Diagram Models in Continuous Business Process
Energy and Resources 39 Sienna Street, Warszawa 00-121
Abstract.In this paper the author proposes a change in business modelling techniques classification, based on the results of research and a review of modern workflow systems. The described amendment has an impact on ap-plicable – commercial approaches to continuous business process improve-ment. It fits the gap between the commercial need for easy understandable tool (based on widely-used diagrams) for analysts and researchers’ prefer-ence for working with well-defined mathematical models.
Keywords: business process, modelling, continuous process improvement, simulation based optimisation.
In the modern globalised, competitive market, it is not enough to design a good product. These are also well designed and implemented business processes that give a competitive advantage – processes which allow the business to reach the customer faster or to do it with lower costs. To survive in this commercial jungle, business processes have to be continuously analysed and improved.
Business process modelling techniques have been reviewed and classified sev-eral times by Kettinger, Teng and Guha  in 1997, Cheung and Bal  in 1998
Figure 1. Business process modelling techniques classification by Tiwari, Vergidis and Majeed 
and Melao and Pidd  in 2000. The approaches classified by them enable the visualization of processes and the analysis of some of their characteristics, like resource utilization or correctness of the structure, but do not provide significant support for process improvement.
The latest classification of business modelling techniques by Tiwari, Vergidis and Majeed  in 2008 focuses more on optimisation, but even they, in 2008, tended to ignore diagrammatic notations and preferred mathematical models.
In this paper, the author reviews approaches to business model description, es-pecially graphical ones. Based on the outputs of experiments and a review of com-mercially available business process management solutions, the author presents arguments for changing the latest classification.
In the latest classification , modelling techniques are divided into the cat-egories of mathematical models (i.e. Hofacker and Vetschera ), business pro-cess languages (i.e. BPEL - Business Propro-cess Execution Language), diagrammatic
Table 1. Classification of business process optimisation methods by Tiwari, Vergidis and Majeed 
Model of business process Classification of the model Types of business process analysis Types of business processes optimi-sation Flowcharts Diagrammatic models Observational RADs Diagrammatic models Observational, Per-formance analysis IDEF Diagrammatic models Observational, Simulation Petri Net Diagrammatic
models, Mathe-matical models, Business process languages Observational, Validation, Verifi-cation, Simulation, Performance analysis Graph reduction Mathematical models Mathematical models Performance anal-ysis, Simulation Algorithmic ap-proaches Business process languages Business process languages Performance anal-ysis Simulation
models (i.e. BPMN - Business Process Modelling Notation) or combinations of those. Figure 1 presents possible 7 categories.
Possible business process analysis and optimisation approaches have been as-signed to classified models, in the way presented in Table 1.
3. Diagrammatic business process modelling approachIn his previous works [6, 7], the author uses Petri nets as a diagrammatic busi-ness process model. This modelling approach is composed not only of Petri net notation objects (arcs, transitions and places), but also contains the definition of how those elements are used to define the process in a non-ambiguous way, so it can be used to implement processes for simulation purposes.
Figure 2. Transition firing
Figure 3. Simple representation of a task
The idea of Petri nets came from the PhD thesis of Adam Petri "Kommu-nikation mit Automaten" , prepared in 1962 at the faculty of mathematics and physics of The Technical University of Darmstadt. A Petri net enables a graphical model of the process to be built. It is composed of places, transitions and arcs join-ing them. Input arcs join places with transitions while output arcs join transitions with places. Places might contain tokens. The actual state of the model is described by the type (if tokens are distinguishable) and number of tokens in each place.
Places are passive elements, transitions are active – they simulate actions or events which change the state of a Petri net.
Transitions can be fired only when all starting conditions are met; there are enough tokens in all input places. When the transitions are fired, tokens are re-moved from the input places and added to the output ones (Fig. 2). The number of tokens removed and added to places depends on the arcs joining them with transitions.
The most frequent way of modelling technological or operational processes is to consider them as sets of tasks. Based on the purposes and goals of simulation more or less complicated task representation can be used, from the simplest (Fig. 3) to the more complicated, like the one applied to work organisation management (Fig. 4).
Figure 4. Representation from "Modelling and analysing workflow using a Petri-net based approach" 
Figure 5. Task with reusable resources
For the purpose of business process optimisation method, the author represents a task as two transitions and a place, stating the status of the task, between them. The first transition simulates the start of the task and its input places simulate the necessary resources. Firing the second transition means the end of the task and its output places simulate task deliverables and returned reusable resources (Fig. 5).
In order to simulate a sequence of tasks, the output place of the first task has to be the input place of the second (Fig. 6).
Alternative tasks competing for a resource have a common input place (Fig. 7) while parallel tasks have input places filled by a common preceding task (Fig. 8).
The same approach is used in other notations i.e. UML notation, where an activity diagram is used to define the process for implementation or BPMN - mod-elling notation which is transferable to BPEL runnable business process language.
Figure 6. Task sequence
Figure 7. Alternative tasks
Figure 8. Parallel tasks
Figure 9. BPMN task sequence
In BPMN notation, processes are composed of tasks and arcs defining the flow between them (Fig. 9). Flow can be split or joined using "XOR" (Fig. 10), "AND" (Fig. 11) or more sophisticated (complex) gates for alternative or parallel tasks.
Properly used business process diagrammatic modelling techniques are unam-biguous definitions of steering flow and become business process languages.
Figure 10. BPMN "XOR" gate
Figure 11. BPMN "AND" gate
In modern business process management systems visual, diagrammatic models of processes are used as executable input for workflow or document-flow engines.
4. Diagrams in modern workflow systems
Modern workflow systems focus a lot on usability and ease of use. The need for a fast response to market conditions make it compulsory to shorten the time to market for new products, services, procedures and processes. Marketing special-ists and business analysts, without programming skills, can now implement these processes using graphical tools. They use diagrammatic models, which are trans-ferable to runnable process definitions - code for business process engines.
According to independent IT market analysts from Gartner  and Forrester , there are 3 leaders of the multiple vendors of business process management systems:
• Pegasystems (with Pega BPM solution)
• Appian (Appian)
One of the key features of the state of these art solutions is the possibility of designing flows and procedures without engaging programmers. They all use graphical editors for BPMN.
Forrester’s analysts state 3 key areas for future business process management suites. One is "empowering business architects to design and execute on process strategy. Business architects are playing a broader role in driving business change initiatives. In many cases, these architects focus on defining the strategy for trans-forming end-to-end business processes. Historically, BPM tools offered very little to help these strategists scope and manage large-scale change projects. This dis-connect between strategy and execution keeps BPM suites isolated to the CIO’s office, without a way to have a greater impact on enterprise strategy."
Gartner experts use 10 criteria to evaluate solutions and 2 of them focus on process modelling:
1. "A model-driven composition environment for designing processes and their supporting activities and process artifacts"
2. "A process component registry/repository for process component leverage and reuse".
Diagrammatic models have become widely commercially used process lan-guages which are available not only for programmers, thus reducing the time needed to develop new data or work flows.
5. Mathematical business process model
The mathematical business process model comes from the works of Hofacker and Vetschera . It was later used by Tiwari, Vergidis and Majeed . The model is composed of sets of:
• physical resourcesBP =(b1...)
• informational resourcesBI =(b1...) and matrixes:
• ri j with binary variables indicating if physical resource bj is available for use by activityai
• gijandgojone dimensional binary constants that indicate which resources belong to global inputs or global outputs.
The model additionally defines constraints to those sets and matrixes, wherexi is a binary variable indicating that activity ai is participating in the process, pi is the start time ofai,diis the duration ofai,IiandOiare the sets of input and output resources ofai,yjis the binary variable indicating resourcebjbecoming available during the process,qj is the timebj becomes available,hi j is the binary variable that indicate thatbj is created byai.
1. All input resources of an activity must be available at some stage of the process if the activity is participating in it (equations 1, 2).
∀i,j:bj∈Ii,b j∈BPxi ≤ri j (1) ∀i,j:bj∈Ii,bj∈BIxi ≤yj (2)
2. The output physical resources must not exceed the sum of initial resources and resources produced during the process (equation 3).
∀j:bj∈BPgoi+ X i ri j≤ Mgij+ X i ti jxi (3)
3. A resource can be available either at the beginning of the process as an initial resource or as an output resource of a participating activity (equation 4).
∀j:bj∈BIyj ≤gij+ X
ti jxi (4)
4. A resource cannot be part of the output without first being available at some stage of the process (equation 5).
5. A participating activity must start only after the time that all its input re-sources have become available (equation 6).
∀i,j:bj∈Iipi ≥qi−M(1−xi) (6)
6. An output resource must become available exactly when the generating ac-tivity has been completed (equations 7, 8).
∀i:bj∈Oiqj ≥ pi+di+M(1−xi) (7) ∀i:bj∈Oiqi≤ pi+di−M(1−xi)−M(1−hi j) (8)
7. A non-participating activity cannot have output resources (equation 9).
∀i,j:bj∈Oihi j ≤ xi (9)
8. When a physical resource does not belong to initial resources, it must be produced during the process in greater or equal amounts to the required re-source inputs of the participating activities (equation 10).
hi j ≥ sumiri j+goi−M(1−yj) (10)
9. Each physical resource that does not belong to the initial resources but ap-pears in the output of a participating activity must be produced at least once (equation 11).
∀i:bj∈BP,gij=0h(i j)≥1−M(1−yj) (11)
6. Mathematical representation of Petri nets
Petri nets also have their mathematical representation. A Petri net is a tuple C=(P;T;I;O), where setP= p1;p2;p3...is a set of places and setT =t1;t2;t3... is a set of transitions. FunctionI : T− > Pis an input function andO: P− > T is an output one. Obviously setsT andPare disjoint. The value of functionI(ti) is a collection of input places for transition ti and the value of functionO(ti) is a collection of output places for transitionti.
The graphical representation of a Petri net is presented earlier in figure 2 bi-partite graph.
For the purpose of further definitions, by #(pi;I(ti)) the author defines the num-ber of arcs between input place piand transitionti, and on an analogy #(pi;O(ti)) the number of arcs between transitiontiand output placepi.
Places contain tokens; and the distribution of tokens among places defines Petri net marking defined with the function m : P− > N. This marking is a vector m = (m(p1);m(p2), ...,m(pn)). The marked Petri net is defined as a tuple Z =(C;m0), whereCis a Petri net andm0:P−>Nis a starting markup. Marking can change by firing the transition, but this can only happen when transition has been enabled.
Transitiontiis enabled in markingm, if in all placespi inI(ti) the number of tokens is not smaller than the number of arcs joining place pi with transition ti, m(pi)>#(pi;I(ti)). Firing transitiontichanges the marking of the net. For eachpi inP:m0(pi) = m(pi)−#(pi;I(ti))+#(pi;O(ti)). This mechanism makes the Petri net an abstract machine inNmspace.
In the same way as mathematical business process model described in a pre-vious section, a marked Petri net is defined by sets of activities and resources and functions defining the mapping between those. The equivalent of the constraints from the mathematical model are rules guarding transition firing and marking changes.
The runnable Petri net model used by the author fulfils all the constraints de-scribed. Other business process languages also fit this mathematical model as they define processes as sets of tasks performed in a defined order. This order (work-flow) defines mapping between activity outputs and other activity inputs in the same way that the described mathematical model of a Petri net does.
Runnable diagrammatic notations are at the same time graphical notations and business process languages, compliant with the mathematical model defini-tion (Fig. 12).
7. Simulation based optimisation
Many optimisation methods assume the exact form of the goal function is known, but in complex problems, the assumption that the value of the goal func-tion can be calculated is incorrect.
In many cases the value of the goal function can only be estimated by simula-tion. Obviously calculations or estimations of the value of the goal function, based on simulation data, does not solve the optimisation problem. The simulator needs
Figure 12. Amended business process modelling technique classification
to be connected with he optimiser - a generator of better solutions. The approach presented in Fig. 13 [12, 13] is known as simulation based optimisation.
The initialiser starts the whole process by generating the primary solution s1 from setS. It is used by the simulator to estimate of the goal function f(s). Based on its value the decision about further steps is made. The solution can be accepted when the solution is good enough and the optimisation process is stopped. In the other case, the process is continued and a new possible solution is generated and passed to the simulator.
In previous work by author, a genetic algorithm is applied as a new solution generator. The tasks’ parameters form a genotype, used for mutation and crossing. The process model is a phenotype used for evaluating the solution.
Positive outputs from the experiments provide another change in the latest clas-sification. Simulation based optimisation can be considered as a business process optimisation approach (Table 2).
The proposed change in the classification of business modelling, analysis and optimisation techniques might have an impact on commercially applicable
contin-Figure 13. Simulation in optimisation
Table 2. Amended latest classification of business process optimisation methods
Model of business process Classification of the model Types of business process analysis Types of business processes optimi-sation Flowcharts Diagrammatic models Observational RADs Diagrammatic models Observational, Per-formance analysis IDEF Diagrammatic models Observational, Simulation Simulation based optimisation Petri Net Diagrammatic
models, Mathe-matical models, Business process languages Observational, Validation, Verifi-cation, Simulation, Performance analysis Graph reduction, Simulation based optimisation Mathematical models Mathematical models Performance anal-ysis, Simulation Algorithmic approaches, Sim-ulation based optimisation Business process languages Business process languages Performance anal-ysis, Simulation Simulation based optimisation
uous business process improvement approaches. It fits the gap between the com-mercial need for an easily understandable tool based on readable diagrams for analysts and researchers preference for working with well-defined mathematical models.
Diagrammatic models are the most frequently used way of defining business processes. They are easy to read and can be executed in modern workflow software. At the same time, commercial continuous improvement is mostly done manually in reaction to bottlenecks rather than proactively. Although project engineering was first mentioned in early sixties and became a popular topic in the 1980s, the first time Gartner (independent IT industry analysts) prepared Magic Quadrant  – the famous market research report for intelligent business process management suites, supporting process optimisation and process analytics was September 2012. Companies and organisations would benefit from scientific guidance and po-tential new algorithms.
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