Enhanced energetic reasoning-based lower bounds for the resource constrained project scheduling problem
Highlights
► We present three classes of lower bounds that are based on the concept of energetic reasoning. ► The first class includes bounds that are based on the so-called revisited energetic reasoning. ► The second class includes bounds that are based on the concept of reduced instances. ► Finally, the last class of bounds are based on discrete dual feasible functions. ► Our computational results provide evidence that a deceptively simple DFF-based lower bound is effective.
Introduction
The resource constrained project scheduling problem (RCPSP) is a central problem in scheduling theory that has great relevance in project management and more specifically to the crucial issue of allocating scarce resources to activities. Formally, the RCPSP is defined as follows: We consider a project that consists of a set of activities to be scheduled nonpreemptively. A set of renewable resources are required for processing these activities. Each resource is continuously available from time zero onwards with resource capacity Bk. The processing of an activity lasts pj units of time, and requires bjk units of resource . Moreover, the activities are interrelated through precedence constraints. These time restrictions are often modeled using an appropriate activity-on-node graph . In this graph, is the set of nodes corresponding to the activities, in addition to two dummy zero-duration activities and that represent the start and the end of the project, respectively. The arcset represents the precedence restrictions (that is, is an immediate predecessor of . The objective is to minimize the project completion time (or, makespan).
The RCPSP is a fundamental scheduling problem that has been extensively investigated in the operations research literature. We refer to the excellent book by Demeulemeester and Herroelen [19] for a comprehensive review of the impressive research effort that has been devoted to the study of the RCPSP. It is well known that the RCPSP is strongly [1]. Moreover, from a computational perspective, the RCPSP turns out to be an extremely hard nut to crack. Indeed, to give an insight of the notorious intractability of the RCPSP, we mention that state-of-the-art exact algorithms fail to solve some well-known 60-activity benchmark instances that are still open about 50 years after their publication (see [2]). Actually, it is widely recognized that the effectiveness of exact enumerative algorithms strongly relies on the performance of the embedded lower bounds. Indeed, the lower bounding procedure that is invoked within the enumerative algorithm should ideally be both effective and efficient (that is, it should yield a tight lower bound while requiring a short computing time). However, a review of the lower bounds that have been proposed for the RCPSP (see Section 2) reveals that these bounds can be roughly classified into two categories. A first category includes fast lower bounds that exhibit an erratic (i.e., nonrobust) behavior. By contrast, the second category includes tight lower bounds that require a substantial computing burden. Clearly, none of these two categories is fully satisfactory with regard to the effective solution of large-scale RCPSPs. The objective of this paper is to propose new lower bounds that would prove both tight and fast. More precisely, we make the following contributions:
- 1.
We propose new lower bounds that are derived through enhancing the energetic reasoning (ER) [3]. These enhancements are achieved through nontrivial generalizations of the so-called revisited energetic reasoning (RER) that was initially developed by Hidri et al. [4] in the context of multiprocessor scheduling.
- 2.
We introduce the concept of reduced instance and show how it could be used to derive a new enhanced lower bound using previously proposed lower bounds.
- 3.
We introduce a further improvement of ER using dual feasible functions (DFFs).
- 4.
We present the results of a comprehensive computational study that provides evidence that the best proposed lower bound exhibits an excellent performance while requiring a modest computing time.
Section snippets
Review of classical lower bounds for the RCPSP
In this section, we briefly review classical lower bounds from the literature. For the sake of brevity, we restrict our attention to the lower bounds that we shall use in the subsequent sections. For a comprehensive and thorough survey of lower bounds for the RCPSP, the reader is referred to Artigues et al. [2].
Revisited energetic reasoning
In this section, we introduce new bounds for the RCPSP that are based on the so-called revisited energetic reasoning (RER). The RER was initially introduced by Hidri et al. [4] in the context of parallel machine scheduling with heads and tails. It aims at strengthening the classical ER by improving the computation of the work through formulating an integer programming (IP) problem. It is noteworthy that a preliminary version of this section has been published in a conference proceedings [8]. In
Improved energetic reasoning
Given a trial value and a time-interval , we define a lower bound on the total processing time of activity during .
Given an RCPSP instance, a trial value and a time-interval we construct an associated reduced instance in the following way:
- •
the set of activities is ,
- •
the processing time of activity is ,
- •
the values of the availabilities of the renewable resources, the resources
Energetic reasoning and dual feasible functions
Definition 1 A function f is said to be discrete dual feasible if for any discrete finite set S of nonnegative integers, we have
Dual feasible functions (DFFs) have been extensively used for deriving lower bounds both for one- and two-dimensional bin packing problems (see [10], [11], [12]). The main idea is based on using a DFF to transform the item height and width and then to compute a lower bound on the transformed instance. Fekete and Schepers [10]
Experimental results
To assess the performance of the proposed lower bounds, we considered PSPLIP the well-known set of benchmark instances proposed by Kolisch et al. [15]. These instances were generated according to three parameters: network complexity (NC), resource factor (RF) and resource strength (RS). For each triplet (NC,RF,RS), Kolish et al. generated 10 instances for a total of 480 instances for 60 and 90 activities (KSD60 and KSD90), respectively, and 600 instances for 120 activities (KSD120). Hence, our
Conclusion
In this paper, we presented three classes of lower bounds that are based on the concept of energetic reasoning. The first class includes bounds that are based on the so-called revisited energetic reasoning that was initially introduced for parallel machine scheduling problems. We proposed several nontrivial generalizations of this concept. In particular, we introduced a new global feasibility test that considers all the resources simultaneously. The second class includes bounds that are based
Acknowledgements
Dr. Mohamed Haouari would like to thank Fatimah Alnijris Research Chair for Advanced Manufacturing Technology for the financial support provided for this research.
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