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...transformation opportunity into a product available for sale (Krishnan and Ulrich, 1997). Frequently, managers of NPD projects are overwhelmed by complicating factors such as stochastic task times, ill-defined specifications, complex interrelationships between tasks, and information dependencies.
Traditional project management methodologies including PERT and CPM are useful tools for handling some of the problems associated with NPD projects. For example, given a list of project tasks with corresponding estimates of completion times and precedence relationships, the critical path listing those tasks crucial in determining the overall project completion time can be easily determined. However, these methodologies fail to take into account many of the complicating factors inherent in managing NPD projects. GERT (Graphical Evaluation and Review Technique), a transform-based method of describing networks algebraically, overcomes some of the limitations of PERT analysis. In principle, GERT analysis can characterize the distribution of project completion time taking into account activity iteration and probabilistic looping. However, it becomes necessary to resort to simulation for all but simple networks.
In their study of several companies, Adler et al. (1996) discuss the failure of these traditional project management methodologies in aiding firms to speed new products to market. Two of the key factors they identify confounding successful project management of NPD processes were ineffective measures of resource utilization, and the iterative nature of NPD processes. To illustrate, they discuss the experiences of a major computer equipment manufacturer that created cross-functional concurrent teams in an attempt to minimize the number of iterations (or rework cycles) needed to bring new products to market. While such a strategy may increase the communication between team members, thereby decreasing the chances of rework, they found that the team itself then became a bottleneck resource which actually limited the company's development speed. Similar experiences bave been reported by several companies, including Motorola, Hewlett Packard, General Electric, AT & T, and Ford.
Over the past decade, an alternative project management tool that explicitly takes into account the iterative nature of NPD projects has been developed; this is the Design Structure Matrix (DSM) approach. While DSM methodologies encompass the iterative nature of design projects, exact analysis of the computational difficulty of problems of this nature has remained relatively unexplored. In this paper, we first provide a mixed-integer linear programming formulation of the numerical DSM problem and highlight the computational difficulties with the approach. We then analyze the numerical DSM method and formally establish its complexity. Moreover, we explicitly address the situation where the task times are stochastic within the DSM framework. Finally, we address the computational aspect of the problem by solving several problem instances using the CPLEX integer programming software. We also examine the performance of some intuitively appealing heuristics, thereby offering managers efficient alternative solution approaches to the original DSM problem.
The remainder of the paper is organized as follows. In Section 2, we provide an overview of the basic DSM problem and summarize the existing literature on this problem. The basic DSM problem and a related problem are described and analyzed in Section 3. In Section 4, computational issues are described. Numerical analysis of these models and other sample heuristics are also shown in Section 4. Finally, the conclusions offer an overview of the implications of this analysis as well as suggestions concerning future directions for research.
2. Overview of the DSM problem and existing literature
Steward (1981) introduced the notion of a DSM which listed the project tasks and showed an explicit relationship between two of the project tasks by placing an "X" in the matrix. Such a method is similar to PERT/CPM techniques which specify precedence relationships, in that entries in the lower-diagonal portion of the DSM represent a precedent task relationship. However, an entry in the upper-diagonal portion of the DSM denotes an iterative process whereby the outcome of a particular activity can impact directly on a previously performed task. Smith and Eppinger (1997) further refined the DSM concept to include the stochastic nature of task iteration by adding probabilities to the off-diagonal entries, and deterministic task completion times to the diagonal entries of the matrix. The probabilities reflect the chance of having to repeat a task given the information obtained during the completion of another task. To illustrate, in a typical numerical DSM the task times [t.sub.i] are listed in the diagonal entries and the probability [p.sub.ij] that a certain task i will have to be iterated again after completion of activity j is listed in the nondiagonal entries. The authors discuss ways to identify the best task ordering utilizing a reward Markov chain approach and note that "no simple closed-form relationship has been found that can determine the minimum length ordering of a 3 x 3 or larger matrix."
Denker et al. (2001) and Eppinger (2001) both offer simple tutorials describing the basic mechanics of the DSM approach. For example, focusing on a particular row of the DSM shows all of the information inputs necessary to complete a task, while focusing on a particular column of the DSM shows all of the information outputs that one provides to other tasks. Similarly, managers should first attempt to determine an appropriate task sequence which creates a lower-diagonal matrix, thereby minimizing the task dependencies. If this is not possible, then these authors advocate that managers attempt to limit the scope of the iteration by arranging all of the dependencies in the top of the matrix as close to the diagonal as possible.
Other authors offer insights into the application of DSM methodologies in a variety of different ways to many diverse industries. Browning (2001) offers an excellent overview of the literature on the DSM approach. Also, Ahmadi et al. (2001) apply an alternate version of the DSM approach to the problem of minimizing the number of iterations necessary to complete large-scale design projects.
3. Analysis of the DSM problem
3.1. The DSM problem description
Consider a set of n activities to be executed sequentially. Each activity takes a certain time for its completion, and in general the completion times of the activities are random variables. We assume that the mean completion times of all the activities can be estimated a priori. There is a specific type of dependence between activity completion times and activity sequence. Let us explain the dependence structure associated with the DSM with respect to activities displayed in Fig. 1.
Activity 1 is completed after a completion time [t.sub.1], and activity 2 is begun. After activity 2 is completed, there are two possible outcomes depending on whether or not activity 1 needs to be reworked. There is a probability [p.sub.12] that activity 1 will have to be iterated and a probability (1 - [p.sub.12]) that we may proceed to the next activity in the sequence, activity 3. Suppose an iteration of activity 1 is required. In that case, after the iteration is completed, there are two possibilities: activity 2 may have to be iterated (with probability [p.sub.21]) or we may proceed to activity 3 (with probability 1 - [p.sub.21]).
Consider the situation when we reach activity j for the first time. After activity j is completed, there are j possibilities: any one of the (j - 1) preceding activities i may have to be repeated with probability [p.sub.ij] (i = 1, 2,..., j - 1), or else we proceed to activity (j + 1). If one of the preceding activities (say activity k)...
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