Applications of Kinetic Bond Graphs in Corporate and Production Management Systems

Copyright © IFAC Control of Industrial Systems, Belfort, France, 1997

APPLICATIONS OF KINETIC BOND GRAPHS IN CORPORATE AND PRODUCTION MANAGEMENT SYSTEMS

Jacques LeFevre

* Sherrington School ofPhysiology

- UMDS - University ofLondon St. Thomas's Campus. London - SE1 TEH - UK EMAJL:[email protected] .. Fax: +44 (0)1719280729 ** LA1L. URA CNRS D1440. Ecole Centrale de Lille. BP 48 5Y651 Villeneuve d'Ascq- Cedex - France. Fax: +33320335418

Abstract: Dynamic models are used for strategic policy analysis by high level executives. In such applications, intuitive graphical model description is a must and Forrester System Dynamics (FSD) is often used for that purpose. However, we show that, in management like in other fields, FSD does not represent well the structure of our mental models. We then describe Kinetic Bond Graphs (KEG), a new method using Bond Graph-like notations to generalise FSD and reproduce these mental structures. Finally, we discuss a few applications: strategic models of production and sales, Just in Time and Kan Ban. Resume: La Dynamique des Systemes (DS) est frequemment utili see par les cadres superieurs d' entreprise pour simuler les effets de leurs strategies de decision. Les modeles utilises pour ce faire doivent etre intuitifs or sur ce point, la DS a de grosses lacunes. Nous decrivons donc les Bond graphs Cinetiques (BGC) utilisant des notations graphiques de type bond graphs pour generaliser la DSF de faeon intuitive. L' interet des BGC est illustre par quelques applications (production et vente, production Just in Time, Kan Ban). Keywords: System Dynamics, Bond Graphs, Production Engineering, Management Systems, Simulation Languages. Mots-Clefs: Dynamique des Systemes, Bond Graphs, Ingenierie des Systemes de Production et Management, Outils de Modelisation.

1. IN1RODUCTION: MODELS IN STRATEGIC MANAGEMENT

models used in production systems might appear suited to this purpose. However, these models arc at an operational or tactical aggregation level. Despite some strategic overtones, they are much too detailed. Strategic issues do not involve specific machines, production operations or queues. Top executives think at the much more global level of the total company situation (stability, growth, decline, trend reversal, retranchment, investment...) or at least in an analysis of its major product groupings and functions coupled to socio-economic models of the market and of the environment. At that level, individual entities and stochastic aspects are lost and

Modern business is highly complex. Indeed, management and decision processes involve not only production but also RJD, procurement, sales and marketing functions interacting with environmental and socio-econornic factors. Due to this complexity, top executives feel now the need for modelling their enterprise and its environment in order to test their mental models and to analyse their strategic decisions (Senge, 1990). When augmented by models of decision processes, the discrete event 121

variables are seen as deterministic and continuous. These systems, being replete with nonlinear feedbacks, may be described by nonlinear differential equations which are too abstract for interdisciplinary work. An intuitive graphical modelling language must thus be used. Forrester System Dynamics (FSD) was created for that purpose (Forrester, 1971).

C(Qi) storing each a given extension Qi' In addition exogenous compartments (grey clouds) represent the exchange of extensions between the system and its environment. Each C(Qi) may send and receive flows (continuous arrows) which have the dimension dQjldt The exogenous compartments either send (sources Si) or receive (sinks or outputs Oil extension flows. Each flow goes only in one direction and is thus always positive or zero. A flow is exchanged between two and only two compartments (its donor and acceptor). Each flow is defined by a rate equation represented by a unidirectional valve (triangles) called also a rate box placed on its arrow. Each extension Qi is obtained at time t by integrating from given initial conditions the instantaneous flow balance of all the flows linked to C(Qi). The equations of the rate boxes define these flows and are computed at each time t in block diagram (BD) using notations similar to classical BD languages like Simulink and represented simply in Fig.1 by white rectangles. These block diagrams and their inputs and outputs form the modulation part of the FSD model. The value of the flow at time t resulting from a BD is imposed to a rate box by an interrupted arrow going from the BD to the rate box. The inputs to this BD are also indicated by interrupted arrows. They are measurements of extensions in compartments, time inputs, constant parameters singled out for special attention (other parameters being implicitly defined by the BDs), and values of other flows (possible algebraic loops). The control block diagrams do not deal with extensions but with signals (interrupted arrows) which do not do not participate in extension balances. These BDs may be decomposed further to represent auxiliaI)' computations or important intermediate variables. However, the work reported here focuses on extensions and we will thus draw BDs like in Fig. I.

2. FORRESTER SYSTEM DYNAMICS (FSD) In the seventies, FSD elicited a wide interest damped later due to excessive rlaims and now, reemerging due to careful reappraisal and availability of good modelling software. For instance, many MBA programs use FSD in their teaching (Coyle 1995) and a recent UK and US bestseller in management is based on qualitative FSD (Senge, 1990). As a reminder, let us describe a simple production process using the main FSD elements. From a strategic management point of view, this model is utterly trivial but it is sufficient to introduce the idea.

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Despite the triviality of Fig. I, it is easy to see why FSD is successful in management and in other semiquantitative fields like ecology, biology and socieeconomy: (i) Its main variables (flows and extensions) are the only one easily measurable in these domains. (ii) Signals provide a good way to represent controls and policies. (ill) The FSD methodology is easily understood by everybody without extensive retraining. (iv) FSD may be used to represent any set of first order differential equations although sometimes in cumbersome ways.

A FSD-model (Fig. I) has two parts: an extension graph (continuous lines, grey elements) and a modulation part (control block diagrams: interrupted lines and white rectangles). The extension graph describes the fate of extensive quantities similar to displacements in bond graphs (BG) and called hereafter "extensions" (e.g. number of orders, number of products, amount of pollutant, populations, capital...) and their flows or timederivatives. Extensions are stored in compartments (grey rectangles) obeying flow continuity. An extension graph may be disconnected. It has several kinds of nodes (grey-filled) and one kind of link (continuous arrows). The main nodes are the grey rectangles which denote endogenous compartments

3. TIIE MAIN PROBLEM OF FSD To be useful in interdisciplinary work, a FSD model must have a topology translating faithfully the mental model of the system structure used by domain specialists. This is a basic need without which any hope of acceptance by non modellers is futile. Our claim is that FSD does not satisfy this requirement.

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For simple systems like in Fig.l, everything is nice and the structure is intuitive. However, let us consider a slightly more complex production system (Fig.2). Two kind of raw materials (RI and R2) enter by procurement processes (PRO) and after delays D I and D2 reach production (PR) which puts them together and gives some losses (1.) and products which go to a transport store T and then to a storage process (ST). From there the products go into a warehouse (W) and then into a sales process (S) which takes orders (0) and satisfy them resulting in payments (money M) and satisfied customers (SC). Money is used in the various processes and in investment and dividends (output to non represented subsystems). Remark that delays are processes too since we may improve them by investing.

this model are correct However, no chemist worth his salt will admit that Fig.3 is a simple and intuitive diagram of the reactions. For comparison. we give in inset a map of these reactions like in Fig.2 (stores: A,B,C; processes: Rl,R2). Undoubtedly, this map is easier to grasp than the FSD model.

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In conclusion. FSD is bad at representing chemical reactions. Since we can represent the production process of Fig.2 by generalised chemical metaphors (see Eq.l), FSD will do a bad job at modelling it. We have described such generalised reactions in many fields (LeFevre, 1996): physiology, biochemistJy, endocrinology, epidemio-Iogy, biotechnology, ecology, socio-economics, management. They all make a heavy use of FSD and, for the above reason. their models are often difficult to grasp. A second problem is even more important: since FSD is very messy, people oversimplify their models. Motivated by these remarks, we have developed a generalised FSD called "Kinetic Bond Graphs (KTG)" representating reactions like in Fig.2 but with a complete model definition. KBG uses bond graphlike symbols but in a purely kinetic framework.

Fig.2. An informal representation of a production process slightly more detailed than in Fig. I. We could continue almost ad infinitum to add supplementary details (e.g. various delays, stores and flows of capacity and workers, decomposition of sales into marketing advertising and selling, services, market, policy feedbacks ...). This is not necessary. Our point right now is just that Fig.2 is clear and intuitive. This structure may be wrong or oversimplified. This is not important. It reproduces naturally our mental model. In contrast, we will not even attempt to draw the corresponding FSD model. It would indeed be a maze of control signals and all the extension flows, so nicely coupled in Fig.2 would be separated like the orders and material flows of Fig.I. The simplicity and intuitiveness of Fig.2 would be lost and using this model in interdisciplinary discussions would be very difficult. Indeed., let us remark that Fig.2 has two kinds of nodes: stores like [RMl] and processes like PR>. All these processes have several input and output stores. They are thus analog to chemical reactions like m RI + n R2 + money~ losses + products (1) products+orders+money ~ sat. cust. + money

4. KINETIC BOND GRAPHS Networks of reactions are the nuts and bolts of biochemists who have thus developed an informal notation called "metabolic maps" to deal with them. We have previously shown that chemical pseudo bond graphs preserve the intuitiveness of metabolic maps. They may thus provide a starting point for the far more complex kinetics encountered in all the above fields (non linear functions and functionals, delays). Since pseudo BGs use notions of pseudo energy, power and effort which are not relevant in our case, we need to define a relaxed BG notation preserving the topology of BG but stripped from all irrelevant features and augmented with new elements to represent general kinetic laws. This will now be done in Fig.4 on a part of the production process ofFig.2.

FSD is unable to represent such chemical reactions simply and intuitively. For instance, Fig.3 shows the FSD model of a simple system of two mass action reactions A+2B~C 'lIld A~B. Due to the signals couplings in the block diagrams, the equations of 123

Extension, flows and signals: Extension and flows are defined like in FSD and BGs. They obey conservation laws and the Kirckhoff flow laws. Signals are defined like in BDs and behave like BG modulations by active bonds.

Stoechiometric transformers TF ({;.t/tp»): A IF imposes fl = J1{fl) but the strength equation is 51 = q>(sl)~ J1 and cp are not necessarily related and their product does not need to be conserved. We often need lFs with strengths related by a delay or a non1inear function. Both functions may be independently modulated by signals.

Bonds (continuous semi-arrows) flows and strengths: Like in BGs, a bond conducts a flow and may be bidirectional. In place of efforts, we define variables called "kinetic strength~ or strengths" which define the strength of influence of an extension on a box determining a flow rate (i.e. the functional dependency of the rate equation on that extension). Strengths are similar to efforts and propagates their causality like efforts in BGs. However, they have no interpretation other than a purely kinetic one. The product (flow *strength) has no meaning and is not conserved. Kirckhofs effort laws do not exist The definition of strengths is a major cause of the simplicity of KBG models. Indeed, in FSD, all functional dependencies of flow rates are expressed by signal links and the signal part of a FSD model is thus complicated. Usually most of the influences on a FSD or KBG rate box come from compartments directly connected to it (donors and acceptors). By transmitting these inOuences by strengths carried by the flow bonds, KBG simplifies enormously the signal part devoted only to distant modulations.

o and 1junctions: Like in BGs,we use 0 and 1 junctions. The 0junction is identical to its BG cousin and not frequently used. Its main role is to express a flow balance in a compartment. Often, we find more convenient to hide it in the notation used for the compartment (see inset, Fig.4.). The I-junction defines flow equality like in BGs but, due to the absence of power interpretation, its strength law is different Its role is indeed to combine strengths coming from different extensions and to transmit their combined influences further in the model. Many functions or even functionals may be used for that role. We use four forms: I: the usual BG law 1 _: do not transmit strengths (blocking causalities). I.: transmit the product of incoming strengths using causality like in BGs. Ig: with n+1 bonds, g is a function of n arguments computing the causally determined strength as a function of the n others (ordering needed). To represent the adaptive structures observed in many management systems, we also use timevarying or conditional junctions. Finally, when a KBG has several similar parts (like the two procurement processes in Fig.4, we may use an array notation similar to vectorial BGs. Strengths causality is indicated by a causal marker I like effort causality in BGs. Strengths may be blocked in some pathways (cf. 1 _). Consequently, flow and effort causality are not always dual and flow causality is marked by o.

Endogenous compartments (e.g. [RMl]: Like in FSD they store extensions and, similarly to C-elements in BGs, they have also a strength defined by an equation !LA=fs(A) where A is the stored extension. In addition, we may also give another equation A*= fe(A) specifying the value of the extension A * avail
Control block diagrams: Defined like in FSD. The inputs are compartments, specific parameters (defined by {p} and singled out for special attention, other parameters being defined implicitly in the BD), time functions defined by (f(1» or other flows (algebraic loops).

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5. DISCUSSION We could have converted the whole production model in a KBG. However, from a strategic management point of view, even this complete model could at first sight still appear oversimplified and, to put it plainly, quite uninteresting. First let us remark that, with KBG, adding complexity is easyl. ~ However, complexity is dangerous. Most our hypotheses are ill«tiDed and the exercise may just become another bad case of the Garbage in -+Garbage out principle. 124

Fig.4 may appear extremely abstract and generate instant rejection. We have thus defined a series of simpler representations encapsulating the KBG model like Russian Matriochka and giving less and less details but more and more intuition. We use these simplified KBGs in interdisciplinary model building or discussion sessions. To promote interdisciplinaly development , we are currently elaborating a progressive methodology incorporating these mattiochka diagrams into the soft system approach ofWolstenholme (1990) and Coyle (1996). Used in brainstorming sessions, these tools and methods truly put KBG in the hand of managers.

For instance, we could add more detailed production steps or include a model of the customer population and more details about sales and marketing. manpower, training and maintenance. One point is specially interesting. The production rate box depends causally on its backward affinity. In biochemistry, this is a well known effect called endproduct inhibition: if a reaction produces too much of a substance in our body, it accumulates and the inhibition of a producing enzyme decreases the production rate and leads to a kind of "Just in Time" metabolic production. Building on this idea, we have worked out a model of a production system using enzyrnatic-like controls. Details will be published elsewhere but one result is quite striking: the simplest biochemical controls (enzymes, co-factors) are similar to the Kan Ban control of Just in time production. This gives a new view on production management inspired from the many sophisticated control procedures found in biochemistry.

KBG is just starting. Its representation of coupled extension flows is now mature and we focuse now on policy representation and synthesis (including representation of the responsibilities and interactions of decision makers), fuzzy characteristics and adaptive topologies. These developments are needed but already in its present state, KBG appears promising. 10 addition to the work presented here, we have indeed investigated applications in biochemistry, physiology, ecology and socioeconomy. In each case, the models were remarkably intuitive and led us naturally to new developments. This would not have been possible with FSD or with equation-oriented methods. Considering the current expansion ofFSD, we believe that the link with bond graphs made here is important. Our hope is that it will also make interdisciplinary cooperation easier.

Moreover the simplicity of Fig.2 is misleading. This model has many loops and we should add its signal feedbacks. With inclusion of delays, excitation and inhibition strengths and I * junctions, it becomes highly nonlinear and its behaviour is quite difficult to forecast. In another work, we have shown that a simpler but generic model of production may produce chaos. Managers usually find this a traumatic but illuminating experience and games based on these models are educationally useful. Fig.2 hides a lot of potential complications and it is wise to begin a study at this or even at a simpler level before to go to more realistic models. Indeed, Fig.2, once transformcd in a KBG, suggests many interesting questions. For instance, we can investigate the effects of different money allocation policies. The list of policy issues which may be analysed is almost infir;tc. One of them is specially remarkable. In Fig.4, we have included signal pathways callcd MIS (management information systems). Indeed. block diagrams are not restricted to simple equations. They may include any well specified algorithm intcrfaced correctly with the rest of the model. For instance, we could have in MIS a rough simulation of the computerised flow of information in the system, measuring various parameters, realising computations and forecasts and sending information and orders at various processes. Obviously, our goal would not be to design the MIS itself, this is best done by software engineering methods. But before to commit resources to a MIS development or modification, we could test on the simulation if the availability of the new data will allow bctter operation, control, and policy. We can thus easily dispel the criticism that our model is oversimplified. In fact, some managers might just have the oppositc reactIOn. Being used to BGs, BDs and FSD, we claim that the KBG language is easy and natural ... Howcver, to overstretched managers,

ACKNOWLEDGEMENTS Without the encouragements received from Dr. G. Dauphin-Tanguy (Ecole Centrale de Lille), this work would not exist. Her moral support is thus gratefully acknowledged.

REFERENCES Coyle, RG. (1996). System Dynamics Modelling, Chapman & Hall, London, UK. Forrester, J.W. (1971). World Dynamics, Wright AlIen Press, Cambridge, Mass. USA. LeFevre, 1. (1996). Bond Graphs are not the proper tools to model the world but Bond + Kinetic Graphs could very well be! Plenary lecture,

Proceedings ofthe 3rd Intemational Conference on Bond Graph Modeling and Simulation, (Dauphin-Tanguy, G. and 1. Granda (Eds.»,

Simul. Series, Vol 29, N°l, pp. 3-9, SCS Publ., San Diego, Calif., USA. Senge, P.M (1990). The Fifth Discipline, Random House, London UK. Wolstenholme, E.F. (1990). System Enquiry, A System Dynamics Approach., Wiley, Chichester, UK.

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