Optimal resource allocation for dynamic product development process via convex optimization

Abstract

Resource allocation is an essential aspect of successful Product Development (PD). In this paper, we formulate the dynamic resource allocation problem of the PD process as a convex optimization problem. Specially, we build and solve two variants of this problem: the budget-constrained problem and the performance-constrained problem. We use convex optimization as a framework to optimally solve large problem instances at a relatively small computational cost. The solutions to both problems exhibit similar trends regarding resource allocation decisions and performance evolution. Furthermore, we show that the product architecture affects resource allocation, which in turn affects the performance of the PD process. By introducing centrality metrics for measuring the location of the modules and design rules within the product architecture, we find that resource allocation decisions correlate to their metrics. These results provide simple, but powerful, managerial guidelines for efficiently designing and managing the PD process. Finally, for validating the model and its results, we introduce and solve two design case studies for a mechanical manipulator and for an automotive appearance design process.

Appendix

In this appendix, we illustrate how we can reduce Problems 1 and 2 to convex optimization problems. We introduce the following notations for cost functions. For the total cost function in (3), we define

$$\begin{aligned} B_k^+(\phi _{k}, \gamma _{k})=\sum _{i=1}^{n} f_i^+(\phi _{i, k})+\sum _{i=1}^{n}\sum _{i\ne j}g_{ij}^+(\gamma _{ij, k}), \\ B_k^-(\phi _{k}, \gamma _{k})=\sum _{i=1}^{n} f_i^+({\bar{\phi }}_{i, k})+\sum _{i=1}^{n}\sum _{i\ne j}g_{ij}^+({\bar{\gamma }}_{ij, k}). \end{aligned}$$

Let us first show that Problem 1 reduces to solving a convex optimization problem. Notice that the optimization problem (4) is equivalent to

$$ {\mkern 1mu} \mathop {{\text{minimize}}}\limits_{{\phi ,\gamma }} {\mkern 1mu} \;\sum\limits_{{i = 1}}^{n} {P_{i} } (T) $$

(7a)

$$\begin{aligned} {{\,\mathrm{\text {subject to}}\,}}&B_k(\phi _{k}, \gamma _{k}) \le {\bar{B}}_k, \end{aligned}$$

(7b)

$$\begin{aligned}&{0<{\underline{\phi }}_{i, k} \le \phi _{i, k} \le {\bar{\phi }}_{i, k},}\end{aligned}$$

(7c)

$$\begin{aligned}&{0<{\underline{\gamma }}_{ij, k} \le \gamma _{ij, k} \le {\bar{\gamma }}_{ij, k},}~{k=1, \dotsc ,T.} \end{aligned}$$

(7d)

Under this notation, we can show that the solution of the budget-constrained problem is given by

$$\begin{aligned} \phi =\exp [x],~\gamma =\exp [y], \end{aligned}$$

(8)

where $\exp [\cdot ]$ is the entrywise exponential function of the variables, and $x=\{x_k\}^T_{k=1}$ and $y=\{y_k\}^T_{k=1}$ solve the following convex optimization problem:

$$ {\mkern 1mu} \mathop {{\text{minimize}}}\limits_{{x,y,\Gamma }} \;\Gamma $$

(9a)

$$\begin{aligned} {{\,\mathrm{\text {subject to}}\,}}&\log B_k^+(x_{k}, y_{k}) \le \log ({\bar{B}}_k+B_k^-), \end{aligned}$$

(9b)

$$\begin{aligned}&\log {\sum _{i=1}^{n}P_i(T)} {\le \Gamma }, \end{aligned}$$

(9c)

$$\begin{aligned}&\log {\underline{\phi }}_{i, k} \le x_{i, k} \le \log {\bar{\phi }}_{i, k},\end{aligned}$$

(9d)

$$\begin{aligned}&\log {\underline{\gamma }}_{ij, k} \le y_{ij, k} \le \log {\bar{\gamma }}_{ij, k}. \end{aligned}$$

(9e)

Let us give a brief proof of this statement. Under Lemma 1, it can easily be seen that (7a), (7b), and (7d) in the budget-constrained problem are equivalent to (9b), (9c), (9d), and (9e). Therefore, the solution of the optimization problem (9) given by (8) is the solution of the budget-constrained problem. Under this equivalence, we show the convexity of the optimization problem (9). It is sufficient to show that constraints (9b) and (9c) are convex if the performance functions (7a), (7b) and the cost function (3) follow Definition 1.

We can similarly reduce Problem 2 to a convex optimization problem. We can specifically show that the solution of the performance-constrained problem is given by (8), where $x=\{x_k\}^T_{k=1}$ and $y=\{y_k\}^T_{k=1}$ solve the following convex optimization problem

$$ \begin{gathered} \mathop {{\text{minimize}}}\limits_{{_{{x,y,\Psi }} }} {\mkern 1mu} \Psi {\mkern 1mu} \hfill \\ {\text{subject to}}{\mkern 1mu} \;\log \sum\limits_{{i = 1}}^{n} {P_{i} } (T) \le \log \bar{P},\; \hfill \\ \log \sum\limits_{{k = 1}}^{T} {B_{k}^{ + } } (x_{k} ,y_{k} ) \le \log \left( {\Psi + \sum\limits_{{k = 1}}^{T} {B_{k}^{ - } } } \right), \hfill \\ \log \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\phi } _{{i,k}} \le x_{{i,k}} \le \log \bar{\phi }_{{i,k}} , \hfill \\ \log \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\gamma } _{{ij,k}} \le y_{{ij,k}} \le \log \bar{\gamma }_{{ij,k}} . \hfill \\ \end{gathered} $$

We omit the proof of this statement because it is similar to the one for the budget-constrained problem.