# Understanding Project Performance with Stochastic Interruption

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## Abstract

**:**

## 1. Introduction

## 2. Model

#### 2.1. Behavior of the Contractor under Discrete–Continuous Multi–Period Setting

#### 2.1.1. Profit Function of the Contractor

#### 2.1.2. Optimal Decision of the Contractor

#### 2.2. Behavior of the Owner under Discrete–Continuous Multi–Period Setting

#### 2.2.1. Benefit Function of the Owner

#### 2.2.2. Optimal Decision of the Owner

#### 2.2.3. Project Selection Rule of the Owner

## 3. Case Study: One-Year and Two-Year Project ($N=2$)

#### 3.1. Optimal Decisions under Two-Year Project

**Lemma**

**1.**

- (i)
- $\Phi \left(\sigma \right)$increases with the increase in the variance of the length of interruption periods σ, i.e.,$$\frac{d\Phi \left(\sigma \right)}{d\sigma}>0$$
- (ii)
- The value of$\Phi \left(\sigma \right)$has the lower and upper bound, such as,$$1\le \Phi \left(\sigma \right)\le \frac{{e}^{r\sigma}+{e}^{-r\sigma}}{2}\le \frac{{e}^{r({\overline{t}}_{2}-{T}_{1})}+{e}^{-r({\overline{t}}_{2}-{T}_{1})}}{2}$$

**Proposition**

**1.**

- (i)
- The expected benefit of the owner is decreased by the existence of interruption periods.
- (ii)
- The increase in the expected interruption length decreases the expected benefit of the two-year project.
- (iii)
- The increase in the variance of the interruption length decreases the expected benefit of the two-year project.

**Proposition**

**2.**

- (i)
- The expected profit of the contractor is increased [decreased] by the existence of interruption if the project value v is sufficiently larger [smaller] than the fixed cost f.
- (ii)
- The expected profit of the contractor is increased [decreased] with the increase in the expected length of interruption if the project value v is sufficiently larger [smaller] than the fixed cost f.
- (iii)
- The expected profit of the contractor is always decreasing with the increase in the variance of interruption length.

#### 3.2. Comparison between Projects

#### Optimal Decision under One-Year Project ($N=1$)

**Lemma**

**2.**

#### 3.3. Project Selection by the Owner

**Proposition**

**3.**

- (i)
- The owner selects the two-year project, if the interruption length is sufficiently shorter, i.e.,${\overline{\zeta}}_{1}<{\zeta}_{[E\left[{V}_{2}^{*}\right]={V}_{1}^{*}]}$.
- (ii)
- The owner selects ones-year project, if the interruption length is sufficiently longer, i.e.,${\overline{\zeta}}_{1}>{\zeta}_{[E\left[{V}_{2}^{*}\right]={V}_{1}^{*}]}$.

**Proposition**

**4.**

- (i)
- When the owner selects the two-year project, the amount of work effort of the two-year project is always larger than that of the one-year project.
- (ii)
- When the owner selects the one-year project,
- (a)
- the amount of work effort of the one-year project is larger than that of the two-year project, if the interruption period is longer than${\zeta}_{[E\left[{x}_{2}^{*}\right]={x}_{1}^{*}]}$, i.e.,${\overline{\zeta}}_{1}>{\zeta}_{[E\left[{x}_{2}^{*}\right]={x}_{1}^{*}]}$.
- (b)
- the amount of work effort of the one-year project is smaller than that of the two-year project, if the interruption period is shorter than${\zeta}_{[E\left[{x}_{2}^{*}\right]={x}_{1}^{*}]}$, i.e.,${\overline{\zeta}}_{1}<{\zeta}_{[E\left[{x}_{2}^{*}\right]={x}_{1}^{*}]}$.

where,$${\zeta}_{[E\left[{x}_{2}^{*}\right]={x}_{1}^{*}]}=\frac{1}{r}\left\{ln({e}^{r({T}_{2}-{T}_{1})}+{e}^{r{T}_{1}}-1)-ln\Phi \left(\sigma \right)\right\}-{T}_{1}$$

**Proposition**

**5.**

#### 3.4. Project Preference of the Contractor

**Proposition**

**6.**

- (i)
- When the owner orders the one-year project and the project is implemented, the contractor preferred the one-year project than the two-year project, i.e.,$E\left[{J}_{2}^{*}(0,{\tilde{\mathbf{t}}}_{1},{\mathbf{T}}_{1})\right]\le {J}_{1}^{*}(0,{t}_{1},{T}_{1}))$.
- (ii)
- When the owner orders the two-year project and the project is implemented, the contractor preferred the two-year [one-year] project, if the project value v is sufficiently larger [smaller] than the fixed cost f.

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Proofs

**Proof**

**of**

**Lemma**

**1.**

**Proof**

**of**

**Proposition**

**1.**

**Proof**

**of**

**Proposition**

**2.**

**Proof**

**of**

**Proposition**

**3.**

**Proof**

**of**

**Proposition**

**4.**

**Proof**

**of**

**Proposition**

**5.**

**Proof**

**of**

**Proposition**

**6.**

**Lemma**

**A1.**

#### Appendix A.2. Numerical Setting for Each Figure

Parameter | Figure 2 | Figure 3 | Figure 4 | Figure 5 | Figure 6 | Figure 7 |
---|---|---|---|---|---|---|

r | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 | 0.003 |

${T}_{1}$ | 28 | 28 | 28 | 28 | 28 | 28 |

${\overline{\zeta}}_{1}$ | Equations (45) and (47) | Equations (45) and (47) | 24 | 36 | 24 | 24 |

${T}_{2}$ | [52, 75] | [52, 70] | 80 | 70 | 80 | 80 |

$\sigma $ | 0 | 24 | [0, 24] | 2 | 24 | [0, 24] |

v | - | - | 1 | Equations (A15), (A17) and (A19) | Equations (A15), (A17) and (A19) | 1 |

c | - | - | 1 | 1 | 1 | 1 |

f | - | - | - | [0, 1.0] | [0, 1.0] | 1 |

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**Figure 2.**Owner’s decision in the deterministic case ($\sigma =0\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{weeks}\right)$).

**Figure 3.**Owner’s decision in the stochastic case ($\sigma =24\phantom{\rule{0.166667em}{0ex}}\left(\mathrm{weeks}\right)$).

**Figure 5.**Contractor’s preference when the owner orders the one-year project ($E\left[{V}_{2}^{*}\left({T}_{2}\right)\right]<{V}_{1}^{*}\left({T}_{1}\right)$).

**Figure 6.**Contractor’s preference when the owner orders the two-year project ($E\left[{V}_{2}^{*}\left({T}_{2}\right)\right]>{V}_{1}^{*}\left({T}_{1}\right)$).

**Figure 7.**Changes in the contractor’s profits with the changes in the variance of interruption length.

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Okubo, K.; Okumura, M. Understanding Project Performance with Stochastic Interruption. *Sustainability* **2022**, *14*, 2964.
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Okubo K, Okumura M. Understanding Project Performance with Stochastic Interruption. *Sustainability*. 2022; 14(5):2964.
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**Chicago/Turabian Style**

Okubo, Kazuaki, and Makoto Okumura. 2022. "Understanding Project Performance with Stochastic Interruption" *Sustainability* 14, no. 5: 2964.
https://doi.org/10.3390/su14052964