Sagawa Land 2018 Representing WLC Mathmod

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<p>IFAC PapersOnLine 51-2 (2018) 825–830</p><p>ScienceDirectScienceDirect</p><p>Available online at www.sciencedirect.com</p><p>2405-8963 © 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.</p><p>Peer review under responsibility of International Federation of Automatic Control.</p><p>10.1016/j.ifacol.2018.04.016</p><p>© 2018, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.</p><p>10.1016/j.ifacol.2018.04.016 2405-8963</p><p>Representing workload control of</p><p>manufacturing systems as a dynamic model</p><p>Juliana K. Sagawa ∗ Martin J. Land ∗∗</p><p>∗ Production Engineering Department, Federal University of São</p><p>Carlos, São Carlos, SP 13565-905 Brazil (e-mail:</p><p>juliana@dep.ufscar.br).</p><p>∗∗ Department of Operations, Faculty of Economics and Business,</p><p>University of Groningen, 9700 AV Groningen, The Netherlands</p><p>(e-mail:m.j.land@rug.nl)</p><p>Abstract: Workload control (WLC) is a methodology that aims to regulate the total workload</p><p>in the production system by controlling the input and output of orders. It leads to more</p><p>predictable throughput times and more accurate delivery date promising. In this paper, we</p><p>propose a representation of the workload control principles by means of a dynamic model</p><p>based on bond graphs, and present the modeling of this system. According to the bond graph</p><p>methodology, the manufacturing entities can be associated to constitutive equations, which</p><p>can be combined to generate a state model. The presented state model provides informations</p><p>about instantaneous levels of work in process of the system, and can show the effect of release</p><p>policies (input control) and capacity adjustments (output control) into these levels of work in</p><p>process. Exploratory simulations were carried out in Simulink. The existing simulation of WLC</p><p>systems usually employs open-loop descriptive models based on discrete events simulation, and</p><p>the parameters of input and output control are empirically/experimentally defined. A gap was</p><p>identified in the literature, specially concerning the parameter setting for output control. In</p><p>the proposed model, input and output control are performed automatically, by the controllers</p><p>implemented to the dynamic model (closed-loop system). Therefore, the proposed approach can</p><p>bring prescriptive directions to the parameter setting of WLC, and we believe it can also bring</p><p>future comparative insights to the existing simulations of this methodology.</p><p>Keywords: dynamic modelling, workload control, bond graphs, production control,</p><p>manufacturing systems.</p><p>1. INTRODUCTION</p><p>Workload Control is a production planning and control</p><p>methodology originally developed for job shops and make-</p><p>to-order (MTS) companies (Zäpfel and Missbauer (1993);</p><p>Stevenson et al. (2005); Land (2006)), whose production</p><p>plans are based on customer orders. It is grounded on</p><p>the principle of input/output control (Plossl and Wight</p><p>(1973)), which states that the input rate to a shop should</p><p>be equal to the output rate. Workload refers to the amount</p><p>of work that must be processed, measured in time units.</p><p>As known, job shops are characterized by a variety of</p><p>products with different production routings; their produc-</p><p>tion usually employs universal machinery arranged in a</p><p>functional layout. According to Land (2006), dealing with</p><p>time-phased capacity requirements is crucial for the con-</p><p>trol of a job shop, since various jobs may compete for the</p><p>capacity of a work station at any time. The control of work</p><p>in process in the shop leads to the control of throughput</p><p>time of the orders and, as a consequence, more predictable</p><p>� The first author would like to thank the São Paulo Research Foun-</p><p>dation (FAPESP) [grant #2017/24716-6] and the Brazilian National</p><p>Council for Scientific and Technological Development (CNPq) [grant</p><p>#20064 8/2015-2]. We also thank the reviewers for the comments.</p><p>completion times and more assertive definition of due</p><p>dates. Various works have already reported benefits of the</p><p>application of WLC (Hendry et al. (2008); Thürer et al.</p><p>(2011)).The methodology also embeds a timing function,</p><p>which helps to increase the due date performance (Land</p><p>and Gaalman (1996); Land (2006)).</p><p>Input control, i.e. the regulation of how much work should</p><p>enter the shop is an important aspect of workload control</p><p>(Land (2006); Fredendall et al. (2010); Soepenberg et al.</p><p>(2012)). According to the WLC concept, this decision is</p><p>taken in three levels: the entry level, which is used to</p><p>control the amount of accepted work; the release level,</p><p>which refers to the control of the amount work in the shop</p><p>floor; and the dispatching level, which affects the progress</p><p>of individual jobs through the work stations (Hendry and</p><p>Kingsman (1991)). The release policy associated to the</p><p>intermediary decision level is named load-oriented order</p><p>release. According to that, orders only should be released</p><p>if they did not exceed a specified limit when their load is</p><p>summed up to the current (existing) load of the shop floor.</p><p>In the literature of the area, this limit is usually referred</p><p>to as workload norm.</p><p>The WLC concept is mostly tested and evaluated in the</p><p>literature by means of discrete events simulation, with</p><p>Proceedings of the 9th Vienna International Conference on</p><p>Mathematical Modelling</p><p>Vienna, Austria, February 21-23, 2018</p><p>Copyright © 2018 IFAC 1</p><p>Representing workload control of</p><p>manufacturing systems as a dynamic model</p><p>Juliana K. Sagawa ∗ Martin J. Land ∗∗</p><p>∗ Production Engineering Department, Federal University of São</p><p>Carlos, São Carlos, SP 13565-905 Brazil (e-mail:</p><p>juliana@dep.ufscar.br).</p><p>∗∗ Department of Operations, Faculty of Economics and Business,</p><p>University of Groningen, 9700 AV Groningen, The Netherlands</p><p>(e-mail:m.j.land@rug.nl)</p><p>Abstract: Workload control (WLC) is a methodology that aims to regulate the total workload</p><p>in the production system by controlling the input and output of orders. It leads to more</p><p>predictable throughput times and more accurate delivery date promising. In this paper, we</p><p>propose a representation of the workload control principles by means of a dynamic model</p><p>based on bond graphs, and present the modeling of this system. According to the bond graph</p><p>methodology, the manufacturing entities can be associated to constitutive equations, which</p><p>can be combined to generate a state model. The presented state model provides informations</p><p>about instantaneous levels of work in process of the system, and can show the effect of release</p><p>policies (input control) and capacity adjustments (output control) into these levels of work in</p><p>process. Exploratory simulations were carried out in Simulink. The existing simulation of WLC</p><p>systems usually employs open-loop descriptive models based on discrete events simulation, and</p><p>the parameters of input and output control are empirically/experimentally defined. A gap was</p><p>identified in the literature, specially concerning the parameter setting for output control. In</p><p>the proposed model, input and output control are performed automatically, by the controllers</p><p>implemented to the dynamic model (closed-loop system). Therefore, the proposed approach can</p><p>bring prescriptive directions to the parameter setting of WLC, and we believe it can also bring</p><p>future comparative insights to the existing simulations of this methodology.</p><p>Keywords: dynamic modelling, workload control, bond graphs, production control,</p><p>manufacturing systems.</p><p>1. INTRODUCTION</p><p>Workload Control is a production planning and control</p><p>methodology originally developed for job shops and make-</p><p>to-order (MTS) companies (Zäpfel and Missbauer (1993);</p><p>Stevenson et al. (2005); Land (2006)), whose production</p><p>plans are based on customer orders. It is grounded on</p><p>the principle of input/output control (Plossl and Wight</p><p>(1973)), which states that the input rate to a shop should</p><p>be equal to the output rate. Workload refers to the amount</p><p>of work that must be processed, measured in time units.</p><p>As known, job shops are characterized by a variety of</p><p>products with different production routings; their produc-</p><p>tion usually</p><p>employs universal machinery arranged in a</p><p>functional layout. According to Land (2006), dealing with</p><p>time-phased capacity requirements is crucial for the con-</p><p>trol of a job shop, since various jobs may compete for the</p><p>capacity of a work station at any time. The control of work</p><p>in process in the shop leads to the control of throughput</p><p>time of the orders and, as a consequence, more predictable</p><p>� The first author would like to thank the São Paulo Research Foun-</p><p>dation (FAPESP) [grant #2017/24716-6] and the Brazilian National</p><p>Council for Scientific and Technological Development (CNPq) [grant</p><p>#20064 8/2015-2]. We also thank the reviewers for the comments.</p><p>completion times and more assertive definition of due</p><p>dates. Various works have already reported benefits of the</p><p>application of WLC (Hendry et al. (2008); Thürer et al.</p><p>(2011)).The methodology also embeds a timing function,</p><p>which helps to increase the due date performance (Land</p><p>and Gaalman (1996); Land (2006)).</p><p>Input control, i.e. the regulation of how much work should</p><p>enter the shop is an important aspect of workload control</p><p>(Land (2006); Fredendall et al. (2010); Soepenberg et al.</p><p>(2012)). According to the WLC concept, this decision is</p><p>taken in three levels: the entry level, which is used to</p><p>control the amount of accepted work; the release level,</p><p>which refers to the control of the amount work in the shop</p><p>floor; and the dispatching level, which affects the progress</p><p>of individual jobs through the work stations (Hendry and</p><p>Kingsman (1991)). The release policy associated to the</p><p>intermediary decision level is named load-oriented order</p><p>release. According to that, orders only should be released</p><p>if they did not exceed a specified limit when their load is</p><p>summed up to the current (existing) load of the shop floor.</p><p>In the literature of the area, this limit is usually referred</p><p>to as workload norm.</p><p>The WLC concept is mostly tested and evaluated in the</p><p>literature by means of discrete events simulation, with</p><p>Proceedings of the 9th Vienna International Conference on</p><p>Mathematical Modelling</p><p>Vienna, Austria, February 21-23, 2018</p><p>Copyright © 2018 IFAC 1</p><p>Representing workload control of</p><p>manufacturing systems as a dynamic model</p><p>Juliana K. Sagawa ∗ Martin J. Land ∗∗</p><p>∗ Production Engineering Department, Federal University of São</p><p>Carlos, São Carlos, SP 13565-905 Brazil (e-mail:</p><p>juliana@dep.ufscar.br).</p><p>∗∗ Department of Operations, Faculty of Economics and Business,</p><p>University of Groningen, 9700 AV Groningen, The Netherlands</p><p>(e-mail:m.j.land@rug.nl)</p><p>Abstract: Workload control (WLC) is a methodology that aims to regulate the total workload</p><p>in the production system by controlling the input and output of orders. It leads to more</p><p>predictable throughput times and more accurate delivery date promising. In this paper, we</p><p>propose a representation of the workload control principles by means of a dynamic model</p><p>based on bond graphs, and present the modeling of this system. According to the bond graph</p><p>methodology, the manufacturing entities can be associated to constitutive equations, which</p><p>can be combined to generate a state model. The presented state model provides informations</p><p>about instantaneous levels of work in process of the system, and can show the effect of release</p><p>policies (input control) and capacity adjustments (output control) into these levels of work in</p><p>process. Exploratory simulations were carried out in Simulink. The existing simulation of WLC</p><p>systems usually employs open-loop descriptive models based on discrete events simulation, and</p><p>the parameters of input and output control are empirically/experimentally defined. A gap was</p><p>identified in the literature, specially concerning the parameter setting for output control. In</p><p>the proposed model, input and output control are performed automatically, by the controllers</p><p>implemented to the dynamic model (closed-loop system). Therefore, the proposed approach can</p><p>bring prescriptive directions to the parameter setting of WLC, and we believe it can also bring</p><p>future comparative insights to the existing simulations of this methodology.</p><p>Keywords: dynamic modelling, workload control, bond graphs, production control,</p><p>manufacturing systems.</p><p>1. INTRODUCTION</p><p>Workload Control is a production planning and control</p><p>methodology originally developed for job shops and make-</p><p>to-order (MTS) companies (Zäpfel and Missbauer (1993);</p><p>Stevenson et al. (2005); Land (2006)), whose production</p><p>plans are based on customer orders. It is grounded on</p><p>the principle of input/output control (Plossl and Wight</p><p>(1973)), which states that the input rate to a shop should</p><p>be equal to the output rate. Workload refers to the amount</p><p>of work that must be processed, measured in time units.</p><p>As known, job shops are characterized by a variety of</p><p>products with different production routings; their produc-</p><p>tion usually employs universal machinery arranged in a</p><p>functional layout. According to Land (2006), dealing with</p><p>time-phased capacity requirements is crucial for the con-</p><p>trol of a job shop, since various jobs may compete for the</p><p>capacity of a work station at any time. The control of work</p><p>in process in the shop leads to the control of throughput</p><p>time of the orders and, as a consequence, more predictable</p><p>� The first author would like to thank the São Paulo Research Foun-</p><p>dation (FAPESP) [grant #2017/24716-6] and the Brazilian National</p><p>Council for Scientific and Technological Development (CNPq) [grant</p><p>#20064 8/2015-2]. We also thank the reviewers for the comments.</p><p>completion times and more assertive definition of due</p><p>dates. Various works have already reported benefits of the</p><p>application of WLC (Hendry et al. (2008); Thürer et al.</p><p>(2011)).The methodology also embeds a timing function,</p><p>which helps to increase the due date performance (Land</p><p>and Gaalman (1996); Land (2006)).</p><p>Input control, i.e. the regulation of how much work should</p><p>enter the shop is an important aspect of workload control</p><p>(Land (2006); Fredendall et al. (2010); Soepenberg et al.</p><p>(2012)). According to the WLC concept, this decision is</p><p>taken in three levels: the entry level, which is used to</p><p>control the amount of accepted work; the release level,</p><p>which refers to the control of the amount work in the shop</p><p>floor; and the dispatching level, which affects the progress</p><p>of individual jobs through the work stations (Hendry and</p><p>Kingsman (1991)). The release policy associated to the</p><p>intermediary decision level is named load-oriented order</p><p>release. According to that, orders only should be released</p><p>if they did not exceed a specified limit when their load is</p><p>summed up to the current (existing) load of the shop floor.</p><p>In the literature of the area, this limit is usually referred</p><p>to as workload norm.</p><p>The WLC concept is mostly tested and evaluated in the</p><p>literature by means of discrete events simulation, with</p><p>Proceedings of the 9th Vienna International Conference on</p><p>Mathematical Modelling</p><p>Vienna, Austria, February 21-23, 2018</p><p>Copyright © 2018 IFAC 1</p><p>Representing workload control of</p><p>manufacturing systems as a dynamic model</p><p>Juliana K. Sagawa ∗ Martin J. Land ∗∗</p><p>∗ Production Engineering Department, Federal University of São</p><p>Carlos, São Carlos, SP 13565-905 Brazil (e-mail:</p><p>juliana@dep.ufscar.br).</p><p>∗∗ Department of Operations, Faculty of Economics and Business,</p><p>University of Groningen, 9700 AV Groningen, The Netherlands</p><p>(e-mail:m.j.land@rug.nl)</p><p>Abstract: Workload control (WLC) is a methodology that aims to regulate the total workload</p><p>in the production system by controlling the input and output of orders. It leads to more</p><p>predictable throughput times and more accurate delivery date promising. In this paper, we</p><p>propose a representation of the workload control principles by means of a dynamic model</p><p>based on bond graphs, and present the modeling of this system. According to the bond graph</p><p>methodology, the manufacturing entities can be associated to constitutive equations, which</p><p>can be combined to generate a state model. The presented state model provides informations</p><p>about instantaneous levels of work in</p><p>process of the system, and can show the effect of release</p><p>policies (input control) and capacity adjustments (output control) into these levels of work in</p><p>process. Exploratory simulations were carried out in Simulink. The existing simulation of WLC</p><p>systems usually employs open-loop descriptive models based on discrete events simulation, and</p><p>the parameters of input and output control are empirically/experimentally defined. A gap was</p><p>identified in the literature, specially concerning the parameter setting for output control. In</p><p>the proposed model, input and output control are performed automatically, by the controllers</p><p>implemented to the dynamic model (closed-loop system). Therefore, the proposed approach can</p><p>bring prescriptive directions to the parameter setting of WLC, and we believe it can also bring</p><p>future comparative insights to the existing simulations of this methodology.</p><p>Keywords: dynamic modelling, workload control, bond graphs, production control,</p><p>manufacturing systems.</p><p>1. INTRODUCTION</p><p>Workload Control is a production planning and control</p><p>methodology originally developed for job shops and make-</p><p>to-order (MTS) companies (Zäpfel and Missbauer (1993);</p><p>Stevenson et al. (2005); Land (2006)), whose production</p><p>plans are based on customer orders. It is grounded on</p><p>the principle of input/output control (Plossl and Wight</p><p>(1973)), which states that the input rate to a shop should</p><p>be equal to the output rate. Workload refers to the amount</p><p>of work that must be processed, measured in time units.</p><p>As known, job shops are characterized by a variety of</p><p>products with different production routings; their produc-</p><p>tion usually employs universal machinery arranged in a</p><p>functional layout. According to Land (2006), dealing with</p><p>time-phased capacity requirements is crucial for the con-</p><p>trol of a job shop, since various jobs may compete for the</p><p>capacity of a work station at any time. The control of work</p><p>in process in the shop leads to the control of throughput</p><p>time of the orders and, as a consequence, more predictable</p><p>� The first author would like to thank the São Paulo Research Foun-</p><p>dation (FAPESP) [grant #2017/24716-6] and the Brazilian National</p><p>Council for Scientific and Technological Development (CNPq) [grant</p><p>#20064 8/2015-2]. We also thank the reviewers for the comments.</p><p>completion times and more assertive definition of due</p><p>dates. Various works have already reported benefits of the</p><p>application of WLC (Hendry et al. (2008); Thürer et al.</p><p>(2011)).The methodology also embeds a timing function,</p><p>which helps to increase the due date performance (Land</p><p>and Gaalman (1996); Land (2006)).</p><p>Input control, i.e. the regulation of how much work should</p><p>enter the shop is an important aspect of workload control</p><p>(Land (2006); Fredendall et al. (2010); Soepenberg et al.</p><p>(2012)). According to the WLC concept, this decision is</p><p>taken in three levels: the entry level, which is used to</p><p>control the amount of accepted work; the release level,</p><p>which refers to the control of the amount work in the shop</p><p>floor; and the dispatching level, which affects the progress</p><p>of individual jobs through the work stations (Hendry and</p><p>Kingsman (1991)). The release policy associated to the</p><p>intermediary decision level is named load-oriented order</p><p>release. According to that, orders only should be released</p><p>if they did not exceed a specified limit when their load is</p><p>summed up to the current (existing) load of the shop floor.</p><p>In the literature of the area, this limit is usually referred</p><p>to as workload norm.</p><p>The WLC concept is mostly tested and evaluated in the</p><p>literature by means of discrete events simulation, with</p><p>Proceedings of the 9th Vienna International Conference on</p><p>Mathematical Modelling</p><p>Vienna, Austria, February 21-23, 2018</p><p>Copyright © 2018 IFAC 1</p><p>826 Juliana K. Sagawa et al. / IFAC PapersOnLine 51-2 (2018) 825–830</p><p>order arrivals, due dates and processing times modelled</p><p>as probability distributions. This technique has several</p><p>advantages, such as its ability to represent scenarios in</p><p>a very detailed level, i.e. high descriptive power. On</p><p>the other hand, efforts have been also employed towards</p><p>the representation and analysis of production systems as</p><p>dynamic models (Scholz-Reiter et al. (2005); Jeken et al.</p><p>(2012); Duffie et al. (2014); Falu and Duffie (2014)). A</p><p>relevant advantage of these models is their ability to</p><p>provide prescriptive solutions or guidelines by means of</p><p>the implementation of controllers grounded on the control</p><p>engineering discipline. They also allow to derive insights</p><p>about the general dynamic behaviour of the productions</p><p>systems.</p><p>Different approaches from the control engineering disci-</p><p>pline have been proposed to depict production systems</p><p>and supply chains, such as the use of flow models, block</p><p>diagrams and transfer functions, optimal control, model</p><p>predictive control, and so on. A recently explored option</p><p>for this purpose was presented in Sagawa et al. (2017),</p><p>where multi-product manufacturing systems are modelled</p><p>as flow systems using bond graphs. This approach is based</p><p>on the pioneer representation of manufacturing entities</p><p>as bond graphs Ferney (2000), in such a way that the</p><p>mathematical formalism of this technique is preserved.</p><p>Considering the context presented, this paper aims to</p><p>represent the workload control concept as a dynamic</p><p>model based on bond graphs, in order to bring new</p><p>possibilities of analysis of this concept, from a dynamic</p><p>perspective. The control of a multi-product system based</p><p>on a pull strategy were already implemented in Sagawa</p><p>and Nagano (2015a,b). Likewise, the idea is to extend this</p><p>approach to other order release strategies of Production</p><p>Planing and Control (PPC), such as, in this case, load-</p><p>oriented order release.</p><p>In the WLC, the release decision is directly based on</p><p>the current state of the shop floor, in terms of existing</p><p>workload. As know, control theory is grounded on the</p><p>concepts of state, state space and state variables. That is</p><p>an additional reason for the interest for integrating these</p><p>two areas, since both of them focus on states.</p><p>2. PROBLEM SETTING AND MODELLING</p><p>The modelling approach adopted in this research is based</p><p>on bond graphs, which are a pictorial representation of</p><p>power/energy transmission among ideal elements. The</p><p>methodology basically employs the generalized variables</p><p>of effort and flow, which correspond to respective power</p><p>variables in different physical domains, such as electrical,</p><p>mechanical, hydraulic, etc. From the pictorial representa-</p><p>tion of a system, it is possible to derive its mathematical</p><p>model, based on the correspondence of each ideal element</p><p>and its constitutive equation, defined according to the laws</p><p>of Physics.</p><p>Although not originally designed for this application,</p><p>adaptations were proposed for the use of bond graphs</p><p>to model manufacturing systems. Ferney (2000) proposed</p><p>the representation of a production station based on analo-</p><p>gies with a resistor, capacitor and source of effort. This</p><p>approach have been recently adopted to model multi-</p><p>product manufacturing systems (Sagawa et al. (2017)).</p><p>The basic physical variable, in this case, is the material</p><p>flow or amount of work, which implies that the resulting</p><p>models are ”fluidised”, i.e. are based on flows and rates,</p><p>such as amount of work per time or material flow rate.</p><p>According to Ferney (2000), if the assumption of buffers</p><p>with unlimited capacity is taken/set, the output flow of</p><p>the i-th production station may be defined as:</p><p>fi = Uimin(1, qi). (1)</p><p>where Ui represents the processing frequency of the station</p><p>(for more details about this formulation and the use of the</p><p>minimum function in this expression, please refer to Ferney</p><p>(2000)). According to this perspective, the material flow</p><p>being processed by a machine is analogous to the electrical</p><p>current in a resistor, to which Ui is the inverse of resistance.</p><p>The rate of material storage q̇i in an intermediary buffer</p><p>of the system is thus given by the difference between the</p><p>station input and output flows, that is:</p><p>q̇i = fei − Uimin(1, qi), (2)</p><p>where</p><p>fei is the input flow of station i, and Ui and qi</p><p>are defined like for (1). In terms of the bond graphs, (2)</p><p>is already a simplification, derived from the assumption</p><p>of unlimited capacity of the buffers (see Sagawa et al.</p><p>(2017)). The bond graph model, however, allows the repre-</p><p>sentation of a more general case and has some advantages,</p><p>such as the easiness of obtaining the state model from</p><p>the pictorial representation. For such reason, this approach</p><p>was kept as the basis for the presented model. Based on</p><p>this basic equation, several stations may be arranged in</p><p>series or in parallel. In Fig. 1 is represented a 4-station</p><p>production system with one source of flow. To this model,</p><p>we added an extra production station that corresponds to</p><p>a pool of orders (and the respective raw materials needed</p><p>to process these orders). The job pool is an important</p><p>element of workload control. It is where the jobs queue</p><p>up before being released to the shop floor. If the increase</p><p>in the existing workload caused by a given job exceeds</p><p>the workload norm, the job is not released and goes back</p><p>to the pool. The workstations and the pool in Fig. 1 are</p><p>the controllable elements (with control variables Ui and</p><p>Upool); the flows in each section of the system are also vari-</p><p>ables, influenced by the applied controls; the percentages</p><p>of split in divergent junctions are parameters calculated in</p><p>function of a fixed production mix; the demand rates for</p><p>each family are parameters which were initially maintained</p><p>fixed, but can vary in time to represent volatile demand;</p><p>m1 and m2 are parameters whose values are defined in</p><p>order to keep the system balanced (see Subsection 2.1).</p><p>More specifically, the variables used for modelling the</p><p>production system are defined as follows:</p><p>q̇pool: rate of release of material/orders in the pool. Corre-</p><p>spond to the difference between the input and output flow</p><p>in the pool;</p><p>q̇i: rate of material storage or consumption in the buffer i.</p><p>In the presented case, i = 1, 2, 3, 4;</p><p>qpool: amount of material (orders waiting for release in the</p><p>pool);</p><p>qi: amount of material stored in the buffer i, i = 1, 2, 3, 4;</p><p>Proceedings of the 9th MATHMOD</p><p>Vienna, Austria, February 21-23, 2018</p><p>2</p><p>Juliana K. Sagawa et al. / IFAC PapersOnLine 51-2 (2018) 825–830 827</p><p>order arrivals, due dates and processing times modelled</p><p>as probability distributions. This technique has several</p><p>advantages, such as its ability to represent scenarios in</p><p>a very detailed level, i.e. high descriptive power. On</p><p>the other hand, efforts have been also employed towards</p><p>the representation and analysis of production systems as</p><p>dynamic models (Scholz-Reiter et al. (2005); Jeken et al.</p><p>(2012); Duffie et al. (2014); Falu and Duffie (2014)). A</p><p>relevant advantage of these models is their ability to</p><p>provide prescriptive solutions or guidelines by means of</p><p>the implementation of controllers grounded on the control</p><p>engineering discipline. They also allow to derive insights</p><p>about the general dynamic behaviour of the productions</p><p>systems.</p><p>Different approaches from the control engineering disci-</p><p>pline have been proposed to depict production systems</p><p>and supply chains, such as the use of flow models, block</p><p>diagrams and transfer functions, optimal control, model</p><p>predictive control, and so on. A recently explored option</p><p>for this purpose was presented in Sagawa et al. (2017),</p><p>where multi-product manufacturing systems are modelled</p><p>as flow systems using bond graphs. This approach is based</p><p>on the pioneer representation of manufacturing entities</p><p>as bond graphs Ferney (2000), in such a way that the</p><p>mathematical formalism of this technique is preserved.</p><p>Considering the context presented, this paper aims to</p><p>represent the workload control concept as a dynamic</p><p>model based on bond graphs, in order to bring new</p><p>possibilities of analysis of this concept, from a dynamic</p><p>perspective. The control of a multi-product system based</p><p>on a pull strategy were already implemented in Sagawa</p><p>and Nagano (2015a,b). Likewise, the idea is to extend this</p><p>approach to other order release strategies of Production</p><p>Planing and Control (PPC), such as, in this case, load-</p><p>oriented order release.</p><p>In the WLC, the release decision is directly based on</p><p>the current state of the shop floor, in terms of existing</p><p>workload. As know, control theory is grounded on the</p><p>concepts of state, state space and state variables. That is</p><p>an additional reason for the interest for integrating these</p><p>two areas, since both of them focus on states.</p><p>2. PROBLEM SETTING AND MODELLING</p><p>The modelling approach adopted in this research is based</p><p>on bond graphs, which are a pictorial representation of</p><p>power/energy transmission among ideal elements. The</p><p>methodology basically employs the generalized variables</p><p>of effort and flow, which correspond to respective power</p><p>variables in different physical domains, such as electrical,</p><p>mechanical, hydraulic, etc. From the pictorial representa-</p><p>tion of a system, it is possible to derive its mathematical</p><p>model, based on the correspondence of each ideal element</p><p>and its constitutive equation, defined according to the laws</p><p>of Physics.</p><p>Although not originally designed for this application,</p><p>adaptations were proposed for the use of bond graphs</p><p>to model manufacturing systems. Ferney (2000) proposed</p><p>the representation of a production station based on analo-</p><p>gies with a resistor, capacitor and source of effort. This</p><p>approach have been recently adopted to model multi-</p><p>product manufacturing systems (Sagawa et al. (2017)).</p><p>The basic physical variable, in this case, is the material</p><p>flow or amount of work, which implies that the resulting</p><p>models are ”fluidised”, i.e. are based on flows and rates,</p><p>such as amount of work per time or material flow rate.</p><p>According to Ferney (2000), if the assumption of buffers</p><p>with unlimited capacity is taken/set, the output flow of</p><p>the i-th production station may be defined as:</p><p>fi = Uimin(1, qi). (1)</p><p>where Ui represents the processing frequency of the station</p><p>(for more details about this formulation and the use of the</p><p>minimum function in this expression, please refer to Ferney</p><p>(2000)). According to this perspective, the material flow</p><p>being processed by a machine is analogous to the electrical</p><p>current in a resistor, to which Ui is the inverse of resistance.</p><p>The rate of material storage q̇i in an intermediary buffer</p><p>of the system is thus given by the difference between the</p><p>station input and output flows, that is:</p><p>q̇i = fei − Uimin(1, qi), (2)</p><p>where fei is the input flow of station i, and Ui and qi</p><p>are defined like for (1). In terms of the bond graphs, (2)</p><p>is already a simplification, derived from the assumption</p><p>of unlimited capacity of the buffers (see Sagawa et al.</p><p>(2017)). The bond graph model, however, allows the repre-</p><p>sentation of a more general case and has some advantages,</p><p>such as the easiness of obtaining the state model from</p><p>the pictorial representation. For such reason, this approach</p><p>was kept as the basis for the presented model. Based on</p><p>this basic equation, several stations may be arranged in</p><p>series or in parallel. In Fig. 1 is represented a 4-station</p><p>production system with one source of flow. To this model,</p><p>we added an extra production station that corresponds to</p><p>a pool of orders (and the respective raw materials needed</p><p>to process these orders). The job pool is an important</p><p>element of workload control. It is where the jobs queue</p><p>up before being released to the shop floor. If the increase</p><p>in the existing workload caused by a given job exceeds</p><p>the workload norm, the job is not released and goes back</p><p>to the pool. The workstations and the pool in Fig. 1 are</p><p>the controllable elements (with control variables Ui and</p><p>Upool); the flows in each section of the system are also vari-</p><p>ables, influenced by the applied controls; the percentages</p><p>of split in divergent junctions are parameters calculated in</p><p>function of a fixed production mix; the demand rates for</p><p>each family are parameters which were initially maintained</p><p>fixed, but can vary in time to represent volatile demand;</p><p>m1 and m2 are</p><p>parameters whose values are defined in</p><p>order to keep the system balanced (see Subsection 2.1).</p><p>More specifically, the variables used for modelling the</p><p>production system are defined as follows:</p><p>q̇pool: rate of release of material/orders in the pool. Corre-</p><p>spond to the difference between the input and output flow</p><p>in the pool;</p><p>q̇i: rate of material storage or consumption in the buffer i.</p><p>In the presented case, i = 1, 2, 3, 4;</p><p>qpool: amount of material (orders waiting for release in the</p><p>pool);</p><p>qi: amount of material stored in the buffer i, i = 1, 2, 3, 4;</p><p>Proceedings of the 9th MATHMOD</p><p>Vienna, Austria, February 21-23, 2018</p><p>2</p><p>Upool</p><p>S01-pool</p><p>fs1</p><p>U2</p><p>U3 U4</p><p>U1</p><p>CNC-machine</p><p>centers</p><p>turning</p><p>(CNC)</p><p>drilling</p><p>turning</p><p>(conventional)</p><p>f24 45,45%</p><p>fF1 54,55%</p><p>fs2</p><p>1,5 units/day</p><p>fF2 53,19%</p><p>fF4 46,81%</p><p>1,25 units/day</p><p>1,1 units/day</p><p>f34 45,83% fs3</p><p>fF3 54,17%</p><p>1,3 units/day</p><p>m2</p><p>m1</p><p>Fig. 1. Four-station production system with a job pool</p><p>U01pool: frequency of the source of material/orders that</p><p>feed the pool. It is the primary input for the whole system;</p><p>Upool: frequency with which the release operation is per-</p><p>formed. In the proposed system, the release is viewed as</p><p>an operation and the pool is a station that works in a way</p><p>similar to a machine station;</p><p>Ui: processing frequency of machine i, which is related to</p><p>its production rate or could be compared to the average</p><p>processing times, in a scenario of discrete simulation.</p><p>The application of (2) to the pool and to the buffer of</p><p>station 2 yields, respectively:</p><p>q̇pool = −Upool min (1, qpool) + U01pool (3)</p><p>q̇2 = U1 min (1, q1)− U2 min (1, q2) . (4)</p><p>The station 2 is connected in series with station 1. There-</p><p>fore, the output of station 1 is the input of station 2,</p><p>as seen in (4). In the system under analysis, 4 different</p><p>products (or families of products) with different customer</p><p>demands are manufactured. Each product or family follows</p><p>its specific production routing, but the general flow of the</p><p>system is unidirectional. The product mix associated to</p><p>the system is calculated based on the demand rates for</p><p>each product and the total demand rate that must be</p><p>fulfilled by the system.</p><p>Table 1 outlines the data about product families, customer</p><p>demands and processing routings. In this table it is shown,</p><p>for instance, that the stations 1 and 2 are shared by</p><p>product families 1 and 2, while the station 4 is shared</p><p>by product families 2 and 4. Considering this and the</p><p>demand rates, it is possible to calculate the percentages</p><p>of material that must flow to each branch of the system,</p><p>in order to produce this specific mix, as shown in Fig. 1.</p><p>These percentages, as shown, are associated to divergent</p><p>and convergent junctions of the manufacturing system.</p><p>Table 1. Product families, process routings and</p><p>demand</p><p>S1 S2 S3 S4 Demand rates (units/day)</p><p>Family 1 x x 1.5</p><p>Family 2 x x x 1.25</p><p>Family 3 x 1.3</p><p>Family 4 x x 1.1</p><p>Convergent junctions may be represented as a 1-junction</p><p>in bond graphs, which establishes the flow conservation,</p><p>that is,</p><p>fsi =</p><p>P∑</p><p>p=1</p><p>fep, (5)</p><p>where fsi is the output flow of a given i-th convergent</p><p>junction and fep, with p = 1, 2, ...P , are the branches of</p><p>incoming flow. On the other hand, the divergent junctions</p><p>of a manufacturing system may be represented by trans-</p><p>formers in the bond graph methodology, such that the</p><p>output flow of the q-th transformer (i.e. fsq) is the overall</p><p>input flow fei multiplied by its module mq:</p><p>fsq = mqfei, (6)</p><p>with</p><p>∑Q</p><p>q=1 mq = 1 and q = 1, 2, ..., Q.</p><p>For the first divergent junction (Fig. 1), which splits the</p><p>flow coming from the pool to stations 1 and 2, we let</p><p>the modules of the transformers as variables m1 and m2</p><p>in order to add more degrees of freedom to the pool.</p><p>Therefore, by applying (6) and (2), respectively, to the</p><p>first divergent junction of the system and to stations 1</p><p>and 3, we obtain</p><p>q̇1 = m1Upool min (1, qpool)− U1 min (1, q1) (7)</p><p>q̇3 = m2Upool min (1, qpool)− U3 min (1, q3) . (8)</p><p>The main idea of this system is that there is a source of</p><p>material flow and respective orders that feeds the pool of</p><p>orders and material with a certain frequency. We assume</p><p>that the orders and respective raw materials necessary</p><p>to produce them are associated, like the kanban cards</p><p>move along with the parts in a manufacturing system. The</p><p>release frequency of the pool must be controlled based on</p><p>the load of the shop floor. Moreover, it is also possible</p><p>to control not only the frequency of release, but also the</p><p>balance of flow in the different branches of the system, by</p><p>adjusting the percentages m1 and m2.</p><p>The remaining divergent junctions of the system have</p><p>definite modules, that is, the flow is split according to</p><p>average percentages that where calculated based on the</p><p>data of Table 1. Thus, the application of the respective</p><p>equations for the convergent and divergent junctions of</p><p>Fig. 1 yields:</p><p>fe4 = f24 + f34 (9)</p><p>f24 = 0.4545U2 min (1, q2) (10)</p><p>f34 = 0.4583U3 min (1, q3) (11)</p><p>q̇4 = 0.4545U2 min (1, q2)</p><p>+ 0.4593U3 min (1, q3)− U4 min (1, q4) . (12)</p><p>The rates of material storage or consumption in the buffers</p><p>represent the state of the system, since the integral of these</p><p>rates correspond to the instantaneous work in process in</p><p>the shop floor and the amount of orders in the pool. The</p><p>throughput time is one important performance measure</p><p>in workload control, as known. Station throughput times</p><p>include job processing time and waiting time in a given</p><p>station. Thus, the performance in terms of throughput</p><p>times is related to the control of queues (Land (2006)),</p><p>and queues in a discrete system are equivalent to the work</p><p>in process in this continuous system. By arranging (3), (7),</p><p>(4), (8) and (12), we obtain the state representation of the</p><p>system shown in (13).</p><p></p><p></p><p>q̇pool</p><p>q̇1</p><p>q̇2</p><p>q̇3</p><p>q̇4</p><p></p><p>=</p><p></p><p></p><p>−1 0 0 0 0 1</p><p>m1 −1 0 0 0 0</p><p>0 1 −1 0 0 0</p><p>m2 0 0 −1 0 0</p><p>0 0 0.4545 0.4583 −1 0</p><p></p><p></p><p></p><p></p><p>Upool min (1, qpool)</p><p>U1 min (1, q1)</p><p>U2 min (1, q2)</p><p>U3 min (1, q3)</p><p>U4 min (1, q4)</p><p>U01pool</p><p></p><p></p><p>(13)</p><p>Proceedings of the 9th MATHMOD</p><p>Vienna, Austria, February 21-23, 2018</p><p>3</p><p>828 Juliana K. Sagawa et al. / IFAC PapersOnLine 51-2 (2018) 825–830</p><p>2.1 Definition of parameters, steady state solution and</p><p>control law</p><p>The presented system may have up to 8 degrees of freedom,</p><p>namely: the material/orders release frequency of the pool</p><p>(Upool), the processing frequencies of the stations 1 to 4 (U1</p><p>to U4), the frequency of the source of material/orders that</p><p>feed the pool (U01pool) and the modules m1 and m2, which</p><p>allow to adjust the split of flow into the two main branches</p><p>of the system. The demand fulfilment is one condition that</p><p>should be considered prior to the definition of control laws</p><p>for these variables. Therefore, the outputs of the system</p><p>in steady state should correspond to the average demand</p><p>rates for each product family, in the medium-term time</p><p>horizon. This leads the additional equations presented as</p><p>follows.</p><p>fF1 = 0.5455U2 min (1, q2) = dF1 = 1.5units/day (14)</p><p>fF2 = 0.5319U4 min (1, q4) = dF2 = 1.25units/day (15)</p><p>fF3 = 0.5417U3 min (1, q3) = dF3 = 1.3units/day (16)</p><p>fF4 = 0.4681U4 min (1, q4) = dF2 = 1.1units/day (17)</p><p>From these additional equations, it is possible to de-</p><p>termine the desired frequencies of the machines in the</p><p>steady state, Uip, that is, the frequencies with which</p><p>the machines should process the material flow in order</p><p>to attend the average demand for each family. Thus,</p><p>U2p = 2.75units/day, U3p = 2.4units/day and U4p =</p><p>2.3500units/day.</p><p>In the steady state, the level of the buffers must be stable,</p><p>i.e. q̇i = 0, and there must be more than one material unit</p><p>in stock, that is, min (1, qi) = 1. Thus, the state model</p><p>becomes:</p><p></p><p></p><p>−1 0 0 0 0 1</p><p>m1 −1 0 0 0 0</p><p>0 1 −1 0 0 0</p><p>m2 0 0 −1 0 0</p><p>0 0 0.4545 0.4583 −1 0</p><p></p><p></p><p></p><p></p><p>Upool−p</p><p>U1p</p><p>U2p</p><p>U3p</p><p>U4p</p><p>U01pool−p</p><p></p><p></p><p>=</p><p></p><p></p><p>0</p><p>0</p><p>0</p><p>0</p><p>0</p><p></p><p> (18)</p><p>Replacing U2p in the third line of the system yields U1p =</p><p>U2p = 2.75units/day. We could presume that, in the</p><p>steady state, the flow coming from the pool is balanced</p><p>between the two branches, i.e. m1 = m2 = 0.5. However,</p><p>with this assumption, the equations corresponding to the</p><p>second and fourth rows of the matrix are not verified.</p><p>Therefore, in order to define the values of these modules,</p><p>the following system should be solved:</p><p>m1Upool−p − U1p = 0</p><p>m2Upool−p − U3p = 0 (19)</p><p>m1 +m2 = 1</p><p>It results in m1 = U1p/(U1p + U3p) = 0.534, m2 =</p><p>U3p/(U1p + U3p) = 0.4660 and Upool = 5.1500. From</p><p>the first row of the matrix in (18), it is also plain to</p><p>see that the frequency of the source of the flow in the</p><p>steady state (U01pool−p) corresponds to the frequency of</p><p>the release in the pool (Upool−p) in the steady state (i.e.</p><p>U01pool−p = Upool−p = 5.1500)</p><p>As it can be seen, the distribution of flow into the two</p><p>branches is slightly unbalanced (i.e. m1 �= m2). It means</p><p>that the current product mix and the current demands, in</p><p>0 10 20 30 40 50 60 70 80 90 100</p><p>time (days)</p><p>0</p><p>2</p><p>4</p><p>6</p><p>8</p><p>10</p><p>m</p><p>at</p><p>er</p><p>ia</p><p>l i</p><p>n</p><p>st</p><p>oc</p><p>k</p><p>(m</p><p>2 )</p><p>Stock levels X time</p><p>q</p><p>pool</p><p>q1</p><p>q2</p><p>q3</p><p>q4</p><p>Fig. 2. Job pool and buffers of the system in the initial</p><p>scenario (without control)</p><p>terms of volume, impose a slightly unbalanced load in the</p><p>two processing branches.</p><p>The processing frequency of each workstation, including</p><p>the pool, is controlled based on the relative error of its</p><p>respective buffer, i.e.:</p><p>ei = (qi − qic)/qic, (20)</p><p>where ei is the relative error of buffer i, qi is the instan-</p><p>taneous level of this buffer (amount of material) and qic is</p><p>the reference level of it.</p><p>In addition, the instantaneous adjustments in the process-</p><p>ing frequencies Ui of the stations are relative, i.e. they</p><p>are applied to the processing frequencies in the steady</p><p>state (Uip), calculated in the previous subsection. These</p><p>adjustments are done using Proportional-integral (PI) con-</p><p>trollers. Thus, the control law is represented by:</p><p>Ui = (1 + Piei(t) + Ii</p><p>∫</p><p>ei(t), dt)Uip, (21)</p><p>where Pi and Ii are the design parameters of the controller,</p><p>set as Pi = 0.1 and Ii = 0.001 for the simulations</p><p>(Section 3); ei is defined like in (20) and Uip are calculated</p><p>according to (14)-(19).</p><p>3. SIMULATIONS</p><p>Exploratory simulations of the presented model were car-</p><p>ried out in order to test its ability to represent the WLC</p><p>concept. The state model presented in (13) was imple-</p><p>mented in Matlab and Simulink. In the initial scenario</p><p>considered, all the buffers are filled with an amount of</p><p>work in process that is approximately equivalent to 5 days</p><p>of stock, considering the demand rates. In the 20-th day, an</p><p>increase in the rate of incoming orders occurs, motivated</p><p>by an increase in the demand of the product family 4.</p><p>However, at a first moment, the release rate of orders in the</p><p>pool is not modified, to avoid an increase in the load of the</p><p>shop floor. All the stations also keep processing material</p><p>with the steady state frequencies calculated in the last</p><p>section. Thus, it is expected that the amount of orders</p><p>waiting for the release in the pool increases. This initial</p><p>situation is shown in Fig. 2. It is assumed that the demand</p><p>increased by a constant percentage for one month; that is</p><p>why the increase of the queue in the job pool is linear. The</p><p>initial levels of the buffers shown in this figure were also</p><p>adopted as reference levels, i.e. it is desired that the work</p><p>in process in each buffer is kept close to these levels, in</p><p>order to avoid an increase in the throughput time of the</p><p>shop floor.</p><p>Proceedings of the 9th MATHMOD</p><p>Vienna, Austria, February 21-23, 2018</p><p>4</p><p>Juliana K. Sagawa et al. / IFAC PapersOnLine 51-2 (2018) 825–830 829</p><p>2.1 Definition of parameters, steady state solution and</p><p>control law</p><p>The presented system may have up to 8 degrees of freedom,</p><p>namely: the material/orders release frequency of the pool</p><p>(Upool), the processing frequencies of the stations 1 to 4 (U1</p><p>to U4), the frequency of the source of material/orders that</p><p>feed the pool (U01pool) and the modules m1 and m2, which</p><p>allow to adjust the split of flow into the two main branches</p><p>of the system. The demand fulfilment is one condition that</p><p>should be considered prior to the definition of control laws</p><p>for these variables. Therefore, the outputs of the system</p><p>in steady state should correspond to the average demand</p><p>rates for each product family, in the medium-term time</p><p>horizon. This leads the additional equations presented as</p><p>follows.</p><p>fF1 = 0.5455U2 min (1, q2) = dF1 = 1.5units/day (14)</p><p>fF2 = 0.5319U4 min (1, q4) = dF2 = 1.25units/day (15)</p><p>fF3 = 0.5417U3 min (1, q3) = dF3 = 1.3units/day (16)</p><p>fF4 = 0.4681U4 min (1, q4) = dF2 = 1.1units/day (17)</p><p>From these additional equations, it is possible to de-</p><p>termine the desired frequencies of the machines in the</p><p>steady state, Uip, that is, the frequencies with which</p><p>the machines should process the material flow in order</p><p>to attend the average demand for each family. Thus,</p><p>U2p = 2.75units/day, U3p = 2.4units/day and U4p =</p><p>2.3500units/day.</p><p>In the steady state, the level of the buffers must be stable,</p><p>i.e. q̇i = 0, and there must be more than one material unit</p><p>in stock, that is, min (1, qi) = 1. Thus, the state model</p><p>becomes:</p><p></p><p></p><p>−1 0 0 0 0 1</p><p>m1 −1 0 0 0 0</p><p>0 1 −1 0 0 0</p><p>m2 0 0 −1 0 0</p><p>0 0 0.4545 0.4583 −1 0</p><p></p><p></p><p></p><p></p><p>Upool−p</p><p>U1p</p><p>U2p</p><p>U3p</p><p>U4p</p><p>U01pool−p</p><p></p><p></p><p>=</p><p></p><p></p><p>0</p><p>0</p><p>0</p><p>0</p><p>0</p><p></p><p> (18)</p><p>Replacing U2p in the third line of the system yields U1p =</p><p>U2p = 2.75units/day. We could presume that, in the</p><p>steady state, the flow coming from the pool is balanced</p><p>between the two branches, i.e. m1 = m2 = 0.5. However,</p><p>with this assumption, the equations corresponding to the</p><p>second and fourth rows of the matrix are not verified.</p><p>Therefore, in order to define the values of these modules,</p><p>the following system should be solved:</p><p>m1Upool−p − U1p = 0</p><p>m2Upool−p − U3p = 0 (19)</p><p>m1 +m2 = 1</p><p>It results in m1 = U1p/(U1p + U3p) = 0.534, m2 =</p><p>U3p/(U1p + U3p) = 0.4660 and Upool = 5.1500. From</p><p>the first row of the matrix in (18), it is also plain to</p><p>see that the frequency of the source of the flow in the</p><p>steady state (U01pool−p) corresponds to the frequency of</p><p>the release in the pool (Upool−p) in the steady state (i.e.</p><p>U01pool−p = Upool−p = 5.1500)</p><p>As it can be seen, the distribution of flow into the two</p><p>branches is slightly unbalanced (i.e. m1 �= m2). It means</p><p>that the current product mix and the current demands, in</p><p>0 10 20 30 40 50 60 70 80 90 100</p><p>time (days)</p><p>0</p><p>2</p><p>4</p><p>6</p><p>8</p><p>10</p><p>m</p><p>at</p><p>er</p><p>ia</p><p>l i</p><p>n</p><p>st</p><p>oc</p><p>k</p><p>(m</p><p>2 )</p><p>Stock levels X time</p><p>q</p><p>pool</p><p>q1</p><p>q2</p><p>q3</p><p>q4</p><p>Fig. 2. Job pool and buffers of the system in the initial</p><p>scenario (without control)</p><p>terms of volume, impose a slightly unbalanced load in the</p><p>two processing branches.</p><p>The processing frequency of each workstation, including</p><p>the pool, is controlled based on the relative error of its</p><p>respective buffer, i.e.:</p><p>ei = (qi − qic)/qic, (20)</p><p>where ei is the relative error of buffer i, qi is the instan-</p><p>taneous level of this buffer (amount of material) and qic is</p><p>the reference level of it.</p><p>In addition, the instantaneous adjustments in the process-</p><p>ing frequencies Ui of the stations are relative, i.e. they</p><p>are applied to the processing frequencies in the steady</p><p>state (Uip), calculated in the previous subsection. These</p><p>adjustments are done using Proportional-integral (PI) con-</p><p>trollers. Thus, the control law is represented by:</p><p>Ui = (1 + Piei(t) + Ii</p><p>∫</p><p>ei(t), dt)Uip, (21)</p><p>where Pi and Ii are the design parameters of the controller,</p><p>set as Pi = 0.1 and Ii = 0.001 for the simulations</p><p>(Section 3); ei is defined like in (20) and Uip are calculated</p><p>according to (14)-(19).</p><p>3. SIMULATIONS</p><p>Exploratory simulations of the presented model were car-</p><p>ried out in order to test its ability to represent the WLC</p><p>concept. The state model presented in (13) was imple-</p><p>mented in Matlab and Simulink. In the initial scenario</p><p>considered, all the buffers are filled with an amount of</p><p>work in process that is approximately equivalent to 5 days</p><p>of stock, considering the demand rates. In the 20-th day, an</p><p>increase in the rate of incoming orders occurs, motivated</p><p>by an increase in the demand of the product family 4.</p><p>However, at a first moment, the release rate of orders in the</p><p>pool is not modified, to avoid an increase in the load of the</p><p>shop floor. All the stations also keep processing material</p><p>with the steady state frequencies calculated in the last</p><p>section. Thus, it is expected that the amount of orders</p><p>waiting for the release in the pool increases. This initial</p><p>situation is shown in Fig. 2. It is assumed that the demand</p><p>increased by a constant percentage for one month; that is</p><p>why the increase of the queue in the job pool is linear. The</p><p>initial levels of the buffers shown in this figure were also</p><p>adopted as reference levels, i.e. it is desired that the work</p><p>in process in each buffer is kept close to these levels, in</p><p>order to avoid an increase in the throughput time of the</p><p>shop floor.</p><p>Proceedings of the 9th MATHMOD</p><p>Vienna, Austria, February 21-23, 2018</p><p>4</p><p>Fig. 3. Relative processing frequencies of the machines</p><p>adjusted by the controller</p><p>Fig. 4. Relative processing frequencies of the machines</p><p>adjusted by the controller</p><p>After that, a controller was implemented in the pool to</p><p>adjust its release rate according to the buffer of awaiting</p><p>orders. In this case, if the processing frequencies of the</p><p>stations are not adjusted, the released orders will queue</p><p>up before the stations 1 and 3, causing an increase in</p><p>the throughput of the shop floor. Thus, controllers were</p><p>implemented to all the stations. This corresponds to out-</p><p>put control in workload control. For the stations 1, 2</p><p>and 3, a traditional pushed production control strategy is</p><p>implemented, since the processing frequency of a station</p><p>is adjusted based on the error in the level of its buffer (the</p><p>buffer that precedes the machine). In station 4, however,</p><p>the control is based on the error of demand fulfilment, cal-</p><p>culated based on the outputs of the system. Proportional-</p><p>integral controllers with gains Pi = 0.1 and Ii = 0.001</p><p>were chosen for this exploratory test. The results of the</p><p>simulation in the depicted scenario are shown in Fig. 3</p><p>and 4. The relative processing frequencies presented in 3</p><p>were calculated in relation to the steady state frequencies</p><p>presented in the Subsection 2.1.</p><p>The results of the simulation (Fig. 3) show that the con-</p><p>troller increases the release frequency of the pool (Upool),</p><p>as expected, to compensate for the increase in frequency</p><p>of incoming orders. This prevents an excessive increase in</p><p>the amount of orders awaiting release, as it can be seen by</p><p>comparing Fig. 2 and Fig. 4. In addition, the processing</p><p>frequency of station 4 also increases, since it is closely</p><p>related to the fulfilment of the demand of family 4. With</p><p>a certain delay, the controllers of the remaining stations</p><p>also adjust the capacities (i.e. processing frequencies of</p><p>the stations) in order to avoid an increase in the work in</p><p>process in the shop floor. The machines in the beginning</p><p>of the system (1 and 2) react faster, since each controller</p><p>considers the error in the level of immediately precedent</p><p>buffer. After a temporary increase, the buffer levels are led</p><p>back to the reference levels (or close to these levels).</p><p>3.1 Discussions</p><p>Although a pushed system was implemented in these</p><p>exploratory simulations, for simplicity, other control laws</p><p>can be defined to represent the WLC principles. It is of</p><p>interest, for instance, that the control of the job pool be</p><p>defined both in terms of the amount of awaiting orders and</p><p>the level of buffers of the system, which correspond to the</p><p>workload in the system. To implement that, both errors</p><p>in the buffer levels must be computed, i.e. the relative</p><p>error in the pool and the relative error in the buffers of all</p><p>stations, altogether. Then, the control could be based on</p><p>the maximum of these two errors. This would provide a</p><p>balance between the throughput time in the pool and the</p><p>throughput time in the system, preventing the increase of</p><p>the gross throughput time, i.e. the sum of both mentioned</p><p>throughput times.</p><p>Based on the proposed model, it is possible to see that</p><p>this balance will lead to a desired decrease in the gross</p><p>throughput time just when coupled with output control,</p><p>i.e. the control of the processing frequencies of the stations.</p><p>In the workload control literature, far less attention has</p><p>been given to output control (i.e. capacity adjustments)</p><p>than to input control (i.e. control of order release) (Land</p><p>et al. (2015); Thürer et al. (2016)). The aforementioned</p><p>authors also highlight that an issue in output control is to</p><p>decide when to start the capacity adjustment, how much</p><p>capacity to provide and for how long. The modelling tech-</p><p>niques usually employed to test the WLC concepts, based</p><p>on discrete events simulation, do not have prescriptive</p><p>power. Thus, the setting of these parameters is usually</p><p>investigated empirically, based on experiments.</p><p>The relative processing frequencies shown in Fig. 3 are,</p><p>in fact, capacity adjustments that can be interpreted as</p><p>overtime or downtime of the stations, when greater or</p><p>lower than one, respectively. In this sense, the design and</p><p>implementation of good controllers (that automatically</p><p>perform these capacity adjustments) represent a prescrip-</p><p>tive tool that can aid in the parameter setting of output</p><p>control.</p><p>The approach used for the formulation of this model is</p><p>not new, and was presented in the references cited in</p><p>Section 2. Thus, the contribution of this work concerns</p><p>the proposition of an analogy between a flow/bond graph</p><p>model and a workload control system, since in the vast</p><p>majority of the cases, open-loop models are employed to</p><p>represent WLC systems instead of closed-loop models, like</p><p>the proposed one. This first draft of this analogy raised</p><p>some relevant issues. The quality of approximation of</p><p>discrete event models by differential equations depends on</p><p>the model structure and the distributions used to model</p><p>service and inter-arrival times. In the presented model,</p><p>the service times are ”constant” for all the jobs, that is,</p><p>since the model is based on flows, it is not possible to</p><p>distinguish jobs and establish different service times for</p><p>each job. The variations imposed by the controllers in the</p><p>processing frequencies of the machines are interpreted as</p><p>Proceedings of the 9th MATHMOD</p><p>Vienna, Austria, February 21-23, 2018</p><p>5</p><p>830 Juliana K. Sagawa et al. / IFAC PapersOnLine 51-2 (2018) 825–830</p><p>capacity adjustments, as mentioned. Therefore, the next</p><p>step to improve the model is to include variable service</p><p>and arrival times, as well as to establish a better corre-</p><p>spondence between the discrete and continuous variables</p><p>of the compared models.</p><p>4. CONCLUSION</p><p>The results of the exploratory simulations indicate that it</p><p>is possible to represent the principles of workload control</p><p>by means of a continuous dynamic model based on bond</p><p>graphs. This representation is based on analogies between</p><p>the elements of both approaches. For instance, the job</p><p>pool of the WLC methodology is represented as a buffer of</p><p>incoming flow and the output control corresponds to the</p><p>adjustments in the processing frequencies of the stations.</p><p>This approach allows the implementation of controllers,</p><p>which adjust these processing frequencies and the release</p><p>frequency in the pool. These capacity adjustments may be</p><p>seen as prescriptive directions for the parameter setting of</p><p>output control in WLC implementations, a topic that was</p><p>identified as a gap in the literature. Thus, the presented</p><p>approach may help to diminish this gap.</p><p>Extensions of this research towards the definition and test</p><p>of different control laws for the release of flow in the</p><p>pool could be proposed. This release could be defined</p><p>as a function of the buffer of awaiting orders (i.e. flow)</p><p>in the pool, as well as the buffer levels of the system,</p><p>considering all the workstations. In addition, some noise</p><p>could be added to the variables of proposed model. The</p><p>simulation models commonly used to study the workload</p><p>control are discrete and stochastic. As the proposed model</p><p>is continuous and deterministic, the addition of random</p><p>noise could bring interesting comparative</p><p>insights.</p><p>REFERENCES</p><p>Duffie, N., Chehade, A., and Athavale, A. (2014). Control</p><p>Theoretical Modeling of Transient Behavior of Produc-</p><p>tion Planning and Control: A Review. Procedia CIRP,</p><p>17, 20–25. doi:10.1016/j.procir.2014.01.099.</p><p>Falu, I. and Duffie, N. (2014). Adaptive Due Date</p><p>Deviation Regulation Using Capacity and Order Release</p><p>Time Adjustment. Procedia CIRP, 17, 398–403. doi:</p><p>10.1016/j.procir.2014.01.073.</p><p>Ferney, M. (2000). 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