2.1 Design, invention and cognitive psychology literature

What connections between technological change and design research can be inferred from the existing literature? Business scholars and economists often view technical change as occurring inside a black box, and have usually avoided examining design activities that are the source of technological change. An important publication that begins to build a bridge between aspects of design research and the economics of technological change is the paper by Baldwin & Clark (Reference Baldwin, Clark, Kahin and Foray2006). These authors (and Luo, Olechowski & Magee Reference Luo, Olechowski and Magee2014) point specifically to a central role for design in achieving economic value. In addition to economic perspectives, another view that somewhat ignores design is the linear model accredited to Bush (Reference Bush1945), which considers technological change occurring through application of science. As a counterview, in his seminal book, The Sciences of the Artificial, Simon (Reference Simon1969, Reference Simon1996) was the first to highlight that design is an activity standing on its own right, like natural sciences, and has its own set of logic, concepts and principles. While the primary goal of natural science is to produce predictive explanations of natural phenomena, the primary goal of design is to create artifacts. The design activity is central to creation and improvement of artifacts in all technological domains and involves cognitive activities such as the use of knowledge, reasoning and understanding. These indisputable cognitive activities have been noted by many scholars who have studied invention and design (Simon Reference Simon1969; Dasgupta Reference Dasgupta1996; Gero & Kannengiesser Reference Gero and Kannengiesser2004; Hatchuel & Weil Reference Hatchuel and Weil2009).

In the context of realizing higher performance from subsequent generations of artifacts, the role of invention, as one outcome of the design process, is a critical one since improvement in performance of artifacts must strongly reflect the inventions. As Vincenti (Reference Vincenti1990, p. 230) puts it, inventive activity is a source of new operational principles, and normal configurations that underlie future normal or radical designs. The operational principles (Polanyi Reference Polanyi1962; Vincenti Reference Vincenti1990) of an artifact describe how its components fulfill their special functions in combining to an overall operation to achieve the function of the artifact.

Models found useful in describing the creative design process include the Geneplore model (Finke, Ward & Smith Reference Finke, Ward and Smith1996), topological structures (Braha & Reich Reference Braha and Reich2003), FBS (function-behavior-structure) theory (Gero & Kannengiesser Reference Gero and Kannengiesser2004), concept–knowledge (CK) theory (Hatchuel & Weil Reference Hatchuel and Weil2009), infused design (Shai, Reich & Rubin Reference Shai, Reich and Rubin2009), analytical product design (Frischknecht et al. Reference Frischknecht, Gonzalez, Papalambros and Reid2009) and other modeling approaches. Although all of these frameworks include – to some degree – the key idea of combining existing ideas (for example in the form of conceptual synthesis, and blending of mental models described in discussion of the Geneplore model), the framework found most helpful in our modeling of performance changes resulting from a cumulative design process is analogical transfer. Although this idea can be traced as beginning with Polya (Reference Polya1945) or earlier, the framework remains an active area in design research (Clement, Mawby & Giles Reference Clement, Mawby and Giles1994; Holyoak & Thagard Reference Holyoak and Thagard1995; Goel Reference Goel1997; Gentner & Markman Reference Gentner and Markman1997; Leclercq & Heylighen Reference Leclercq, Heylighen and Gero2002; Dahl & Moreau Reference Dahl and Moreau2002; Christensen & Schunn Reference Christensen and Schunn2007; Linsey, Wood & Markman Reference Linsey, Wood and Markman2008; Tseng et al. Reference Tseng, Moss, Cagan and Kotovsky2008; Linsey, Markman & Wood Reference Linsey, Markman and Wood2012; Fu et al. Reference Fu, Chan, Cagan, Kotovsky, Schunn and Wood2013). Scholars of analogical transfer (Gentner & Markman Reference Gentner and Markman1997; Holyoak & Thagard Reference Holyoak and Thagard1995; Weisberg Reference Weisberg2006) explain analogical transfer as involving the use of conceptual knowledge from a familiar domain (base) and applying it to create knowledge in a domain with similar structure (target): analogical transfer exploits past knowledge in both the base and target domains. The analogies utilized can be local, regional, or remote, depending on surface and structural similarities between objects involved in the base and target domains. Weisberg discusses the example of the Wright brothers using several analogical transfers to first recognize and solve the problem of flight control. First, they viewed flying as being similar to biking in which the rider has to be actively involved in controlling the bike, an application of regional analogy. Interestingly, many others attempting to design artifacts for flying did not access this regional analogy and thus did not even identify the key control problem. Second, the Wright brothers studied birds to see how they controlled themselves during flight, and learned that they adjusted their position about the rolling axis using their wing tips. From this insight, they had the idea of using similar moving surfaces, another instance of using regional analogy. Lastly, they developed the idea of warping the wings, demonstrated by using a twisted cardboard box, to act like vanes of windmills to make the airplane roll. The use of three analogical transfers in combination to see and solve the flight control problem is a clear case of analogical transfer but there is also evidence (cited earlier in this paragraph) of much wider applicability.

There are more abstract versions of combinatorial analogical transfer that have been proposed in the wider literature. Based on an extensive historical study of mechanical inventions and drawing insights from Gestalt psychology, Usher (Reference Usher1954) proposed a cumulative synthesis approach for creation of inventions. The notion of bisociation (Koestler Reference Koestler1964; Dasgupta Reference Dasgupta1996) develops the cumulative synthesis approach further and says that a new inventive idea is ideated combining disparate ideas. More recently, Fleming (Reference Fleming2001), Arthur (Reference Arthur2007) have respectively used the same combinatorial notions of invention in studying technological change. Other research in the technological change literature also discusses a related concept that is usually called ‘spillover’. Rosenberg (Reference Rosenberg1982) showed that such technological spillover greatly impacted the quantity and quality of technological change in the United States in the 20th century – a result supported by Nelson & Winter (Reference Nelson and Winter1982), Ruttan (Reference Ruttan2001). Indeed, a recent paper by Nemet & Johnson (Reference Nemet and Johnson2012) states that ‘one of the most fundamental concepts in innovation theory is that important inventions involve the transfer of knowledge from one technical area to another’. We note that these descriptions do not always make a clear distinction regarding whether the transfer is occurring at the idea level or at the artifact level. They are silent regarding how and from where designers or inventors get their disparate ideas to combine and regarding details about the complexities of transfer and combination.

Analogical transfer of ideas as a broad mechanism and expertise as the foundation of ideas (Weisberg Reference Weisberg2006) provides adequate specificity for modeling science and invention in this paper. Weisberg contends that analogical transfer is utilized in generation of both scientific and technological knowledge. Vincenti (Reference Vincenti1990), Mokyr (Reference Mokyr2002) take the view that scientific and technological knowledge can be classified into descriptive (Understanding) and prescriptive (Operations) knowledgeFootnote ¹ regimes. The Understanding regime can be seen as a body of ‘what’ knowledge and includes scientific principles and explanations, natural regularities, materials properties and physical constants. A unit of understanding (UOU) is falsifiable (Popper Reference Popper1959) and enables explanation and prediction about specific phenomena, including behavior of artifacts. The Operations regime, on the other hand, can be viewed as a body of ‘design knowledge’, which suggests how to leverage natural ‘effects’ (Arthur Reference Arthur2007; Vincenti Reference Vincenti1990) to achieve a technological advantage or purpose. It includes operating principles, design methods, experimental methods and tools (Vincenti Reference Vincenti1990; Dasgupta Reference Dasgupta1996). Based on this distinction, understanding enables generation of operational knowledge, which ultimately contributes towards design of some artifact. However, operations is not entirely based upon existing understanding and in fact innovations in know-how can and often do occur before any understanding of related natural effects is available.

An important aspect of design and invention is the cooperative interaction between Understanding and Operations regimes (Musson Reference Musson and Musson1972; Musson & Robinson Reference Musson and Robinson1989). Using a historical perspective, Mokyr (Reference Mokyr2002) has carefully observed that a synergistic exchange between the two has been occurring, where each enables the other. The contribution of Understanding to Operations is well known: it provides principles, and regularities of natural effects, including new ones, in the form of predictive equations, and descriptive facts, such as material properties. Fleming & Sorenson (Reference Fleming and Sorenson2004) provide evidence that understanding helps inventors by providing a richer map to search for operating ideas, which can be combined together. Understanding also provides insight about where new technological opportunities may be found (Klevorick et al. Reference Klevorick, Levin, Nelson and Winter1995). Beyond these contributions, there is the more general view, discussed in the initial paragraph of this section, that new operational ideas can be derived from new understanding.Footnote ² What is less discussed is the multi-faceted contributions of Operations to the Understanding regime. In his paper, Sealing wax and string, de Solla Price (Reference de Solla Price1986), a physicist, and historian of science, highlighted that instruments (an output of the Operations regime) were a dominant force in enabling scientific revolutions. He states: ‘changes in paradigm that accompany great and revolutionary changes (in science) were caused more often by application of technology to science, rather than changes from ‘putting on a new thinking cap’. ‘Operations provide tools and instruments to make measurements, and to make new discoveries. In his book, The Scientist: A History of Science Told Through the Lives of its Greatest Inventors, Gribbin (Reference Gribbin2002), a British astrophysicist, and science writer, has described how the ability to grind eyeglass lenses made it possible to make better telescopes, and hence paved the way for astronomers to make new discoveries. New or improved observational techniques are still a major driver of progress in science. Gribbin has aptly summarized the enabling exchange between the two regimes: ‘new scientific ideas leading…to improved technology and new technology providing scientists with the means to test new ideas to greater and greater accuracy’. In addition, the Operations regime provides new problems for the Understanding regime to study, and has led to birth of new fields in Understanding (Hunt Reference Hunt2010). Based upon these insights and with our focus on explaining performance improvement arising from continuing streams of inventions, our model treats mutual exchange between Understanding and Operations.

In design of artifacts, Simon (Reference Simon1962) introduced the notion of interactions in his essay on the complexity of artifacts. When a design of an artifact is changed from one state to another (with differences between the two states as defined by multiple attributes, say D1, D2 and D3) by taking some actions (say A1, A2 and A3), in many cases, any specific action taken may affect more than one attribute, thus potentially manifesting as interactions of the attributes. The same notion of interaction/conflicts is captured by the concept of coupling of functional requirements (Suh Reference Suh2001), or dependencies between characteristics (Weber & Deubel Reference Weber and Deubel2003), which can occur when two or more functional requirements are influenced by a design parameter. Theoretically it seems ideal to have one design parameter controlling one functional requirement to achieve a fully decomposable (modular) design (Suh Reference Suh2001; Baldwin & Clark Reference Baldwin and Clark2000). However, Whitney (Reference Whitney1996, Reference Whitney2004) has argued that, in reality, how decomposable a design of an artifact can be depends on the physics involved or additional constraints, such as permissible mass. These are reflected as component-to-component, and component-to-system interactions, or as a need to have multi-functional components. Consequently, Whitney argues, complex electro-mechanical-optical (CEMO) systems, primarily designed to carry power, cannot be made as decomposable as very-large-scale integration (VLSI) systems primarily designed to transmit and transform information. For example, in energy applications, the impedance of transmitting and receiving elements has to be matched for maximum power transfer, thus making the two elements coupled. Further, CEMO systems typically need to have multi-functional components in order to keep the artifact size reasonable, creating coupling of attributes at the component level. Another type of interaction Whitney has identified are the side effects, such as waste heat in computers, and corrosion of electrodes in batteries – that occur in artifacts, which in some electro-mechanical systems can consume significant portion of the design effort for their mitigation. The presence, and thus the resolution, of these different interactions cause significant delay, consume significant engineering resources and potentially stop applications of some concepts, thus making the level of interactions of a technological domain a potentially strong factor influencing its rate of improvement. Based upon Whitney’s work, the effect of interactions on rates of improvement was suggested qualitatively by Koh & Magee (Reference Koh and Magee2008) and a quantitative model of the effect was developed by McNerney et al. (Reference McNerney, Farmer, Redner and Trancik2011) – see Section 2.3.

The influence of design parameters on artifact performance is an essential part of design knowledge. Many technological domains have complex mathematical equations relating some aspects of performance with design parameters. Indeed, the so-called engineering science literature has such equations for many aspects affecting the design of artifacts of perhaps all technological domains. Simpler relationships concerning the geometrical scale of artifacts are also available and generally give performance metrics as a function of a design variable raised to a power. Use of power-law relationships can be found in: (1) Sahal (Reference Sahal1985) who studied scaling in three different sets of artifacts – airplanes, tractors and computers; and (2) Gold (Reference Gold1974) who demonstrated that doubling the size of a blast furnace reduces their cost by about 40 %. The constant percent change per doubling in size results from the power law (assumed by Gold) between performance/cost and geometrical variables such as volume.

2.2 Technological change literature

What descriptive models and theories help us understand why technologies improve and how the improvement patterns are structured? Schumpeter (Reference Schumpeter1934) introduced the idea that entrepreneurs, whose primary role is to provide improved products and services through innovation, drive economic progress. These innovations, which Schumpeter describes as industrial mutations, displace competing products and services from the economy. However, they, too, are displaced by higher performing innovations that follow, thus perpetuating the cycle of creative destruction. Building upon Schumpeter’s notion, Solow (Reference Solow1956) recognized and incorporated technological change as the key element in his quantitative explanatory theory of economic growth. The basic conclusion that technological change is the foundation of sustained economic growth has stood the test of time. Later, theorists of economic growth (Arrow Reference Arrow1962; Romer Reference Romer1990; Acemoglu Reference Acemoglu2002) have attempted to deal with the more complex problem of embedding technological change within the economy (endogenous to different degrees). Although the later theories are important, the issues are outside the scope of this paper and will not be covered here. A related question of demand-pull and technology push does have more relevance.

What drives technological innovation? Some early explanations emphasized pure demand push (Carter & Williams Reference Carter and Williams1957, Reference Carter and Williams1959; Baker, Siegman & Rubenstein Reference Baker, Siegman and Rubenstein1967; Myers & Marquis Reference Myers and Marquis1969; Langrish et al. Reference Langrish, Gibbons, Evans and Jevons1972; Utterback Reference Utterback1974) where the needs of the economy at a given time dictate technological direction. Mowery & Rosenberg (Reference Mowery and Rosenberg1979) reanalyzed the data and methodology in this early work and arrived at a strong role for science/technology push (the discoveries of scientists and inventors primarily determine technological direction). Taking a balanced view, Dosi (Reference Dosi1982) argued that both market pull (customer needs and potential for profitability) and technology push (in the form of promising new technology, and the underpinning procedures) are equally important for being sources of innovation.

Tushman & Anderson (Reference Tushman and Anderson1986) discuss discontinuities as having large socio-technical effects and note that such discontinuities are an essential element of technological change. In another highly referenced paper, Henderson & Clark (Reference Henderson and Clark1990) emphasize the importance of architectural change of artifacts – as opposed to component change – having large effects on the firm-level impact of change. Christensen & Bower (Reference Christensen and Bower1996), on the other hand, view technological change occurring as a series of disruptive product innovations that start in a niche market catering to different functional requirements, but then rapidly improve towards the requirements of mainstream performance. The disruptive technology surpasses the mature market leaders (by achieving the necessary performance in smaller, cheaper artifacts), and displaces them.

All of the concepts of technological change described in the preceding paragraphs – at least implicitly – depend upon relative rates of change of performance. This is the focus of our modeling effort, so we will now briefly review concepts related to trends in performance of designed artifacts, and what patterns they have followed. We first review two established frameworks – generalizations of Wright’s early research, and Moore’s Law – for describing trends in technological performance. In 1936, Wright (Reference Wright1936) in his seminal paper ‘Factors affecting the Cost of Airplanes’ for the first time introduced the idea of measuring technological progress of artifacts. From his empirical study of airplane manufacturing, he demonstrated that labor cost or total cost of specific airplane designs decreased as a power law against their cumulative production. This relationship is expressed as:

(1) $$\begin{eqnarray}C=C_{0}P^{-w},\end{eqnarray}$$

where $C_{0}$ , and $C$ are unit cost of the first, and subsequent airplanes respectively, and where $P$ and $w$ are cumulative production and its exponent that relates it to unit cost. Wright explains that labor cost reductions are realized as shop floor personnel gain experience with the manufacturing processes, and material usage and have access to better production tools. Since Wright’s work, this approach has been used to study production of airplanes and ships during World War II, and extended to private enterprises (Yelle Reference Yelle2007). It should be noted that Wright did not look at improvement due to new designs; instead he only considered improved manufacturing of a fixed design.

Moore (Reference Moore1965) presented the second approach – using time as the independent variable and investigating a series of newly designed artifacts – in his seminal paper that describes improvement of integrated circuits. He observed that the number of transistors on a die was doubling roughly every 18 months (modified to 2 years in 1975). This exponential relationship between the number of transistors on a die and time, famously knownFootnote ³ as Moore’s Law, can be mathematically expressed as:

(2) $$\begin{eqnarray}Q_{\!J}(t)=Q_{\!J}(t_{0})\exp \{K_{\!J}(t-t_{0})\},\end{eqnarray}$$

where $Q_{\!J}(t_{0})$ and $Q_{\!J}(t)$ are the number of transistors per die (a measure of performance) at time $t_{0}$ and time $t$ , and $K_{\!J}$ is the rate of improvement (annual if time is in years). For integrated circuits, the exponential relationship has held broadly true for five decades. Others (Girifalco Reference Girifalco1991; Nordhaus Reference Nordhaus1996; Koh & Magee Reference Koh and Magee2006, Reference Koh and Magee2008; Lienhard Reference Lienhard2008) utilized this temporal approach to study performance of different technologies, and have demonstrated that many technologies exhibit exponential behavior with time. More recently, Magee et al. (Reference Magee, Basnet, Funk and Benosn2014) extended the study to 73 different performance metrics in 28 different technology domains. The performance curves have continued to demonstrate exponential behavior, although annual rates vary widely across domains but not across different metrics for a single domain. We note that Moore and all others who used his framework basically compared the performance of different designs over time differentiating the Wright and Moore frameworks. However, it is also possible to use the Wright framework for different designs, but only if the amount produced increases exponentially with time (Sahal Reference Sahal1979; Nagy et al. Reference Nagy, Farmer, Bui and Trancik2013; Magee et al. Reference Magee, Basnet, Funk and Benson2016).

In order to clarify for readers the nature of empirical performance data, we present performance data for two sample domains, magnetic resonance imaging (MRI) and electric motors (Figure 1a), and a summary of improvement rates for 28 domains (Figure 1b) from Magee et al. Reference Magee, Basnet, Funk and Benson2016. The exponential trend for each domain can be described by Eq. (2), where $Q_{\!J}(t)$ and $Q_{\!J}(t_{0})$ are the intensive performance of an artifact in domain $J$ at time $t$ and $t_{0}$ , and $K_{\!J}$ is the annual rate of improvement of the domain in question.

Figure 1. (a) Exponential growth of performance in sample domains – electric motor and magnetic resonance imaging (MRI). Adapted from Magee et al. Reference Magee, Basnet, Funk and Benson2016. (b) Annual rate of performance improvement, $K_{\!J}$ , for 28 domains. Adapted from Magee et al. Reference Magee, Basnet, Funk and Benson2016.

A recent paper (Benson & Magee Reference Benson and Magee2015a ) has empirically investigated the variation of the improvement rates in these 28 domains. The work has important relationships to the current work, so we describe it to not only note the relationships but to also clarify the fundamental differences. Benson and Magee found strong correlations between specific meta-characteristics of the patents in the 28 domainsFootnote ⁴ and the improvement rate in the domains. These authors found that patent meta-characteristics reflecting the importance (citations per patent by other patents), recency (age of patents in a domain) and immediacy (the average over time of the usage of current new knowledge in the domain) are all correlated with the improvement rate. They found a particularly strong correlation $(r=0.76,p=2.1\times 10^{-6})$ with a metric that combines immediacy and importance (the average number of citations that patents in the domain receive in their first 3 years). The findings (and associated multiple regressions) are robust over time and with domain selection and are of practical importance in predicting technological progress in domains where performance data is not available (Benson & Magee Reference Benson and Magee2015a ). Nonetheless, the conceptual basis for the findings is observed attributes of the inventive output from a technological field (importance, recency and immediacy of a patent set) and not the process of invention, design knowledge or other technical aspects of designed artifacts in the domain. The aim of the work reported in the present paper is to develop a model that yields insights about the pace of change without recourse to concepts based upon observation of the output over time. If fully successful, we would be able to judge the potential for change based only upon the nature of the design knowledge and we might even be able to find new approaches that might achieve technological goals at more rapid improvement rates.

2.3 Literature on quantitative modeling of technological change

What research has attempted to model the technological performance trends that we just discussed? Muth (Reference Muth1986) and Auerswald et al. (Reference Auerswald, Kau, Lobo and Shell2000) have developed models to explain Wright’s results by introducing the notion of search for technological possibilities. Each paper assumes that random search, a key element of technological problem solving, for a better technique is made within a fixed population of possibilities. Considering a case of a single manufacturing process, Muth (Reference Muth1986) developed a model to capture the idea of substituting manufacturing sequences with better ones. He argues that shop personnel improve the process by learning through experience and making random search for new techniques, which enable improvement of processes leading to cost reductions. Muth demonstrated that the notion of fixed possibilities easily leads to fewer and fewer improvements that can be realized and he argues that the data (for fixed designs) shows a leveling off and eventual stoppage as the model suggests. Building on Muth’s idea of random search within a set of fixed design possibilities, Auerswald et al. modeled a multi-process system, in which different processes can be combined to create diverse recipes, and for the first time introduced the notion of interactions by allowing adjoining processes to affect each other’s cost.

Following similar reasoning as Muth and Auerswald et al., McNerney et al. (Reference McNerney, Farmer, Redner and Trancik2011) have developed a stochastic model to explain how the cost reduction of a multi-component system is influenced by component interactions, which they refer to as connectivity between components. McNerney et al. operationalized the notion of interactions as outlinks representing influence of a component on other components. When a specific component in a domain artifact changes by introducing a new operational idea, the change affects the design of all the components it influences. If the performance of the artifact (influencing and influenced components) as a whole improves, then McNerney et al. consider the interactions to be resolved and the operating idea is considered successful. The McNerney et al. paper demonstrates that artifacts with more interactions improve more slowly than artifacts with less interactions.

Using agent-based modeling, Axtell et al. (Reference Axtell, Casstevens, Hendrey, Kennedy and Litsch2013) have developed a competitive micro-economic model of technological innovation utilizing the notion of technological fitness. Although they do not discuss or cite Moore’s law or his work, they have demonstrated that cumulative technological fitness of all agents increases exponentially overtime. This is different from other researchers who have predominantly been focused on Wright’s framework. Consistently, Axtell et al. consider new designs and not just process optimization.

Using a simulation approach, Arthur & Polak (Reference Arthur and Polak2006) have modeled how new generations of artifacts arise by combining currently available artifacts. The artifacts considered are electronic logic gates. New designs (combinations) are more complex logic gates that can then also be combined into even more complex logic gates. In their model, Arthur and Polak specify several design goals towards which the logic gates evolve. They have demonstrated that designs with higher levels of complexity cannot be attained without realizing design configurations with intermediate levels of complexity, and new designs with higher functionality substitute for current designs with inferior functionality. This model is much richer than other models in representing the artifact part of the design process; however, it does not consider performance improvement, as do the other models. It is also limited to developing pre-specified artifacts and is thus a specific process; consequently it is not open-ended or general which are characteristics necessary for modeling performance trends for general technological domains.

Although some are more explicit than others, one feature common to all these models is that all utilize the notion of building upon the performance (in the form of cost) or designs of the past, a key feature of cumulative processes included in the model presented here. On the other hand, they do not consider three aspects we believe essential for answering our research question; thus, these three factors differentiate our model from this relevant past work. First, none of them discusses or includes the influential role played by exchange between science and technology. In this paper, we treat the design process and the exchange between science and technology as important elements for understanding the change in performance over time that in turn is essential to understanding technological change. Second, none consider the design process or operating principles as part of combinatorial analogical transfer – they instead look at combinations at the artifact level instead of combination of ideas. In this paper, we consider both the idea and artifact regimes in developing our model. Third, no prior model has considered (or modeled) the role of scaling of design parameters on performance. In this paper, scaling is introduced and the resulting model results in scaling having an important predicted influence on the rate of performance improvement.

3.1 Conceptual basis of model

We utilize two sets of mechanisms from design to construct the overall model. The first set, which gives rise to exponential trends, includes growth of knowledge – understanding and operations – using combinatorial analogical transfer aided with mutual exchange between the two. The second set, which gives rise to variation in improvement rates, includes component interactions and scaling of design variables. Since the goal of the model is to develop an explanatory and quantitative predictive model, while modeling these mechanisms we have, where necessary, simplified (removed details) and utilized abstraction to keep the model tractable.

Figure 2. Model of exchange between Understanding and Operations regimes and modulation of IOI assimilation by interaction $(d_{J})$ and scaling $(A_{\!J})$ parameters of domain $J$ .

The overall architecture of the model is shown in Figure 2. Based on the work of Vincenti (Reference Vincenti1990) and Mokyr (Reference Mokyr2002) that we discussed earlier, we classify scientific and technical knowledge into Understanding and Operations regimes. We further split the Operations regime into idea and artifact sub-regimes where non-physical representation of artifacts are in the idea sub-regime. The idea sub-regime, represented as an ideas pool, consists of individual operating ideas (IOI). The IOI concept is an abstraction and generalizes the idea of operating principle introduced by Polanyi (Reference Polanyi1962) and includes any ideas, including operating principles, invention claims, design structures, component integration tricks, trade secrets and other design knowledge that lead to performance improvement of artifacts. An IOI is different from a UOU which includes scientific principles, and factual information. An example of a UOU is the principle of total internal reflection, which describes how a beam of light undergoes reflection inside a dense medium, when the angle of incidence is above a critical value (see Figure 3). This principle accurately describes a natural effect, but it does not prescribe how we can use it to transmit information. On the other hand, a pair of parallel surfaces (or a fiber) enclosing a dense medium and utilizing the principle of total internal reflection provides a mechanism – an operating principle – to make a ray of light travel down the length of the medium (see Figure 3). Such a mechanism is an example of an IOI. Unlike artifacts, which belong to a specific technological domain, we model IOI in the ideas (IOI) pool as being non-domain specific and available to all technological domains. For instance, the operating principle of total internal reflection is utilized in fiber optic telecommunications, fluorescent microscopy, and fingerprinting, very distinct technological domains. In the idea sub-regime, designers/inventors source existing ideas (IOI) using analogical transfer and combine them probabilistically to create new ideas (IOI). Once new IOI are successfully created through probabilistic combination, they become part of the IOI pool, thus enlarging the number of ideas (IOI) in the pool for combination. It is important to clarify that model considers combinations at the ideas level rather than combination of components, with the former being fundamental and allowing combination of ideas from different fields using analogical transfer according to the ideas of Weisberg described earlier.

Figure 3. Examples of unit of understanding (UOU) and incremental operating idea (IOI).

Growth in the explanatory reach of the Understanding regime also occurs by the analogical transfer process described by, and also applied to, scientific creativity by Weisberg. The Understanding regime is conceptualized to consist of UOU. The units of understanding (UOU) from different fields within the understanding regime participate to create a new UOU that potentially (probabilistically) has a greater level of explanatory and predictive power. Following the treatment in Axtell et al. (Reference Axtell, Casstevens, Hendrey, Kennedy and Litsch2013), we model the explanatory and predictive power of a field of Understanding as a fitness parameter, $f_{i}$ . If the new UOU has a greater fitness value, it replaces the UOU with the smallest fitness value. Since our primary focus is on performance – the output of the Operations regime, we simulate the Understanding regime only at this higher abstraction level.

Although both regimes – Understanding and Operations – evolve independently, they cannot do so indefinitely. We model the de Solla Price and Gribbin insights by having each regime act as a ‘barrier breaker’ for the other regime. When each regime hits a barrier, the other can eventually aid in breaking the barrier: infusion of understanding enables creation of important IOI in the Operations regime; and infusion of new operational tools enable new discoveries in the Understanding regime.

The performances of the artifacts in technological domains are improved by a series of designs/inventions (IOI) over time. IOI enable designers to change specific components in the domain artifact leading to a potential improvement. Following McNerney et al.’s treatment, the IOI in question is assimilated only if the performance of the artifact improves after accounting for interactions.

Another, and final, factor that we model is scaling, a property inherent in the physics of the design of the artifact.Footnote ⁵ Footnote ⁶ The successfully assimilated IOI, which we refer to as $\text{IOI}_{S}$ , effect improvement of the domain artifact by enabling favorable change of a relevant design parameter. The design parameter is increased or decreased such that it leads to improved performanceFootnote ⁷ . Scaling refers to how change in a design parameter relates to relative change in the performance of an artifact. The formulation we use in the model is that relative performance change is related to design parameters raised to some power, in other words scaled. As covered in Section 2.1, this is the most widely used functional relationship with decent empirical support and theoretical justification in some cases (Barenblatt Reference Barenblatt1996).

3.2 Mathematical summary

A performance (intensive) metric of a domain, labeled $Q_{\!J}$ , is a function of a set of design parameters $(s_{1},s_{2},s_{3})$ of a domain artifact and time, but for simplicity here we consider only a single parameter $(s)$ . The design parameter is changed by $\text{IOI}_{s}$ (successfully assimilated IOI into domain artifacts), which in turn are assimilated from $\text{IOI}_{C}$ (number of accumulated operating ideas in the IOI pool shown in Figure 2). $\text{IOI}_{C}$ is a function of time. Equations describing these nested variables in logarithmic form are:

(3) $$\begin{eqnarray}\displaystyle & \ln Q_{\!J}=f_{1}(\ln s);\quad \ln s=f_{2}(\ln \text{IOI}_{SC}); & \displaystyle \nonumber\\ \displaystyle & \ln \text{IOI}_{SC}=f_{3}(\ln \text{IOI}_{C});\quad \ln \text{IOI}_{C}=f_{4}(t). & \displaystyle\end{eqnarray}$$

Assuming that the functions are continuous and all dependence is through the named variables, the chain rule is applied and yields

(4) $$\begin{eqnarray}\displaystyle d\ln Q_{\!J}/dt & = & \displaystyle d\ln Q_{\!J}/d\ln s\cdot d\ln s/d\ln \text{IOI}_{SC}\nonumber\\ \displaystyle & & \displaystyle \cdot \;d\ln \text{IOI}_{SC}/d\ln \text{IOI}_{C}\cdot d\ln \text{IOI}_{C}/dt.\end{eqnarray}$$

The first term on the right hand side represents relative impact of design variable change on performance change, which will be shown in Section 4.5 to be equal to the scaling parameter $(A_{\!J})$ when $Q_{\!J}$ follows a power law in $s\,:\,d\ln Q_{\!J}/d\ln s=A_{\!J}$ . The second term is the ‘smaller-is-better/larger-is-better’ factor, and captures the notion whether a design variable has to be increased or decreased in order to improve performance. We capture this dependence using an abstraction and equate $d\ln s/d\ln \text{IOI}_{SC}=\pm 1$ . Thus, Eq. (4) becomes

(5) $$\begin{eqnarray}d\ln Q_{\!J}/dt=A_{\!J}\cdot (\pm 1)\cdot d~\ln \text{IOI}_{SC}/d~\ln \text{IOI}_{C}\cdot d~\ln \text{IOI}_{C}/dt.\end{eqnarray}$$

The third term on the right of Eq. (5) represents ‘difficulty of implementing ideas’ in specific domains, and thus relates the domain specific successful $\text{IOI}_{SC}$ to the $\text{IOI}_{C}$ in the pool: we will show in Section 4.4 – following McNerney et al. – that $d\ln \text{IOI}_{SC}/d~\ln \text{IOI}_{C}=1/d_{J}$ , where $d_{J}$ is the interaction parameter introduced by McNerney et al. for technological domain $J$ . Finally, the fourth term represents the rate of idea production. $K=d~\ln \text{IOI}_{C}/dt$ is arrived at by a simulation of combinatorial analogical transfer which is presented in the first (following) section of the results.

Figure 4. Combination of individual operating ideas (a) basic and derived IOI (b) accumulation of IOI through feedback.