Algorithm for Finding Overlapping Schedule in Project Activities

Mα»™t phαΊ§n của tΓ i liệu Develop a project planning method based on building information model (bim) to optimally reduce activity overlaps and time cost (Trang 44 - 49)

Calculate total project duration Activity on node is illustrated as follow:

𝐸𝑆𝑖 𝐸𝑆𝑖 𝐿𝑆𝑖

i

𝐸𝐹𝑖 𝐷𝑖 𝐿𝐹𝑖

Where:

i: Activity number i 𝐷𝑖: Duration of activity i 𝐸𝑆𝑖: Early start of activity i 𝐸𝐹𝑖: Early finish of activity i 𝐿𝑆𝑖: Late start of activity i 𝐿𝐹𝑖: Late finish of activity i 𝑇𝐹𝑖: Total float of activity i Determine π‘¬π‘Ίπ’Š and π‘¬π‘­π’Š

Table 3.2: Determine ES and EF

Case Description ES, EF

1 If activity i has no relationship with the predecessor(s) 𝐸𝑆𝑖 = 0

𝐸𝐹𝑖 = 𝐸𝑆𝑖+ 𝐷𝑖

2 If activity i has FS relationship with activity j predecessors 𝐸𝑆𝑖 = 𝐸𝐹𝑗+ 𝐹𝑆 𝐸𝐹𝑖 = 𝐸𝑆𝑖+ 𝐷𝑖

3 If activity i has SF relationship with activity j predecessors 𝐸𝑆𝑖 = 𝐸𝐹𝑗+ 𝑆𝐹 𝐸𝐹𝑖 = πΈπ‘†π‘–βˆ’ 𝐷𝑖 4 If activity i has SS relationship with activity j predecessors 𝐸𝑆𝑖 = 𝐸𝐹𝑗+ 𝑆𝑆 𝐸𝐹𝑖 = 𝐸𝑆𝑖+ 𝐷𝑖

5 If activity i has FF relationship with activity j predecessors 𝐸𝑆𝑖 = 𝐸𝐹𝑗+ 𝐹𝐹 𝐸𝐹𝑖 = πΈπ‘†π‘–βˆ’ 𝐷𝑖

6 If activity i has FS and SS relationships with predecessors 𝐸𝑆𝑖 = π‘€π‘Žπ‘₯(𝐸𝑆𝑖) 𝐸𝐹𝑖 = 𝐸𝑆𝑖+ 𝐷𝑖

7 If activity i has SF and FF relationship with predecessors 𝐸𝐹𝑖 = π‘€π‘Žπ‘₯(𝐸𝐹𝑖) 𝐸𝑆𝑖 = πΈπ‘†π‘–βˆ’ 𝐷𝑖

8 If activity i has FS and SF, FS and FF, SF and SS, SS and FF and more than 3 relationships with predecessors

𝐸𝐹𝑖 = π‘€π‘Žπ‘₯(𝐸𝐹𝑖) 𝐸𝑆𝑖 = π‘€π‘Žπ‘₯(𝐸𝑆𝑖)

Determine π‘³π‘Ίπ’Š and π‘³π‘­π’Š

Table 3.3: Determine LS and LF

Case Description ES, EF

1 If activity i has no relationship with the successor(s) 𝐿𝑆𝑖 = π‘€π‘Žπ‘₯(𝐸𝐹𝑖) 𝐿𝐹𝑖 = πΏπ‘†π‘–βˆ’ 𝐷𝑖

2 If activity i has FS relationship with activity j successors 𝐿𝑆𝑖 = πΏπΉπ‘—βˆ’ 𝐹𝑆 𝐿𝐹𝑖 = πΏπ‘†π‘–βˆ’ 𝐷𝑖

3 If activity i has SF relationship with activity j successors 𝐿𝑆𝑖 = πΏπΉπ‘—βˆ’ 𝑆𝐹 𝐿𝐹𝑖 = 𝐿𝑆𝑖+ 𝐷𝑖

4 If activity i has SS relationship with activity j successors 𝐿𝑆𝑖 = πΏπ‘†π‘—βˆ’ 𝑆𝑆 𝐿𝐹𝑖 = 𝐿𝑆𝑖+ 𝐷𝑖

5 If activity i has FF relationship with activity j successors 𝐿𝐹𝑖 = πΏπΉπ‘—βˆ’ 𝐹𝐹 𝐿𝑆𝑖 = πΏπΉπ‘–βˆ’ 𝐷𝑖

6 If activity i has FS and FF relationships with successors 𝐿𝐹𝑖 = 𝑀𝑖𝑛(𝐿𝐹𝑖) 𝐿𝑆𝑖 = πΏπΉπ‘–βˆ’ 𝐷𝑖

7 If activity i has SF and SS relationship with successors 𝐿𝑆𝑖 = 𝑀𝑖𝑛(𝐿𝑆𝑖) 𝐿𝐹𝑖 = 𝐿𝑆𝑖+ 𝐷𝑖

8 If activity i has FS and SF, FS and FF, SF and SS, SS and FF and more than 3 relationships with successors

𝐿𝑆𝑖 = 𝑀𝑖𝑛(𝐿𝑆𝑖) 𝐿𝐹𝑖 = 𝑀𝑖𝑛(𝐿𝐹𝑖)

After the initial schedule for a project is established, it becomes possible to quantitatively calculate the level of overlapping between activities. Schedule overlapping occurs when two or more activities are scheduled to occur simultaneously or at the same time (Figure 3.17).

A

B

BASE SCHEDULE

REFERENCE ACTIVITY OVERLAPPING

DURATION

SOR of A activity =

SOR of B activity =

Duration of activity A

Duration of activity B SOR(i): Schedule overlapping ratio of activity i

OVERLAPPING DURATION

Figure 3.17: Typical condition of schedule overlapping

To detect overlapping activities, it is necessary to compare all activities with a base activity. The base activity serves as the reference point for the analysis of overlapping. In Figure 3.18, the activity A, which has the earliest schedule, is initially selected as the first base activity for assessing overlaps with activities B-F. Once the overlapping check for activity A is completed, the activity E, with the second earliest schedule, becomes the next base activity for analyzing overlaps with activities B-F.

To begin, the start and finish dates are established for all activities. Subsequently, the start date for each activity is identified through a search process. The activity with the earliest start date is then referred to as the first base activity, as follows:

π‘†π‘‘π‘Žπ‘‘π‘’π‘– = 𝐸𝑆 (5)

πΉπ‘‘π‘Žπ‘‘π‘’π‘– = 𝐸𝐹 π‘œπ‘Ÿ π‘†π‘‘π‘Žπ‘‘π‘’π‘– + π·π‘’π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘›π‘– (6) 𝐢𝑒𝑛𝑑𝑖 = (π‘†π‘‘π‘Žπ‘‘π‘’π‘– + πΉπ‘‘π‘Žπ‘‘π‘’π‘–)/2 (7)

Where π‘†π‘‘π‘Žπ‘‘π‘’π‘– is the start date of activity i; πΉπ‘‘π‘Žπ‘‘π‘’π‘– is the finish date of activity i; and 𝐢𝑒𝑛𝑑𝑖 is the center date of activity i.

The values of ES, EF, LS, and LF for all activities are calculated, followed by the use of a discriminant to identify schedule overlapping by checking the overlapping status sequentially. By using equations (8) and (9), it is possible to examine the occurrence of overlapping between a base activity and other activities. These equations facilitate the identification of overlapping instances between the base activity and all other activities within the project.

πΆπ‘Žπ‘ π‘’ 1: πΆπ‘’π‘›π‘‘π‘˜ βˆ’ 𝐢𝑒𝑛𝑑𝑖 < (π·π‘’π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘›π‘˜ + π·π‘’π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘›π‘–)/2 (8)

πΆπ‘Žπ‘ π‘’ 2: πΆπ‘’π‘›π‘‘π‘˜ βˆ’ 𝐢𝑒𝑛𝑑𝑖 β‰₯ (π·π‘’π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘›π‘˜ + π·π‘’π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘›π‘–)/2 (9)

In Case 1, the existence of schedule overlapping is confirmed, while in Case 2, it is determined that there is no schedule overlapping.

To determine if there is schedule overlapping between two activities, their center dates are compared to the sum of their durations divided by 2. In Case 1, if the difference between the center dates of two activities is smaller than half the sum of their durations, it indicates that there is overlapping between the activities. Conversely, in Case 2, if the difference exceeds half the sum of the durations, it signifies the absence of overlapping, and no further checks are necessary. The equations (8) and (9) illustrate this process.

A B C D E F

A Activity B Activity C Activity D Activity E Activity F Activity

Center date of activity A

Center date of activity B

< Scheduling by early start > < Schedule overlapping check by

pairwise comparison analysis > < Schedule overlapping determination condition >

The formula for schedule overlapping

determination The evaluation conditions for schedule overlapping dates Use Equation (11), (12), (13), (14), and (15) below Use CASE 1, 2, 3, 4, and 5 below

Figure 3.18: The process of checking for schedule overlapping for each activity

Following a sequence of computational steps, all activities in the project are analyzed to identify any instances of overlapping. The duration of overlap and the schedule overlapping ratio are then calculated for the activities that overlap. The schedule overlapping ratio, denoted as (𝑆𝑂𝑅𝑖) for activity i, is determined by dividing the total duration of overlap between activity i and the reference activities (with respect to the base activity) by the duration of the base activity. This calculation involves summing up the durations of overlap for each reference activity and dividing it by the duration of the base activity. This is illustrated in equation (10).

𝑆𝑂𝑅𝑖 = βˆ‘ π‘‚π‘£π‘’π‘Ÿπ‘™π‘Žπ‘π‘π‘’π‘‘ π·π‘’π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘›π‘˜

π·π‘’π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘›π‘–

π‘›π‘˜=1 (10)

The schedule overlapping ratio of the base activity i is represented by 𝑆𝑂𝑅𝑖. The duration of the base activity is denoted by π·π‘’π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘›π‘–. On the other hand,

π‘‚π‘£π‘’π‘Ÿπ‘™π‘Žπ‘π‘π‘’π‘‘ π·π‘’π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘›π‘˜ stands for the duration of overlapping between the base activity i and activity k.

The calculation of overlapping duration and schedule overlapping ratio (SOR) for two activities can be performed in five different ways, depending on the relationship between their start and finish dates, represented by (π‘†π‘‘π‘Žπ‘‘π‘’π‘–, π‘“π‘‘π‘Žπ‘‘π‘’π‘–) and (π‘†π‘‘π‘Žπ‘‘π‘’π‘˜, π‘“π‘‘π‘Žπ‘‘π‘’π‘˜), respectively. These five cases are described below:

1. If π‘†π‘‘π‘Žπ‘‘π‘’π‘– ≀ π‘†π‘‘π‘Žπ‘‘π‘’π‘˜ and π‘“π‘‘π‘Žπ‘‘π‘’π‘– ≀ π‘“π‘‘π‘Žπ‘‘π‘’π‘˜ then Case 1 𝑆𝑂𝑅𝑖 =π‘“π‘‘π‘Žπ‘‘π‘’π‘–βˆ’π‘†π‘‘π‘Žπ‘‘π‘’π‘˜

π·π‘’π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘›π‘– (11)

2. If π‘†π‘‘π‘Žπ‘‘π‘’π‘– ≀ π‘†π‘‘π‘Žπ‘‘π‘’π‘˜ and π‘“π‘‘π‘Žπ‘‘π‘’π‘– β‰₯ π‘“π‘‘π‘Žπ‘‘π‘’π‘˜ then Case 2 𝑆𝑂𝑅𝑖 =π‘“π‘‘π‘Žπ‘‘π‘’π‘˜βˆ’π‘†π‘‘π‘Žπ‘‘π‘’π‘˜

π·π‘’π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘›π‘– (12)

3. If π‘†π‘‘π‘Žπ‘‘π‘’π‘– β‰₯ π‘†π‘‘π‘Žπ‘‘π‘’π‘˜ and π‘“π‘‘π‘Žπ‘‘π‘’π‘– ≀ π‘“π‘‘π‘Žπ‘‘π‘’π‘˜ then Case 3

𝑆𝑂𝑅𝑖 = 1 (13)

4. If π‘†π‘‘π‘Žπ‘‘π‘’π‘– β‰₯ π‘†π‘‘π‘Žπ‘‘π‘’π‘˜ and π‘“π‘‘π‘Žπ‘‘π‘’π‘– β‰₯ π‘“π‘‘π‘Žπ‘‘π‘’π‘˜ then Case 4 𝑆𝑂𝑅𝑖 =π‘“π‘‘π‘Žπ‘‘π‘’π‘˜βˆ’π‘†π‘‘π‘Žπ‘‘π‘’π‘–

π·π‘’π‘Ÿπ‘Žπ‘‘π‘–π‘œπ‘›π‘– (14)

5. If π‘†π‘‘π‘Žπ‘‘π‘’π‘– β‰₯ π‘“π‘‘π‘Žπ‘‘π‘’π‘˜ or π‘“π‘‘π‘Žπ‘‘π‘’π‘– ≀ π‘†π‘‘π‘Žπ‘‘π‘’π‘˜ then Case 5

𝑆𝑂𝑅𝑖 = 0 (15)

This refers to the start and finish dates of the two overlapping activities, which are represented by π‘†π‘‘π‘Žπ‘‘π‘’π‘– and π‘“π‘‘π‘Žπ‘‘π‘’π‘– for activity i, and π‘†π‘‘π‘Žπ‘‘π‘’π‘˜ and π‘“π‘‘π‘Žπ‘‘π‘’π‘˜ for activity k.

The schedule overlapping ratio (SOR) indicates the ratio of overlapping time to the total duration of activities and gives a comparative measure of the overlapping that helps to detect potential bottlenecks and assist project scheduling. The cumulative value of all SOR determines the overall level of overlapping for the project. The project’s overlapping level is utilized in the Genetic Algorithm analysis to identify and reduce schedule overlapping risks. After evaluating the overlapping status of all activities, a risk analysis process is conducted specifically for those activities with overlaps. In this research, the risk associated with overlapping activities indicates the level of risk regarding schedule overlaps. To quantify such risks, a fuzzy analysis approach is applied.

Mα»™t phαΊ§n của tΓ i liệu Develop a project planning method based on building information model (bim) to optimally reduce activity overlaps and time cost (Trang 44 - 49)

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