Applying complexity science to new product development: Modeling considerations, extensions, and implications

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Contents lists available at ScienceDirect

Journal of Engineering and Technology Management journal homepage: www.elsevier.com/locate/jengtecman

Applying complexity science to new product development: Modeling considerations, extensions, and implications Kyle Oyama a,*, Gerard Learmonth b, Raul Chao c a

Air Force Institute of Technology, 2950 Hobson Way, AFIT/ENV, Wright-Patterson AFB, OH 45433, USA Department of Systems and Information Engineering, University of Virginia, PO Box 400747, Charlottesville, VA 22904-4747, USA c Darden School of Business Administration, University of Virginia, Box 6550, Charlottesville, VA 22906-6550, USA b

A R T I C L E I N F O

A B S T R A C T

JEL classification: C63 D83 O3 O31 O32

We extend the popular NK model of complex landscapes to incorporate two realities of NPD: (1) complementary vs. conflicting dependencies in a project and (2) predominantly incremental design changes to components in evolutionary NPD projects. We show, through stylized projects that the nature of dependencies among system elements moderate the effect of system complexity. Our study highlights that NPD development times may be longer than the original NK model suggests. We offer a modeling framework that can be used to test hypotheses regarding actual systems. Finally, we discuss promising directions for future research. Published by Elsevier B.V.

Keywords: New product development (NPD) Complexity Product architecture NK model Simulation

Introduction The development of new products is a source of competitive advantage for firms. New product development (NPD), however, is often viewed as a ‘‘messy and complex’’ process involving many engineers and managers responsible for designing components that interact to perform a desired set

* Corresponding author. Tel.: +1 937 260 7296. E-mail addresses: kyle.oyama@afit.edu, [email protected] (K. Oyama), [email protected] (G. Learmonth), [email protected] (R. Chao). http://dx.doi.org/10.1016/j.jengtecman.2014.07.003 0923-4748/Published by Elsevier B.V.

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of functions. Such complex systems are comprised of ‘‘a large number of parts that interact in nonsimple ways’’ (Simon, 1969, p. 195). In Simon’s definition, ‘‘non-simple’’ implies that interactions among elements of the system are uncertain and, as a result, often produce surprising, unanticipated outcomes at the system level. We, therefore, do not view NPD as a linear set of activities (which many models and tools used in NPD and project management often assume, e.g. Gantt and PERT charts). Rather, we see NPD as a complex adaptive system (McCarthy et al., 2006) in which macro scale properties of a system/process cannot be inferred from properties of its constituent parts and rules of their interaction. As an example of complexity in NPD, consider the development of a new aircraft. Such a project typically involves hundreds of engineers and managers working in teams. The efforts of these varied teams (e.g. fuselage, flight controls, wing assembly, etc.) can be conceptualized as a complex system in that decisions made by an agent (an individual or a team) not only affect their own component’s design but often influence the relative performance of other components due to interactions and dependencies between and among components. Thus, the development efforts of each component are both variable and interdependent. This coexistence of variation and dependency is a hallmark of complex systems across many domains (Shalizi, 2006). Early research on NPD attempted to characterize product development and management as a logical and ordered process (Zaltman et al., 1973; Cooper, 1990) that took inputs (requirements) and used resources (engineers and managers) to produce output(s) in the form of new products (Clark and Wheelwright, 1993). However, researchers and practitioners have concluded that NPD is typically more complex. Thus, NPD efforts instead follow a less deterministic path consisting of rework, restarts, iterations and changes (Leonard-Barton, 1988; Cheng and Van De Ven, 1996; Smith and Eppinger, 1997; McCarthy et al., 2006) as agents, teams, and decisions interact. In order to better understand the dynamics of NPD projects and the organizations that engage in NPD, recent research has looked to the relatively nascent field of complexity science for insight and understanding (e.g. Anderson, 1999; McCarthy et al., 2006; Baumann and Siggelkow, 2013; Akgu¨n et al., 2014). In this paper, we investigate the phenomenon of complex NPD project dynamics using the framework of the popular NK model. To aid our understanding, we incorporate and explicitly model two important contextual realities of the NPD process to specifically account for (1) varying degrees of complementary and conflicting dependencies within NPD projects and (2) predominantly incremental component level design changes in NPD projects. This study produces three insights. First, we find our extended NK model and simulation results suggest the nature of dependencies between system elements can moderate the effect of system complexity: when a system has a low degree of complementary dependencies, system performance is relatively unaffected by complexity, but when that same system contains a moderate to high degree of complementary dependencies, system performance increases with increased complexity. This insight is counterintuitive in that it is widely believed that more dependencies in a system have a universally deleterious effect. Second, our study highlights that NPD development times may be longer than the original NK model suggests. When we model component changes using a skew triangular distribution, as opposed to the uniform distribution specified in the original NK model, we find development times are longer and system improvements are more incremental. This is an important insight because the NK model’s popularity and widespread use in the field of management science may lead some to draw conclusions regarding NPD applications when, in fact, conclusions could be additionally informed by this research which highlights a key difference regarding development times. Finally, the extensions to the NK model in the present study uncover the tension between development time and product quality that is inherent in NPD, whereas previous studies using the NK model have reported little regarding the relationship development time and complexity. Specifically, we find the level of complementary dependencies in a system not only moderates the effect of complexity on system performance, but also moderates how complexity impacts the trade-off between system development time and system performance: when complementarities are few, as complexity increases, system performance declines but system development time is reduced; however, when complementarities are many, both system performance and system development time increase as a function of increased complexity. Please cite this article in press as: Oyama, K., et al., Applying complexity science to new product development: Modeling considerations, extensions, and implications. J. Eng. Technol. Manage. (2014), http://dx.doi.org/10.1016/j.jengtecman.2014.07.003

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The article is structured as follows. Section ‘Theory’ reviews the relevant literature regarding complex NPD phenomena, as well as the application of the NK model to organizational decision making and NPD. Section ‘Model and methods’ briefly reviews the NK model, including its mathematical foundations, and describes our extended NK model and methodology. Section ‘Results’ presents the results of our simulation experiments, both on stylized NPD projects and on a real-world application from the NPD literature. Section ‘Discussion’ discusses our findings and broader implications for NPD theory and practice as well as potential directions for future research. Theory In the domain of complex NPD projects, previous research efforts have significantly aided our understanding of complex NPD phenomena. For instance, we conceptually understand that interdependent decisions lead to subsystem design instability (Alexander, 1964; Thomke, 1997) and long system convergence times (Terwiesch and Loch, 1999). In addition to schedule delays, complexity in NPD reveals itself in the form of unmet specifications and budget overruns (e.g. Morris and Hugh, 1987; Tatikonda and Rosenthal, 2000). We also have a conceptual understanding that interdependencies within a system give rise to complexity in the design process and, thus, a heuristic has been proposed which states that it dependencies between components should be kept within subsystems to the extent possible (Maier and Rechtin, 2002). To better understand how complexity is related to organizational decision making such as that found in NPD, many researchers have employed a stochastic combinatorial model, known as the NK model (Kauffman, 1993). The NK model is a mathematical model of ‘‘tunably rugged’’ fitness1 landscapes in which the parameter N describes the number of components in a system, and the parameter K is a proxy for complexity and describes the number of other components that affect the performance of each component. Using these two tuning parameters, the NK model generates landscapes of varying complexity which agents search, attempting to find the best configuration of decisions. Management and strategy researchers have employed the NK model in various ways to explore questions regarding organizational design (Levinthal and Warglien, 1999; Rivkin and Siggelkow, 2003); interdependent decisions (Gavetti and Levinthal, 2000); industry dynamics (Lenox et al., 2007); product portfolio selection (Chao and Kavadias, 2008); and new product development (Mihm et al., 2003; Sommer and Loch, 2004). As an indication of the NK model’s popularity, Ganco and Hoetker’s review (2009) of the NK model and its use in the management literature identifies 30 papers published in leading management and strategy journals. A study illustrating how NK models can be used to model fitness of different combinations of a manufacturing strategy is found in McCarthy (2004). Nevertheless, previous studies have focused almost exclusively on how the search behavior of agents (Sommer and Loch, 2004; Frenken, 2006; Lenox et al., 2007; Chao and Kavadias, 2008; Baumann and Siggelkow, 2012) and the structure of dependencies (Rivkin and Siggelkow, 2007; Ghemawat and Levinthal, 2008) impacts performance of the organization. Search behaviors and the structure of dependencies have shed substantial light on the dynamics of organizational decision making such as that found in the context of NPD. However, in the present study we are also interested in how two contextual realities inherent in the NPD process shape the search landscape and affect NPD outcomes. Namely, we are interested in the likelihood of incremental vs. abrupt changes to components; and the degree of complementary vs. conflicting dependencies within the NPD project. Therefore, in the present study we operationalize each of these contextual realities in an extended NK model in order to develop a refined model of the relationship between complexity and NPD project outcomes of quality and development time.

1 The term ‘‘fitness’’ is a metric of relative performance and, in the NK model, can take values on the unit interval (0, 1). To avoid confusion, it is important to note that, in the NK model, there are component fitness values as well as an overall system fitness value.

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Linking the NK model to NPD It should be noted that Kauffman’s original NK model was developed in the study of evolutionary biology to investigate the role of gene interactions and mutations in the evolution of genomes. Typically, we think of evolution in a biological sense, but the development of new products is also evolutionary, in that firms search, through trial and error, for solutions to a complex search problem defined by interactions among components. For instance, in an attempt to find competitive advantage, firms constantly engage in efforts to develop new and better product offerings. The NPD literature reveals that firms undertake many types of activities in search for competitive advantage including R&D spending (Chen and Miller, 2007), capital investment (Greve, 2003a), and the search for innovation (Greve, 2003b). Central to these activities is new product development (NPD), which is essentially a search for winning products in the marketplace. Indeed, previous research efforts have used the NK model to investigate interdependent decision making within organizations (see Ganco and Hoetker, 2009 for an excellent survey of these efforts) and McCarthy et al. (2006) suggest the NK model could be used to further investigate NPD projects as complex adaptive systems. We argue, however, the NPD literature indicates that there are at least two important aspects of NPD that are not accounted for in Kauffman’s original NK model: (1) complementary and conflicting dependencies2 exist in varying proportions in different NPD projects and (2) changes to components are subject to uncertain, but not completely random, results. The nature of dependencies in NPD The first aspect of the NK model that we modify in our investigation is the nature of dependencies between components. Previous NPD and innovation studies using the NK model have found that the structure of dependencies within a system/product affect the outcome of the NPD search process (e.g. Ethiraj and Levinthal, 2004; Rivkin and Siggelkow, 2007; Ghemawat and Levinthal, 2008). However, in these previous efforts, the dependencies between components within a system are modeled simply as binary. No explicit treatment is given to the nature of the dependencies between components. Thus, it is not known how sensitive the NPD search is to different distributions of complementary and conflicting dependencies. The notion of complementary dependencies has been examined in the NPD literature from a product architecture standpoint (Maier and Rechtin, 2002) as well as from a product strategy perspective (Milgrom and Roberts, 1990; Camuffo et al., 2008), with general consensus that complementarities within and between products are desirable. However, the NPD research which has used the NK model as a framework does not explicitly model complementary or conflicting dependencies. Furthermore, while it is generally agreed that complementary dependencies are desired, it is not known how the mixture of complementary and conflicting dependencies in an NPD project influence outcomes such as performance or development time. Therefore, in this paper, we introduce a modeling construct that allows the NK model to more explicitly represent the nature of dependencies in NPD projects so that we can investigate how different mixtures of dependency types (conflicting vs. complementary) impact NPD outcomes. Component-level changes in NPD We further propose that adopting the NK model in our investigation of NPD dynamics should account for how design decisions regarding components are made, and how the outcomes of such changes are modeled. In previous NPD related research that employed Kaufmann’s original NK model, changes to components are modeled as random draws from a uniform distribution (e.g. Ethiraj and Levinthal, 2004; Sommer and Loch, 2004; Lenox et al., 2007). This modeling construct suggests that a

2 Complementary dependencies describe a relationship between two components of a system in which an improvement to component A leads to an improvement in component B (but is not necessarily a bi-directional relationship), whereas a conflicting dependency has the opposite effect and can be thought of as a tradeoff. The use of the term complementary is adapted from the economic literature used to describe complementarities within economic systems (Milgrom and Roberts, 1990).

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change to a component can result in incremental or radically different outcomes, each with equal likelihood. However, technological changes are generally characterized as incremental because firms apply similar techniques related to a new technology or related technological application (Dosi et al., 1988; Nelson and Winter, 1982). There is also a large body of literature that argues search is boundedly rational (Simon, 1969) and ‘‘local’’ in nature, rather than exhaustive (Greve, 1998). In NPD terms, this implies firms search for new products that are similar to those they already produce because they already possess the necessary resources, costs are generally lower for incremental improvements, there are fewer technical risks, and there is willingness to satisfice (Cyert and March, 1963; Simon, 1979). This search process is equivalent to an adaptive walk (Kauffman and Levin, 1987). Additionally, firms tend to pursue incremental design changes and upgrades, because it has been shown that a set of cumulative incremental changes generate very large price performance gains over the long term (Usher, 1954; Sahal, 1981). Therefore, in this paper we model component-level technological changes as more likely to result in improvements that are incremental. By employing this modeling construct, we account for the fact that NPD is a complex adaptive system that is goal-directed; in other words, changes made to system components in the NPD process are calculated decisions that account for the risk of an undesirable outcome (lower fitness), but have a relatively good chance of success (incremental fitness improvement). This is in contrast to the NK model’s treatment of component level changes in which component fitness may radically change (up or down) due to new fitness values being modeled as random draws from a uniform distribution. In summary, it is understood that structure of interdependencies between components and subsystems in NPD projects and the search behavior of agents/design teams are important drivers of complexity and the resulting phenomenon of iterations, instability and long development times. It is also recognized that the NK model is a framework for modeling the evolutionary search of a landscape that is characterized by interdependencies. We posit that NPD is such an evolutionary search process that is also governed and characterized by interdependencies. However, there are at least two salient gaps in applying the NK model to NPD projects.3 Therefore, to more fully link the NK model to NPD, we build upon the original NK model by altering the way in which interdependencies are represented in the model, as well as how design changes are modeled. The modeling constructs we use to extend the original NK model are described in the next section. Model and methods In this paper, our focus is developing and analyzing an extended NK model that helps us better understand NPD dynamics by incorporating two important aspects of NPD management: (1) varying degrees of complementary vs. conflicting dependencies within an NPD project and (2) incremental changes to components rather than uniformly random changes. However, we first provide the reader with a brief overview of Kauffman’s NK model. We describe how we implemented and validated the NK model of complex landscapes. We then discuss the two modeling constructs that help us adapt the NK model for our objective of investigating NPD as a complex system of interdependent design decisions. Finally, we discuss how we used our extended NK model to investigate complex NPD dynamics. NK model overview In The Origins of Order (1993), Kauffman outlines the NK model which has two primary features. The first feature is a stochastically generated fitness landscape, where ‘‘higher peaks’’ correspond to better solutions or combinations of elements. The second feature is the agent(s) that search a given landscape in an effort to improve their ‘‘fitness’’ (performance). 3 There are other fitness models that address interdependencies (e.g. McCarthy, 2004 and Hordijk and Kauffman, 2005). We limit our discussion to the NK model here.

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The first feature of the NK model—a fitness landscape—is stochastically generated using two parameters, N and K. The parameter N represents the number of individual components in a given system and is characterized as a string of N binary digits.4 The value of each digit in the string describes a specific variant of each component. Thus, for a system comprised of N components, the number of possible configurations of the system is 2N. The second parameter K defines the degree of dependence between the components of a system. Specifically, K represents the number of other components in the system that affect the performance or fitness value of each component. The value of K can range from 0 to N  1 and each component can assume one of 2K+1 fitness values. In Kauffman’s model, fitness values are uniformly distributed random variates on the interval (0, 1). The overall fitness value, F, of the system is the arithmetic mean of the values assigned to each of the N components: F¼

N 1X f N c¼1 c

(1)

If K = 0, then all components in the system are independent and, as K increases, the components become more dependent on one another. Higher values of K result in more local peaks in the landscape and performance differences are more pronounced when a change is made to any one of the N components (Kauffman, 1993; Levinthal, 1997). Therefore, K serves as a measure of system complexity because, as K increases, the web of dependencies thickens and the ‘‘ruggedness’’ of the landscape increases. The second salient feature of the NK model involves an agent(s) searching on fitness landscapes such as those described above. An agent searching a given fitness landscape does so using trial-anderror, in which one component is mutated at random. If, after the mutation, the agent finds the new system fitness value is greater than its current fitness value, the agent ‘‘moves’’ to the new configuration (position on the landscape). If the mutation does not yield a higher system fitness value, the agent retains its previous configuration and mutates a different randomly selected component. Search stops when no greater fitness values are available by mutating one component in the system. Simon notes that this type of local trial-and-error search is congruent with how biological evolution and selection occurs through mutation (Simon, 1969). Kauffman’s original intent in developing the NK model was to build a model of complexity that could be tuned with a single parameter. The parameter K, the degree of interaction between components, is the parameter used to adjust the ‘‘shape’’ or ‘‘topography’’ of the landscape. In the extreme case, where K = 0, each component is independent of all other components in the system and, thus, there is one globally optimal configuration of components. However, the search becomes nontrivial for values of K > 1, as searches will often terminate at a local optimum due to changes in one bit causing changes in K other bits which frustrate the search. Perhaps most significantly, Kauffman found that the highest mean fitness values for systems of size N > 8 tended to exist in systems where K was approximately equal to 3, even as N increased to larger and larger sizes. This implies that there are advantages at a moderately low level of complexity. In other words, complete system decomposition was not found to be optimal, and complete integration of system components (K = N  1) leads to a ‘‘complexity catastrophe’’. A complexity catastrophe is the effect of higher complexity (K) leading to increasingly rugged landscapes with local peaks that proliferate in number and are less differentiated from the overall landscape (Kauffman, 1993). For clarity within this article, Kauffman’s NK model described in this section is hereafter referred to as ‘‘the original NK model’’. Implementing the original NK model Our approach begins by first coding Kauffman’s original NK model of fitness landscapes in the R statistical computing language (R Development Core Team, 2013). To generate the fitness landscapes, 4 Each digit can theoretically assume A possible ‘‘alleles’’, but Kauffman restricts his analysis, without loss of generality, to systems in which A = 2, or systems that can be described as a binary string of N digits.

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we follow the details and pseudo-code given by Altenberg (1997). To obtain a non-trivial sized fitness landscape we selected a system size of N = 12 which yields 212 possible system configurations. In an NPD context, each configuration represents a particular set of components that comprise the overall product design. For example, automobile designers may have the option to use a gasoline or diesel engine. Thus, in a given configuration, this choice could be represented either as 0 (gasoline) or 1 (diesel). Again, without loss of generality, we model the case in which each component within a product has two possible options. Following Kauffman’s original model specification, interactions within the system are defined by a local (adjacent neighbor) pattern of interaction. For example, if K = 2, each component in the system is affected by its two nearest neighbors. In other words, each component is dependent upon K other components in determining its fitness. To further draw the connection between the NK model and NPD, the interactions within the system can be represented in the form of a Design Structure Matrix (DSM) (Steward, 1981; Baldwin and Clark, 2000). In the NPD literature, the DSM is a tool used to depict design dependencies within a system. More specifically, the DSM is a square matrix (N  N) that has an entry in the ith row, jth column if component j has an influence on the performance of component i. In this sense, the dependency exists between the ith component and the jth component. For example, the battery life of a laptop (element i) is dependent on processor speed (element j), and potentially other design elements. The local pattern of interaction used by Kauffman in his original model represents a specific type of interaction structure, but in practice a particular interaction pattern (either existing or proposed) could be used to study how changes to a given product architecture could influence system-level outcomes. In this study, agents search for superior configurations via adaptive local search. Additional search strategies can certainly be modeled in which agents search less locally by expanding the number of components they change at a given time (Gavetti and Levinthal, 2000; Rivkin and Siggelkow, 2007), or by dynamically adapting their search behavior (Battiti et al., 2008). However, in this exploratory study, we maintain the local search strategy which has been empirically shown to closely represent human problem solving behavior (Billinger et al., 2010). Extended NK model constructs Complementary and conflicting dependencies As discussed in section ‘Theory’, one assumption of the original NK model (Kauffman, 1993) that does not align well with NPD reality is the treatment of complementary and conflicting dependencies between components. Dependencies describe a relationship between a pair of components. There are two types of dependencies that we identify and address in this research— complementary and conflicting dependencies. Specifically, if a complementary dependency exists between two components, then the performance of component 1 depends on the performance of component 2. In other words, if the fitness (performance) of component 2 increases, then the fitness of component 1 also improves. Note that dependencies are one-way, in that the relationship is not necessarily bi-lateral (e.g. increased fitness in component 2 that leads to increased fitness in component 1 does not imply a reciprocal relationship). Conflicting dependencies between components are the opposite of complementary dependencies. Components can be dependent upon multiple other components, and it is these multiple dependencies for a given component that give rise to the complexity of a landscape. In the original NK model, as changes are made to a selected component, referred to as the focal component,5 the effect on other, dependent components is completely random. In NPD, however, there is generally some knowledge regarding how changes to one component will impact other components; however, some dependencies are not known to exist and can be the source of slowed search or inferior performance (Ethiraj and Levinthal, 2004). In either case, there are dependencies within a NPD project, some of which are known and other that must be discovered. Therefore, we 5

As opposed to the dependent components which are affected by the change to the focal component.

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implement the following function to assign values to the dependent components as a function of the focal component change6: 8 ðx0  xÞ > > ð1  yÞ for complememntary ðpositiveÞ dependency; x0 > x yþ > > ð1  xÞ > < ðx0  xÞ (2) y0 ¼ > ðyÞ for conflicting ðnegativeÞ dependency; x0 > x yþ > > ð1  xÞ > > : y for conflicting or conflicting dependency; x0  x In Eq. (2) y’ is the new value of the affected component, y is the previous value of the affected component, x’ is the new value of the focal component and x is the previous value of the focal component. This method of assigning fitness values to the dependent components specifies that changes in the dependent components are proportional to the change in the focal component, and follows the notion of structured dependence between components introduced by Solow et al. (1999). Proportionality, in their model, is defined based on the ratio of the actual change to the maximum possible change toward either extreme (zero or one). For example, if the focal component improves from a fitness value of 0.4– 0.7, it has moved 50% of the distance toward the maximum value of 1 and, therefore, the fitness of all dependent components will also move half the distance toward the maximum value of 1. Negative changes are handled in a similar fashion. Using this method of assigning fitness values to dependent components, Solow et al. (1999) showed that increases in the value of K do not lead to a complexity catastrophe. A complexity catastrophe is the effect of higher complexity (K) leading to increasingly rugged landscapes with local peaks that proliferate in number and are less differentiated from the overall landscape (Kauffman, 1993). However, Solow et al. (1999) only examined uniformly positive or negative influences; that is, if the focal component improves, all dependent components also improve. Likewise, if the focal component declines in fitness, each dependent component also declines in fitness value. In our research, we model the reality found in NPD that some dependencies are complementary while others impose a conflicting constraint which is what often leads to rework and iterations in the NPD process. In our model, we maintain the concept of proportional changes in the dependent components so that we can compare our results to those of previous researchers (Solow et al., 1999). First, we examine two extreme cases of component dependencies, one in which all changes to dependent components are complementary (AllComp) and one in which examine what happens when all improvements to a focal component result in changes to dependent components that are inversely proportional or conflictling (NoComp). We then extend the NK model by recognizing that interactions between component pairs in NPD projects are neither completely unknown (as in the original NK model), nor are the interactions all mutually reinforcing (Solow et al., 1999) or all characterized as design conflicts. Rather, in an NPD project some dependencies between components are complementary, while others are characterized by conflicts or tradeoffs. Therefore, we explicitly model different mixtures of complementary and conflicting interactions. In this study we use three treatment levels: Low Complementarity (LC), Moderate Complementarity (MC), and High Complementarity (HC) in which 25%, 50% and 75%, respectively, of the dependencies between components are complementary. Assignment of component-level fitness values As discussed, another assumption of the original NK model that does not align well with evolutionary NPD processes is that design changes to a focal component result in a new fitness value drawn from the uniform distribution on the unit interval. This assumption implies the outcome of a design change is completely random, whereas design changes are typically incremental. In the NPD context, changes to components (often referred to as engineering change orders) do not always result in an improved design; however, if the probability of improvement were not at least 50%, few 6 It should be observed from Eq. (2) that when the focal component, x, decreases in fitness, the dependent components retain their previous values because we assume that unsuccessful local changes to components are deliberately not implemented in the system (or product). We make this assumption because a characteristic of complex systems is that individual elements in the system are ignorant of the behavior of larger system and only respond to what is available locally (Cilliers, 1998).

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managers would be inclined to approve the change. We model incremental changes, along with the possibility that some changes will not result in an improvement, by introducing the following method of updating the fitness value, x, of focal the component: 1. Assign a probability, p, that the modified component will result in an improved design. 2. Model the updated fitness value of the focal component, x0 , using a triangular distribution with parameters: min, max, mode, p, with the mode equal to the current value of the focal component, x, and the max is set to half the distance from the current fitness value to the theoretical maximum of 1. 3. From the mode and max and, the value of p, we can compute the min parameter as min = (mode  max * (1  p))/p. By using the triangular distribution with the parameters just described for assigning fitness values to the changed focal component, we capture the fact that design changes made during the NPD process are subject to uncertainty, but are also more likely to be incremental, rather than uniformly random as implied by the original NK model. For this study, we nominally set the probability of success, p, equal to 0.75. In terms of the triangular distribution, this means that 75% of the triangle’s area lies to the ‘‘right’’ of the mode. (For simplicity, p is assumed to be time stationary and apply globally to each component in the system). In the interest of readability, a more detailed mathematical example of how our extended model works is not undertaken here, but is provided at Appendix A. Model investigations In this study we are interested in how our extensions to the NK model help us understand complex NPD projects. To this end, we collect and investigate two measures of performance that are important in an NPD context: (1) the maximum overall system fitness F achieved and (2) the number of candidate changes considered before stopping as a result of reaching a local optimum at which no further one-component changes can improve the system fitness. These two metrics directly relate to NPD measures of product quality and development time7. To control for variance due to differences in initial conditions (a hallmark of complex system behavior), each experimental run is conducted with the same set of initial starting configurations. Table 1 highlights the specific modifications and parameter settings for our modeling approach and simulation experiments. Results Having described the theoretical foundations and methodological framework for extending the original NK model to the NPD domain, we now describe the results of our simulations. In our study we are primarily interested in two measures of NPD performance: product quality (computed as the mean of component fitness values) and development time (measured in terms of the number of simulation steps, where each step represents a consideration of a component design change). Product quality is important for obvious reasons, while the development time metric is important because managers of NPD teams and projects have finite development time horizons and faster development times can lead to ‘‘first-mover advantage’’ (Lieberman and Montgomery, 1998). In each of our experiments we initialize our simulation (t = 0) by randomly defining a product configuration, v = (x1, x2, . . ., xN), comprised of 12 components (N = 12). During each time period (t = 1, 2, 3, . . .), a change to one of the N components is considered, resulting in a new product 7 Additionally, cost is acknowledged to be an important measure in the NPD context, but it is not explicitly modeled in this study as costs are very specific to each industry and individual project. However, in practice, costs associated with components and change orders (if available) could be modeled via probability distributions, similar to how fitness values are modeled in this study, either in scaled or absolute terms using appropriate probability distributions.

Please cite this article in press as: Oyama, K., et al., Applying complexity science to new product development: Modeling considerations, extensions, and implications. J. Eng. Technol. Manage. (2014), http://dx.doi.org/10.1016/j.jengtecman.2014.07.003

Focal component modeling via triangular distribution

B. AllComp NK

C. NoComp NK

D. Low complementarity

E. Moderate complementarity

F. High complementarity

G. Low complementarity

H. Moderate complementarity

I. High complementarity

Focal component fitness Dependent component fitness

Uniform (0,1)

Uniform (0,1)

Uniform (0,1)

Uniform (0,1)

Uniform (0,1)

Uniform (0,1)

Triangular

Triangular

Triangular

Uniform (0,1)

Proportional to focal

Inversely proportional to focal

Complementarities

N/A

100%

0%

Mixture of proportional/ inverse proportional 25%

Mixture of proportional/ inverse proportional 50%

Mixture of proportional/ inverse proportional 75%

Mixture of proportional/ inverse proportional 25%

Mixture of proportional/ inverse proportional 50%

Mixture of proportional/ inverse proportional 75%

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Mix complementary and conflicting constraints

A. Original NK model

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Table 1 Comparison of models and parameters used in this study. Changes from the original NK model are highlighted as the model is incrementally modified (from left to right).

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configuration, v0 . If F(v0 ) > F(v), then the new product configuration is adopted. This process of change, selection, and adoption is repeated in each time period until no further improvement to the product can be made by changing a single component. In generating the fitness values for each component, we do not generate the entire fitness landscape a priori. That is, we do not assign fitness values for each possible combination of components, as this would require 2k+1  N numbers. Instead, we dynamically generate the fitness values for each component as they are encountered in the search as suggested by Altenberg (1997). However, after a fitness value for a particular combination of components has been generated it is stored in memory such that if the same combination of components is revisited during the search, it will have the same value as previously encountered. By dynamically generating the landscapes, we significantly improve our ability to simulate larger systems because the landscape size (possible stagrows exponentially in N). Baseline results—original NK model Table 2 and Fig. 1 illustrate how system performance varies with complexity (K) under the original NK model (Kauffman, 1993). Those familiar with Kauffman’s NK model will recognize the inverted ‘‘Ushaped’’ curve, indicating the average maximum system fitness is greatest when the value of K is moderately low (K = 2, in this case). This is well-known result in the literature and it provides a baseline for the remainder of our results. Table 3 and Fig. 2 present a result of the original NK model not discussed in the extant literature concerning the time (simulation steps) required to reach a local or global optimum. As Fig. 2 shows, a general pattern of shorter development time is associated with increasing complexity in the original NK model. We note, here, that this result does not align with intuition regarding Table 2 Comparison of results obtained for average maximum fitness obtained and associated standard deviation of the found local maxima. Each result in the table above is an average over 2000 iterations. To control for variance due to differing starting configurations between simulations, the same set of starting configurations was used for each model. Peak system fitness A. Original NK

K=0 K=1 K=2 K=3 K=4 K=5

B. All complementarities

C. No complementarities

Mean

St Dev

Mean

St Dev

Mean

St Dev

0.6645 0.6949 0.7028 0.7026 0.6994 0.6946

0.06996 0.05776 0.05427 0.0517 0.05084 0.04852

0.6645 0.7406 0.7913 0.828 0.8535 0.8761

0.06996 0.06905 0.06976 0.06996 0.0668 0.06466

0.6645 0.6085 0.522 0.5054 0.4991 0.4981

0.06996 0.06307 0.06979 0.07837 0.08463 0.08591

Table 3 Comparison of results obtained for average number of design changes considered (simulation steps) and associated standard deviation for the number of design changes. Each result in the table above is an average over 2000 iterations. To control for variance due to differing starting configurations between simulations, the same set of starting configurations was used for each model. Steps to solution A. Original NK

K=0 K=1 K=2 K=3 K=4 K=5

B. All complementarities

C. No complementarities

Mean

St Dev

Mean

St Dev

Mean

St Dev

51.42 46.47 41.52 37.2 34.3 31.89

12.918 14.654 14.623 13.497 12.33 11.874

51.42 48.41 43.22 39.33 35.95 33.98

12.918 13.282 12.088 11.875 10.873 10.522

51.42 47.97 24.47 16.38 13.71 13.17

12.918 18.663 15.274 8.959 4.025 1.824

Please cite this article in press as: Oyama, K., et al., Applying complexity science to new product development: Modeling considerations, extensions, and implications. J. Eng. Technol. Manage. (2014), http://dx.doi.org/10.1016/j.jengtecman.2014.07.003

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Fig. 1. Mean maximum fitness for original NK model, All Complementarities Model and No Complementarities Model. Each point is an average over 2000 simulation iterations using the same set of initial starting configurations.

development times in NPD: more complex systems typically take longer to develop. However, when we consider the search strategy described by an adaptive walk (local search), the result, though not necessarily realistic in an NPD context, makes sense because higher complexity in the NK model generates more ‘‘rugged’’ landscapes which are characterized by more local maxima, leading to shorter ‘‘walks’’. Extreme results—all complementary and no complementary dependencies We now examine the results of two extreme extensions of the NK model, namely the models in which all changes to dependent components are either positively or negatively proportional to changes in the focal component. Table 2 and Fig. 1 illustrate the effect of these extreme extensions on system performance (fitness). In the AllComp model in which all dependencies are complementary (positively correlated with the focal component), we observe that the performance of the system increases, in a logarithmic fashion, Please cite this article in press as: Oyama, K., et al., Applying complexity science to new product development: Modeling considerations, extensions, and implications. J. Eng. Technol. Manage. (2014), http://dx.doi.org/10.1016/j.jengtecman.2014.07.003

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Fig. 2. Development time (steps) for original NK model, All Complementarities Model and No Complementarities Model. Each point is an average over 2000 simulation iterations using the same set of initial starting configurations.

with respect to complexity (K) (Fig. 1). However, in the NoComp model where all dependencies are conflicting (negatively correlated with improvements to the focal component), each interaction is a conflicting constraint and we observe that fitness values decline as complexity increases (Fig. 1). In fact, as K increases, the maximum fitness values regress to the mean. In terms of system development time, we again observe there is a pattern of shorter development times as complexity increases (Fig. 2). In the NoComp model (all conflicting dependencies and no complementary dependencies) each conflicting dependency places an additional constraint on opportunities for system-wide fitness improvement and, therefore, the search for better product configurations terminates faster, much as in the original NK model, but at an even faster rate because system-wide improvements are highly unlikely. On the other hand, in the AllComp model (all complementary dependencies) development time decreases as a function of K because, but for a different reason: when all dependencies are complementary, the average improvement at each step is greatly increased, resulting in a more rapid convergence to a final product configuration solution. Please cite this article in press as: Oyama, K., et al., Applying complexity science to new product development: Modeling considerations, extensions, and implications. J. Eng. Technol. Manage. (2014), http://dx.doi.org/10.1016/j.jengtecman.2014.07.003

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NPD extensions Having briefly examined the original NK model, and two extreme cases of complementary and conflicting dependencies, we now examine the results of two extensions to the original NK model grounded in the contextual realities of NPD in an attempt to gain insight to NPD dynamics. Mixture of complementary and conflicting dependencies The results from our experiments using the original NK model and the models which have positively or negatively correlated dependencies are helpful in understanding that complementarities (and a lack thereof) have a definite influence on the fitness of a system as it evolves. However, in the context of NPD, dependencies between components of a system are almost always a mixture of complementarities and conflicts. Thus, in this research, we are primarily concerned with what happens when mixtures of complementary and conflicting dependencies are present in an NPD project. Table 4 and Fig. 3 (dashed lines) show how system fitness is influenced by the presence of both complementary and conflicting dependencies. When complementary (positively correlated) dependencies are low (25% in this study), system fitness declines as K increases, but at a slower rate than when there are no complementarities (see Fig. 3, Panel A, dashed line). It is interesting to note that when complementary dependencies are low, system fitness (for all values of K > 0) is less than when K = 0. This was an unexpected result, for it was believed that even a low number of complementary dependencies would be better than a system in which all components are independent. However, it is not until complementary dependencies account for at least 50% of the total number of dependencies that we see a benefit to additional complexity. Namely, when we increased the level of complementary dependencies to a moderate level (50%), we found that the number of complementary dependencies was sufficient to yield an increase in system fitness when compared to the case where K = 0 (see Fig. 3, Panel B, dashed line). However, as K increases, the degree to which system fitness improves effectively ‘‘plateaus’’. This occurs because, as K increases, the

Table 4 Comparison of results obtained for average maximum fitness by varying the 2 parameters of interest in this study—the mix of complementary and conflicting dependencies; and the method of updating the focal component fitness value. Also reported are the standard deviations for each result. Each result is an average over 2000 iterations (20 iterations on each of 100 different landscapes.). Uniform Mean Panel 1. Low complementarity K=0 0.6533 K=1 0.6299 K=2 0.6058 K=3 0.5867 K=4 0.5652 K=5 0.5547 Panel 2. Moderate complementarity K=0 0.6533 K=1 0.6673 K=2 0.6795 K=3 0.6853 K=4 0.6829 K=5 0.6855 Panel 3. High complementarity K=0 0.6533 K=1 0.6998 K=2 0.74 K=3 0.7615 K=4 0.7812 K=5 0.7913

Triangular

t-test of means

St Dev

Mean

St Dev

0.07171 0.06864 0.06326 0.06335 0.06095 0.06012

0.5453 0.5477 0.5459 0.5424 0.5329 0.5282

0.0776 0.07143 0.06654 0.06183 0.06169 0.06089

p = 2.2e16 p = 2.2e16 p = 2.2e16 p = 2.2e16 p = 2.2e16 p = 2.2e16

0.07171 0.07359 0.06753 0.06628 0.06839 0.07292

0.5453 0.5729 0.6034 0.6273 0.6413 0.6543

0.0776 0.07166 0.06546 0.05595 0.05125 0.04805

p = 2.2e16 p = 2.2e16 p = 2.2e16 p = 2.2e16 p = 2.2e16 p = 2.2e16

0.07171 0.0737 0.0706 0.06866 0.07113 0.07274

0.5453 0.6007 0.6647 0.7263 0.7764 0.8082

0.0776 0.07232 0.06579 0.05928 0.05108 0.04473

p = 2.2e16 p = 2.2e16 p = 2.2e16 p = 2.2e16 0.01556 p = 2.2e16

Please cite this article in press as: Oyama, K., et al., Applying complexity science to new product development: Modeling considerations, extensions, and implications. J. Eng. Technol. Manage. (2014), http://dx.doi.org/10.1016/j.jengtecman.2014.07.003

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Fig. 3. Mean maximum fitness for varying degrees of complementary dependencies. Panel A shows results for low complementarities (25% of dependencies). Panel B shows results for moderate complementarities (50% of dependencies). Panel C shows results for high complementarities (75% of dependencies). In each panel, the results from two different methods of assigning new fitness values to the focal component are also shown—the dashed line represents new fitness values drawn from the uniform distribution as in the original NK model; the solid line represents new fitness values drawn from the triangular distribution described and proposed in this paper. Each point represents the average of 2000 simulation runs generated from using 100 unique landscapes and a set of 20 starting configurations on each landscape.

expected number of components that will benefit from an improvement in one component becomes nearly equal to the expected number that will experience a decline in fitness thus creating an upper limit on the average maximum system fitness. Finally, when we model a high level of complementary dependencies (75%) in the system (Fig. 3, Panel C, dashed line), we observe that system fitness increases quite rapidly with complexity. This result follows intuition that when complementary dependencies outnumber conflicting dependencies, system fitness increases with the number of total dependencies because complementarities can be exploited to a greater extent as K increases. With regard to development time for an NPD project, when we combine complementary and conflicting dependencies into the same model in varying percentages (models D, E, and F in Table 1) we obtain results that, again, share the same general trend—decreasing development time as a function of K. The most notable decrease, as function of K, is found in the model with low complementarities. This more pronounced decrease, in the low complementarity model, results from the fact that when few complementary dependencies exist, an increased number of total dependencies creates a situation in which improvements to a given component is almost always outweighed by the conflicts with other components, which leads to the search becoming frustrated very quickly. Fig. 4 (dashed lines) and Table 5 illustrate this phenomenon. Focal component fitness changes Modeling changes to the focal component using the triangular distribution, as discussed in section ‘Assignment of Component-Level Fitness Values’ allows us to account for the fact that design changes Please cite this article in press as: Oyama, K., et al., Applying complexity science to new product development: Modeling considerations, extensions, and implications. J. Eng. Technol. Manage. (2014), http://dx.doi.org/10.1016/j.jengtecman.2014.07.003

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Fig. 4. Mean development time (steps) for varying degrees of complementary dependencies. Panel A shows results for low complementarities (25% of dependencies). Panel B shows results for moderate complementarities (50% of dependencies). Panel C shows results for high complementarities (75% of dependencies). In each panel, the results from two different methods of assigning new fitness values to the focal component are also shown—the dashed line represents new fitness values drawn from the uniform distribution as in the original NK model; the solid line represents new fitness values drawn from the triangular distribution described and proposed in this paper. Each point represents the average of 2000 simulation runs generated from using 100 unique landscapes and a set of 20 starting configurations on each landscape.

are stochastic but not completely random in their result, and that improvements to components tend to be incremental rather than drastic in nature (Dosi et al., 1988; Nelson and Winter, 1982). The results of our extension to the original NK model using the triangular distribution for new focal component fitness values when a change is made (models G, H, and I in Table 1) are shown in Table 4 and Fig. 3 (solid lines). We first observe that, compared to the models in which fitness values for the focal component are distributed uniformly, system fitness decreases more slowly in K when complementarities are low (25%) and increases more rapidly when complementarities are moderate and high (50% and 75%, respectively). In other words, the slope of the line which plots fitness as a function of K is greater, in all cases, when changes to the focal component are distributed according to the triangular distribution. In our study, the probability of improvement, p, equals 0.75 which gives a higher probability of a component change resulting in improved fitness when compared to the original NK model in which the probability of a component change resulting in improvement is equal to: 1  current fitness of component. We could debate whether p = 0.75 is an appropriate parameter setting, but the parameter p can be adjusted to account for varying levels of technological uncertainty and/or risk seeking/avoidance preferences. Ultimately, the result of this increased probability of component improvement is that there are more opportunities for component changes to result in component and system wide fitness improvement. Therefore, when complementary dependencies are low, system fitness still regresses as K increases, but at a slower rate. And, when complementary dependencies are equal to or greater than 50%, the increased probability of component changes resulting in system improvement yields a greater marginal benefit for each increase in K. Please cite this article in press as: Oyama, K., et al., Applying complexity science to new product development: Modeling considerations, extensions, and implications. J. Eng. Technol. Manage. (2014), http://dx.doi.org/10.1016/j.jengtecman.2014.07.003

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Table 5 Comparison of results obtained for average number of design changes considered (simulation steps) by varying the 2 parameters of interest in this study—the mix of complementary and conflicting dependencies; and the method of updating the focal component fitness value. Also reported are the standard deviations for each result. Each result is an average over 2000 iterations (20 iterations on each of 100 different landscapes.). Uniform Mean Panel 1. Low complementarity K=0 51.88 K=1 47.75 K=2 37.34 K=3 32.6 K=4 28.38 K=5 26.07 Panel 2. Moderate complementarity K=0 51.88 K=1 47.09 K=2 43.43 K=3 40.28 K=4 38.15 K=5 37.15 Panel 3. High complementarity K=0 51.88 K=1 47.16 K=2 45.29 K=3 41.7 K=4 40.47 K=5 38.72

Triangular St Dev

Mean

t-test of means St Dev

13.2 16.77 15.34 15.35 14.54 12.48

71.03 60.66 50.4 46.97 40.03 36.72

11.81 18.08 20.08 23.06 22.15 21.22

p = 2.2e16 p = 2.2e16 p = 2.2e16 p = 2.2e16 p = 2.2e16 p = 2.2e16

13.2 15.46 14.92 15.67 15.64 14.57

71.03 69.04 72.15 76.04 78.05 75.04

11.81 18.04 21.26 27.24 29.49 29.67

p = 2.2e16 p = 2.2e16 p = 2.2e16 p = 2.2e16 p = 2.2e16 p = 2.2e16

13.2 14.65 14.1 14.1 13.97 13.27

71.03 78.41 92.76 106.14 113.49 113.67

11.81 17.42 23.4 28.62 32.44 31.95

p = 2.2e16 p = 2.2e16 p = 2.2e16 p = 2.2e16 p = 2.2e16 p = 2.2e16

However, in Table 4 and Fig. 3 (solid lines) we observe that, in using the triangular distribution to model focal component changes, system fitness is, in almost all cases, less than when using the uniform distribution to model component changes in the original NK model. To understand why this happens, we note that in the triangular distribution we have chosen, the maximum improvement is limited to half the difference between a component’s current fitness and the theoretic maximum value of 1. And, due to the peaked shape of the triangular distribution, component improvements are more likely to be incremental because new component fitness values are more likely to be close to mode of the distribution which, in our study, is the current component fitness value. We now consider how using the triangular distribution in our extended NK model impacts development time. When we changed the distribution of new fitness values for focal component changes, we obtained results more aligned with what we would expect regarding the development time of complex systems. As shown in Table 5 and Fig. 4 (solid lines), development time is longer in comparison to the original NK model specification (dashed lines) when we model component changes as being (1) more likely to succeed (p = 0.75) and (2) more incremental by using the triangular distribution. This is because the increased probability of change success, coupled with incremental gains, leads to more possible paths for exploration as well as slower convergence to a solution, resulting in longer and more gradual adaptive walks. Additionally, in Fig. 4, Panel C (solid line) we note that when complementary dependencies account for a high percentage (75% in this study) of the total dependencies in the system, the length of the project development time increases with complexity. This occurs because, when there is a sufficiently high percentage of complementary dependencies in a system, adding more dependencies (increasing K) creates a situation in which additional dependencies increase the likelihood of improved system fitness when a change is made to a component. This, in turn, lengthens the overall search process because new paths for improvement are more likely to continue to be found. This result is in stark contrast to what the original NK model would suggest—that increasing complexity always leads to shorter development times, regardless of the mixture of complementary and conflicting dependencies (shown by dashed lines in Fig. 4). Please cite this article in press as: Oyama, K., et al., Applying complexity science to new product development: Modeling considerations, extensions, and implications. J. Eng. Technol. Manage. (2014), http://dx.doi.org/10.1016/j.jengtecman.2014.07.003

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Changed performance landscapes It is important to note that our modeling extensions in this paper—varying mixtures of complementary and conflicting dependencies, and modeling changes as more likely to be incremental—have an impact on the structure of the NPD search problem. That is, the characteristics of the performance landscapes in the original NK model have been changed. To investigate how the performance landscapes have changed as a result of our modeling choices, we examine the variability (reported as standard deviation) in the found local maxima for each of our models. In Table 2, it can be observed that in the original NK model, the variance in the found maxima exhibits heteroskedasticity; specifically, the variance in the found maxima decreases as complexity increases. In other words, as complexity increases, fitness tends to decrease, but there is less variability in the found solutions, whereas at lower levels of complexity, the NK model suggests that higher levels of system fitness can be found, but with the tradeoff of higher variance. On the other hand, in our extended model, the relationship between the fitness and variance of the found maxima is again a function of the mixture of dependency types in the system. In particular, Table 4 (triangular distribution results) shows that in systems with a low percentage of complementarities, there is again a tradeoff between the mean found maxima and the variance—higher levels of fitness also have higher variance. However, as systems increase in the percentage of complementary dependencies, the situation improves and the tradeoff between system fitness and variance no longer exists. And, notably, for systems with a high percentage of complementary dependencies, system fitness improves with additional dependencies, as discussed earlier, and the variance in the mean found maxima decreases which reduces risk exposure to inferior solutions. With regard to the development time (steps) outcome, Table 3, Panel A shows that, in the original NK model, performance landscapes require less time to search as complexity increases, and the variability in search time is fairly uniform in absolute terms. However, as complexity of the performance landscape increases variance of the development time actually increases as a percentage of the average development time. The net result is that increased complexity again leads to a tradeoff in the original NK model, this time between average development and variance in development time (when measured as a percentage of development time). However, when we examine how the performance landscapes change as a result of our modeling extensions (Table 5, triangular distribution results) in this study, we find that variance in the development time remains relatively constant, as a percentage of development time, as complexity increases, regardless of the mixture of dependency types. Discussion Theoretical implications A number of previously published studies have used the NK model to investigate and gain insight into the interactions among multiple decisions within an organization. Due to small sample sizes, costs, and time, insights at the theoretical level regarding outcomes resulting from new product development decisions are difficult to develop and test in the real world. However, with regard to the relationship between product complexity, product performance, and development time, our extensions to the NK model offer insights for NPD theory and opens new avenues for further research and model development. The objective of this study was to model and investigate NPD phenomenon using the canonical NK model as a starting point, due to its ability to model dependencies between components, which is not unlike the dependencies found between elements of an NPD project. The first phenomenon we studied was the relationship between the mixture of complementary and conflicting dependencies in an NPD project and project outcomes of product quality (fitness) and development time. Our research revealed an important theoretical implication for NPD management: the nature of dependencies— whether they are complementary or not—within an NPD project matter as much or more than sheer number of dependencies. Several NPD research efforts suggest that interdependencies and coupling between components are undesirable and should be minimized (McKelvey, 1999; Schlick, 2007; Maier and Rechtin, 2002), and the NK model which has been used extensively in the NPD domain even suggests that, in general, an increased number of interdependencies leads to more complexity and reduced system fitness. However, our research shows that it is not necessarily the number of Please cite this article in press as: Oyama, K., et al., Applying complexity science to new product development: Modeling considerations, extensions, and implications. J. Eng. Technol. Manage. (2014), http://dx.doi.org/10.1016/j.jengtecman.2014.07.003

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dependencies that matter, but rather the type of dependencies play an important role in determining NPD project outcomes. For instance, Fig. 3 (solid lines) and Fig. 4 (solid lines) show that if we hold K constant, the system performance and development times vary widely, depending on the nature of the interactions within the system. In other words, the degree to which dependencies are complementary vs. conflicting moderates the effect of complexity (K) on system performance and development time. Therefore, with this insight, we may be able to design dependencies and dependency structures that can be exploited for more successful NPD. In addition, this insight highlights the importance of complementarities and also the importance of avoiding conflicting dependencies within a product. For instance, in Fig. 3, Panel A (solid line) we see that when there is a low level of complementary dependencies, system performance (fitness) monotonically decreases as complexity increases, suggesting that conflicting dependencies should be avoided, especially as product complexity increases. However, it is often the case, especially in large complex projects, that the presence and type of interdependencies are beyond a designer’s or firm’s control. When this is the case, the implication from our research is that trial-and-error learning in NPD projects can and should be exploited in order to learn where dependencies exist; and more importantly, the implication from this research is that learning the types of dependencies is more important than merely discovering their presence. Previous NPD research using the NK model would suggest that more dependencies (higher K) should be avoided. However, our research suggests that as complementary dependencies are discovered, they can serve as opportunities for future exploration paths in a project, while discovering conflicting dependencies can help steer a team or firm away from certain future paths. The second NPD reality that we studied and modeled was how design changes occur in NPD projects, which led to another important insight concerning the development time of an NPD project. In Fig. 4, we observe that our extensions to the NK model suggest the development time of NPD projects is consistently longer than under the original NK model due to the incremental nature of design changes and improvements. We argue that this dynamic of higher probability of incremental performance improvement more accurately models how NPD projects typically evolve. This is an important insight because applying the original NK model ‘‘off-the-shelf’’ may lead to inaccurate conclusions regarding how NPD projects evolve over time, whereas our extended model offers a more realistic framework for modeling actual projects in practice. Our study also reveals another relationship that has an important theoretical implication that differs from the original NK model of complexity. Namely, Figs. 3 and 4 (solid lines) show that, in our extended NK model of NPD, performance and development time are often negatively correlated. In other words, as system performance increases (favorable), development time also increases (unfavorable). This result is in contrast to the original NK model which suggests that there is a moderately low level of complexity (K) at which system performance is higher than at other values. It has not received much attention in the extant literature, but at this critical value of K the adaptive walk length is shorter than at lower values of K, suggesting that we are able to have dual benefits—faster development and better system performance. However, our model calls attention to the fact that there is a constant tension and trade-off between product quality and development time. A summary of our results regarding the tradeoffs between system performance and development time, as complexity increases, are presented in Table 6. In Table 6, it can be seen that the effect of complexity on measures Table 6 Summary of results highlighting tradeoffs, as complexity increases, in systems characterized by low and high percentages of complementary dependencies. As complexity (K) increases. . .

System fitness

Decreases

Increases

Variance in system fitness (as percentage of system fitness)

Remains constant

Decreases

Development time

Decreases

Increases

Variance in development time (as percentage of development time)

Increases

Remains constant

Percentage of complementary dependencies Low High

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of system fitness and development time are clearly moderated by the types of dependencies in the system as opposed to just the number of dependencies. Specifically, in low complementarity systems, there is a tradeoff in that as complexity increases, system performance declines, on average, but the time to arrive at a solution is typically faster because the search becomes frustrated quickly and terminates at a local maximum. Therefore, the implication is that in these types of systems, fewer dependencies are advantageous. On the other hand, in systems with a high percentage of complementarities, system fitness increases with interaction complexity and the variance in system performance decreases, thereby reducing the likelihood of inferior solutions; however, these advantages come at the expense of longer development times. Therefore, the implication is that in systems that are either known to have or are found to have a high percentage of complementarities, higher interaction complexity can actually improve system performance, and reduce risk, but is accompanied by longer development times. Another, perhaps more important, result of our study is that our extended NK model of NPD offers practitioners a relatively easy tool for exploring various real world product architectures via Monte Carlo simulation. For instance, the removal or addition of a specific component interaction can be explicitly modeled via the interaction matrix which is familiar to practitioners due the popularity and prevalence of the DSM tool (Steward, 1981). Additionally, sensitivity to uncertainty in the probability of design change success can be modeled using the extensions described in this study in conjunction with Monte Carlo simulation techniques. Finally, by virtue of being a computational model, the assumptions and parameters in the model can be easily modified in order to test specific hypotheses regarding candidate product design alternatives, while the model itself remains parsimonious. Limitations and future research The model we have presented is exploratory in nature. We have described and illustrated two significant ways in which the NK model can be adapted to more realistically simulate and explore the complex process of NPD. However, we recognize this model has limitations which create opportunities for further research in this domain. First, this model remains relatively abstract in nature. For example, the metric of fitness, measured on a scale of 0–1, is a proxy for system performance, but such a metric is rather abstract. Therefore, further research in the applied domain could investigate how to map component performance and system performance to more meaningful metrics via methods such as conjoint analysis (Green and Srinivasan, 1978). Another limitation of our study is that we did not implement more sophisticated search behaviors in our model. However, our intent was not to find optimal search strategies. Our objective was to develop and explore a new modeling methodology for NPD based on the popular NK model. Testing hypotheses regarding alternative search behaviors to the adaptive walk described in this paper could prove to be a fruitful avenue for future research. Additionally, our model assumes a time homogeneous fitness landscape. In other words, the nature of the interactions and the probability for improvement at each step of the simulation does not change as the system evolves. Further research on shifting fitness landscapes presents an opportunity to explore how periodic—and often exogenous—changes to the interaction structure of a product impact the development process. Another extension to this study could involve further characterizing the nature of dependency interactions within a system by modeling dependency strengths. In other words, rather than characterize each dependency as complementary or conflicting (positive or negative effect), future research could explicitly model the fact that some dependencies are stronger than others by scaling dependencies on the interval (1, 1). Finally, in addition to the finding that the nature of interactions within a system are important, exploration of the how interactions are configured could be an important research opportunity, combining the concepts in this study with the notion of patterned interactions introduced by Rivkin and Siggelkow (2007). Conclusion In summary, our study examined how contextual realities of NPD could be modeled within the framework of an existing model of complexity to better understand NPD phenomenon that arise from Please cite this article in press as: Oyama, K., et al., Applying complexity science to new product development: Modeling considerations, extensions, and implications. J. Eng. Technol. Manage. (2014), http://dx.doi.org/10.1016/j.jengtecman.2014.07.003

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interdependent decisions, a hallmark of complex systems. Specifically, we were interested in how the nature of interdependencies and the outcomes of design changes affect complex NPD projects. To this end, we critically analyzed the NK model and several of the assumptions that underpin this popular model of complexity, in the context of the model’s application to NPD, and found that the NK model neatly captures one of the central problems found in NPD organizations—dependencies between components—but could be extended for modeling NPD by addressing two aspects of NPD that are not captured within the NK model: the nature of dependencies within a system, and design changes that have inherent uncertainty, but are not completely random in their outcomes. We then proposed and implemented two modeling constructs that address these two contextual realities regarding NPD as it relates to the NK model. We call attention to how the simple, yet powerful, NK model can be adapted and extended to model the complex process of NPD while avoiding over-parameterization in the model. A central result of our study is that the nature of dependencies within a system moderates the effect of complexity on NPD project outcomes of quality and development time. Our study also revealed that applying the original NK model to the NPD domain could lead to misleading conclusions regarding how NPD projects evolve and perform over time. Specifically, we call attention to our result which suggests the original NK model under-estimates the time required for NPD project development; instead, the results from our model presented in this study show that development times are longer than the original NK model would suggests, because we have modeled changes as being more likely to be incremental. This added reality will allow practitioners to use our model to more accurately represent and simulate the development of real world projects using the NK modeling framework. Finally, perhaps the most important aspect of this study is the development of a modeling framework that can be used to gain further insight into the general domain of new product development. For instance, our extended NK model for NPD can support the development of specific system-level hypotheses about the outcomes of real world projects which can then be tested with real world parameters regarding component-level dependencies and component-level change probabilities. Appendix A. Mathematical example of extended NK model for NPD Here, we summarize the implementation of our extended model via an example using N = 6, K = 2, and low complementarities (25%). A particular instance of this example system could be represented as shown in Fig. 5. Note that when N = 6 and K = 2, there are a total of 12 dependencies in the system (other than the self-referential dependencies shown on the main diagonal). The complementary or conflicting nature of the dependencies in this example system are shown with either a ‘‘+’’ or ‘‘’’, respectively. In this case, there are 3 complementary dependencies because the level of complementary dependencies is set at ‘‘low’’.

[(Fig._5)TD$IG]

Fig. 5. Example system in which N = 6, K = 2, and complementary dependencies are ‘‘low’’ (25% of overall number of dependencies). Complementary dependencies are indicated by a ‘‘+’’ while conflicting dependencies are shown with a ‘‘’’. Note that the matrix is not symmetrical, indicating that dependencies do not necessarily exist in both directions, nor are they necessarily bi-directional in nature.

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To simulate the evolution of this example NPD project, we begin with an initial configuration of the N = 6 components. Let us assume, in this example, the initial configuration is given by

v¼000000 We also assign initial fitness values to each of the individual components, and we employ a table of fitness values, such as the one shown in Table 7 to model the fitness landscape. In general, our table of fitness values needs to be of size N  2(K+1). The N rows represent the possible fitness values for each of the N components, whereas the 2(K+1) columns represent the possible configurations for the ith component and the K = 2 other components on which its fitness depends. Initially, we do not populate the entire table of fitness values; rather, to keep track of the fitness values, we use a bitmask to identify the component dependencies, where the bitmask is of length K + 1 and indicates the positions of the dependencies for each components. For example, reading across the rows in Fig. 5, the bitmask for component 2 is ‘‘1 2 3’’ and the bitmask for component 1 is ‘‘1 2 6’’. In our initial configuration, v = 0 0 0 0 0 0, none of the component configurations are equal to 1, and thus each comparison against the bitmask results in ‘‘false’’ for all 3 positions and, therefore, we assign all initial fitness values to the first column labeled ‘‘F F F’’ because the equivalent to ‘‘0 0 0’’ in binary is 0 in decimal. (Note: the columns are necessarily indexed by +1 of the decimal equivalent as there is no 0th column index). In this example, our initial system fitness is the mean of the initial component fitness values: 0:2655 þ 0:3721 þ 0:5729 þ 0:9082 þ 0:2107 þ 0:8984 ¼ 0:538 6 However, if we change the initial configuration from v = 0 0 0 0 0 0 to v0 = 0 1 0 0 0 0, then the values of the first three components change. In our new configuration, v0 = 0 1 0 0 0 0, the second component now has a configuration of ‘‘1’’. Therefore, the bitmask for component 2 [1 2 3] now has a ‘‘true’’ value in the second position, resulting a bitmask comparison of ‘‘F T F’’ or ‘‘0 1 0’’ which equals 2 in binary. So, we assign a new fitness value to row 2, column 3 (2 + 1, again because there is no 0th column index) for component 2. This new value for component 2 is drawn from the triangular distribution as described in section ‘Assignment of Component-Level Fitness Values’ and, in this example equals 0.6008. Additionally, when component 2 is changed, the fitness values of component 1 and component 3 are impacted. And, reading down column 2 in Fig. 5, we see that the fitness value of component 1 will decrease and the fitness value of component 3 will also decrease, due to the positive change in component 2. The bitmask of component 1 [1 2 6] now has a ‘‘true’’ condition in the second position (‘‘F T F’’ = ‘‘0 1 0’’ = 2, so the new value will be placed in row 1 column 3, whereas the bitmask of component 3 [2 3 4] now has a ‘‘true’’ condition in the first position (‘‘T F F’’ = 1 0 0 = 4), so the new fitness value for component 3 will be placed in row 3, column 5). The new values of components 1 and 3, in this example, are 0.1688 and 0.3642, respectively. The dynamically populated fitness value table, at this step of the simulation, is shown in Table 8. These new component fitness values are calculated

Table 7 Example initial fitness value table for dynamically building a stochastic landscape. The rows represent the possible fitness values for each of the N components. Note that there are 2K+1 = 22+1 = 8 columns of possible fitness values for each component because the fitness value of each component depends on its on configuration and the configuration of K = 2 other components resulting in 8 possible configurations of these 3 components. The particular column used for each component fitness value is determined using a bitmask which compares the state of the K + 1 components to the dependency matrix.

1 2 3 4 5 6

‘‘FFF’’

‘‘FFT’’

‘‘FTF’’

‘‘FTT’’

‘‘TFF’’

‘‘TFT’’

‘‘TTF’’

‘‘TTT’’

0.2655 0.3721 0.5729 0.9082 0.2107 0.8984

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

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Table 8 Example fitness value table for dynamically building a stochastic landscape after considering a change to component 2. Note that a new fitness value has been assigned to component 2 as a result of this change, as well as its dependent components (1 and 3). The dependencies for components 1 and 3 were conflicting and therefore resulted in decreased fitness from their previous values. The fitness values for each component, once assigned in the table, are saved so that if a particular configuration is revisited during the search process, the same fitness values are given.

1 2 3 4 5 6

‘‘FFF’’

‘‘FFT’’

‘‘FTF’’

‘‘FTT’’

‘‘TFF’’

‘‘TFT’’

‘‘TTF’’

‘‘TTT’’

0.2655 0.3721 0.5729 0.9082 0.2107 0.8984

0 0 0 0 0 0

0.1688 0.6008 0 0 0 0

0 0 0 0 0 0

0 0 0.3642 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

0 0 0 0 0 0

according to Eq. (2). Finally, the new system fitness value is equal to the mean of the new component fitness values:

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