تجزیه و تحلیل ریسک هزینه پروژه: یک روش شبکه های بیزی برای مدل سازی وابستگی بین اقلام هزینه

Uncertainty of cost items is an important aspect of complex projects. Cost uncertainty analysis aims to help decision makers to understand and model different factors affecting funding exposure and ultimately estimate the cost of project. The common practice in cost uncertainty analysis includes breaking the project into cost items and probabilistically capturing the uncertainty of each item. Dependencies between these items are important and if not considered properly may influence the accuracy of cost estimation. However these dependencies are seldom examined and there are theoretical and practical obstacles in modeling them. This paper proposes a quantitative assessment framework integrating the inference process of Bayesian networks (BN) to the traditional probabilistic risk analysis. BNs provide a framework for presenting causal relationships and enable probabilistic inference among a set of variables. The new approach explicitly quantifies uncertainty in project cost and also provides an appropriate method for modeling complex relationships in a project, such as common causal factors, formal use of experts' judgments, and learning from data to update previous beliefs and probabilities. The capabilities of the proposed approach are explained by a simple example.

To make cost estimates, project managers use cost analysis; a discipline that attempts to forecast the ultimate cost of a project. The difficulty about this analysis, especially for complex projects, is that there are a lot of uncertainties about cost items such as technology, productivity of human resources, economic conditions, market conditions, prices, inflation and other future risks and events. In general uncertainty occurs for a number of reasons: • Uniqueness (no similar experience) • Variability (trade-off between performance measure like time, cost, and quality) • Ambiguity (lack of clarity, data, structure, and bias in estimates) Cost uncertainty analysis is an important aspect of cost estimation that helps decision makers to understand not only the potential funding exposure but also the nature of risks for a particular project or program and possible responses to them. Without considering the uncertainty involved, there is a high risk that the actual cost of a project exceeds what it was originally anticipated, which in turn causes several other risks such as delays and performance problems (Elkjaer, 2000 and Lai et al., 2008). Several techniques including Regression modeling, Artificial Neural Networks (ANNs), feature-based method (FBM) and case-based reasoning (CBR) are proposed for modeling risk and uncertainty in project cost analysis. Section 2 briefly reviews a number of notable techniques. However none of these techniques capture the dependencies between project cost items consistent to real word conditions. Simulation-based techniques are the state-of-the-art approach that are adopted by many project management software tools and are arguably the best practice available (Arena et al., 2006, Chapman and Ward, 2011, Chou et al., 2009 and Cooper et al., 2005). Monte Carlo simulations (MCSs) treat cost uncertainty with the input distributions derived from elicitation sessions with technical experts or, occasionally, from historical data. In particular, the ultimate goal is to estimate a cumulative distribution function (equivalently, a probability distribution function) for the final cost, which in principle contains all of the uncertainty information and allows the computation and comparison of different project riskiness. Steadily improving software and the wide availability of powerful desktop computers make this approach straightforward to be implemented in practice. It is also attractive to analysts because of its conceptual simplicity and the widespread use of simulation technology elsewhere in the business world (Cooper et al., 2005). However, a debate continues over the inclusion of explicit correlation in simulated quantities. The cost items may not be independent—that is, the high cost of one element may affect the cost of another because of shared technology or manufacturing resources. Such correlation—if it exists—must be captured, or the final distribution will not accurately represent the uncertainty in the final cost (Book and Young, 1997 and Vose, 2000). The difficulty of doing so is an obstacle to applying quantitative risk methods. However, modeling this correlation is known to be much more difficult for human subjects to do without using various conditioning arrangements (Kadane and Wolfson, 1998). This problem is compounded by the fact that correlations between random variables, which can only take positive values (as duration and cost can), are not unconstrained. Unlike normal distributions, where the correlations between unlimited sets of normal random variables can be arbitrarily specified, subject to an overall condition of a positive definite correlation matrix, the constraints on correlated positive random variables are stronger and far less intuitive. As is the common practice of combining probability distribution, MCS employs a correlation coefficient matrix to model dependencies between distributions of cost items (Chapman and Ward, 2011). Usually, the same project participants that estimated the cost ranges will estimate the correlation. It ranges between + 1 and − 1. A correlation coefficient of + 1 signifies a perfect positive relationship, while − 1 shows a perfect negative one. Practically, it is difficult to estimate these correlation coefficients because the estimator has to specify all the joint distributions among cost items, which is highly impractical when the number of items is large (Yang, 2011). Moreover, matrix theory implies that a correlation matrix will not have any negative determinants in real life. Yet, with several people sitting around in a room, the correlation matrix that results may not be feasible. Adjusting the coefficients allows the user to ensure that the correlation matrix is at least not demonstrably impossible. This is a minimal test and does not ensure us that the correlation coefficients are “right” in any sense. The dependencies between project cost items have seldom been examined (Chou et al., 2009). In order to consider the dependencies and correlations between the cost items, this paper proposes a quantitative assessment framework integrating the inference process of a Bayesian network to the traditional probabilistic risk analysis. Bayesian networks (BNs) are a framework that presents probabilistic relationships and enables probabilistic inference among a set of variables (Heckerman et al., 1995). BNs provide an intuitively compelling approach for handling causal relationships and external influences. The application of BNs as a tool for mapping causal dependencies between frequencies and severities of risk events is demonstrated for modeling financial institutions' operational risk (Mittnik and Starobinskaya, 2010). The proposed model of this paper handles the correlation problem by modeling the uncertainty of common characteristics and performance indicators which affect cost items. The motivation behind this paper is derived from the fact that despite a causal relationship between uncertainty sources and cost items; this causality is not modeled in current state-of-the-art project cost risk analysis techniques (such as simulation techniques). The purpose of the proposed approach is to improve the assessment of likelihood distribution of project cost. The new approach explicitly quantifies uncertainty in project cost and also provides an appropriate method for modeling complex relationships and factors in project, such as: causal relation between variables, common causal factors, formal use of expert's judgments, and learning from data to update previous beliefs and probabilities. The remainder of this paper is organized as follows. In Section 2, we briefly review previous approaches for project cost estimating. Section 3 describes the concept of correlation and dependency between cost items. An overview of BN methodology and its application to modeling dependency, and developing the proposed model is covered in Section 4. For clarification purposes, a simple example is studied in Section 5. The paper ends with the conclusion.

Project cost analysis is one of the main challenges of project managers. In recent decades, through advances in computer technology and availability of simulation software, different methods are developed and used for cost risk analysis. However simulation based methods have theoretical and practical limitations and are not able to model complex cause and effect relationships among variables. More advanced techniques are required to capture different aspects of uncertainty in project cost estimation. This paper has proposed a new approach that makes it possible to incorporate uncertainty and causality in project cost estimating. The approach brings the full weight and power of BN analysis to bear on the problem of project cost risk analysis. This makes it possible to: • Express uncertainty about the final cost for each cost item and the whole project with full probability distributions • Model the dependency among various cost items using the causal relationships and conditional dependency among them • Perform complex “what-if?” analysis • Learn from data so that the predictions become more relevant and accurate. The application of the approach was explained by the use of a simple example. The aim of this paper is to demonstrate how advanced artificial intelligence tools such as BN can be employed to capture complex issues such as uncertainty in project cost analysis. The model proposed here is not the ultimate model. In fact, one of the advantages of BN modeling is that the network can be easily (although it needs some knowledge of probability) tailored and modified to fit to the problem. For example the same method can be used to model schedule uncertainty. Also when required (e.g. for critical phases of project) the model can be expanded by adding more variables to quantify decision making scenarios such as management reactions to cost overrun. The framework proposed in this paper can help us move to a new generation of cost risk assessment tools that are better informed by available knowledge and data and hence, more valid and useful. However, the presented case study is not intended to be comprehensive. More empirical research is required in order to justify the applicability of the model in real size projects (Pitchforth and Mengersen, 2013). When studying several hundred cost items and numerous common causes, the BN becomes too complex in term of efficiency of inference process as well as constructing the model. For our future work, the so-called Object Oriented Bayesian network (OOBN) approach (Koller and Pfeffer, 1997) would be sought. By using OOBNs, complex models can be constructed easily using inter-related objects. This not only facilitates the process of building the model but also speeds up the inference process.