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The purpose of this study is to consider a stochastic workflow in the product development process with both random activity durations and predefined resource constraint and to develop the method of estimating the completion time in the product development process. We first present a stochastic workflow considering resource constraints and then develop a method to estimate the parameters of completion time in the product development process using a Markovian model. We present a practical case study to show that our method can be effectively used in practice.

A workflow has been traditionally defined in office terms—moving the paper, processing the order, or issuing the invoice. The workflow management coalition defines the workflow as the automation of a business process, in whole or part, during which documents, information or tasks are passed from one participant to another for action, according to a set of procedural rules (WfMC, 1999). A workflow consists of a set of activities, while an activity is a description of a piece of work that forms one logical step within a process. An activity requires human and/or machine resources to support process execution. It can be manually executed, or it can be automated. Nowadays, the workflow receives much attention for its capability to support today's complex business processes, such as product development process especially. Product development process is the process of transforming customer needs into an economically viable product that satisfies them. There are many research spans on product development process ranging from engineering to management, and recent research on product development process has focused on the approaches for reducing lead time, cutting costs, and improving product quality. In addition, there has been also an important research issue—estimating the overall duration of a project or product development process. In estimating the duration of a project or product development process, an activity-on-arc (AoA) directed acyclic graph method, which has one start node s and one finish node t, has been widely used. This type of graph is often called a stochastic activity network or program evaluation and review technique (PERT) network (Malcolm et al., 1959). In a stochastic activity network, durations of all activities are positive random variables with known probability distribution. The completion time of a project or product development process is also a random variable whose realization can be determined, but whose exact distribution function is very difficult to calculate for most stochastic activity networks. The difficulty in obtaining the distribution function is that there are a large number of paths in stochastic activity networks and interdependencies among them. In practice PERT or the product development process is usually carried out with limited resources. However, most researches ignore the resource constraints, and the need for proper resource-constrained stochastic activity network problem is now being needed. To the best of our knowledge, few of the existing techniques clearly consider this subject from structural viewpoints. The main research directions to explore the distribution function of a project or product development process are discussed next.

Our intent in this paper is to concentrate on the estimation of the stochastic workflow completion time. Although practical workflows or projects are usually carried out with limited resources, many recent researches only focus the various algorithms on stochastic activity networks without limited resources. In this paper we model stochastic workflow network considering resource constraints with independent and exponentially distributed activity durations by CTMC with upper triangular rate matrix. The state space of this CTMC is dependent on the structure and resource allocation of the stochastic workflow network. Finally, there is need for an effective approach to evaluate the interaction between changes in the parameters of individual activities and project completion time, which are the issues related to activity criticality and the sensitivity of the mean and variance of project completion time to the changes in the mean and variance of individual activities, considering resource constraints.