Formal Specification of DEMO Process Model and Its Submodel

Abstract

This paper discusses a specification and merge operation over submodels of a given Process Model (PM) in Design and Engineering Methodology for Organizations (DEMO). In general, a submodel is a part of a given model. An earlier work proposed how submodels of a given DEMO Construction Model (CM) can be attained by a set-theoretic formalization. However, it remains unclear how to expand the formalism to the notion of submodels of a given PM. Since the given PM should align with the corresponding CM, a submodel of the given PM should not only be a PM, but also conform to the corresponding submodel of the CM. These two independent constraints indicate the desired definition and formalization of submodels of PMs. The proposed approach is shown to be applicable to a common demonstration case. Through the formalization, this paper shows the closure, commutativity, and associativity of the merge operation over submodels of a given PM. Moreover, it is found that the consistency between CMs and PMs is preserved during the merge operation.

Appendix: Proofs

Proof of Proposition 1 . Suppose $\left\langle T,V,W\right\rangle$ is an arbitrary member of the family of sub-TPS-nets $\wp \left( \left\langle T^{\circ },V^{\circ },W^{\circ }\right\rangle \right)$. Considering that $\left\langle T^{\circ },V^{\circ },W^{\circ }\right\rangle$ is a PSD, for any $t_{i}$ and $t_{j}$ in $T^{\circ }$, if $f_{Tname}\left( t_{i}\right) =f_{Tname}\left( t_{j}\right)$ then $t_{i}=t_{j}$. Now that $T\subseteq T^{\circ }$ by Definition 9, for any $t_{i}$ and $t_{j}$ in $T\subseteq T^{\circ }$, if $f_{Tname}\left( t_{i}\right) =f_{Tname}\left( t_{j}\right)$ then $t_{i}=t_{j}$.

Proof of Theorem 3 . Since $\left\langle T_{X},V_{X},W_{X}\right\rangle$ and $\left\langle T_{Y},V_{Y},W_{Y}\right\rangle$ are members of the family of sub-PSDs of a given PSD $\left\langle T^{\circ },V^{\circ },W^{\circ }\right\rangle$, we have $T_{X}\subseteq T^{\circ }$, $T_{Y}\subseteq T^{\circ }$, $V_{X}\subseteq V^{\circ }$, $V_{Y}\subseteq V^{\circ }$, $W_{X}\subseteq W^{\circ }$, and $W_{Y}\subseteq W^{\circ }$ by Definition 9. Then, because $T_{X}\cup T_{Y}\subseteq T^{\circ }$, $V_{X}\cup V_{Y}\subseteq V^{\circ }$, and $W_{X}\cup W_{Y}\in W^{\circ }$ hold, it is obvious with Definition 14 that $\left\langle T_{X},V_{X},W_{X}\right\rangle \sqcup \left\langle T_{Y},V_{Y},W_{Y}\right\rangle$ is a sub-TPS-net of $\left\langle T^{\circ },V^{\circ },W^{\circ }\right\rangle$. Next, for any $\left( \left( t,i\right) ,t^{\prime }\right)$ in $V_{X}\cup V_{Y}$, if $\left( \left( t,i\right) ,t^{\prime }\right)$ is in $V_{X}$ [resp. $V_{Y}$], $t\in T_{X}$ and $t^{\prime }\in T_{X}$ [resp. $t\in T_{Y}$ and $t^{\prime }\in T_{Y}$] holds, hence $t\in T_{X}\cup T_{Y}$ and $t^{\prime }\in T_{X}\cup T_{Y}$. Thus, any $\left( \left( t,i\right) ,t^{\prime }\right)$ in $V_{X}\cup V_{Y}$ satisfies the second property of Definition 10. Similarly, any $\left( \left( t,i\right) ,\left( t^{\prime },i^{\prime }\right) \right)$ in $W_{X}\cup W_{Y}$ satisfies the second property of Definition 10. Note that the first property of Definition 10 is satisfied by Proposition 1 because $\left\langle T_{X},V_{X},W_{X}\right\rangle \sqcup \left\langle T_{Y},V_{Y},W_{Y}\right\rangle$ is a TPS-net. Hence, $\left\langle T_{X},V_{X},W_{X}\right\rangle \sqcup \left\langle T_{Y},V_{Y},W_{Y}\right\rangle$ is a PSD. Therefore, by Definition 12, $\left\langle T_{X},V_{X},W_{X}\right\rangle \sqcup \left\langle T_{Y},V_{Y},W_{Y}\right\rangle$ is a sub-TPS-net of $\left\langle T^{\circ },V^{\circ },W^{\circ }\right\rangle$ and a PSD, thus a sub-PSD of the given PSD $\left\langle T^{\circ },V^{\circ },W^{\circ }\right\rangle$.

Proof of Theorem 4 . The commutativity and associativity of merge operation $\sqcup$ are obvious from those of set-theoretic union of sets.

Proof of Theorem 5 . Since $\left\langle T_{X},V_{X},W_{X}\right\rangle$ is matched to $\left\langle \mathcal {A}_{X},\mathcal {T}_{X}\right\rangle$ and $\left\langle T_{Y},V_{Y},W_{Y}\right\rangle$ is matched to $\left\langle \mathcal {A}_{Y},\mathcal {T}_{Y}\right\rangle$, Definition 13 gives $\mathcal {T}_{X}=T_{X}$, $\mathcal {T}_{Y}=T_{Y}$, “${\forall t,t^{\prime }\in \mathcal {T}_{X},}{(\exists a\in \mathcal {A}_{X},}{f_{ex}\left( t\right) =a=f_{in}\left( t^{\prime }\right) )}\iff {(\exists i\in \mathbb {I},}{\left( \left( t,i\right) ,t^{\prime }\right) \in V_{X})}$”, and “${\forall t,t^{\prime }\in \mathcal {T}_{Y},}{(\exists a\in \mathcal {A}_{Y},}{f_{ex}\left( t\right) =a=f_{in}\left( t^{\prime }\right) )}\iff {(\exists i\in \mathbb {I},}{\left( \left( t,i\right) ,t^{\prime }\right) \in V_{Y})}$”. Thus, ${\mathcal {T}_{X}\cup \mathcal {T}_{Y}}{=T_{X}\cup T_{Y}}$. It also holds that ${\forall t,t^{\prime }\in \mathcal {T}_{X}\cup \mathcal {T}_{Y},}{(\exists a\in \mathcal {A}_{X}\cup \mathcal {A}_{Y},}{f_{ex}\left( t\right) =a=f_{in}\left( t^{\prime }\right) )}\iff {(\exists i\in \mathbb {I},}{\left( \left( t,i\right) ,t^{\prime }\right) \in V_{X}\cup V_{Y})}$. Therefore, by Definition 13, the PSD of $\left\langle T_{X},V_{X},W_{X}\right\rangle \sqcup \left\langle T_{Y},V_{Y},W_{Y}\right\rangle$ is matched to the OCD $\left\langle \mathcal {A}_{X},\mathcal {T}_{X}\right\rangle \nabla \left\langle \mathcal {A}_{Y},\mathcal {T}_{Y}\right\rangle$.