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A Simulation Model for the Closed-Loop Controlof a Multi-Workstation Production System
Juliana Keiko Sagawa1 Michael Freitag2
1Production Engineering Department, Federal University of São Carlos, Brazil (e-mail: [email protected])2BIBA - Bremer Institut für Produktion und Logistik, Faculty of Production Engineering, University of Bremen,
Germany (e-mail: [email protected])
AbstractIn this paper, we propose a simulation model with a PIcontroller to analyze and control the dynamics of a multi-workstation production system. The formulation is basedon dynamic modelling and control theory, and the modelwas implemented in Matlab and Simulink. Exploratorytests were carried out, and the results indicated some re-lationships between the values of the parameters of thecontroller and the values of the output variables, that is,the levels of work in process. They also showed that theproposed model has the potential of providing manage-rial directions on how to dynamically adjust the capac-ity, aiming to smooth the operation of the shop floor andto keep the work in process close to the desired levels.Keywords: production systems, control theory, dynamics,simulation, planning and scheduling
1 IntroductionAs known, the increasing computational capacity engen-dered a sound evolution of the operations managementarea, since it allowed the development of several tools tocope with large amounts of data related to the planningprocess and to ground the analysis of the decision mak-ers. In this sense, the use of dynamic modeling and sim-ulation techniques complements the static approaches forplanning optimization of production systems and supplychains. Control theory is also a correlated area whose the-ories and tools have been applied to production and sup-ply chain management. Some steps in these directions areenumerated in the literature review section of this paper.
Considering the aforementioned approaches, wepresent in this paper a simulation model based on stateequations to depict the dynamics of a multi-workstationmanufacturing system. A proportional-integral (PI) con-troller is applied to the model, and exploratory tests arecarried out.
The model basically deals with the work in process(WIP) and capacity allocation variables (represented bythe processing frequency of the stations), in a plant withjob shop configuration. In general terms, the controlof work in process is a classical concern in the opera-tion of job shops, since it generates more predictable cy-cle/throughput times, which lead to better promises and
fulfilment of delivery dates, a more stable coordination ofthe shop floor, and more flexibility to attend changes inthe customer demand. These effects are highlighted in theliterature related to various methodologies in productionengineering, such as just-in-time, quick response manu-facturing, workload control, factory physics, and others.
The results obtained with the simulation of the pro-posed model provided some indications of how its param-eters influence the WIP levels, and demonstrated the po-tential of the proposed approach to depict the dynamic re-lations between capacity allocation, work in process andoperations smoothness in production systems.
2 Literature ReviewThe effort of evolving from the static to the dynamic anal-ysis of production and supply chain systems relies on sys-tem dynamics and control theory, as mentioned. In abroader sense, system dynamics may be defined as an areaof knowledge that deals with the time-varying behavior ofa system (Doebelin, 1998). This includes not only me-chanical, electrical, fluid and thermal, but may also in-clude biological, manufacturing, social and hybrid sys-tems. Control theory, on its turn, has different subareasand a range of tools for the analysis and design of closed-loop systems, where the information of the outputs is fedback to the system in order to lead it to desired goals. Inthe literature reviews concerning the application of con-trol theory to production and supply chain, the models areclassified according to the area of application [(Ortega andLin, 2004), (Sagawa and Nagano, 2015b)], the underlyingcontrol methodology (Sarimveis et al., 2008), the type ofanalysis that is carried out (i.e. robustness, stability, etc.)(Ivanov and Sokolov, 2013), or according to mixed criteria(Åström and Kumar, 2014).
In Table 1, we present some applications of control the-ory to production and supply chain systems, classified ac-cording to subareas of application and the methods under-lying the models. Our intention here is not to present anextensive review, but rather to enumerate different possi-bilities and to provide few references in each category, asexamples.
Other applications based on model predictive control orrobust optimal control are not listed here, but can be foundin (Sarimveis et al., 2008). Also, there are some alterna-
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459DOI: 10.3384/ecp17142459 Proceedings of the 9th EUROSIM & the 57th SIMSSeptember 12th-16th, 2016, Oulu, Finland
Table 1. Some Applications of control theory in production sys-tems and supply chains
Area/type of appli-cation
Appliedmethodolo-gies andtools
References(examples)
single-productproduction-inventory modelsand extensions tosupply chain
classicalcontrol the-ory, blockdiagrams,transferfunctions
(Towell,1982; Zhouet al., 2006;Spiegleret al., 2016)
production-inventory modelswith single andmultiple-machines(with or without ad-ditional constraints)
dynamic pro-grammingand optimalcontrol
(Scarf, 1960;Boukas andLiu, 2001;Gharbi andKenne, 2003)
multi-echelonproduction-inventory mod-els using bills ofmaterial as input
input-outputanalysis,Laplace orz-transform,probabilitydistributions,NPV
(Axsäter,1976; Grubb-ström andMolinder,1994; Grubb-ström et al.,2010)
multiple-machineand multi-productsystems based onflow models
flow mod-els, blockdiagrams,transfer func-tions, bondgraphs
(Wiendahland Brei-thaupt, 2000;Kim andDuffie, 2006;Sagawa andNagano,2015a)
supervisory/processcontrol of contin-uous productionsystems and itsintegration withinthe hierarchicalproduction planning
mixed inte-ger dynamicoptimization(MIDO),mixedinteger non-linear pro-gramming(MINLP)
(Monfaredand Yang,2007; Mu-nawar andGudi, 2004;Du et al.,2015)
production and sup-ply chain modelswith autonomouscontrol/decentral-ized agents
queue lengthestimator(QLE),pheromone,heuristicmethods,RFID
(Scholz-Reiterand Fre-itag, 2007;Wang andLin, 2009;Barenjiet al., 2014;Schukraftet al., 2016)
tive formulations out of the control theory area, based, forinstance, in queueing systems, which are out of the scopeof this paper.
In the following subsection, we present the mathemat-ical model that was adopted as basis for the simulationmodel proposed in this paper.
2.1 Dynamic multi-workstation model basedon electrical components
A dynamic model based on the ideal properties of elec-trical components is proposed in (Sagawa and Nagano,2015a) to depict a multi-workstation system that can man-ufacture different families of products. The model is ba-sically composed by machines, buffers and junction ele-ments.
The machines are compared to resistors and their pro-cessing frequency Ui correspond to 1
R , where R is an idealresistance. Similarly, the buffers are seen as ideal ca-pacitors with capacitance C, which corresponds to theirstorage capacity (Ferney, 2000). The junctions are usedto couple these manufacturing elements and to depict theconfiguration of the production flow in the system, i.e. torepresent the different process routings of each productor product family (Sagawa and Nagano, 2015a). Whena given machine outputs flow to m workstations down-stream, it is coupled to these workstations by means ofa divergent junction that imposes the conservation of flow.Similarly, the upstream flows coming from different work-stations to a given workstation are merged by means of aconvergent junction that conserves the total flow (Sagawaand Nagano, 2015a; Ferney, 2000). The discussed modelis continuous and deals with 3 variables: the productionflow f , the production volume q and the effort e. Theproduction volume corresponds to the integral of the flow,and the effort is used as an auxiliary variable, for cou-pling a machine and its precedent buffer, as well for theapproximation of a discrete system as a continuous sys-tem (Ferney, 2000). The basic equation of the model isderived from the well-known constitutive equations of theideal electrical components previously mentioned, and isshown in (1). The variables and parameters of this equa-tion were already mentioned in the text. The index i de-notes a given workstation, the index s applied to the flowor effort variables denotes the output of this station, andthe index e denotes its input. Eq. (2) is based on the afore-mentioned integral relation between the production vol-ume q and the flow variable f , likewise the electric chargestored in a capacitor is defined as a function of the integralof the electric current. In the context of manufacturing,qi(t) is interpreted as a rate of material storage or con-sumption, expressed as the difference between the inputand output flow of a workstation.
fsi =Ui
[qi(t)Ci
+min1,qi(t)− esi(t)]
(1)
qi(t) = fei(t)−Ui min1,qi(t) (2)
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- Set the parameters ofthe simulation
- Call the Simulinkmodel
- Calculate theperformance measures
- Plot the results
Main routine
- Call the embeddedfunction
- Save the instantaneousoutputs of the simulationinto matrix or arrays
- integrate signals(integration blocks)
Simulink model
Used elements:- Timmer, multiplexers
and demultiplexers, integration blocks,
linking elements, sinks, embedded function
- calculate theinstantaneous errors
-implement thedifferential equations
- implement thecontrol laws
Embbeded function
Figure 1. Schematics of the simulation model
The assumption of buffers with unlimited capacity allowssimplifying (1), and the combination of (1) and (2) yieldsthe basic state equation of a workstation, presented in (3).
qi(t) = fei(t)−Ui min1,qi(t) (3)
This basic equation and the constitutive equations of thejunctions were applied to a 11-workstation production sys-tem, as presented in (Sagawa and Nagano, 2015a), result-ing in the state model shown in (4).
q1q2q3q4q5q6q7q8q9q10q11
=
−1 0 0 0 0 0 0 0 0 0 01 −1 0 0 0 0 0 0 0 0 00 0.4300 −1 0 0 0 0 0 0 0 00 0.2508 0.5118 −1 0 0 0 0 0 0 00 0.1538 0.3140 0 −1 0 0 0 0 0 00 0.0170 0.0347 0 0 −1 0 0 0 0 00 0.0175 0.0020 0.2811 0.2811 0 −1 0 0 0 00 0.0087 0.0010 0.1399 0.1399 0 0 −1 0 0 00 0.0161 0.0018 0.2588 0.2588 0 0 0 −1 0 00 0.0200 0.0023 0.3204 0.3204 0 0 0 0 −1 00 0 0 0 0 0 0.2407 0.2407 0.2407 0.2407 −1
U1 min1,q1U2 min1,q2U3 min1,q3U4 min1,q4U5 min1,q5U6 min1,q6U7 min1,q7U8 min1,q8U9 min1,q9
U10 min1,q10U11 min1,q11
+
U010000000000
(4)
3 Simulation Model with a PI Con-troller
A simulation model based on the presented state equa-tions was implemented using Matlabr and Simulinkr. Itincluded a proportional-integral (PI) controller, as men-tioned. The structure of this model, as well as the executedinstructions, are shown in Fig. 1. As it can be seen, the rel-evant parameters of the simulation are defined in the mainroutine. After that, this routine calls the Simulink model(Fig. 2), which contains the block diagram of the dynamicmodel with the controller. The computation of the stateequations is performed by a user-defined function embed-ded in the Simulinkr model.
After the iterations are carried out for the total simu-lated time, the main routine compiles the results and calcu-lates the performance measures. For the implementationof a proportional-integral controller, integration blocks(1/s) of the first level should be applied to the instanta-neous material storage rates qi, while second level integra-tions should be applied to the relative errors in the stocklevels, as it can be seen in Fig. 2.
Figure 2. Schematics of the model build in Simulinkr
Inputs
• reference processing frequency of the machines Uip (steady state)
• reference levels of the buffers qjc
• initial levels of the buffers qi0
• coefficients of the state model matrix
• parameters of the PI controller (gains)
Outputs
• instantaneous rate of material storage
• instantaneous processing frequency of the machines Ui(t)
• relative errors ej(t)
Processing
• Step 1: calculation of the relative errors ej(t) = (qj(t)– qjc)/ qjc
• Step 2: application of the control law, i.e., calculation of the instantaneous
frequencies Ui(t) according to the equations shown
• Step 3: calculation of the instantaneous rate of material storage , i.e. implementation of the state equations
Embedded function
()
( + 1)
Figure 3. User-defined function implemented in the simulationmodel
As mentioned, the state model is implemented bymeans of a user-defined function. The inputs of this func-tion are those parameters defined in the main routine, andpresented in Fig. 3.
With these inputs, the function calculates, at each timet, the relative errors of the stock levels, shown in (5), andimplements the control law. For an integral controller,this control law is shown in (6). With the values of theinstantaneous processing frequencies of the machines Ui,resulting from the implementation of the control law, theinstantaneous rates of material storage qi(t + 1) are thencalculated. These rates are integrated in the integrationblocks and fed back to the model
epj(t) =−e j(t) =q jc−q j(c)
q jc(5)
Ui(t) =Uip
(1+ kpepj(t)+ ki
∫epj(t)dt
)(6)
where q j(t) is the instantaneous amount of material storedin buffer j, epj(t) is the relative error considering the actuallevel q j(t) and the reference level q jc;Uip is the referencefor the processing frequency of machine i, considering thecustomer demand fulfillment in the steady state; kp is theproportional gain of the controller; and ki is the integralgain. The presented equations apply for the case where thebuffer j immediately succeeds the machine i. If machine iis succeeded by more than one buffer, the minimum valueof e j is computed.
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Figure 4. Processing frequencies of the machines for a test car-ried out with kp = 0.05 and ki = 0.05 (whitout saturation limits).
4 Test and ResultsThe proposed simulation model was applied to the 11-workstation system presented in (Sagawa and Nagano,2015a). For an initial analysis, it was of interest to con-sider the warm-up of the manufacturing system and itstransition to the regular operation, that is, to consider thesituation where all the buffers are empty (qi0 = 0 for alli) and the system starts to work, aiming to attend the cus-tomer demand and to reach the desired levels of work inprocess. This starting condition is somewhat similar tothe application of a step input, conventionally used for thestudy of the response in dynamic systems. In our case,however, each buffer has a different reference level, sincethese levels were defined as a multiple of the amount ofmaterial cumulated in each buffer when the system wassimulated without control, i.e., when the open-loop sys-tem was simulated. In order to allow comparisons, weadopted a multiplication factor of 100 times, as in (Sagawaand Nagano, 2015a). As output variables of the tests, weanalyzed the values of the processing frequencies of themachines Ui(t) (the controlled variables); the relative pro-cessing frequencies, i.e. (Ui(t)−Uip)/Uip; and the rel-ative errors in the buffer levels (ei) over time. This lastmeasure indicates the variation of the work in process inthe system. Depending on the selected values of the gains,the machines are led to operate with processing frequen-cies above the reference frequencies, in order to fulfillthe buffers. In Fig. 4, the source of material works witha processing frequency that is 60% greater than the fre-quency that attends the customer demand in the mediumterm, i.e. in the steady state. This control command gen-erates a surplus of material in the buffers. When the con-troller receives this information, it reduces the processingfrequency of the machines. This reaction, however, is ex-cessive, so that the machines are shutdown. The saturationof the integral controller could be a relevant parameter ofinfluence for the control of the processing frequencies ofthe machines. Due to the saturation in PI controllers, theintegrator may drift to undesirable values, since it tendsto produce progressively larger control signs. This effectis known as windup of the integral controller (F Franklin
0 50 100 150 200 250 300time (days)
-5
0
5
10
15
20
rela
tive
erro
rs
Relative errors of stock levels X time
e1e2e3e4e5e6e7e8e9e10e11
Figure 5. Evolution of the relative errors in stock levels (WIP),for saturation limits of ±10, kp = 0.05 and ki = 0.001.
0 50 100 150 200 250 300time (days)
-2
0
2
4
6
8
10
12
rela
tive
erro
rs
Relative errors of stock levels X time
e1e2e3e4e5e6e7e8e9e10e11
Figure 6. Evolution of the relative errors in stock levels (WIP),for saturation limits of ±1, kp = 0.05 and ki = 0.001.
et al., 1994; Moreno-Valenzuela, 2008). Therefore, addi-tional tests were performed with the establishment of sat-uration limits for the integral controller. Some results areshown in Fig. 5-8. The values of the gains were keptconstant and different saturation limits were tested.
The presented results indicate that, with narrower sat-uration limits, the overshoots in the WIP (represented bythe relative errors) were significantly reduced. In Fig. 5,the overshoot is of 20 times the reference level and inFig. 5, it is of 10 times. Although punctual instabilitiesin the control of some machines arose (Fig. 7), the oper-ation of the system also became smoother with the estab-
0 50 100 150 200 250 300time (days)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Pro
cess
ing
freq
uenc
y (d
ay -1
)
104 Processing frequency of the machines X time
U1U2U3U4U5U6U7U8U9U10U11U01
Figure 7. Evolution of the processing frequency of the ma-chines, for saturation limits of ±10, kp = 0.05 and ki = 0.001.
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0 50 100 150 200 250 300time (days)
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Pro
cess
ing
freq
uenc
y (d
ay -1
)104 Processing frequency of the machines X time
U1U2U3U4U5U6U7U8U9U10U11U01
Figure 8. Evolution of the processing frequency of the ma-chines, for saturation limits of ±1, kp = 0.05 and ki = 0.001.
lishment of saturation limits.The exploratory tests have also shown that the over-
shoot in the processing frequencies of the machines tendto increase with the increase of the integral gain. Hence,the results could be further improved by means of system-atic experiments and the application of control tools forthe optimization of these parameters (gains and saturationlimits).
From the managerial perspective, the controller canlead the system to undesired operating points, dependingon parameters set, because working either too far above orbelow the regular capacity imply higher operational costs.Moreover, there are well-known relations among WIP, cy-cle times and throughput rate. An increase in WIP mayengender a relevant increase in the cycle time (lead time),without producing any effect in the throughput rate of theproduction system (Hopp and Spearman, 2001). Longercycles times usually result in delayed jobs, rush orders,and difficulty of coordination. Based on the aforemen-tioned reasons, it is of interest, in our model, to find theparameters that enable to reduce the overshoot of WIPand, meanwhile, to reduce the oscillations in the process-ing frequencies of the machines.
The variations in the processing frequencies represent,in fact, capacity additions or reductions. Thus, the pro-posed simulation model may provide insights on howmuch capacity should be increased or decreased over timein order to: 1. guarantee adequate levels of WIP, to ab-sorb fluctuations; 2. avoid an excessive increase in thecycle/throughput times; 3. smooth operations, so that theoperational costs stay low. These capacity adjustments canbe implemented in various ways, i.e. overtime, subcon-tracting, adding/renting extra resources, or reducing theutilization of the machines.
In practice, the goal of production managers is to keepthe operations stable, as much as possible, so that no extracosts are incurred (although this pursuit of stability shouldnot unreasonably compromise the flexibility to attend cus-tomers). In MRP systems, the short term variations in theplans are usually smoothed or prevented by applying timefences to demand, planning and/or order release. This so-lution is efficient to keep the costs under control, but dis-
regards the dynamics of the production systems, so thatbackorders and stock outs in the short term may occur. Inthis sense, the use of the dynamic modeling and simula-tion seems to be an interesting alternative or complementto other methodologies used in the production engineeringarea. The presented formulation is also relatively simpleand easy to implement, which is an advantage in terms ofuse.
In order to implement it, the production system un-der consideration must be first modeled according to themethodology proposed in (Sagawa and Nagano, 2015a),which requires data related to the production routings, thehistorical demand for the end products, the product mixand the capacity of the machines (for more details, pleaserefer to (Sagawa and Nagano, 2015a)). The mentionedmethodology presents a generalization capability due toits modularity. The basic manufacturing entities, each oneassociated to its respective constitutive equation, may bearranged to represent different shop floor configurations,such as single machine, parallel machines, flow shop, jobshop and open shop. The resulting state model will be acombination of the expressions that represent the involvedentities. After this model is defined and implemented ina software for dynamic simulation of continuous systems,the adequate parameters of the controller must be defined,aiming to reduce the WIP levels and to smooth the os-cillations. The generalization of the presented model re-quires an endeavor in this direction, since for each partic-ular manufacturing system, a different type of controllerwith specific tuning could provide the best results. Thus,the control synthesis for different manufacturing systemsis still an issue to be tackled.
The analysis of the results, especially in terms of the rel-ative processing frequencies of the machines, show howmuch and when the capacity of each workstation shouldbe increased or reduced, in order to achieve the desiredlevels of WIP. One practical limitation of the model refersto the level of aggregation of the data. Therefore, it canindicate that, for a certain period of time, a given stationshould work 5% above its regular capacity, but it does notgive indications regarding the detailed scheduling level,i.e. indications about which specific jobs/products to pro-cess with this extra capacity, in which sequence, and soon. In other words, the model is suitable for the planninglevel, instead of for the detailed execution level. In orderto overcome this limitation, it could be used together withdiscrete event simulation models, or future efforts couldbe undertaken towards incorporating variables that con-cern the scheduling level, such as set up times of the ma-chines or processing times of individual jobs.
5 Final RemarksIn this paper, we proposed a closed-loop simulation modelwith a PI controller to depict the dynamics of multi-workstation production system. The model was imple-mented and simulated in Matlabr and Simulinkr con-
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sidering the warm-up of the system, when the initiallyempty buffers should be filled to desired levels, while themedium-term customer demand is fulfilled.
The results of exploratory tests showed that the satura-tion limits of the integral controller exert a relevant influ-ence in the reduction of the work in process, but may alsointroduce some punctual instabilities. In future works, thisparameter and the gains of the controllers could be simul-taneously optimized, by means of the application of con-trol theory tools and the execution of systematic experi-ments.
In terms of operations management, the proposed sim-ulation model has the potential to give prescriptive direc-tions about the dynamic adjustment of the capacities, inorder to keep the WIP in the desired levels and the pro-duction costs relatively low, when the smoothing of capac-ity variations is set as a goal. In conventional productionplanning and control systems, based on MRP, the shortterm variations in production are avoided by means of theimplementation of time fences to demand, planning or or-der release, disregarding the dynamics of the system andits ability to react to disturbances. In this sense, the pre-sented tool can complement the existing tools for analysisand control of production systems and supply chains, al-lowing to take the perspective of the dynamics into con-sideration.
AcknowledgementsJ.K.S would like to thank CNPq (Brazilian National Coun-cil for Scientific and Technological Development) for sup-porting this research (Grants 200648/2015-2).
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EUROSIM 2016 & SIMS 2016
465DOI: 10.3384/ecp17142459 Proceedings of the 9th EUROSIM & the 57th SIMSSeptember 12th-16th, 2016, Oulu, Finland
