A stochastic bi-objective project scheduling model under failure of activities

Abstract

In this paper, the research and development project scheduling problem (RDPSP) under uncertain failure of activities is formulated where an activity’s failure results in the project’s overall failure. A scenario-based bi-objective model to maximize the expected net present value (eNPV) and to minimize the NPV’s risk by conditional value-at-risk (CVaR) measurement is presented. For this purpose, different modes of failure or success of activities have been considered as a stochastic parameter by a set of scenarios. To formulate the problem, a nonlinear model is first presented, then a mixed-integer programming (MIP) model of the problem is developed by piecewise approximation. Some valid inequalities are presented to improve the performance of the MIP model. A sequential sampling procedure is also used to approximate the solution of the MIP model with a large number of scenarios. The experimental results have shown that the sequential sampling procedure attains high-quality solutions in a reasonable CPU time.

Appendix: Mixed integer programming model (Model M5)

$$ Min\,\,\;CVaR = VaR + \frac{1}{{\left( {1 - \alpha } \right)}}\mathop \sum \limits_{s\epsilon S} \pi_{s} *\gamma_{s} $$

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$$ Max\,\,\user2{ } eNPV = \pi^{*} *CF*\mathop \sum \limits_{v} e^{{a_{{\left( {n + 1} \right)v}} }} \lambda_{{\left( {n + 1} \right)v}} - \mathop \sum \limits_{s\epsilon S} \pi_{s} .\mathop \sum \limits_{i = 0}^{n + 1} c_{i} .\mathop \sum \limits_{v} e^{{a_{iv} }} *T\left( {s,i,v} \right) $$

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$$ \gamma_{s} \ge \mathop \sum \limits_{i = 0}^{n + 1} c_{i} .\mathop \sum \limits_{v} e^{{a_{iv} }} *T\left( {s,i,v} \right) - {\text{VaR}}\;\;\;\;\forall \left( {s \in S\backslash s^{*} } \right) $$

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$$ \gamma_{s} \ge - \pi^{*} *CF*\mathop \sum \limits_{v} e^{{a_{{\left( {n + 1} \right)v}} }} \lambda_{{\left( {n + 1} \right)v}} + \mathop \sum \limits_{i = 0}^{n + 1} c_{i} .\mathop \sum \limits_{v} e^{{a_{iv} }} *T\left( {s,i,v} \right) - {\text{VaR}}\;\;\;\;\forall \left( {s = s^{*} } \right) $$

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$$ - rx_{i} = \mathop \sum \limits_{v} a_{iv} \lambda_{iv} \;\;\;\; \forall \left( {i \in N} \right) $$

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$$ \mathop \sum \limits_{v} \lambda_{iv} = 1\;\;\;\; \forall \left( {i \in N} \right) $$

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$$ T\left( {s,i,v} \right) \le \lambda_{iv} \;\;\;\;\forall \left( {i \in N; s \in S;{ }v \in V} \right) $$

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$$ T\left( {s,i,v} \right) \le M*\Delta_{i}^{s} \;\;\;\;\forall \left( {i \in N; s \in S; v \in V} \right) $$

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$$ T\left( {s,i,v} \right) \ge \lambda_{iv} - M*(1 - \Delta_{i}^{s} )\;\;\;\;\forall \left( {i \in N; s \in S; v \in V} \right) $$

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$$ \mathop \sum \limits_{{\begin{array}{*{20}c} \scriptstyle{j = 0} \\ \scriptstyle{ j \ne i} \\ \end{array} }}^{n + 1} y_{ji} - 1 + \Delta_{i}^{s} \ge \mathop \sum \limits_{{\begin{array}{*{20}c} \scriptstyle{j = 0} \\ \scriptstyle{ j \ne i} \\ \end{array} }}^{n + 1} y_{ji} *W_{j}^{s} \;\;\;\;\forall \left( {i \in N; s \in S} \right) $$

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$$ x_{0} = 0 $$

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$$ x_{n + 1} \le \delta_{n + 1} $$

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$$ x_{i} + d_{i} \le x_{j} + M\left( {1 - y_{ij} } \right)\;\;\;\; \forall \left( {i,j \in N, j \ne i} \right) $$

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$$ y_{ij} = 1 \;\;\;\; \forall \left( {\left( {i,j} \right) \in A} \right) $$

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$$ y_{ij} + y_{jk} \le y_{ik} + 1 \;\;\;\; \forall (i,j,k \in N,k > j > i) $$

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$$ x_{i} \ge 0\;\;\;\; \forall \left( {i \in N} \right) $$

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$$ \gamma_{s} \ge 0 \;\;\;\; \forall \left( {s \in S} \right) $$

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$$ \lambda_{iv} \ge 0 \;\;\;\; \forall \left( {i \in N,\forall v \in V} \right) $$

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$$ T\left( {s,i,v} \right) \ge 0\;\;\;\; \forall \left( {i \in N; s \in S; v \in V} \right) $$

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$$ VaR\;\;\;\;{\text{Free}}\;{\text{variable}} $$

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