Information hiding in product development: the design churn effect

Abstract

Execution of a complex product development project is facilitated through its decomposition into an interrelated set of localized development tasks. When a local task is completed, its output is integrated through an iterative cycle of system-wide integration activities. Integration is often accompanied by inadvertent information hiding due to the asynchronous information exchanges. We show that information hiding leads to persistent recurrence of problems (termed the design churn effect) such that progress oscillates between being on schedule and falling behind. The oscillatory nature of the PD process confounds progress measurement and makes it difficult to judge whether the project is on schedule or slipping. We develop a dynamic model of work transformation to derive conditions under which churn is observed as an unintended consequence of information hiding due to local and system task decomposition. We illustrate these conditions with a case example from an automotive development project and discuss strategies to mitigate design churn.

Appendices

Appendix 1

Specifications of work transformation matrices (W^L, W^S, W^LS, W^SH, W^SL)

The specification of the work transformation matrices is based on the assumption that only work that is done in the previous period is considered to create rework as a normal course of operation. Let ${\Omega ^{{\rm{L}}} = {\left( {\alpha ^{{\rm{L}}}_{{ij}} } \right)}}$ be the local DSM. The work completion coefficient ${\alpha ^{{\rm{L}}}_{{ii}} \equiv \alpha ^{{\rm{L}}}_{i} }$ is the local autonomous completion rate for task i at each iteration step. The coupling coefficient ${\alpha ^{{\rm{L}}}_{{ij}} }$ (for ${i \ne j}$) is the amount of rework created for local task i per unit of work done on local task j. Consequently, the elements of the work transformation matrix W^L become ${{\rm{w}}^{{\rm{L}}}_{{ii}} = 1 - \alpha ^{{\rm{L}}}_{i} }$ and ${{\rm{w}}^{{\rm{L}}}_{{ij}} = \alpha ^{{\rm{L}}}_{{ij}} \alpha ^{{\rm{L}}}_{{jj}} }$ (for ${i \ne j}$). The system DSM ${\Omega ^{{\rm{S}}} }$ and work transformation matrix W^S are defined similarly.

The interaction between the local and system teams is captured by the intercomponent dependency matrices ${\Omega ^{{{\rm{LS}}}} = {\left( {\alpha ^{{{\rm{LS}}}}_{{ij}} } \right)}}$ and $\Omega ^{{{\text{SL}}}} = {\left( {\alpha ^{{{\text{SL}}}}_{{ij}} } \right)}.$ The coupling coefficient ${\alpha ^{{{\rm{LS}}}}_{{ij}} }$ is the amount of rework created for system task i per unit of work done on local task j. Similarly, the coupling coefficient ${\alpha ^{{{\rm{SL}}}}_{{ij}} }$ is the amount of rework created for local task i per unit of work done on system task j. Consequently, the elements of the work transformation matrix W^LS are ${\text{w}}^{{{\text{LS}}}}_{{ij}} = \alpha ^{{{\text{LS}}}}_{{ij}} \alpha ^{{\text{L}}}_{{jj}} .$ The matrix W^SL is defined as ${\text{w}}^{{{\text{SL}}}}_{{ij}} = \alpha ^{{{\text{SL}}}}_{{ij}} \alpha ^{{\text{L}}}_{{jj}} .$ Finally, the "holding" matrix W^SH is defined as W^SH=W^SL.

Appendix 2

Proof of Lemma 1, Theorem 1, and Corollary 1

See Richards (1983).

Proof of Theorem 2

From Theorem 1, x(k) is a solution of the linear periodic system described by Eq. 7 if and only if ${y(k) = P^{{ - 1}} (k)x(k)}$ is a solution of the linear autonomous system described by Eq. 8. The matrix P(k) is nonsingular and periodic. Thus, the stability of the linear periodic system (Eq.7) is equivalent to the stability of the associated linear autonomous system (Eq. 8). Consequently:

• 1. If the largest magnitude eigenvalue of B (i.e., the largest magnitude Floquet exponent) is less than 1, then every solution x(k) of Eq. 7 satisfies $\lim _{{k \to \infty }} x(k) = 0.$

• 2. If the largest magnitude eigenvalue of B is less than or equal to 1, then every solution y(k) of Eq.8 remains bounded for k≥0.

• 3. (Only if part). Assume that the largest magnitude eigenvalue of B is greater than 1. Then there is a solution y(k) of Eq. 8 such that $\lim _{{k \to \infty }} x(k) = \infty ,$ and the zero solution is unstable.

Corollary 2

Since the eigenvalues of B are the T ^th roots of the eigenvalues of the monodromy matrix C, corollary 2 immediately follows.

Proof of Theorem 3

From Theorem 1, the general solution x(k) of Eq. 7 may be written as ${x(k) = P(k)y(k)}$ where y(k) is the general solution of the linear autonomous system (Eq. 8). For the linear autonomous system, it can be verified that the general solution can be written as $y(k) = B^{k} S_{B} g,$ where S _B is the eigenvector matrix of B and $g = (g_{1} ,g_{2} ,...,g_{n} )^{'} \in R^{n} .$ The powers of B can be found by $B^{k} = S_{B} \Lambda ^{k}_{B} S^{{ - 1}}_{B} ,$ where ${\Lambda _{B} }$ is a diagonal matrix of the eigenvalues of B. Consequently,

$${y(k) = B^{k} S_{B} c = S_{B} \Lambda ^{k}_{B} g = }$$

$${[\xi _{1} ,\xi _{2} ,...,\xi _{n} ]{\left[ {\matrix{ {{\lambda ^{k}_{1} }} & {{}} & {0} & {{}} \cr {{}} & {{\lambda ^{k}_{2} }} & {{}} & {{}} \cr {{}} & {{}} & { \ddots } & {{}} \cr {{}} & {0} & {{}} & {{\lambda ^{k}_{n} }} \cr } } \right]}{\left[ {\matrix{ {{g_{1} }} \cr {{g_{2} }} \cr { \vdots } \cr {{g_{n} }} \cr } } \right]} = [\lambda ^{k}_{1} \xi _{1} ,\lambda ^{k}_{2} \xi _{2} , \cdots ,\lambda ^{k}_{n} \xi _{n} ]{\left[ {\matrix{ {{g_{1} }} \cr {{g_{2} }} \cr { \vdots } \cr {{g_{n} }} \cr } } \right]}}$$

where ${[\xi _{1} ,\xi _{2} ,...,\xi _{n} ]}$ is the eigenvector matrix for B.

Hence the general solution x(k) of Eq. 7 may be given by

$${x(k) = P(k)y(k) = [\lambda ^{k}_{1} P(k)\xi _{1} ,\lambda ^{k}_{2} P(k)\xi _{2} , \cdots ,\lambda ^{k}_{n} P(k)\xi _{n} ]{\left[ {\matrix{ {{g_{1} }} \cr {{g_{2} }} \cr { \vdots } \cr {{g_{n} }} \cr } } \right]}}$$

(B.1)

From Eq. B.1 we see that the general solution x(k) of Eq. 7 may be given by $x(k) = \Phi (k)g,$ i.e., each of the column vectors of ${\Phi (k)}$ is a nontrivial solution of Eq. 7. Let ${\hat{x}(k) = \lambda ^{k}_{i} P(k)\xi _{i} }$ be such a nontrivial solution. We have

$${\hat{x}(k + T) = \lambda ^{{k + T}}_{i} P(k + T)\xi _{i} = \lambda ^{T}_{i} \lambda ^{k}_{i} P(k)\xi = \lambda ^{T}_{i} \hat{x}(k)}$$

(B.2)

Notice that ${\lambda ^{k}_{i} }$ is an eigenvalue of the monodromy matrix C, i.e., ${\lambda ^{T}_{i} }$ is a Floquet multiplier of the linear periodic system (Eq. B.1). Thus, there exists a solution ${\hat{x}(k)}$ of the linear periodic system (Eq. B.1) such that $\hat{x}(k + T) = \lambda ^{T}_{i} \hat{x}(k),$ and this is the reason we call ${\lambda ^{T}_{i} }$ a multiplier. Now,

(i):

If the matrix C has an eigenvalue equal to 1, then ${\lambda ^{T}_{i} = 1}$ and from Eq. B.2 there exists a periodic solution of period T.

(ii):

If the matrix C has an eigenvalue equal to −1, then ${\lambda ^{T}_{i} = - 1}$ and from Eq. B.2 there exists a periodic solution of period 2T.

(iii):

Let the local and system work transformation matrices be coupled and non-negative. Consequently, the monodromy matrix C will be coupled and non-negative. Thus, in many applications, C ^L>0 for some power L (i.e., C is primitive) for L>0. By the Perron-Frobenius theorems for primitive matrices one of its eigenvalues ${\lambda ^{*}_{C} }$ is positive real and strictly greater (in absolute value) than all other eigenvalues, and there is a positive eigenvector corresponding to that eigenvalue. Since $\lambda ^{*}_{B} = \sqrt[T]{{\lambda ^{*}_{C} }},$ according to Eq. B.1, the largest magnitude eigenvalue of B is also positive real, and there is a positive eigenvector corresponding to that eigenvalue. Therefore, the long-term behavior of the system has the form

$$x(k) \sim c_{1} (\lambda ^{*}_{B} )^{k} P(k)\xi $$

(B.3)

If the largest eigenvalue of C is equal to 1, then it follows from Eq. B.3 that the long-term behavior of the system is periodic of period T.

Proof of Theorem 4

Since T is the least common multiple of T ₁ ,T ₂ ,...,T _m , it follows that there are integers a ₁ ,a ₂ ,...,a _m such that T=a _i T _i for 1≤i≤m. Let k≥0 be any time point. Assume that at time point k the system team provides updates only to the local teams i ₁ ,i ₂ ,...,i _j . From the information release policy it follows that there are integers b ₁ ,b ₂ ,...,b _n such that ${k = b_{{i_{{\ell }} }} T_{{i_{{\ell }} }} }$ for ${i_{{\ell }} \in \{ i_{1} ,i_{2} , \cdots ,i_{j} \} }$ and ${k = b_{{i_{{\ell }} }} T_{{i_{{\ell }} }} + e_{{i_{{\ell }} }} }$ for ${i_{{\ell }} \notin \{ i_{1} ,i_{2} , \cdots ,i_{j} \} }$ where $0 < e_{{i_{{\ell }} }} < T_{{i_{{\ell }} }} .$ Consider time point k+T.For $i_{{\ell }} \in \{ i_{1} ,i_{2} , \cdots ,i_{j} \} ,$ ${k + T = b_{{i_{{\ell }} }} T_{{i_{{\ell }} }} + a_{{i_{{\ell }} }} T_{{i_{{\ell }} }} = (b_{{i_{{\ell }} }} + a_{{i_{{\ell }} }} )T_{{i_{{\ell }} }} }$For $i_{{\ell }} \notin \{ i_{1} ,i_{2} , \cdots ,i_{j} \} ,$ ${k + T = b_{{i_{{\ell }} }} T_{{i_{{\ell }} }} + e_{{i_{{\ell }} }} + a_{{i_{{\ell }} }} T_{{i_{{\ell }} }} = (b_{{i_{{\ell }} }} + a_{{i_{{\ell }} }} )T_{{i_{{\ell }} }} + e_{{i_{{\ell }} }} }$Thus, we conclude that at time point k+T the system team will provide updates only to the local teams i ₁,i ₂,...,i _j . Consequently, the fundamental period of the linear system (Eq. 11) is T.