## Numerical examples

Two numerical examples serve to illustrate the methodology and associated software:

1. The example from Lanza's paper (slightly adapted);

2. An actual case from the construction industry.

Path 2

Path 2

Task |
Minimum |
Optimistic |
Expected |
Pessimistic |
Path 1 |
Path 2 |

Duration |
Margin |
Margin |
Margin | |||

P |
12 |
0 |
1 |
13 |
13 | |

Q |
2 |
0 |
1 |
2 |
3 | |

R |
4 |
1 |
4 |
5 |
8 | |

S |
5 |
1 |
3 |
5 |
8 | |

T |
1 |
0 |
1 |
11 |
2 | |

Total |
19 |
15 |

Example from Lanza's paper

Figure 3.9 shows the network Lanza used in his paper.

In this network there are two paths, Path 1 (St-Q-R-S-Fi) and Path 2 (St-P-T-Fi). Table 3.1 shows the minimum duration required anyway for each activity and three margin estimates:

• A most likely estimate (best guess) for the required margin on top of the minimum duration;

• A most pessimistic estimate for that margin, defined as having a 10% probability of being exceeded;

• A most optimistic estimate for that margin, defined also as having a 10% probability of being achieved.

Calculating with the minimum durations and most likely required margins (best guesses), Path 1 turns out to be the Critical Path (Primary Path). Activities on this path would get the most attention from the project manager. Activities on Path 2 (Secondary Path) would get less attention because, based upon this calculation, this path has 2 weeks more slack than the Critical Path.

The following mathematical model describes how such mitigations can be allowed for in the (probabilistic) network planning.

Mathematical model

For each of the n activities Ai (i = 1,2,... n) three estimates for the duration ti are made:

1. Pessimistic estimate (probability of 0.1 that reality will be worse);

2. Best guess;

3. Optimistic estimate (probability of 0.1 that reality will be better). Running the network planning three times will, in general, yield:

1. Run with pessimistic estimates: completion too late;

2. Run with optimistic estimates: completion on time;

3. Run with best guesses: low probability of completion on time, say p = 0.03.

The latter is unacceptable, so measures have to be taken to increase the probability of completion on time. Extra resources have to be mobilised, in particular trustworthy project managers (from the main contractor, the subcontractors or other sources). These are limited in number.

Let us assume that m < m0 measures can be taken. Each measure results in reducing activity times by At, at cost Q, i = 1,2,... m.

In each Monte Carlo run, a Linear Programming optimisation is conducted:

subject to:

te < t0 (te = throughput time; t0 = target completion time) (3.11)

with relaxed activity durations: ti = ti — At;, i = 1,2,... l.

If no solution can be found, the constraint te of the completion duration has to be relaxed. This can be done in steps until a solution becomes feasible.

A counter keeps track of how often an activity duration is relaxed by corrective measures (Figure 3.16). We then assume that, say, the most frequent four are indeed carried out.

a frequency

Figure 3.16 Output graph showing which activities needed corrective measures most frequently activities A

Table 3.3 Mitigations Ci, Ati and required skills S

C2 At2 x x

With these activities reduced durations, a new Monte Carlo simulation is conducted. If the probability of completion turns out to increase to, say, over p = 0.5 the measures can be considered to be sufficient.

This procedure can be repeated at later stages of the project to assess what mitigations are then desirable.

In this way the probability of timely completion can be kept at an acceptable level during the whole execution of the project or, when this becomes infeasible, the target completion time can be relaxed at an early stage.

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