On Calculating Activity Slack in Stochastic Project Networks

Abstract

Problem statement: Identifying critical tasks in a project network is easily done when task times are deterministic, but doing so under stochastic task times is problematic. The few methods that have been proposed contain serious drawbacks which lead to identifying critical tasks incorrectly, leaving project managers without the means to (1) identify and rank the most probable sources of project delays, (2) assess the magnitude of each source of schedule risk, and (3) identify which tasks represent the best opportunities for successfully addressing schedule risk? Approach: In this study we considered the problem of identifying the sources of schedule risk in a stochastic project network. We developed general expressions for determining a task's late starting and ending time distributions. We introduced the concept of stochastic slack and develop a number of metrics that help a project manager directly identify and estimate the magnitude of sources of schedule risk. Finally, we compared critical tasks identified using the activity criticality index to those found using stochastic slack metrics. Results:We have demonstrated that a task may have non-zero probability of negative stochastic slack and that expected total slack for a task may be negative. We also found that while the activity criticality index is effective for calculating the probability that a task is on a critical path, the stochastic slack based metrics discussed in this paper are better predictors of the extent to which a delay in a task will result in a project delay. Conclusion/Recommendations: Project managers should consider using stochastic slack based metrics for assessing project risk and establishing the most likely project schedule outcomes. Given the calculation complexity associated with theoretically exact stochastic slack metrics, effective heuristics are required.

Scheduling Critical Activities on Multi-objectives Stochastic Projects

“…In [11], the sources of schedule risk in a stochastic project network is identified, the general expression for determining an activity's late starting and ending time distributions are developed, to identify the critical activities using the activity critically index to those found using stochastic activity metrics.…”

“…..... (10) The early and late of the starting time distributions are used to calculate expected total late for activity as follows: (11) In this paper, we are constructing an expression for determining the total expected lateness, given that the total latest finishing time is greater than or equal to the known due date D, by considering the total interruption time, as follow, respectively:…”

“…As we believe that, no theoretical difficulties are raised, we perform our approach, on a simple illustrative problem, taken from [11], and generated randomly, -to construct an extension one, by convention, tasks 1 & 9 have zero activity duration with probability 1.0. Therefore, any delay introduced into the project schedule will be introduced by one of the remaining tasks 2-8, independed on both their locations within the project network and their activity distributions extracted from the implementation this project several times, at different states, given in the table below: Given the project due date = 4 ℎ , if we consider three objectives, the first one is to minimize the total stopping times 1 , the second objective is to minimize the total slack times 2 and the third objective is to minimize the total lateness 3 .…”

“…Also the initial processing time is considered to be known, however, the length of the interrupted and the final processing time are uncertain. Based on [11], who developed general expressions for determining activities late starting and ending time distributions, we built an approach for identifying and scheduling the critical activities, according to the type of the weighted cost per unit time, constructed by evaluating the total expected interruption, and slackness times, respectively, for an activity, and the total expected lateness, given that the total latest finishing time is greater than or equal to the given due date. Some theories and experimental results are presented to point out the simplicity of our approach.…”

“…= + ...................................................(1)Where ( ) is the processing time before the interruption and ( ) is the processing time during the interruption. Notations If we define the following notations, see[11] By assuming the early start schedule distributions to be continuous distributions, we can define the following:, ( ) = ∏ , ( ) ...................................... (2) , ( ) = , ( ) ............................................. (3) , ( ) = ∫ ( − 1 ) , 0 ( 1 ) 1 ................ (4) , ( ) = , ( ) .............................................. (5) 1 ) , ( 2 ) 2 1 .................................. (6) , ( ) = , ( ) ............................................. (7) , ( ) = ∏ {1 − , ( )} .............................. (8) , ( ) = , ( ) ............................................. (9)…”