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A stochastic model and algorithms for determining efficient time–cost tradeoffs for a project activity

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We consider a stochastic model for the time–cost tradeoffs of an activity. In this model the activity can be undertaken by using several different resources, and the resource in use may change according to the way the activity is evolving. We present two algorithms for identifying strategies that are in a predefined neighborhood of the efficient set: one of them is based on a tree structure and the other is based on dynamic programming. Both algorithms take advantage of some mathematical properties of the model in order to reduce their running time and memory requirements. We present the results of some computational tests, as well as an application example. We conclude that the dynamic programming algorithm performs quite well, although it is sometimes necessary to adjust the parameters related to the neighborhood of the efficient set to be able to have reasonable running times.

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  1. 1.

    The proofs of the mathematical properties are given in the "Appendix".


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Correspondence to Pedro Godinho.

Electronic supplementary material

Below is the link to the electronic supplementary material.

Supplementary material 1 (DOCX 151 kb)

Appendix: Mathematical proofs

Appendix: Mathematical proofs

Proof of Property 1

Denote by \(\bar{S}\) a full set of \(\left( {\varepsilon_{T} ,\varepsilon_{C} } \right) - {\text{efficient}}\) strategies and consider a strategy \(s \in \bar{S}\). Assume there is a strategy \(s^{\prime} \in S\left( 1 \right)\) such that \(C_{C} \left( {s^{\prime}} \right) < C_{C} \left( s \right) - \varepsilon_{C}\). From part b of Definition 2, we know that there is a strategy \(s_{1} \in \bar{S}\) such that \(C_{C} \left( {s_{1} } \right) \le C_{C} \left( {s^{\prime}} \right) + \varepsilon_{C}\) and \(T_{C} \left( {s_{1} } \right) \le T_{C} \left( {s^{\prime}} \right) + \varepsilon_{T}\). This means that \(C_{C} \left( s \right) > C_{C} \left( {s_{1} } \right)\) and since, from part a of Definition 2, \(s_{1}\) does not dominate \(s\), it must be that \(T_{C} \left( s \right) < T_{C} \left( {s_{1} } \right)\), so \(T_{C} \left( {s^{\prime}} \right) > T_{C} \left( s \right) - \varepsilon_{T}\). Similarly, if \(T_{C} \left( {s^{\prime}} \right) < T_{C} \left( s \right) - \varepsilon_{T}\), it is possible to show that \(C_{C} \left( s \right) < C_{C} \left( {s^{\prime}} \right) + \varepsilon_{C}\). Therefore \(s\) is \(\left( {\varepsilon_{T} ,\varepsilon_{C} } \right) - {\text{efficient}}\).□

Proof of Property 2

This proof follows the same lines of the Proof of Property 1.□

Proof of Property 3

For the sake of simplicity, we prove this property assuming that \(p_{i}^{T, - } \left( \theta \right) = p_{i}^{T, + } \left( \theta \right) = p_{i}^{T} \left( \theta \right)\) (time-adjusted probabilities are independent of whether a large or small activity advance occurred before). The complete proof can be obtained by considering separately the different combinations of the signs that \(\left[ {T\left( {s^{\prime - } ,\left\langle - \right\rangle } \right) + t_{i,j1} } \right] - \left[ {T\left( {s^{\prime + } ,\left\langle + \right\rangle } \right) + t_{i,j2} } \right]\) and \(\left[ {T\left( {s^{\prime\prime - } ,\left\langle - \right\rangle } \right) + t_{i,j1} } \right] - \left[ {T\left( {s^{\prime\prime + } ,\left\langle + \right\rangle } \right) + t_{i,j2} } \right]\) may have in case c (defined below). The complete proof is available as Supplementary Material.

Step 4 of the procedure guarantees that no dominated strategies from \(U_{i} \left( {\theta ,x} \right)\) are included in \(\bar{S}_{i} \left( {\theta ,x} \right)\), therefore part a of Definition 4 holds. Let us now prove that part b also holds. To prove that part b of Definition 4 holds we should prove that for each \(s^{\prime} \in S_{i} \left( x \right)\) there exists \(s \in \bar{S}_{i} \left( {\theta ,x} \right)\) such that:

$$T\left( {s,\theta } \right) - T\left( {s^{\prime},\theta } \right) \le \varepsilon_{T} \cdot x$$
$$C\left( {s,\theta } \right) - C\left( {s^{\prime},\theta } \right) \le \varepsilon_{C} \cdot x$$

Let us consider 3 cases:

  1. (a)

    \(0 < x \le a_{i}\);

  2. (b)

    \(a_{i} + b_{i} \ge x > a_{i}\);

  3. (c)

    \(1 \ge x > a_{i} + b_{i}\).

In order to prove that (13) and (14) hold, we will show that they hold in case (a), and that in cases (b) and (c) they will hold for \(x\) if they also hold for \(x^{\prime}\) such that \(x^{\prime} \le x - a_{i}\). This way we prove by induction that they hold in all cases.

Let us consider case (a). In this case \(s = \left( {i,\varnothing ,\varnothing } \right)\) is the only strategy belonging to \(\bar{S}_{i} \left( {\theta ,x} \right).\)\(s\) also belongs to \(S_{i} \left( x \right)\), and other strategies \(s^{\prime} \in S_{i} \left( x \right)\) must have the same or high times and costs. So, for all \(s^{\prime} \in S_{i} \left( x \right)\), \(T\left( {s,\theta } \right) \le T\left( {s^{\prime},\theta } \right)\) and \(C\left( {s,\theta } \right) \le C\left( {s^{\prime},\theta } \right)\), thus proving (13) and (14) for this case.

Let us now consider case (b). Assume \(s^{\prime} = \left( {i,s^{\prime - } ,\varnothing } \right)\) and that (13) and (14) hold for \(x^{\prime} \le x - a_{i}\), for all resources \(i \in N\). Let us show that they hold for a partial strategy \(s \in \bar{S}_{i} \left( {\theta ,x} \right).\) Let \(j\) be the first resource to be used in strategy \(s^{\prime - }\). By the induction hypothesis, there is \(s^{\prime\prime} = \left( {i,s^{\prime\prime - } ,\varnothing } \right) \in U_{i} \left( {\theta ,x} \right)\) such that \(s^{\prime\prime - } \in \bar{S}_{j} \left( {\left\langle - \right\rangle ,x - a_{i} } \right)\), and (13) and (14) hold for \(s^{\prime - }\) and \(s^{\prime\prime - }\).

$$\begin{aligned} T\left( {s^{\prime\prime},\theta } \right) - T\left( {s^{\prime},\theta } \right) &= \left[ {t_{i} + \left( {1 - p_{i}^{T} \left( \theta \right)} \right) \cdot \left[ {T\left( {s^{\prime\prime - } ,\left\langle - \right\rangle } \right) + t_{i,j} } \right]} \right] \hfill \\ \, &\quad - \left[ {t_{i} + \left( {1 - p_{i}^{T} \left( \theta \right)} \right) \cdot \left[ {T\left( {s^{\prime - } ,\left\langle - \right\rangle } \right) + t_{i,j} } \right]} \right] \hfill \\ \, &= \left( {1 - p_{i}^{T, - } \left( \theta \right)} \right) \cdot \left[ {T\left( {s^{\prime\prime - } ,\left\langle - \right\rangle } \right) - T\left( {s^{\prime - } ,\left\langle - \right\rangle } \right)} \right] \hfill \\ \, &\quad \le \left( {1 - p_{i}^{T, - } \left( \theta \right)} \right) \cdot \varepsilon_{T} \cdot \left( {x - a_{i} } \right) \hfill \\ \, &\quad \le \varepsilon_{T} \cdot \left( {x - a_{i} } \right) \hfill \\ \end{aligned}$$

We would similarly show that \(C\left( {s^{\prime\prime},\theta } \right) - C\left( {s^{\prime},\theta } \right) \le \varepsilon_{C} \cdot \left( {x - a_{i} } \right)\). A strategy \(s \in U_{i} \left( {\theta ,x} \right)\) will be included in \(\bar{S}_{i} \left( {\theta ,x} \right)\) such that \(T\left( {s,\theta } \right) - T\left( {s^{\prime\prime},\theta } \right) \le \varepsilon_{T} \cdot a_{i}\) and \(C\left( {s,\theta } \right) - C\left( {s^{\prime\prime},\theta } \right) \le \varepsilon_{C} \cdot a_{i} .\) So:

$$\begin{aligned} T\left( {s,\theta } \right) - T\left( {s^{\prime},\theta } \right) \le \varepsilon_{T} \cdot \left( {x - a_{i} } \right) + \varepsilon_{T} \cdot a_{i} \le \varepsilon_{T} \cdot x \hfill \\ C\left( {s,\theta } \right) - C\left( {s^{\prime},\theta } \right) \le \varepsilon_{C} \cdot \left( {x - a_{i} } \right) + \varepsilon_{C} \cdot a_{i} \le \varepsilon_{T} \cdot x \hfill \\ \end{aligned}$$

Let us now consider case (c). Assume \(s^{\prime} = \left( {i,s^{\prime - } ,s^{\prime + } } \right)\) and that (13) and (14) hold for \(x^{\prime} \le x - a_{i}\), for all resources \(i \in N\). Let us show that they hold for \(s \in \bar{S}_{i} \left( {\theta ,x} \right)\). Let \(j^{ - }\) and \(j^{ + }\) be the first resources to be used in strategies \(s^{\prime - }\) and \(s^{\prime + }\), respectively. By the induction hypothesis, there is \(s^{\prime\prime} = \left( {i,s^{\prime\prime - } ,s^{\prime\prime + } } \right) \in U_{i} \left( {\theta ,x} \right)\) such that \(s^{\prime\prime - } \in \bar{S}_{j1} \left( {\left\langle - \right\rangle ,x - a_{i} } \right)\), \(s^{\prime\prime + } \in \bar{S}_{j2} \left( {\left\langle + \right\rangle ,x - a_{i} - b_{i} } \right)\), and (13) and (14) hold for \(s^{\prime - }\) and \(s^{\prime\prime - }\), and for \(s^{\prime + }\) and \(s^{\prime\prime + }\). Let us now consider (13). We have:

$$\begin{aligned} &T\left( {s^{\prime\prime},\theta } \right) - T\left( {s^{\prime},\theta } \right) \\ \, &= \left[ {t_{i} + \left( {1 - p_{i}^{T} \left( \theta \right)} \right) \cdot \left[ {T\left( {s^{\prime\prime - } ,\left\langle - \right\rangle } \right) + t{}_{{i,j^{ - } }}} \right] + p_{i}^{T} \left( \theta \right) \cdot \left[ {T\left( {s^{\prime\prime + } ,\left\langle + \right\rangle } \right) + t{}_{{i,j^{ + } }}} \right]} \right] \\ \, &\quad - \left[ {t_{i} + \left( {1 - p_{i}^{T} \left( \theta \right)} \right) \cdot \left[ {T\left( {s^{\prime - } ,\left\langle - \right\rangle } \right) + t{}_{{i,j^{ - } }}} \right] + p_{i}^{T} \left( \theta \right) \cdot \left[ {T\left( {s^{\prime + } ,\left\langle + \right\rangle } \right) + t{}_{{i,j^{ + } }}} \right]} \right] \hfill \\ \, &= \left( {1 - p_{i}^{T} \left( \theta \right)} \right) \cdot \left[ {T\left( {s^{\prime\prime - } ,\left\langle - \right\rangle } \right) - T\left( {s^{\prime - } ,\left\langle - \right\rangle } \right)} \right] + p_{i}^{T} \left( \theta \right) \cdot \left[ {T\left( {s^{\prime\prime + } ,\left\langle + \right\rangle } \right) - T\left( {s^{\prime + } ,\left\langle + \right\rangle } \right)} \right] \hfill \\ \, &\quad\le \left( {1 - p_{i}^{T} \left( \theta \right)} \right) \cdot \varepsilon_{T} \cdot \left( {x - a_{i} } \right) + p_{i}^{T} \left( \theta \right) \cdot \varepsilon_{T} \cdot \left( {x - a_{i} - b_{i} } \right) \hfill \\ \, &\quad\le \varepsilon_{T} \cdot \left( {x - a_{i} } \right) \hfill \\ \end{aligned}$$

We would similarly show that \(C\left( {s^{\prime\prime},\theta } \right) - C\left( {s^{\prime},\theta } \right) \le \varepsilon_{C} \cdot \left( {x - a_{i} } \right)\). According to step 4 of the procedure defining \(\bar{S}_{i} \left( {\theta ,x} \right)\), a strategy \(s \in U_{i} \left( {\theta ,x} \right)\) will be included in \(\bar{S}_{i} \left( {\theta ,x} \right)\) such that \(T \left( {s,\theta } \right) - T \left( {s^{\prime\prime},\theta } \right) \le \varepsilon_{T} \cdot a_{i}\) and \(C \left( {s,\theta } \right) - C \left( {s^{\prime\prime},\theta } \right) \le \varepsilon_{C} \cdot a_{i}\). So:

$$\begin{aligned} T \left( {s,\theta } \right) - T \left( {s^{\prime},\theta } \right) &\le \varepsilon_{T} \cdot \left( {x - a_{i} } \right) + \varepsilon_{T} \cdot a_{i} \hfill \\ \, &\le \varepsilon_{T} \cdot x \hfill \\ \end{aligned}$$
$$\begin{aligned}C \left( {s,\theta } \right) - C \left( {s^{\prime},\theta } \right) &\le \varepsilon_{C} \cdot \left( {x - a_{i} } \right) + \varepsilon_{C} \cdot a_{i} \hfill \\ \, &\le \varepsilon_{C} \cdot x \hfill \\ \end{aligned}$$

So, for each \(s^{\prime} \in S_{i} \left( x \right)\) there exists a \(s \in \bar{S}_{i} \left( {\theta ,x} \right)\) such that (13) and (14) hold, completing the proof.□

Proof of Property 4

Since \(\forall s \in \bar{S}_{i} \left( {\theta ,x^{\prime}} \right),x\left( s \right) \ge x^{\prime\prime}\), we know that \(\bar{S}_{i} \left( {\theta ,x^{\prime}} \right) \subset S\left( {x^{\prime\prime}} \right)\). Part a of Definition 4 is independent of x, so if it holds for \(x^{\prime}\) it will also hold for \(x^{\prime\prime}\). As for part b, note that \(x^{\prime\prime} > x^{\prime} \Rightarrow S\left( {x^{\prime\prime}} \right) \subset S\left( {x^{\prime}} \right)\), since a strategy that completes a fraction \(x^{\prime\prime}\) of the activity will also allow completion of a fraction \(x^{\prime} < x^{\prime\prime}\). So, if part b holds for all \(s^{\prime} \in S\left( {x^{\prime}} \right)\), it will also hold for all strategies \(s^{\prime\prime} \in S\left( {x^{\prime\prime}} \right) \subset S\left( {x^{\prime}} \right)\), thus completing the proof.□

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Godinho, P., Costa, J.P. A stochastic model and algorithms for determining efficient time–cost tradeoffs for a project activity. Oper Res Int J 20, 319–348 (2020).

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  • Stochastic modeling
  • Activity management
  • Multicriteria analysis
  • Time–cost tradeoff

Mathematics Subject Classification

  • 90-08
  • 90B36
  • 90B50