A simulation-based method to evaluate the impact of product architecture on product evolvability

Abstract

Products evolve over time via the continual redesigns of interdependent components. Product architecture, which is embodied in the structure of interactions among components, influences the ability for the product to be subsequently evolved. Despite extensive studies of change propagation via inter-component connections, little is known about the specific influences of product architecture on product evolvability. Related metrics and methods to assess the evolvability of products with given architectures are also under-developed. This paper proposes a simulation-based method to assess the isolated effect of product architecture on product evolvability by analyzing a design structure matrix. We define product evolvability as the ability of the product’s design to subsequently generate heritable performance-improving variations, and propose a quantitative measure for it. We demonstrate the proposed method by using it to investigate a wide spectrum of model-generated DSMs representing products with varied architectures, and show that modularity and inter-component influence cycles promote product evolvability. Our primary contribution is a repeatable method to assess and compare alternative product architectures for architecture selection or redesign for evolvability. A second contribution is the simulation-based evidence about the impacts of two particular product architectural patterns on product evolvability. Both contributions aim to aid in designing for evolvability.

Appendix

The network generation model is revised on the basis of a mathematical model that replicates ecological networks (Williams and Martinez 2000). The simulation model uses three input parameters, which may represent different architectural properties of a product.

• (1) Network size (N): the total number of components in the product.

• (2) Interaction density (K): the average number of components that each component influences. For a product with N components and M influence links, K = M/N. In the context of product design, it is an indicator of product integrality or reverse indicator of product modularity.

• (3) Influence diversity (D): the scope of other components that an average component can influence in an ideal and hypothesized product hierarchy (Fig. 6a) or a “serial design chain,” as Sosa et al. (2013) put it. Its reverse concept is “influence specificity,” which indicates the degree to which a component’s influences concentrate on a subset of components that are proximate to each other in the product hierarchy. As influence diversity increases, inter-influence relationships among components will gradually deviate from the pure serial or upstream–downstream manner.

  Fig. 6

  

  Network rewiring model

1.1 Baseline Scenario (D = 0)

We begin by creating an ideal and hypothesized sequential (upstream–downstream) influence or dependence relationship between components in their interaction network, which will be rewired later to generate more general and cyclic networks. To do this, each of the N components is assigned to a uniformly distributed random position λ _i , along an axis ranging from 0 to 1 (Fig. 6a). Consider a focal component i with position value λ _i , the entire downstream interval for component i has a length (1 − λ _i ). The component’s “influence niche” range r _i is the interval containing the components that it can influence, as defined by

$$r_{i} = X(1 - \lambda_{i} )$$

(4)

where X is a random variable between 0 and 1, and the probability distribution of X is component independent (to be set up later).

The focal component’s influence niche range can be located anywhere downstream. The position parameter b _i fixes the location of component i’s niche range by defining its left most point. b _i is assumed to be uniformly distributed between λ _i and (1 − r _i ). Networks generated this way are strictly acyclic; there is no influence cycle among any components. Thus, they can be used as the basis for later rewiring to introduce cycles.

The niche range of a particular component r _i , as defined in Eq. (4), is a random variable whose statistical properties are affected by the number of components (N) and interaction density (K) of the product. Now we explain this association further. First, the density of components on the entire segment is N. Because the distribution of these components is uniform, the expected number of components in the niche for component i is as follows:

$$E(K_{i} ) = N \cdot E(r_{i} )$$

(5)

For the entire system, excluding the rightmost component, the sum of the expected number of component that each component can influence is as follows:

$$E(M) = \sum\limits_{i = 1}^{N - 1} {E(K_{i} ) = N} \sum\limits_{i = 1}^{N - 1} {E(r_{i} )}$$

(6)

In addition, the expected average number of components that each components influences is simply

$$E(K) = \frac{E(M)}{N} = \sum\limits_{i = 1}^{N - 1} {E(r_{i} ) = } \sum\limits_{i = 1}^{N - 1} {E(1 - \lambda_{i} )} E(X) = \frac{(N - 1)}{2}E(X)$$

(7)

Thus, the random variable X is not only constrained to be between 0 and 1, but its expected value is

$$E(X) = \frac{2E(K)}{N - 1}$$

(8)

E(K) is given as the input variable K to the network construction model. Note that, although K _i is component-specific and randomly distributed, K is the average and an empirically measurable property of a given product’s component network. To generate a network, we need to choose an appropriate functional form for the distribution of X and then impose the constraint of Eq. (8). For computational ease, a beta-distribution with parameters (1, β) is used for the random variable X. This allows E(X) to be in a computationally convenient form, 1/(1 + β). Given K and N as inputs, β will be determined by

$$\beta = \frac{N - 1}{2K} - 1$$

(9)

Then, a random niche range constrained by (4) can be given to each of the aforementioned array of components randomly organized between 0 and 1 on the axis. The focal component is then linked to each component in its influence niche.

1.2 Hybrid (0 < D < 1): Random rewiring

When “influence diversity (D)” is greater than zero (D > 0), a portion D of influence links of a component, which assumedly go into its hypothesized niche, become nonspecific and are wired to components anywhere in the product hierarchy (including the preassigned niche).⁶ Figure 6b demonstrates a hybrid configuration after rewiring. Thus, influence diversity D is operationalized as the percentage of a component’s influence links that can deviate away from its predefined niche of components as given in the baseline scenario, and is the control for the extent of rewiring. Now, cycles can emerge to the degree of rewiring determined by D.

Tuning parameters N, K and D, the model generates random networks with gradually varied cyclic degrees (C). We statistically assessed the basic regularity in the relationship between N, K, D and C via simulations⁷ before using the simulated networks to investigate evolvability. First, the cyclic degree appears almost unaffected by changes in N when N is sufficiently large. Second, the cyclic degree is an increasing function of both K and D. The average cyclic degree of randomly generated network samples as a function of the inputs K and D (when N = 100) is plotted in Fig. 7. In particular, because the relationship between K or D and C is monotonic, one can infer the nominal “influence diversity” of an actual product from its empirically measurable interaction density (K) and cyclic degree (C).

Fig. 7

Impact of influence diversity and interaction density on cycle degree