# Using Stochastic Decision Networks to Assess Costs and Completion Times of Refurbishment Work in Construction

^{*}

## Abstract

**:**

## 1. Introduction

- The complicated course of design work and the unpredictable process of obtaining legal opinions and approvals of proposed refurbishment solutions, the lack of approval of design solutions by architectural conservation authorities.
- Difficulty in determining the scope of work (a high probability of additional or replacement types of work) due to the limited identification of the structure of an existing building, e.g., the technical condition of foundations uncovered while performing construction work can require different forms of reinforcing them.
- In the case of historical buildings, it is possible that discoveries will be made during construction work, resulting in additional archaeological digs, as well as more construction and conservation work [4], e.g., the replacement of existing plasters on a historical structure can result in the discovery of polychromes underneath.

#### 1.1. Stochastic Networks—Literature Review

#### 1.2. Decision Networks—Literature Review

#### 1.3. Research Aims

- An innovative approach to planning the restoration of buildings, allowing for the consideration of various completion scenarios, the occurrence of which can be both random and decision-based. The following will be required for this goal to be achieved:
- Defining the stochastic-decision structure of the network model containing an appropriate topology of vertices with deterministic, stochastic and decision emitters (as a directed, non-cyclical graph, with one initial vertex and numerous final end vertices).
- Developing a nonlinear one-and multi-criteria binary programming model for the purpose of optimising (in the time-cost aspect) various scenarios of the carrying out of the project that is being modelled by the network.
- As a result, the decision-maker, by specifying their preferences as to the planned result of a project and their risk aversion, will receive an optimal (in terms of expected time and costs) scenario of carrying out their project. In the case of a multi-criteria analysis (time and cost), the decision-maker will also be able to specify different values of weights for the expected time and cost of the planned project in the goal function. As a part of the results, the decision-maker will also obtain information on the type of technical solutions or the manner of carrying out the work that should be included in the plan of the optimal scenario of carrying out the project.

- Developing a digital application of the approach and performing a calculation experiment within which the effectiveness of the stochastic decision network will be demonstrated in relation to the traditional approach.

## 2. Method

#### 2.1. Definition of the Structure of a Stochastic Decision Network

**Receivers**, defining the conditions for achieving a given state (receiver activation);**Emitters**, specifying the conditions for the carrying out of specific arcs that originate from it (Table 1).

#### 2.2. Optimisation Model

- The shortest expected completion time of the scenario of the planned project;
- The lowest expected cost of the carrying out of the scenario of the planned project;
- The shortest completion time and lowest cost of the carrying out of the scenario of the planned project.

**Symbols concerning network structure:**

- $s$—starting vertex, $s\in Y$
- $k$—end vertex, $s\in Y$
- $r$—any other vertex, $r\in Y$
- $D$—set of permissible solutions
- ${\Gamma}_{r}$—set of direct successors $r$
- ${\Gamma}_{r}^{-}$—collection of direct predecessors of r
- ${\mathsf{\pi}}^{+}r$—out-degree (number of actions (arcs) exiting a vertex r)
- ${\mathsf{\pi}}^{-}r$—in-degree (number of actions (arcs) entering a vertex $r$)
- ${\lambda}_{ij}$—binary decision variable determining the existence of the i-j action

**Symbols concerning network parameters**:

- ${p}_{ij}$—probability of i-j action
- ${\alpha}_{ij}$—accumulated probability of the occurrence of i-j action (taking into account the probability of the occurrence of preceding tasks)
- ${\delta}_{r}$—probability of the occurrence of vertex $r$
- ${c}_{ij}$—the cost of action i-j
- ${d}_{ir}$—duration of action i-j${w}_{t},{w}_{c}$—weights for the criteria of time and cost, respectively
- ${\delta}_{k\_req}$—required probability of occurrence of the final node $k$
- ${T}_{r}$—expected completion time of vertex $r$
- ${C}_{r}$—expected cost of completing vertex r

**GOAL FUNCTION—EXPECTED TIME**

**Goal Function—Expected Cost**

**TWO-CRITERIA GOAL FUNCTION (meta-criterion function)**

**CONSTRAINT CONDITIONS CONCERNING POSSIBLE STRUCTURES**

**“deterministic”**emitter) start node [30]

**“and”**receiver,

**“deterministic”**emitter) [31]

**“and”**receiver,

**“ decision”**emitter) [31]

**“or”**receiver,

**“deterministic”**emitter)

**“or”**receiver,

**“decision”**emitter)

**CONSTRAINT CONDITIONS CONCERNING TIME ANALYSES**

**“and”**receiver)

**“or”**receiver)

**CONSTRAINT CONDITIONS CONCERNING COST ANALYSES**

**“and”**receiver)

**„or”**receiver)

## 3. Calculation Experiment

#### 3.1. Construction of a Stochastic Decision Network Model

#### 3.2. Analysis of the Network Model

- Brute force;
- APOPT solver (for Advanced Process OPTIMIZER) is a software package for solving large-scale optimisation problems (http://apopt.com/). The program is used to solve linear problems (LP), square (QP), non-linear (NLP) and mixed problems (MIP, MILP, MINLP). The APOPT solver was used with the APMonitor service [37].

## 4. Analysis of Results

## 5. Conclusions and Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**A stochastic network model for the renovation of a building’s foundations. Values in brackets signify: “p” probability of occurrence of activities, “d” duration and “c” cost of their implementation, source: based on [25].

**Figure 2.**Stochastic decision network model for the renovation of a building’s foundations with introduced vertices with decision emitters. The values in brackets signify, respectively: “p” probability of occurrence of activities, “d” duration and “c” cost of carrying them out.

**Figure 3.**The expected cost of carrying out the 11th vertex at different probability levels for different types of optimisation.

**Figure 4.**The expected completion time of the 11th vertex at different probability levels for different types of optimisation.

**Figure 5.**Value of the meta-criterion of carrying out vertex 11 at different probability levels for different types of optimisation.

**Figure 6.**The expected cost of the carrying out of vertex 17 at different probability levels for different types of optimisation.

**Figure 7.**The expected completion time of vertex 17 at different probability levels for different types of optimisation.

**Figure 8.**Value of the meta-criterion for the carrying out of vertex 17 at different probability levels for different types of optimisation.

**Figure 9.**Two-criteria optimisation results for the completion of vertex 11 at different probability levels and different combinations of criterion weights.

**Figure 10.**Two-criteria optimisation results for the completion of vertex 17 at different probability levels and different combinations of criteria weights.

**Figure 11.**The sub-network ${\left\{{\mathsf{\lambda}}_{\mathrm{ij}}\right\}}_{7}$, that defines/determines the plan of carrying out the renovation of foundations (reaching vertex no. 17) with a probability equal to min. 0.8 with the minimum expected time and cost for the weighted preferences of the decision-maker at a level of 0.3 for the expected costs and 0.7 for the expected project completion time.

**Table 1.**Graphical representation of the logical forms of receivers and emitters, as well as their reception and emission conditions in the stochastic decision network.

The Name of the Receiver/Emitter | Graphical Representation of the Form of Logical Reception and Emission of Activities (Arches) | Conditions for the Reception and Emission of Activities (arcs) within the Structure of the Network |
---|---|---|

Receiver “AND” | The “AND” receiver of vertex y will be activated if and only if all the actions of ${u}_{1}\dots {u}_{n}$ entering it will be completed. | |

Receiver “inclusive-or IOR” | The receiver “or” of vertex y will be activated if and only if at least one of the actions ${u}_{1}\dots {u}_{n}$ entering it will be completed. | |

Emitter “deterministic” | The deterministic emitter of vertex y enables the carrying out of all actions ${u}_{1}\dots {u}_{n}$ from it, provided that the vertex has been activated. | |

Emitter “stochastic” | The stochastic emitter of vertex y allows the performance of only one of the actions ${u}_{1}\dots {u}_{n}$ from it with a certain probability, provided that the node has been activated. At the same time, the condition $\sum _{j\in {\Gamma}_{\mathrm{i}}}}\text{}{p}_{ij}=1$ must be met where: ${\Gamma}_{\mathrm{i}}$ is a set of direct successors. | |

Emitter “decision” | The y-vertex decision emitter only allows one of the actions ${u}_{1}\dots {u}_{n}$ that are outbound from it, provided that the vertex has been activated. The decision maker determines which task/action will be carried out. |

Receiver | AND | Inclusive-Or IOR | |
---|---|---|---|

Emitter | |||

deterministic | |||

stochastic | |||

decision |

**Table 3.**The D set of possible solutions for the stochastic decision network of the refurbishment of the foundations of a building.

$\mathbf{\left\{}{\mathit{\lambda}}_{\mathit{i}\mathit{j}}\mathbf{\right\}}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

i-j | |||||||||||||||||||||

1–2 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

2–3 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

3–4 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

3–5 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

4–6 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

4–11 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

5–9 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | |

5–9 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

6–7 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

6–8 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

7–8 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | |

7–11 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | |

7–14 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | |

8–12 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | |

8–12 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | |

8–12 | 0 | 0 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | 0 | 0 | 1 | |

9–10 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

10–16 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

12–13 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

13–16 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | |

14–15 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | |

15–16 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | |

16–17 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

**Table 4.**Comparison of the results obtained using the brute force method with the results obtained with APOPT solver for different levels of probability of reaching the vertices No. 11 and No. 17. (* Solutions were found by increasing probability by a small margin).

Results of Cost Optimisation at Different Probability Levels | Results of Time Optimisation at Different Probability Levels | The Results of Two-Criteria Optimisation at Different Probability Levels | |||||
---|---|---|---|---|---|---|---|

Probability | APOPT | Brute Force | APOPT | Brute Force | APOPT | Brute Force | |

Vertex 11 | >0 | 18,000.00 | 18,000.00 | 6.00 | 6.00 | 0.794872 | 0.794872 |

≥0.1 | 18,000.00 | 18,000.00 | solution not found | 6.00 | 0.794872 | 0.794872 | |

≥0.2 | 18,000.00 | 18,000.00 | 6.00 | 6.00 | 0.794872 | 0.794872 | |

Vertex 17 | >0 | 8555.56 | 8555.56 | 10.33 | 10.33 | 0.485979 | 0.485979 |

≥0.1 | 8555.56 | 8555.56 | 10.33 | 10.33 | 0.485979 | 0.485979 | |

≥0.2 | 8555.56 | 8555.56 | 10.33 | 10.33 | 0.485979 | 0.485979 | |

≥0.3 | solution not found * | 8555.56 | 10.33 | 10.33 | 0.485979 | 0.485979 | |

≥0.4 | 8555.56 | 8555.56 | solution not found * | 10.33 | 0.485979 | 0.485979 | |

≥0.5 | 8555.56 | 8555.56 | 10.33 | 10.33 | 0.485979 | 0.485979 | |

≥0.6 | solution not found * | 8555.56 | 10.33 | 10.33 | 0.485979 | 0.485979 | |

≥0.7 | 8555.56 | 8555.56 | 10.33 | 10.33 | 0.485979 | 0.485979 | |

≥0.8 | solution not found * | 11,841.67 | 12.37 | 11.78 | 0.683898 | 0.683426 | |

≥0.9 | 11,841.67 | 11,841.67 | 12.37 | 11.78 | 0.683898 | 0.683426 | |

=1 | 11,841.67 | 11,841.67 | 12.37 | 12.37 | 0.683898 | 0.683898 |

**Table 5.**Comparison of the values of expected completion times and costs of carrying out the final vertices for specific probabilities of carrying them out in the case of the stochastic network model and the stochastic decision network of the planned project of renovating foundations.

Stochastic Network | Stochastic Decision Network | |||||
---|---|---|---|---|---|---|

No. of the Final Vertex | Probability of Being Carried Out | Expected Cost of Completion [Monetary Units] | Expected Completion Time [Days] | The optimal Expected Completion Cost [Monetary Units] | The Optimal Expected Completion Time [Days] | Metacriterion for the Weights: ${\mathit{w}}_{\mathit{t}}\mathbf{,}{\mathit{w}}_{\mathit{c}}\mathbf{=}\mathbf{0.5}$ |

11 | 0.155 | 21290.32 | 9.58 | 18,000.00 for sub-networks: ${\left\{{\lambda}_{ij}\right\}}_{1}$ | 6.00 for sub-networks: ${\left\{{\lambda}_{ij}\right\}}_{1}$ | 0.795 for sub-networks: ${\left\{{\lambda}_{ij}\right\}}_{1}$ |

17 | 0.845 | 11891.62 | 14.85 | 11,841.00 for sub-networks: ${\left\{{\lambda}_{ij}\right\}}_{20}$ | 11.78 for sub-networks: ${\left\{{\lambda}_{ij}\right\}}_{7}$ | 0.683 for sub-networks: ${\left\{{\lambda}_{ij}\right\}}_{8}$ |

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**MDPI and ACS Style**

Śladowski, G.; Szewczyk, B.; Sroka, B.; Radziszewska-Zielina, E. Using Stochastic Decision Networks to Assess Costs and Completion Times of Refurbishment Work in Construction. *Symmetry* **2019**, *11*, 398.
https://doi.org/10.3390/sym11030398

**AMA Style**

Śladowski G, Szewczyk B, Sroka B, Radziszewska-Zielina E. Using Stochastic Decision Networks to Assess Costs and Completion Times of Refurbishment Work in Construction. *Symmetry*. 2019; 11(3):398.
https://doi.org/10.3390/sym11030398

**Chicago/Turabian Style**

Śladowski, Grzegorz, Bartłomiej Szewczyk, Bartłomiej Sroka, and Elżbieta Radziszewska-Zielina. 2019. "Using Stochastic Decision Networks to Assess Costs and Completion Times of Refurbishment Work in Construction" *Symmetry* 11, no. 3: 398.
https://doi.org/10.3390/sym11030398