Fuzzy critical chain method for project scheduling under resource constraints and uncertainty
Introduction
Nowadays, non-routine projects (e.g., research and development projects, new construction projects) are dramatically increasing in many countries. These projects are often executed under resource constraints (e.g., limited number of workers), and uncertainty (e.g., uncertain productivity of workers, bad weather, etc.). In today’s competitive environment, the demand to finish a project in a shorter time is increasing. This situation forces managers to reduce project execution, and becomes critical by the combined effect of uncertainty and resource constraints. Moreover, the project is often disturbed by many delays/interruptions, so it is necessary to manage project schedule during execution. Hence, project managers strongly need an effective method that considers resource constraints and uncertainty for project scheduling, and provides a systematic mechanism for managing schedule during project execution.
In non-routine projects, activity durations may not be known (uncertain). Uncertainty in the activity durations can be modeled by two groups of methods such as probability-based methods, and fuzzy set-based methods [1], [2] which depend on the situation and the project manager’s preference.
For the case of using probability-based methods (e.g., PERT [3]), the first step is to represent the activity duration by a random variable. The second step is to characterize the random variable by a probability distribution function (PDF); this allows for calculating the statistics of the random variable such as the mean value and variance, and then calculating the statistics of the project completion time and its PDF. However, due to the uniqueness of some activities in the above projects, and lack of historical data about activity durations, a project manager may not correctly characterize these random variables [4]. In these cases, if activity durations are estimated by the subjective judgment of the experts, fuzzy set-based methods are introduced for managing these projects. Especially during project execution, many unexpected events may occur, and the manager will have to rely on his/her subjective knowledge about actual situations to estimate delays/interruptions, so fuzzy set theory is a more reliable alternative in this situation.
For these reasons, we believe that fuzzy set-based methods provide viable alternatives in such situations. Many fuzzy set-based methods have been proposed for project scheduling [5], [6], [7], [8], [9], [10], [11], [12], [13], [14]. However, these methods did not provide an effective method for managing schedule during project execution.
Recently, Goldratt [15] introduced the critical chain project management (CCPM) method which is based on the probability theory. CCPM uses a deterministic schedule integrated by a buffer mechanism to deal with both resource constraints and uncertainty. CCPM removes the hidden safety in activity durations to protect from starting late on activities (Student Syndrome) and keeping busy for the entire activity duration (Parkinson’s Law). Then these safety times are placed at key points as buffers to absorb delays. All activities are scheduled as late as possible so that costs are not incurred earlier than necessary. A critical chain is the longest chain of activities under both precedence and resource dependencies. A project buffer is placed at the end of the critical chain to protect against exceeding the project deadline, and feeding buffers are placed at the intersections between any non critical chains and the critical chain to protect it against disturbances. The buffer size is set as 50% of the chain length. During execution, each activity is started as soon as its predecessors are completed. Project control is performed by monitoring the penetration level in buffers. CCPM has attracted considerable attention [16], [17], [18], [19], [20].
However, CCPM based on the probability theory cannot be applied for non-routine projects because of lack of statistical data. Resource-constrained project scheduling (RCPSP) is not adequately solved in CCPM [16], [17]. “As late as possible” approach in CCPM may not be efficient for managing projects in many cases Zwikael [20]. Furthermore, during project execution, feeding buffers may fail to act as a real proactive protection mechanism in traditional CCPM [16], [17].
To overcome these difficulties, a fuzzy critical chain method is developed by incorporating some beneficial features of CCPM for project scheduling under resource constraints and uncertainty.
Section snippets
Description of resource constrained-project scheduling problem
In this paper, to make a project schedule, the project was modeled as an activity-on-node network, where each activity is represented by a node, and each directed arc is the symbolic representation of a precedence requirement between two activities.
In many solution methods for resource-constrained project scheduling (RCPSP), both the duration of each activity and its resource requirements are assumed to be known and fixed. However, in many cases of RCPSP, only work contents (e.g., total amount
Solution methodology
The method is implemented in the three logically interactive and iterative stages.
- 1.
In the first stage for RCPSP, the proposed method solves a scheduling problem with time/resource trade-off to provide a minimum-makespan deterministic schedule which will generally be a good starting point for project execution.
- 2.
In the second stage for dealing with uncertainty, based on the deterministic activity durations in the first stage and expert knowledge, the proposed method uses fuzzy numbers to model
Numerical example
As an example, we consider a project with 20 activities. The data of each activity and the relationship between activities are shown in columns (1)–(5) in Table 1. The total availability of resource ResA(k,t) is 45 workers per day.
Conclusions
This paper presents the fuzzy critical chain method to manage project scheduling. By creating a deterministic schedule under resource constraints, and then adding a project buffer at the end of the selected critical chain to cope with uncertainty, the proposed method is practical and useful for scheduling under resource constraints and uncertainty at both planning and execution stages.
In project planning, for creating a desirable deterministic schedule, the method considers both durations and
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