The formalization process has been briefly described in section 4.6, where the semantics of the MBOM concept were constrained by the application of an axiom (see section 4.6.4.1). In this section, the definitions of specialized classes of MBOM e.g. MBOMs, MBOMh, and MBOMi are formally captured. The formalization process starts with the declaration of properties. For example, MBOMs property has been declared in KFL as follows:
:Prop MBOMs :Inst Type :sup MBOM
:name "Manufacturing Bill of Materials based on Stark, (2011)'s definition"
:rem "MBOMs has assembly component list as well as auxiliary materials list."
Figure 5.2c: UML based lightweight representation of MBOMi
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The name and remark directives are optional and can be added to facilitate the modeller. Similarly two other MBOM properties can be declared using the same KFL format. Now the properties: assembly component list and auxiliary materials list represent the list of assembly components and auxiliary materials.
They have been specialized under the ULO property “list” where the latter describes a series of particulars. For instance, the declaration of auxiliary material list in KFL is as follows:
:Prop AuxiliaryMaterialList :Inst Type
:sup List
Similarly the other list related properties: assembly component list, AD assembly component list, and AR assembly component list have been declared in KFL. Other concepts such as auxiliary material and assembly component are also captured and can be found in appendix B.1.
Most of the relations associated with MBOM are instances of binary relation.
For example, MBOMs has the “hasAuxiliaryMaterialList” relation with auxiliary material list which is actually a binary relation. The relation can be declared as follows:
:Rel hasAuxiliaryMaterialList :Inst BinaryRel
:Sig BOM AuxiliaryMaterialList
However the relation between all the instances of property “list” and instances of properties like assembly component and auxiliary material has variable arity.
For example, an assembly component list may have any number of assembly components and the assembly components may vary depending upon the
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product for which the MBOM has been created. The only variable arity relation in KFL is the relation “item” which would now be used instead of relations like
“hasAssemblyComponent”. Due to this limitation, the relations
“hasAssemblyComponent”, and “hasAuxiliaryMaterial” have been replaced with the ULO relation “item”.
Axioms have been applied to constrain and capture the semantics of concepts already declared as properties. For example, the following two axioms have been applied to capture and constrain the semantics of the concept “assembly component list”: at least one assembly component within the list."
(=> (AssemblyComponentList ?l) (not (exists (?other)
(and (item ?l ?other)
(not (AssemblyComponent ?other)))))) :IC hard "Every assembly component list should consist of exclusively assembly components that make up the list."
The first axiom dictates that an instance of assembly component list should have at least one instance of assembly component. It implies that the system would not accept any list which does not have at least one assembly component. However there are still possible chances that the system accepts a list which has one or more instances of assembly component as well as instances of other concepts like auxiliary material. The second axiom averts
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such attempts and makes the system accept instances of assembly components only for the property “assembly component list”.
The concepts “AD assembly component list” and “AR assembly component list”
are mutually exclusive meaning that no single instance of AD assembly component can be found in AR assembly component list and no single instance of AR assembly component list should be found in AD assembly component list.
This can be captured by applying the following axioms.
(=> (ADAssemblyComponentList ?l) (not (exists (?x)
(and (ARAssemblyComponent ?x) (item ?l ?x)))))
:IC hard "Every AD assembly component list should not consist of AR assembly components."
(=> (ARAssemblyComponentList ?l) (not (exists (?x)
(and (ADAssemblyComponent ?x) (item ?l ?x)))))
:IC hard "Every AR assembly component list should not consist of AD assembly components."
The first axiom (mentioned above) says that AD assembly component list cannot have any AR assembly component whereas the second axiom dictates that AR assembly component list cannot have AD assembly component list.
Similarly semantics related to auxiliary material list has been captured using these kinds of axioms.
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So far, constraints have been applied to assembly component lists and auxiliary material list. The constraints shown in the UML diagrams in figure 5.2 can also be represented in KFL by using axioms. A summary of such axioms is shown in figure 5.3.
Figure 5.3: Example of four different axioms applied to constrain the meanings of MBOMs, MBOMh and MBOMi.
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The first axiom attaches a constraint on MBOMs that it should always have auxiliary material list. This suggests that whenever a user will try to populate an instance of MBOMs, the system will demand an assertion of auxiliary material list as well. It is important to note that a constraint was applied on MBOM (parent class of MBOMs) which dictated that whenever MBOM exists, it should have an assembly component list (see section 4.6.4.1). This implies that the same constraint should also be applicable on MBOMs and other specialized classes of MBOM. Specifically with reference to MBOMs, it should have an assembly component list and an auxiliary material list.
The second axiom dictates that the MBOMh concept should not have an auxiliary material list. This axiom would not allow the users to assert instances of auxiliary materials in the MBOM knowledge base. In the same fashion, the third axiom averts any attempt to assert instances of AR assembly components.
This complies with the definition of MBOMh that it should not have auxiliary materials and AR components as discussed in the previous sections.
The fourth axiom applies a constraint on MBOMi that it should not accept any instance of auxiliary material. This suggests that MBOMi consists of AD and/or AR assembly component lists.
Once the semantics of all the MBOM concepts are formally defined, they can be experimentally evaluated by instantiation. The experimental investigation of these concepts is described in detail in chapter 6.