6

P7P9

P10

Equipment

Buffer






5 Conclusions



Data Availability




Acknowledgments




P1 P2 P3 P4 P5 P6 P7 P8 P9 P101 0875 0400 0875 0625 0538 0763 0429 0542 0365 06252 0677 0706 0687 0809 0925 0736 0659 0980 0937 08363 0675 0833 0738 0639 0206 0755 0482 0196 0286 05694 0565 0686 0737 0579 0198 0859 0783 0837 0691 08965 0763 0902 0849 0782 0390 0491 0509 0359 0425 07396 0803 0285 0729 0832 0849 0729 0492 0401 0839 06857 0586 0339 0605 0572 0449 0839 0837 0938 0729 08178 0806 0839 0358 0609 0362 0937 0885 0829 0582 0606



P1 P2 P3 P4 P5 P6 P7 P8 P9 P101 0875 0400 0875 0625 0538 0763 0429 0542 0365 06252 0677 0706 0687 0809 0925 0736 0659 0980 0937 08363 0875 0952 0896 1000 0984 1255 0482 1000 0325 10004 0865 0977 0904 0892 0795 1389 0886 0949 0892 09165 0763 0902 0849 0782 1286 0695 0616 0479 0495 07396 0933 0858 0786 0897 1497 0881 0797 0771 0739 06577 0266 0217 0605 0572 1930 0780 0633 0814 0796 08178 0766 0583 0658 0509 3362 0837 0775 0626 0594 0316




References






























Ti


⋮ ⋱ ⋮


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (9)




Pp Pp + ΔPp


ϑ ϑ + Δϑ

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(10)






xi(t + 1) (1 minus ϑ)f xi(t)( 1113857 +ϑ1113936

Nj1jne i aij + rij1113872 1113873

k(i)

1113868111386811138681113868111386811138681113868111386811138681113868

1113868111386811138681113868111386811138681113868111386811138681113868 (11)



τ αC + βK + cL (12)




BSN Xi(t)ge τ1113864 1113865









Order change lowast






Route change lowast




Proficiency lowast





Fi 1113944T

t1limΔt⟶0

xi(t + 1) minus xi(t)1113868111386811138681113868

1113868111386811138681113868

Δt (14)





P

0382 0387 0288 0 0

0522 0330 0120 0120 0

0450 0270 0350 0120 0

0 1 0 0 0

0 0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(15)



N

6 6 4 0 0

6 3 2 1 0

3 2 3 0 0

0 2 0 0 0

0 0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)


a 04 03 02 01 001113858 1113859 (17)


1113954p a middot p 0382 0380 0201 0027 01113858 1113859 (18)














accuracy (TP + TN)

(TP + FP + TN + FN) (19)



P1

P2

P

P9 P10

P3

P4

P5 P6

P8

P7

Operator 1

Workshop 1

Operator n

Equipment 1

Equipment n

Workshop 1

Operator 1

Operator n

Equipment 1

Equipment 1



Quality check






Internaldisturbance







Weight 00565 06386 01037 02579



Ai Di Fi Si


P1P2

P3

P4P8

P5

P6

P7P9

P10

Equipment

Buffer






5 Conclusions



Data Availability




Acknowledgments




P1 P2 P3 P4 P5 P6 P7 P8 P9 P101 0875 0400 0875 0625 0538 0763 0429 0542 0365 06252 0677 0706 0687 0809 0925 0736 0659 0980 0937 08363 0675 0833 0738 0639 0206 0755 0482 0196 0286 05694 0565 0686 0737 0579 0198 0859 0783 0837 0691 08965 0763 0902 0849 0782 0390 0491 0509 0359 0425 07396 0803 0285 0729 0832 0849 0729 0492 0401 0839 06857 0586 0339 0605 0572 0449 0839 0837 0938 0729 08178 0806 0839 0358 0609 0362 0937 0885 0829 0582 0606



P1 P2 P3 P4 P5 P6 P7 P8 P9 P101 0875 0400 0875 0625 0538 0763 0429 0542 0365 06252 0677 0706 0687 0809 0925 0736 0659 0980 0937 08363 0875 0952 0896 1000 0984 1255 0482 1000 0325 10004 0865 0977 0904 0892 0795 1389 0886 0949 0892 09165 0763 0902 0849 0782 1286 0695 0616 0479 0495 07396 0933 0858 0786 0897 1497 0881 0797 0771 0739 06577 0266 0217 0605 0572 1930 0780 0633 0814 0796 08178 0766 0583 0658 0509 3362 0837 0775 0626 0594 0316




References






























Fi 1113944T

t1limΔt⟶0

xi(t + 1) minus xi(t)1113868111386811138681113868

1113868111386811138681113868

Δt (14)





P

0382 0387 0288 0 0

0522 0330 0120 0120 0

0450 0270 0350 0120 0

0 1 0 0 0

0 0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(15)



N

6 6 4 0 0

6 3 2 1 0

3 2 3 0 0

0 2 0 0 0

0 0 0 0 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(16)


a 04 03 02 01 001113858 1113859 (17)


1113954p a middot p 0382 0380 0201 0027 01113858 1113859 (18)














accuracy (TP + TN)

(TP + FP + TN + FN) (19)



P1

P2

P

P9 P10

P3

P4

P5 P6

P8

P7

Operator 1

Workshop 1

Operator n

Equipment 1

Equipment n

Workshop 1

Operator 1

Operator n

Equipment 1

Equipment 1



Quality check






Internaldisturbance







Weight 00565 06386 01037 02579



Ai Di Fi Si


P1P2

P3

P4P8

P5

P6

P7P9

P10

Equipment

Buffer






5 Conclusions



Data Availability




Acknowledgments




P1 P2 P3 P4 P5 P6 P7 P8 P9 P101 0875 0400 0875 0625 0538 0763 0429 0542 0365 06252 0677 0706 0687 0809 0925 0736 0659 0980 0937 08363 0675 0833 0738 0639 0206 0755 0482 0196 0286 05694 0565 0686 0737 0579 0198 0859 0783 0837 0691 08965 0763 0902 0849 0782 0390 0491 0509 0359 0425 07396 0803 0285 0729 0832 0849 0729 0492 0401 0839 06857 0586 0339 0605 0572 0449 0839 0837 0938 0729 08178 0806 0839 0358 0609 0362 0937 0885 0829 0582 0606



P1 P2 P3 P4 P5 P6 P7 P8 P9 P101 0875 0400 0875 0625 0538 0763 0429 0542 0365 06252 0677 0706 0687 0809 0925 0736 0659 0980 0937 08363 0875 0952 0896 1000 0984 1255 0482 1000 0325 10004 0865 0977 0904 0892 0795 1389 0886 0949 0892 09165 0763 0902 0849 0782 1286 0695 0616 0479 0495 07396 0933 0858 0786 0897 1497 0881 0797 0771 0739 06577 0266 0217 0605 0572 1930 0780 0633 0814 0796 08178 0766 0583 0658 0509 3362 0837 0775 0626 0594 0316




References































accuracy (TP + TN)

(TP + FP + TN + FN) (19)



P1

P2

P

P9 P10

P3

P4

P5 P6

P8

P7

Operator 1

Workshop 1

Operator n

Equipment 1

Equipment n

Workshop 1

Operator 1

Operator n

Equipment 1

Equipment 1



Quality check






Internaldisturbance







Weight 00565 06386 01037 02579



Ai Di Fi Si


P1P2

P3

P4P8

P5

P6

P7P9

P10

Equipment

Buffer






5 Conclusions



Data Availability




Acknowledgments




P1 P2 P3 P4 P5 P6 P7 P8 P9 P101 0875 0400 0875 0625 0538 0763 0429 0542 0365 06252 0677 0706 0687 0809 0925 0736 0659 0980 0937 08363 0675 0833 0738 0639 0206 0755 0482 0196 0286 05694 0565 0686 0737 0579 0198 0859 0783 0837 0691 08965 0763 0902 0849 0782 0390 0491 0509 0359 0425 07396 0803 0285 0729 0832 0849 0729 0492 0401 0839 06857 0586 0339 0605 0572 0449 0839 0837 0938 0729 08178 0806 0839 0358 0609 0362 0937 0885 0829 0582 0606



P1 P2 P3 P4 P5 P6 P7 P8 P9 P101 0875 0400 0875 0625 0538 0763 0429 0542 0365 06252 0677 0706 0687 0809 0925 0736 0659 0980 0937 08363 0875 0952 0896 1000 0984 1255 0482 1000 0325 10004 0865 0977 0904 0892 0795 1389 0886 0949 0892 09165 0763 0902 0849 0782 1286 0695 0616 0479 0495 07396 0933 0858 0786 0897 1497 0881 0797 0771 0739 06577 0266 0217 0605 0572 1930 0780 0633 0814 0796 08178 0766 0583 0658 0509 3362 0837 0775 0626 0594 0316




References































Ai Di Fi Si


P1P2

P3

P4P8

P5

P6

P7P9

P10

Equipment

Buffer






5 Conclusions



Data Availability




Acknowledgments




P1 P2 P3 P4 P5 P6 P7 P8 P9 P101 0875 0400 0875 0625 0538 0763 0429 0542 0365 06252 0677 0706 0687 0809 0925 0736 0659 0980 0937 08363 0675 0833 0738 0639 0206 0755 0482 0196 0286 05694 0565 0686 0737 0579 0198 0859 0783 0837 0691 08965 0763 0902 0849 0782 0390 0491 0509 0359 0425 07396 0803 0285 0729 0832 0849 0729 0492 0401 0839 06857 0586 0339 0605 0572 0449 0839 0837 0938 0729 08178 0806 0839 0358 0609 0362 0937 0885 0829 0582 0606



P1 P2 P3 P4 P5 P6 P7 P8 P9 P101 0875 0400 0875 0625 0538 0763 0429 0542 0365 06252 0677 0706 0687 0809 0925 0736 0659 0980 0937 08363 0875 0952 0896 1000 0984 1255 0482 1000 0325 10004 0865 0977 0904 0892 0795 1389 0886 0949 0892 09165 0763 0902 0849 0782 1286 0695 0616 0479 0495 07396 0933 0858 0786 0897 1497 0881 0797 0771 0739 06577 0266 0217 0605 0572 1930 0780 0633 0814 0796 08178 0766 0583 0658 0509 3362 0837 0775 0626 0594 0316




References































5 Conclusions



Data Availability




Acknowledgments




P1 P2 P3 P4 P5 P6 P7 P8 P9 P101 0875 0400 0875 0625 0538 0763 0429 0542 0365 06252 0677 0706 0687 0809 0925 0736 0659 0980 0937 08363 0675 0833 0738 0639 0206 0755 0482 0196 0286 05694 0565 0686 0737 0579 0198 0859 0783 0837 0691 08965 0763 0902 0849 0782 0390 0491 0509 0359 0425 07396 0803 0285 0729 0832 0849 0729 0492 0401 0839 06857 0586 0339 0605 0572 0449 0839 0837 0938 0729 08178 0806 0839 0358 0609 0362 0937 0885 0829 0582 0606



P1 P2 P3 P4 P5 P6 P7 P8 P9 P101 0875 0400 0875 0625 0538 0763 0429 0542 0365 06252 0677 0706 0687 0809 0925 0736 0659 0980 0937 08363 0875 0952 0896 1000 0984 1255 0482 1000 0325 10004 0865 0977 0904 0892 0795 1389 0886 0949 0892 09165 0763 0902 0849 0782 1286 0695 0616 0479 0495 07396 0933 0858 0786 0897 1497 0881 0797 0771 0739 06577 0266 0217 0605 0572 1930 0780 0633 0814 0796 08178 0766 0583 0658 0509 3362 0837 0775 0626 0594 0316




References






























References






























mathematical model construction of the production workshop

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