3.1 The relationship between model and target system

Discussions of models in philosophy of science often focus on the theme of representation. The structuralist view of modelling has been an influential perspective, where models are seen as structures (entities and their relations) which represent their target system (Frigg Reference Frigg2003). Simply speaking, proponents of this view regard models as isomorphic representations of the structure of their target system (see e.g., Suppe Reference Suppe1977). However, in more recent discussions, there has been a shift away from this structuralist view and its limitation to a dyadic relationship between model and target system towards a triadic relation that involves either users or interpretation (Knuuttila Reference Knuuttila2005). While isomorphism has the advantage of enabling precise formalisation, it falls short when the real-world system is not a structure in an obvious way. Isomorphism also implies a symmetric relation between model and target system, which does not fit actual modelling practice in science, where models are meant to be representations of their target system, and not vice versa. Including the modeller’s intention, or the purpose, establishes directionality in the relationship between representation and target system.

Other, more pragmatic attempts at describing representation take account of the limitations of isomorphism and claim, for example, that representation is based on some form of similarity (Giere Reference Giere2004). However, this can still be seen as limiting due to the focus on representation, which is only one of the conceivable uses of models (Knuuttila Reference Knuuttila2005). Morrison & Morgan (Reference Morrison, Morgan, Morrison and Morgan1999) adopt a wider perspective by describing models as mediators, “autonomous agents …[who] function as instruments of investigation” (p. 10). According to them, it “is precisely because models are partially independent of both theories and the world that they have this autonomous component and so can be used as instruments of exploration in both domains” (p. 10). They emphasise the various ways models can support problem solving and thus hint at their epistemic value in, but also beyond, representing their target system. Knuuttila (Reference Knuuttila2005) extends this perspective by describing models as epistemic artefacts, which provide us with knowledge in many other ways than just abstract representation. Such representation does not only consist of the model as an artefact in itself but also includes an intentional relation of representation, connecting the artefact with the target system. This implies that the model can be detached from this relation and therefore allow for different interpretations and thus different representations.

The relationship between target system and models used in engineering, and in particular engineering design, can differ from the sciences. While scientific models usually aim to represent some real target system, often with the goal of isomorphism or similarity, engineers can be thought of as actively intervening with the world. “Instead of depicting an already existing world, the engineering sciences aim at theories and models that provide understanding of artificially created phenomena” (Boon & Knuuttila Reference Boon, Knuuttila and Meijers2009, p. 688). This is an important observation and, despite targeting the engineering sciences rather than design, equally hints at some of the challenges encountered when describing the relationship between model and target system in engineering design. Boon & Knuuttila (Reference Boon, Knuuttila and Meijers2009) advocate a pragmatic account of models as epistemic tools rather than representations of a real target system. While they focus on models in engineering science, aiming at scientific understanding of system behaviour or material properties, it is argued here that the same applies to engineering design. When modelling for design, assumptions and choices have to be made regarding how to present an artificial system according to a particular purpose. So, models in engineering design share some properties with scientific models but they are also less straightforward in certain respects; their target system is often less clearly defined, highlighting the significance of purpose, scope and choices regarding model granularity.

3.2 Modelling as abstraction from reality

Before focusing on the role of abstraction in modelling, it is worth recapitulating what the term modelling describes in the scope of this research. With reference to Turnitsa (Reference Turnitsa2012), who distilled the following definition of modelling from various sources in the modelling and simulation literature, Tolk & Turnitsa (Reference Tolk and Turnitsa2012) state: “Modelling is the purposeful process of abstracting and theorizing about a system, and capturing the resulting concepts and relations in a conceptual model” (p. 2). The emphasis on conceptual models results from their focus on computational models, which might not be appropriate in all contexts. For instance, sketches, drawings, and physical prototypes are all models used in engineering design but are beyond the scope of this article. Still, there are a number of insights that can be drawn from this definition. First, it highlights that purpose is central to modelling activities. Second, it states that modelling is a matter of abstracting and is based on theory. Third, the goal is to capture both concepts and relations, or dependencies, in the emerging model.

The importance of abstraction for modelling and simulation is discussed in a variety of fields, ranging from discrete-event simulation (Zeigler, Praehofer & Kim Reference Zeigler, Praehofer and Kim2000) to artificial intelligence (Saitta & Zucker Reference Saitta and Zucker2013). In many accounts abstraction is seen as the crucial step in representing the real-world target system in a (conceptual) model (e.g., Frantz Reference Frantz1995). This process results in a particular level of abstraction of the developed model, which is usually reached from either a bottom-up direction or a top-down direction. Bottom-up generally consists of aggregating elements of the target system, leading to a more abstract description. For instance, a number of specific tasks could be aggregated to describe the overall activity. Conceptually, the term ‘aggregation’ can be thought of as a subset of ‘abstraction’ (Fishwick Reference Fishwick1988). On the other hand, top-down approaches involve decomposition of a more abstract description into smaller, more concrete parts. For instance, the description of a car could be decomposed into descriptions of engine, transmission, body etc.; and a description of an engine into descriptions of crankshaft, pistons, cylinder heads etc. Depending on the context, the purpose and the data available both bottom-up and top-down approaches can be employed in modelling.

So, different levels of abstraction can be obtained in a model depending on how, or to what degree, the target system is abstracted. Floridi (Reference Floridi2008) notes that while a level of abstraction formalises the scope or granularity of a model, a gradient of abstraction presents a way of varying the level of abstraction so observations can be made at differing levels. Different levels of abstraction usually imply an underlying hierarchical structure, which relates different levels with each other. The value of multilevel abstraction has been discussed in disciplines like complex systems design (Alfaris et al. Reference Alfaris, Siddiqi, Rizk, de Weck and Svetinovic2010) and business process modelling (Eshuis & Grefen Reference Eshuis and Grefen2008). The overall motivation is the ability to provide consistent models on different levels, depending on purpose and stakeholders. This also highlights some of the challenges associated with abstraction in general and multilevel models in particular. There is an epistemological issue regarding how multiple levels of abstraction are derived. Both aggregation and decomposition are employed in modelling products and processes, but it depends on factors like data availability, knowledge and the type of target system to determine an appropriate approach. Also, there is a range of practical issues, like ensuring and maintaining consistency across abstraction levels (Smirnov et al. Reference Smirnov, Reijers, Weske and Nugteren2012), choosing appropriate levels (Eshuis & Grefen Reference Eshuis and Grefen2008) and abstraction techniques (Frantz Reference Frantz1995) as well as the impact the different levels can have on analysis (Chiriac et al. Reference Chiriac, Hölttä-Otto, Lysy and Suh2011).

4.1 Model granularity and related concepts

In addition to a smaller number of explicit approaches towards the topic of model granularity, a range of related issues are discussed in the literature. In particular, discussions around model abstraction, decomposition, levelling, complexity, clustering and hierarchies are useful when approaching the topic of model granularity. The most relevant concepts are briefly presented here (see also: Maier, Eckert & Clarkson Reference Maier, Eckert and Clarkson2016) and grouped in three categories:

• ∙ Concepts describing model granularity or similar notions (Table 1).

• ∙ Concepts describing activities that impact model granularity (Table 2).

• ∙ Concepts describing model properties that may result from their granularity (Table 3).

Some of the concepts could be attributed to multiple categories, which is illustrated in Figure 1 and briefly discussed at the end of this section. Apart from ‘granularity’ the terms are ordered alphabetically and the order of definitions does not imply a ranking of their relevance.

Granularity refers to a property of the model itself and is characterised more or less formally, depending on the discipline. Mathematical definitions are based on set theory and consider cardinality, size and interaction of subsets (Yao & Zhao Reference Yao and Zhao2012). In engineering design, the size and detail of model elements determine its granularity, which is commonly understood to result from (hierarchical) decomposition (Chiriac et al. Reference Chiriac, Hölttä-Otto, Lysy and Suh2011; AlGeddawy & ElMaraghy Reference AlGeddawy and ElMaraghy2013). We refer to the granularity of a model as a manifestation of the level of detail in which it represents its target system. In particular, granularity may be used to describe the size and information content of model elements as well as the nature of relationships between model elements. Granularity can also relate to the resolution of output obtained through analysis based on a model.

Even though abstraction is frequently mentioned with regards to modelling and simulation, formal definitions are harder to find. The term abstraction is mostly used in a more conceptual manner (Cartwright Reference Cartwright1999) when describing modelling activities or resulting levels of abstraction (e.g., Fishwick Reference Fishwick1988). Abstraction is a fundamental part of most modelling endeavours (Frigg Reference Frigg2003), leads to a particular level of abstraction of a model and thereby drives its granularity. So, abstraction in the modelling process determines model granularity.

Complexity is a topic that is extensively discussed in various fields and even emerged as a research field on its own: complexity science (Ladyman, Lambert & Wiesner Reference Ladyman, Lambert and Wiesner2013). Without going into further detail, a complex system can be said to consist of multiple components interacting in non-simple ways (Simon Reference Simon1962). While the notion of complexity is generally used to describe real-world systems, it is also used for models (see definitions in Table 1). With respect to model granularity, this concept is therefore important in two respects. First, capturing a more complex system may require a more detailed, fine-grained model. Second, complexity can also refer to the model itself, describing either the number of elements and their connectedness or the difficulty to understand and work with them (Edmonds Reference Edmonds1999). In many cases a larger, more fine-grained model will also be considered more complex. Related, yet distinct, is the notion of complicatedness (or cognitive complexity), which refers to the observer-dependent, subjective dimension of complexity (e.g., Ramasesh & Browning Reference Ramasesh and Browning2014). While the term complicatedness is used to describe systems generally, it also has implications for models and can provide a proxy for understandability. A fine-granular, complex model may not be perceived as complicated by a user if it is architected and displayed in a simplified or abstracted way, thereby making it more easily understandable. Tang & Salminen (Reference Tang and Salminen2001) refer to this as ‘architected complexity’.

The organisation of complex systems is often characterised as multilevel hierarchies of systems and subsystems (Simon Reference Simon1962; Alexander Reference Alexander1964; Ladyman et al. Reference Ladyman, Lambert and Wiesner2013). This can be observed in many aspects of the natural world (organisms, ecosystem or the cosmos itself) but also in man-made systems. Hierarchical structures are important when reflecting on model granularity. Depending on the chosen approach, aggregation or decomposition both lead to hierarchical structures, which strongly influence the resulting model granularity. In addition to such part-whole relations, hierarchies can also result from specialisation and generalisation, indicating ‘is a’ relations (Brachman Reference Brachman1983). The resulting hierarchical structures are referred to as taxonomies and play an important role in fields such as requirements and software engineering. Interactions between parts of the system on different levels of the hierarchy contribute to complex behaviour. Models that offer different levels of granularity are often based on a hierarchical structure – of data or model architecture.

Model resolution is very closely linked with model granularity and is in many cases used to describe similar ideas. It is often associated with the amount of detail a model includes to represent its target system (Tolk Reference Tolk and Tolk2012). Some authors also use the term to describe the precision of a model’s output (Weld Reference Weld1992). Resolution is perhaps the closest related concept to granularity given that both are directly proportional – a high resolution requires a fine granulation, be it of pixels in an image or elements in a model.

Table 2. Definitions of terms describing activities that influence granularity.

Aggregation and decomposition are often used as opposite directions of constructing models, the latter also sometimes being referred to as disaggregation (Benjamin et al. Reference Benjamin, Erraguntla, Delen and Mayer1998). Aggregation refers to a bottom-up approach where elements of a model are grouped together and described on a higher level of abstraction (Iwasaki & Simon Reference Iwasaki and Simon1994). In many cases, this leads to a loss of information, although the goal of model aggregation generally is to conserve information from more detailed levels. Aggregation results in coarser grained models. Decomposition denotes a top-down approach where system elements are broken into a set of smaller elements or subsystems (Alfaris et al. Reference Alfaris, Siddiqi, Rizk, de Weck and Svetinovic2010). This usually adds information about the decomposed systems but also requires additional effort to elicit the required information. Similarly, granulation describes the decomposition of a universe into subsets (although Yao (Reference Yao2003) also uses it to indicate clustering of elements into groups, which is elsewhere distinguished as ‘organization’ (Zadeh Reference Zadeh1997)). We suggest using Zadeh’s (Reference Zadeh1997) original definition of the term granulation, referring to the decomposition of a set, to avoid confusion. Decomposition generally leads to more fine-grained models. For both concepts a range of drivers and approaches exist that are employed depending on the context and purpose of the model.

Clustering aims to group elements of a model that are strongly interconnected into clusters, while minimising the connectivity outside of the clusters (Browning Reference Browning2001). Clustering is a representative example of a type of analysis that forms subsets based on the relationships between model entities, which may also focus on the sequence of design tasks or the responsibility of (teams of) engineers. In engineering design such approaches are often used to modularise product structures, where tightly coupled components are grouped into modules, which are then more loosely coupled with the rest of the system (Chiriac et al. Reference Chiriac, Hölttä-Otto, Lysy and Suh2011). Models can also be clustered on multiple levels, which leads to hierarchical structures. Clustering can be used to aggregate a model, thereby transforming it to a coarser grained instance. Also, the size and number of clusters depend on the granularity of the original model and the desired granularity of the clustered model.

Table 3. Definitions of terms describing model properties that may result from its granularity.

Figure 1. Relationships between the notion of model granularity and related concepts.

Model accuracy can describe the degree to which the model constructs or its output conform to reality (Gross et al. Reference Gross1999). Similarly, precision can describe both a quality of the model itself as well as the results that are obtained from it (Gross et al. Reference Gross1999). A clear distinction between the terms can be made, as emphasised in literature on measurement systems analysis and Six Sigma. Here, ‘accuracy is the closeness of average measurements to reference values’ and ‘precision is the closeness of measurements to each other’ (Sleeper Reference Sleeper2005). According to this definition a model could be precise (consistent results with little variation) but not accurate (results do not conform to reality) and vice versa. It can be argued that both accuracy and precision often depend on the granularity of the model – certain aspects of reality can only be accurately or precisely described by going into a larger amount of detail. On the other hand, a fine-granular model is not automatically accurate/precise and an accurate/precise model does not necessarily have to be fine grained. The term validation typically describes an assessment of model accuracy with respect to its intended purpose (Sargent Reference Sargent2005).

The discussion of model granularity is closely related with the more specific notion of model fidelity, which is also sometimes described with reference to accuracy (Tolk Reference Tolk and Tolk2012, p. 17). In the context of modelling and simulation fidelity has been defined as the ‘degree to which a model or simulation reproduces the state and behaviour of a real world object’ in a report that provides a comprehensive account on model fidelity (Gross et al. Reference Gross1999, p. 3). While this definition hints at the impact model granularity has on fidelity, it is important to note that the two are not the same (even if they are sometimes used to describe similar phenomena). Fidelity describes the quality of a model to represent the real world and thus refers to the relationship between model and target system. On the other hand, granularity describes a property of the model itself, which may be influenced by the characteristics of the target system and impact the model’s fidelity. Weisberg (Reference Weisberg2007) distinguishes two types of fidelity criteria: dynamical and representational fidelity criteria. Dynamical fidelity criteria indicate how close the model output must be to its real-world counterpart. Representational fidelity criteria are more complex and specify, for instance, to what extent the model structure has to match the real-world phenomenon’s causal structure.

To provide an overview of relevant concepts and how they relate, Figure 1 depicts a relationship diagram, picking up the three categories defined in the beginning of this section. To keep it concise, the diagram focuses on how the concepts relate to granularity and only presents a selection of relationships without too much detail. Some of the terms could be placed at different points in the diagram. For example, the term abstraction can describe an activity that impacts model granularity but also a model property that is closely related to granularity. Likewise, resolution can refer to granularity of the model as well as the output. Also, the modularity of a model may depend on its granularity but the modularity of the target system can in turn influence model granularity. These terms are therefore drawn across categories.

4.2 Measuring granularity

Measures offer ways to quantify model granularity and related concepts to help modellers navigate challenges associated with large, hierarchical or multilevel models. Some metrics that aim to capture granularity have been suggested in various fields and a range of metrics exists for related concepts, such as complexity (e.g., Summers & Shah Reference Summers and Shah2010). Due to the number of relevant contributions and the challenges associated with quantifying granularity, this section only provides a brief introduction to the topic, including a number of references for further reading. The presented metrics are not intended as recommendations, which would require a focused study, but rather as representative examples. Granularity metrics and the underlying theories offer an important perspective on the characterisation of model granularity because they formalise concepts in an explicit way and allow for quantification. In a contribution to rough set theory, Yao (Reference Yao2003) proposes an information-theoretic granularity metric $G$ based on the Shannon entropy to quantitatively characterise partitions:

(1) $$\begin{eqnarray}\displaystyle G(\unicode[STIX]{x1D70B})=\mathop{\sum }_{i=1}^{m}\frac{|A_{i}|}{|U|}\log |A_{i}| & & \displaystyle\end{eqnarray}$$

$G(\unicode[STIX]{x1D70B})$ denotes the granularity of partition $\unicode[STIX]{x1D70B}=\{A_{1},A_{2},\ldots ,A_{m}\}$ of a universe $U$ , where $|\cdot |$ denotes the cardinality of a set. The value of $G(\unicode[STIX]{x1D70B})$ ranges from 0 (‘atomic’ granularity; as fine grained as the universe) to $\log |U|$ (where the model only comprises one granule). Holschke, Rake & Levina (Reference Holschke, Rake, Levina and Dayal2009) adapt this metric to quantify the granularity of business process models, noting the need for a ‘baseline’ in order to compare different models (Holschke Reference Holschke, Hull, Mendling and Tai2010). Indeed, knowledge about the cardinality of the universe (or target system) is necessary to quantify the granularity of a partition (or model). Yao & Zhao (Reference Yao and Zhao2012) take a measurement-theoretic contribution in proposing a new class of measures for granularity of partitions. They distinguish between the granularity of a set, which depends on the cardinality of the set, and the granularity of a partition, which depends on the expected granularity of its blocks. Additionally, they include a critical review of existing information-theoretic and interaction based granularity measures. Dai & Tian (Reference Dai and Tian2013) define the concept of knowledge granulation and knowledge granularity measure for set-valued information systems. Further discussions of the term granularity and approaches to measuring it exist in fields such as information theory, cognitive informatics and granular computing but are not reviewed here.

In the engineering design domain, AlGeddawy (Reference AlGeddawy2014) proposes a ‘granularity index’ to quantify the quality of Design Structure Matrix (DSM) (e.g., Steward Reference Steward1981; Browning Reference Browning2001) clustering in the environment of product families. The collective, numeric DSMs used are combined from individual, binary product variant DSMs.

(2) $$\begin{eqnarray}\displaystyle \text{Granularity Index }=P-Z-\mathop{\sum }_{i=1}^{m}(N_{i}-1)\cdot (A_{i_max}-A_{i_min}) & & \displaystyle\end{eqnarray}$$

$P$ is the sum of positive matrix elements inside DSM clusters and $Z$ is the number of zero elements inside DSM clusters. The summation term is the component appearance variability for all clusters and equals the sum of component appearance variability across modules. $N_{i}$ is the number of components in a specific module and $A_{i_max}/A_{i_min}$ is the maximum/minimum number of appearances of any component in the module. This Granularity Index is calculated for each granularity level of a hierarchical clustering obtained through cladistics (AlGeddawy & ElMaraghy Reference AlGeddawy and ElMaraghy2013). The clustering with the highest Granularity Index is then recommended for product family module composition.

Apart from metrics explicitly targeting granularity, others can also be used to quantify certain aspects of model detail. For example, in their investigation of the impact of model granularity on modularity, Chiriac et al. (Reference Chiriac, Hölttä-Otto, Lysy and Suh2011) do not use an explicit metric of granularity. Instead they use three modularity metrics to quantify the effects of granularity changes, which are quantified in terms of the size and density of the DSM, focusing on the structural aspects of granularity. Complexity metrics have been used to quantify the complexity of design problems, products and processes (Ameri et al. Reference Ameri, Summers, Mocko and Porter2008). Building up this, Summers & Shah (Reference Summers and Shah2010) propose measuring three different aspects of complexity: size, coupling and solvability. Size complexity is based on entropic measures and thus comparable to granularity metrics (e.g., Yao & Zhao Reference Yao and Zhao2012). Integrating modularity and additional details about connectivity is suggested to better capture system complexity (Tamaskar, Neema & DeLaurentis Reference Tamaskar, Neema and DeLaurentis2014). Complexity metrics generally focus on design processes and products themselves rather than models thereof. They also include aspects that go beyond granularity, such as solvability of design problems or modularity of product architectures. Real-world complexity may drive but also provide a baseline for assessing model granularity.

Quantitative analysis of granularity provides the opportunity to quickly assess and compare models, which can ultimately contribute to making better modelling choices. However, measuring model granularity is a challenging endeavour. One question is whether absolute or relative measures are more appropriate. Most information-theoretic metrics define granularity relative to a baseline but still have an absolute scale (Yao & Zhao Reference Yao and Zhao2012). This seems to make sense but requires quantifying the target system itself, which may not be straightforward (Holschke Reference Holschke, Hull, Mendling and Tai2010). Relative measures can be useful to compare models of the same target system but make less sense for different target systems. Also, applying existing metrics without clarifying what exactly they represent may not lead to meaningful results (Fenton Reference Fenton1994). In order for a metric to be a useful measure of model granularity, it has to capture underlying empirical relations (Roberts Reference Roberts1985). However, this requires a theoretical foundation, clarifying which aspects of granularity are measured and how this relates to observable model properties. This research aims to provide such a foundation by developing a framework that classifies different dimensions of model granularity (Section 5), rather than to derive measures, which would merit a more comprehensive and focused discussion than is feasible in the scope of this article.

4.3 Approaches towards model granularity in engineering design

Despite the limited attention in the engineering design community, there are a number of approaches directly targeting model granularity, some of which are presented in this section. Other contributions deal with the topic more or less directly, including work on clustering, modularity, decomposition, hierarchy and others (see also Section 4.1). Most of this work focuses on the product domain although it can be argued that it is just as important for the process domain. A difference is that process models are not centred on an artefact, which could provide a certain baseline. Even though it is less explicitly discussed in the design process modelling literature, the choice of granularity is a salient problem in the area, as it can affect the behaviour of such models significantly. For instance, coarser models may combine several tasks into one, which is problematic if it obscures iteration common in the real-world process. Also, given that task sequence influences iteration, rework and thus the overall process performance, it is important for process models to capture relevant process architecture in sufficient detail without overcomplicating the model. A good example to study and illustrate granularity in the engineering design domain are network/dependency-type models, where the topic has so far received most attention. Depending on the modelling approach such models can be used in both domains and share enough properties to capture both in terms of granularity. For instance, models can capture the dependencies between components of a product or the relationships in an activity network model of a process. Due to the scope of this article, we focus on a few examples of such models, directly targeting granularity.

Noting that past research does not consider the effects of granularity on architectural analysis, Chiriac et al. (Reference Chiriac, Hölttä-Otto, Lysy and Suh2011) focus on how the degree of modularity of a product is affected by the level of granularity it is represented in. They represent a range of idealised systems and one real-world complex system in DSMs on two different levels of granularity and analyse them for their degree of modularity using three established metrics. Concluding that the results of architectural analysis can be distorted by the level of granularity of its components, they advise to be cautious when defining a particular system decomposition for analysis tasks. In a similar approach, Suh et al. (Reference Suh, Chiriac and Hölttä-Otto2015) show that the representation of a system can vary significantly depending on the system architect’s decomposition perspective, impacting on system architecture development and resulting modularity.

Cladistics is a classification method that hierarchically groups entities into discrete sets and subsets (Hennig Reference Hennig1966; ElMaraghy, AlGeddawy & Azab Reference ElMaraghy, AlGeddawy and Azab2008). It has been employed to determine an optimum granularity level for design architecture models of modular products using component-based DSMs (AlGeddawy & ElMaraghy Reference AlGeddawy and ElMaraghy2013). Cladistics analysis yields a hierarchical clustering, which can be visualised in a cladogram. For each level of this clustering a modularity index is calculated, which enables choosing the best modularity configuration and its respective granularity (AlGeddawy & ElMaraghy Reference AlGeddawy and ElMaraghy2013).

To address the need for more sophisticated product representations for architectural analysis, Tilstra, Seepersad & Wood (Reference Tilstra, Seepersad and Wood2012) present the High Definition Design Structure Matrix (HDDSM), which captures interactions between components in a product and employs a hierarchical modelling method to include higher levels of detail where needed. The approach builds upon DSM research and allows modular construction and assembly of highly detailed product models and submodels, facilitating the distribution of modelling tasks. Similar issues have been addressed in process modelling literature, where the integration of disciplines and respective process models play an important role (Grose Reference Grose1994). Tilstra et al. (Reference Tilstra, Seepersad and Wood2012) note the potential for analysis on different levels of detail with the potential of including strategic clustering.

The presented approaches show that model granularity has received increasing attention in the field of engineering design. However, while more approaches exist, especially when including ones that deal with related topics, the vast majority focuses on a particular purpose or issue and gives little attention to the wider challenges of model granularity. Also, because the contributions differ in their use of terminology, comparisons and attempts to synthesise are complicated. A clearer view of how model granularity and related challenges and opportunities can be categorised is required to advance understanding of the field and spark further, necessary discourse on the topic.

5.1 Synthesising relevant concepts to characterise model granularity

The structuralist view of models in philosophy of science allows to derive some insights related to model granularity. In this context a “structure $S=[U,R]$ consists of a non-empty set $U$ of objects (the domain of the structure), and a non-empty indexed set $R$ of relations on $U$ . This definition does not assume anything about the nature of the objects in $U$ ” (Frigg Reference Frigg2009). Similarly, relations do not have properties other than formal ones. While this view of models as structures falls short in describing all aspects of modelling practice in engineering design, it provides a basis for the description of model granularity. The domain (or universe) $U$ of the structure $S$ consists of objects, whose quantity and size depend on the degree of abstraction of the model. The number of relations is partly dependent on the degree of abstraction but also on the nature of relations they represent. Based on this, it is proposed that the structural granularity of a model describes the decomposition of model elements as well as the density of relations between them. The density here denotes the number of relations relative to the number of possible relations. This is in line with the use of the term density in complex networks and graph theory (e.g., Newman Reference Newman2003). However, reliance on density alone would be an insufficient metric as it fails to capture the internal size and information content of relations.

The structural dimension outlined in the previous paragraph does not cover information about the content and nature of granules and their interactions. Depending on the granularity of the description the variety of possible states or configurations of the model can change. In a different context, Ashby’s (Reference Ashby1956) law of requisite variety states that a control system needs a certain level of internal variety to respond to external variety, which is also relevant for the modelling domain (Conant & Ashby Reference Conant and Ashby1970). The variety of a model with a given structure can be thought of as dependent on the amount of detail in the description of its elements and their relationships. Furthermore, it is not clear how some aspects of simulation, like temporal and dynamic properties, can be accommodated. In databases, the concept of time, or temporal, granularity is used to describe how finely the time domain is represented (Bettini et al. Reference Bettini, Dyreson, Evans, Snodgrass, Wang, Etzion, Jajodia and Sripada1998). For example, the collections ‘days’, ‘weeks’, and ‘months’ are all granularities where the decomposition determines the size of the granules. Rosen, Saunders & Guharay (Reference Rosen, Saunders and Guharay2014) refer to granularity as a driver for the resolution, or complicatedness, of the response surface of the simulation. This is proportional to the number of simulation runs in the experimental design. We therefore suggest a second dimension, information granularity, which captures the intra-granule information content as well as information resolution related to analysis.

Two other important aspects that are closely related to model granularity, but do not in themselves constitute forms of granularity, are purpose and scope of a model. To a large extent they drive model granularity but do not necessarily prescribe a particular level. They influence decisions about model granularity more (scope) or less (purpose) directly and thus play an important role when discussing the topic.

5.2 A classification framework for model granularity

Model granularity can be characterised in different ways, depending on which aspects of the model are described. Figure 2 illustrates the resulting (sub-)categories in a framework based on the derivation in the previous section. On the first level it distinguishes between structural and information granularity. Structural granularity encompasses the level of (dis)aggregation of model elements. This describes the amount to which the elements in the model have been (dis)aggregated relative to the target system. A fine granularity here means that there are many little elements, whereas coarse granularity indicates fewer, bigger elements. The other type of structural model granularity concerns the relationships between those elements, which is partly dependent on their level of aggregation. However, even on a fixed level of element aggregation, the degree of description of relationships can vary. Even though these two subcategories are often not independent, they can be and describe different aspects of the model, making this distinction a sensible one.

Figure 2. Different dimensions of model granularity.

Information granularity encompasses the information that is contained in the model elements, which is not taken into account by the structural part. Here, granularity relates to the type and amount of information that is associated with these elements. So, the more information content, or different types of information, are included in the model elements, the finer the information granularity within granules. Analysis resolution indicates the level of detail exhibited by analysis based on a model. This could refer to the temporal resolution of a dynamic simulation or the degree of discretisation of a process model. It is harder to define clear boundaries for analysis resolution as it partly overlaps with the other previously defined categories and relates specifically to analysis. However, given that the results obtained from simulation and other types of analysis inform decisions in many fields and engineering design in particular it is important to capture this dimension of model granularity as well. In the context of simulation modelling a related concept is model fidelity, which indicates how well it reproduces real-world behaviour. In particular, dynamic fidelity (Weisberg Reference Weisberg2007) is often proportional to the simulation resolution as it is characterised here (but also other aspects of model granularity). Nevertheless, higher simulation resolution does not automatically lead to higher model fidelity, although it might enable it in many cases.

It is important to note that these dimensions of the framework are not necessarily orthogonal. For instance, decomposing a system further leads to a finer structural granularity, which can also affect the information content of the granules. However, distinguishing between the structure and information dimension is important nonetheless because it allows to capture a wider spectrum and to account for special properties of a particular modelling approach (e.g., keeping the information content of a granule constant but disaggregating further). The two dimensions can also be traded off. For instance, the structural granularity could be increased while decreasing the information content per granule, thereby keeping the overall information content of the model constant. Generally, the degree of orthogonality between structural granularity and information granularity depends on the modelling approach (see Section 5.3 for an illustration), so both dimensions should be considered for a comprehensive characterisation of model granularity. Purpose and scope are very general considerations and affect many aspects of modelling and granularity. Even though they are not explicitly included in Figure 2, purpose and scope can influence all dimensions of granularity. The types of granularity can be said to get more specific from top to bottom.

5.3 Illustrating the classification framework

To clarify and demonstrate the concepts and framework presented in the previous section, a popular modelling approach in engineering design, the Design Structure Matrix (DSM), is used as an illustration and discussed with respect to different manifestations of model granularity. Figure 3 illustrates structural and information granularity, including the scope but not purpose. Simulation resolution, however, is not illustrated in this figure because of its dynamic nature (see Figure 4).

Figure 3. Structural and information granularity of a DSM (including scope).

Figure 4. Temporal simulation resolution.

The two main axes in Figure 3 represent the two main aspects as discussed in the previous section and Figure 2. It is important to note that the structural granularity axis encompasses both (dis-)aggregation and relationships while the other axis only displays the information per granule (but not simulation resolution). Increasing structural granularity leads to a finer decomposition of the modelled system and more relationships between its elements. Even though these two characteristics of structural granularity are not entirely independent they do not necessarily have to correlate as displayed in Figure 3. The system boundaries stay the same unless the scope is increased or decreased, as indicated by the grey cells. This can be done irrespective of the granularity of the model. Increasing the information per granule is illustrated here by shifting from a binary dependency, to a low-medium-high indication of dependency strength and finally numeric values for two different factors. For instance, this can indicate impact and likelihood of change propagation between elements (Clarkson, Simons & Eckert Reference Clarkson, Simons and Eckert2004). Also, other types of models can be used to illustrate the concepts developed in this article. The DSM was chosen because of its simplicity and popularity in engineering design.

Figure 4 illustrates a form of simulation resolution, in particular temporal resolution. Here, a lower resolution is equivalent with fewer data points to describe system behaviour. With a higher resolution system behaviour can be described in more detail. This may yield a better description of the behaviour of the target system, although not necessarily. In any case, more data points can be used to analyse model behaviour. A higher resolution therefore means a finer granularity of the simulation output; the data granules describe a smaller fraction of the observed behaviour. The illustration here is a simple but representative example depicting a particular form of simulation resolution. Depending on the model, other forms of simulation resolution are conceivable. An example is spatial resolution, which can have a trade-off relationship with temporal resolution, for instance in particle physics. Due to the focus on models in engineering design rather than dynamic physical systems this relationship is not discussed in more detail in this article.

While an extensive demonstration of the framework to real-world models would go beyond the scope of this article, a few examples from the literature are mentioned here as a first step. The general challenge in using finished models is that the choices regarding granularity have usually already been made, concealing the underlying trade-offs. In their study on the impact of model granularity on modularity, Chiriac et al. (Reference Chiriac, Hölttä-Otto, Lysy and Suh2011) vary the structural granularity by describing the same target system with different numbers of model elements, which leads to varying amounts (and density) of dependencies between them. The modularity is also varied while keeping the granularity constant. They use only binary DSMs, so information granularity is fixed. Similarly, Tilstra et al. (Reference Tilstra, Seepersad and Wood2012) focus on structural granularity by further decomposing parts of the model, a binary DSM. Other approaches extend the information granularity, for example by assigning values for rework probability and impact to relationships between design tasks (Cho & Eppinger Reference Cho and Eppinger2005). Maier et al. (Reference Maier, Wynn, Biedermann, Lindemann and Clarkson2014) test the sensitivity of their design process simulation to different information granularities of the input model and outline in a later publication how the simulation resolution could be adjusted to account for changes in the structural granularity of the input model (Maier et al. Reference Maier, Eckert and Clarkson2015).