New concepts for activity float in resource-constrained project management

Abstract

The concepts of float and critical path are central to analyzing activity networks in project management. In resource-constrained projects, schedule multiplicity makes it difficult to calculate float and identify critical activities accurately. In this work, new concepts of float and critical activity are developed to ascertain critical activities more precisely without reference to activity start and end times in specific schedules. The notions of float, group float, float set, negative float and zero critical activity are introduced, which the project manager can use to deal more effectively with critical activities, duration uncertainty, activity buffering, and resource allocation than the currently available tools in literature. For practical implementation, algorithms are provided and tested to calculate the new measures on the PSPLIB benchmark instances, specifically the J30, J60 and J120 test sets, for the resource constrained project management, illustrating the effectiveness of the proposed concepts in helping to identify flexibility in scheduling activities.

Introduction

Although the notions of float and critical activity are key to analyzing project networks, float cannot be easily interpreted as the difference between the earliest start time (EST) and latest start time (LST) of an activity once resource limitations come into play. This leads to difficulties in network analysis, for example, when the critical path needs to be ascertained. The application of scheduling analysis depends on a good representation of the float of an activity and hence various methods have been proposed to calculate float [1], [2] and identify critical activities [1], [3], [4], [5], [6], [7].

Measures for calculating float to deal with resource-constrained projects have been explored by Raz and Marshall [1] and Bowers [2]. Raz and Marshall [1] used the classical definition of float and calculated it as the difference between the late schedule time and corresponding early schedule time. Since the standard notion of float is not unique, Bowers recognized that there were different schedules with identical completion time for a resource-constrained project, and float varied according to schedules. To resolve this, he redefined float into three types. However, Bowers’ notions of float were calculated as the difference between the EST and LST of all possible schedules. This is computationally prohibitive, since all possible schedules with identical durations must be found, while on the other hand, the difference between the EST and LST cannot capture the meaning of float for resource-constrained projects as we show in Section 2.

Recognizing that it was misleading to apply the traditional critical path method (CPM) to resource-constrained projects, Wiest [4] pursued a new representation of critical activities as a critical sequence in resource-constrained projects. A critical sequence is defined as the set of activities determined not only by topological ordering, but also by resource constraints. Different approaches have been developed to identify critical sequences [1], [3], [5], [6], [7]. The use of a critical sequence relies heavily on techniques used in the scheduling of resource-constrained projects, where activities identified as critical may be different for different schedules. This poses an inherent weakness in the approach.

In Goldratt [8], the idea of critical chain was proposed as a new approach for project scheduling, which is defined as a chain of activities satisfying both precedence and resource constraints. The chain of activities was critical in the sense that delaying any activities in the chain would delay the project. There is, however, little difference between the concepts of critical chain and critical sequence, both of which are schedule based, except for a form of buffer management. In critical chains, different feasible schedules might yield different critical chains, resulting in an activity being critical in one schedule, but not in another. Detailed discussions on critical chains can be found in the work by Raz et al. [9] and Herroelen and Leus [10].

In view of the need for a suitable method to identify critical activities for resource-constrained projects, Rivera and Duran [11] recently proposed the concepts of critical sets and critical clouds while pointing out the shortcomings of earlier approaches to identify critical activities. It turns out, however, that these definitions allow sets of critical activities to be left out, as shown in Section 2. Moreover, the exact algorithm proposed which uses these notions is unlikely to be practical for identifying all critical sets since even for medium-sized projects, there can be an exponential number of critical sets.

The resource-constrained project scheduling problem (RCPSP), first introduced by Wiest [4], is NP-hard in strong sense [12] and methods have been developed to tackle this problem. These include branch-and-bound methods [13], [14], [15], [16], [17], constraint-propagation-based cutting planes [18], heuristic X-pass methods [19], [20], [21], tabu search [22], [23], simulated annealing [24], [25] and genetic algorithms [26], [27], [28], [29]. A detailed survey on the methods for the RCPSP problem can be found in Demeulemeester and Herroelen [30], Kolisch and Padman [31], Kolisch and Hartmann [32], where a standard benchmark library, PSPLIB, was used to evaluate different methods [33].

A review of the limitations of current approaches for calculating float and critical activities is given in Section 2. New notions are proposed in Section 3 that address the limitations, and can be applied effectively to resource-constrained network project management. In Section 4, group float is defined, and a generalized definition of a critical set is given. In order to implement the new concepts, algorithms are developed in Section 5, and in Section 6, we discuss how critical sets and group sets can be found. In Section 7, float graphs illustrate how the new concepts can be used in managing projects. The ideas are developed further by the introduction of a negative float which complements the use of a float. Correspondingly, negative critical activity, negative group float, negative float set, negative critical set, and negative float graph are introduced. The idea of zero critical activity – which is both critical and negatively critical – is highlighted since it is particularly useful as a marker for activities to which project completion time is highly sensitive. For practical implementation of the analytical tools, algorithms developed to calculate float, critical sets and float sets are tested in numerical experiments described in Section 8. Our work is summarized in Section 9.