A new approach for quantitative risk analysis

Project risk management aims to provide insight into the risk profile of a project as to facilitate decision makers to mitigate the impact of risks on project objectives such as budget and time. A popular approach to determine where to focus mitigation efforts, is the use of so-called ranking indices (e.g., the criticality index, the significance index etc.). Ranking indices allow the ranking of project activities (or risks) based on the impact they have on project objectives. A distinction needs to be made between activity-based ranking indices (those that rank activities) and risk-driven ranking indices (those that rank risks). Because different ranking indices result in different rankings of activities and risks, one might wonder which ranking index is best. In this article, we provide an answer to this question. Our contribution is threefold: (1) we set up a large computational experiment to assess the efficiency of ranking indices in the mitigation of risks, (2) we develop two new ranking indices that outperform existing ranking indices and (3) we show that a risk-driven approach is more effective than an activity-based approach.

This article has benefited from our collaboration with the Belgian Building Research Institute (BBRI) and was funded by IWT-grant 070665.

Authors and Affiliations

1. IESEG School of Management, Rue de la digue 3, 59000, Lille, France

   Stefan Creemers

2. Department of Decision Sciences and Information Management, Research Center for Operations Management, KU Leuven, Naamsestraat 69, 3000, Leuven, Belgium

   Erik Demeulemeester & Stijn Van de Vonder

Corresponding author

Correspondence to Erik Demeulemeester.

Appendix A

The following is a list of notation used in this article:

• \(( \cdot )_{j}^{(E)}\): ranking value of an activity-based ranking index (⋅) when activities are subject to a set of risks E: E⊆R.

• \(( \cdot )_{e}^{(E)}\): ranking value of a risk-driven ranking index (⋅) when activities are subject to a set of risks E: E⊆R.

• A={(i,j)|i,j∈N}: set of arcs.

• A ^(N(⋅)): subset of N that contains all activities in N that are ranked highest by activity-based ranking index (⋅).

• B ^(E(⋅)): subset of E that contains all risks in E that are ranked highest by ranking index (⋅).

• c: deterministic project completion time.

• C: random variable that represents the project completion time.

• c={c ₁,c ₂,…,c _q }: vector of random variates of random variable C.

• \(\mathbf{c}^{(E)} = \{ c_{1}^{(E)}, c_{2}^{(E)}, \ldots, c_{q}^{(E)} \}\): vector of random variates of the project completion time when activities are subject to a set of risks E: E⊆R.

• \(c_{p}^{(E)}\): project completion time during a simulation iteration p when activities are subject to a set of risks E: E⊆R.

• d _j : deterministic duration of an activity j.

• D _j : random variable that represents the duration of an activity j.

• d _j ={d _j,1,d _j,2,…,d _j,q }: vector of random variates of random variable D _j .

• \(\mathbf{d}_{j}^{(E)} = \{ d_{j,1}^{(E)}, d_{j,2}^{(E)}, \ldots, d_{j,q}^{(E)} \}\): vector of random variates of the duration of an activity j when subject to a set of risks E: E⊆R.

• \(\hat{\mathbf{d}}_{j}^{(E)} = \{ \hat{d}_{j,1}^{(E)}, \hat {d}_{j,2}^{(E)}, \ldots, \hat{d}_{j,q}^{(E)} \}\): estimator of \(\mathbf{d}_{j}^{(E)}\).

• \(d_{j,p}^{(E)}\): duration of an activity j during a simulation iteration p when subject to a set of risks E: E⊆R.

• \(\hat{d}_{j,p}^{(E)}\): estimator of \(d_{j,p}^{(E)}\).

• δ _j : binary variable that equals 1 if activity j is critical in the deterministic early-start schedule and 0 otherwise.

• \(\delta_{j,p}^{(E)}\): binary variable that equals 1 if activity j is critical in \(\mathfrak{s}_{p}^{(E)}\) and 0 otherwise.

• Δ ^(E): the expected project delay when activities are subject to a set of risks E: E⊆R.

• \(\delta_{l,v,w,\alpha}^{ ( \cdot )_{x} }\): binary variable that equals 1 if \(H_{l,v,w}^{ ( \cdot )_{x} }\) is rejected at an α level of significance and 0 otherwise.

• \(\varDelta^{ ( \cdot )_{x}}\): the expected project delay after mitigation of x risks using ranking index (⋅).

• E: subset of R.

• f _j : earliest deterministic finish time of an activity j.

• F _j : random variable that represents the earliest finish time of an activity j.

• f _j ={f _j,1,f _j,2,…,f _j,q }: vector of random variates of random variable F _j .

• \(\mathbf{f}_{j}^{(E)} = \{ f_{j,1}^{(E)}, f_{j,2}^{(E)}, \ldots, f_{j,q}^{(E)} \}\): vector of random variates of the earliest finish time of an activity j when activities are subject to a set of risks E: E⊆R.

• \(f_{j,p}^{(E)}\): earliest finish time of an activity j during a simulation iteration p when activities are subject to a set of risks E: E⊆R.

• G=(N,A): graph that consists of a set of nodes N={1,2,…,n} and a set of arcs A={(i,j)|i,j∈N}.

• \(H_{l,v,w}^{ ( \cdot )_{x} }\): null hypothesis of equal expected project completion time if v and w simulation iterations are used to compute the project completion time of a project l.

• L: set of all project networks in the PSPLIB J120 data set.

• λ _t : weight assigned to a scenario π _t .

• M={M _j,e |j∈N∧e∈R}: set of risk impacts.

• M _j,e : random variable that represents the risk impact of a risk e on the duration of an activity j.

• m _j,e ={m _j,e,1,m _j,e,2,…,m _j,e,q }: vector of random variates of M _j,e .

• m _j,e,p : impact of a risk e on an activity j during a simulation iteration p.

• m _e ={m _e,1,m _e,2,…,m _e,q }: vector of total risk impacts.

• m _e,p : total risk impact of a risk e over all activities j∈N during a simulation iteration p.

• MEI^(⋅): Mitigation Efficiency Index of a ranking index (⋅).

• \(\mu_{l,v}^{ ( \cdot )_{x}}\): expected project completion time after mitigation of x risks using ranking index (⋅) when v simulation iterations are used to compute the project completion time.

• N={1,2,…,n}: set of nodes.

• ω: maximum deviation between real risk impacts and their estimates.

• \(p_{l,v,w}^{ ( \cdot )_{x} }\): probability of rejecting \(H_{l,v,w}^{ ( \cdot )_{x} }\).

• π _t : scenario evaluated at a step t that is characterized by a weight λ _t and a set of risks E ^t: E ^t⊆R.

• Π: the set of all scenarios.

• q: number of simulation iterations.

• R={1,2,…,r}: set of all risks.

• \(\mathrm{RRD}^{ (E(\cdot)_{x} )}\): the Relative Residual Delay after mitigation of x risks using ranking index (⋅).

• \(\rho_{v,w,\alpha}^{ ( \cdot )}\): the proportion of projects for which the null hypothesis of equal means is rejected at an α level of significance.

• s _j : earliest deterministic start time of an activity j.

• S _j : random variable that represents the earliest start time of an activity j.

• s _j ={s _j,1,s _j,2,…,s _j,q }: vector of random variates of random variable S _j .

• \(\mathbf{s}_{j}^{(E)} = \{ s_{j,1}^{(E)}, s_{j,2}^{(E)}, \ldots, s_{j,q}^{(E)} \}\): vector of random variates of the earliest start time of an activity j when activities are subject to a set of risks E: E⊆R.

• \(s_{j,p}^{(E)}\): earliest start time of an activity j during a simulation iteration p when activities are subject to a set of risks E: E⊆R.

• \(\mathfrak{s}\): vector of earliest start times when activity durations are deterministic.

• \(\mathfrak{s}_{p}^{(E)}\): vector of earliest starting times during a simulation iteration p when activities are subject to a set of risks E: E⊆R.

• \(\sigma_{l,v}^{ ( \cdot )_{x}}\): standard deviation of the project completion time after mitigation of x risks using ranking index (⋅) when v simulation iterations are used to compute the project completion time.

• \(\mathrm{TF}_{j,p}^{(E)}\): the total float of an activity j during a simulation iteration p when activities are subject to a set of risks E: E⊆R.

• U: a random variable that is uniformly distributed with minimum −1 and maximum 1.

• u _p : a random variate of a random variable U.

• υ _e : proportion of a risk e that cannot be mitigated.

• \(\mathbf{y}_{j}^{(E)} = \{ \delta_{j,1}^{(E)}, \delta _{j,2}^{(E)}, \ldots, \delta_{j,q}^{(E)} \}\): vector of random variates of the criticality of an activity j when activities are subject to a set of risks E: E⊆R.

Appendix B

The efficiency of a ranking index may be seen as its ability to correctly identify those risks that have the largest impact on project objectives. As such, for any good ranking index the following holds: \(\mathrm{RRD}^{ (\cdot )_{x-1}} - \mathrm{RRD}^{ (\cdot )_{x}} \geq \mathrm{RRD}^{ (\cdot )_{x}} - \mathrm{RRD}^{ (\cdot )_{x+1}}\) (i.e., \(\mathrm{RRD}^{ (\cdot )_{x}}\) has to be convex in the interval x∈[1,r]). We illustrate this logic in Fig. 13. \(\mathrm{RRD}^{ (\cdot )_{x}}\) is convex if:

Because \(( \mathrm{RRD}^{ (\cdot )_{r}} = 0 )\) and \(( \mathrm{RRD}^{ (\cdot )_{0}} = 1 )\), the condition translates into:

Illustration of mitigation efficiency

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To assess the mitigation efficiency of a ranking index, we want to evaluate the level of convexity of \(\mathrm{RRD}^{ (\cdot )_{x}}\). For this purpose, we develop the Mitigation Efficiency Index:

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which corresponds to the surface of the gray area in the graph presented in Fig. 13. In order to obtain a relative measure, we divide \(\widehat{\mathrm{MEI}}^{ (\cdot )}\) by \(( \frac{r-1}{2} )\):

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